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CLARKSON UNIVERSITY Detection and Parameter Extraction of Low Probability of Intercept Radar Signals Using the Reassignment Method and the Hough Transform

A Dissertation By Daniel Lee Stevens Department of Electrical and Computer Engineering Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy, Electrical and Computer Engineering

November 22, 2010

Accepted by the Graduate School

_________, _______________________ Date, Dean of the Graduate School

UMI Number: 3437800

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The undersigned have examined the dissertation entitled ‘Detection and Parameter Extraction of Low Probability of Intercept Radar Signals Using the Reassignment Method and the Hough Transform’ presented by Daniel Lee Stevens, a candidate for the degree of Doctor of Philosophy, Electrical and Computer Engineering and hereby certify that it is worthy of acceptance.

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___________________________________ Dr. Stephanie A. Schuckers

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___________________________________ Dr. Robert J. Schilling

______________________ Date

___________________________________ Dr. Abdul J. Jerri

______________________ Date

___________________________________ Dr. Jeremiah J. Remus

______________________ Date

___________________________________ Dr. Andrew J. Noga

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Abstract Current radar systems employ new types of signals which are difficult to detect and characterize. These signals are known as low probability of intercept radar signals, which have as their goal ‘to see and not be seen’ by intercept receivers, devices which are designed to detect and extract information from radar emissions.

Digital intercept

receivers are currently moving away from Fourier-based analysis and towards classical time-frequency analysis techniques, based on the Wigner-Ville distribution, ChoiWilliams distribution, spectrogram, and scalogram, for the purpose of analyzing low probability of intercept radar signals (e.g. triangular modulated frequency modulated continuous wave and frequency shift keying). Although these classical time-frequency techniques are an improvement over the Fourier-based analysis, they still suffer from a lack of readability, due to poor time-frequency localization, cross-term interference, and a mediocre performance in low signal-to-noise ratio environments. This lack of readability may lead to inaccurate detection and parameter extraction of these radar signals, making for a less informed (and therefore less safe) intercept receiver environment. In this dissertation, the reassignment method, because of its ability to smooth out cross-term interference and improve time-frequency localization, is proposed as an improved signal analysis technique. In addition, the Hough transform, due to its ability to suppress crossterm interference, separate signals from cross-term interference, and perform well in a low signal-to-noise ratio environment, is also proposed as an improved signal analysis technique. With these qualities, both the reassignment method and the Hough transform have the potential to produce better readability and consequently, more accurate signal detection and parameter extraction metrics. Simulations are presented that compare timeiii

frequency representations of the classical time-frequency techniques with those of the reassignment method and the Hough transform. Two different triangular modulated frequency modulated continuous wave low probability of intercept radar signals and two frequency shift keying low probability of intercept radar signals (4-component and 8component) were analyzed. The following metrics were used for evaluation of the analysis: percent error of: carrier frequency, modulation bandwidth, modulation period, time-frequency localization, and chirp rate. Also used were: percent detection, number of cross-term false positives, lowest signal-to-noise ratio for signal detection, and plot (processing) time.

Experimental results demonstrate that the ‘squeezing’ and

‘smoothing’ qualities of the reassignment method, along with the cross-term interference suppressing/separating and low signal-to-noise ratio performance qualities of the Hough transform, do in fact lead to improved readability over the classical time-frequency analysis techniques, and consequently, provide more accurate signal detection and parameter extraction metrics than the classical time-frequency analysis techniques. In addition, the Hough transform was utilized to detect, extract parameters, and properly identify a real-world low probability of intercept radar signal in a low signal-to-noise ratio environment, where classical time-frequency analysis failed. In summary, this dissertation provides evidence that the reassignment method and the Hough transform have the potential to outperform the classical time-frequency analysis techniques, which are the current state-of-the-art, cutting edge techniques for this arena. An improvement in performance can easily translate into saved equipment and lives. Future work will include automation of metrics extraction process, analysis of additional low probability of

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intercept radar waveforms of interest, and analysis of other real-world low probability of intercept radar signals utilizing more powerful computing platforms.

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Acknowledgements I’d like to thank the following people, who all played a part in this major accomplishment:

My ‘better-half’, Donna, my loving wife, who lives the life of a Proverbs 31 woman.

Abraham, Solomon, Joy, Judah, Christian, and Glory, my six blessings from the Lord, who continually radiate life, and who are always an inspiration to me.

My Mom, Judy, whose prayers through the years have kept me on the ‘straight and narrow’.

My sister Cathy (Snorque), and my brothers Mike and Jake, who are undoubtedly smarter than I am, but whose heads aren’t quite as large.

My Father, Roger, who was himself a brilliant man, and who is currently part of that ‘great cloud of witnesses’ that we will once again see one day.

Prof. Stephanie Schuckers, who has, in both a personal and professional way, guided me through this process, all the while believing in this ‘non-traditional’ (i.e. old  ) student. She has been as much a friend as an advisor, as we are at similar stages in our lives.

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Dr. Dick Wiley, Dr. Phil Pace, Dr. Steven Kay, and Dr. G. Faye Boudreaux-Bartels: for allowing me to bounce ideas and questions off of their sharp minds.

My committee members: Prof. Bob Schilling (who helped me get the rust out early on), Prof. Abdul Jerri (who always has a smile on his face), Prof. Jeremiah Remus (one of the nicest people around), and Dr. Andy Noga (aka Dr. Demodulation - who pushed all the right buttons to get the banana!) – Thank-you for your feedback and your helpful suggestions.

Prof. Tom Ortmeyer: for introducing me to the program.

John Grieco, Steve Johns, Ed Starczewski, and Dr. John Maher: for their in-house support.

Charlie Estrella: for his MATLAB wisdom (Go Raiders!).

Finally, I would like to thank Jesus, because apart from Him I can do nothing (John 15:5), but with Him I can do all things (Phil 4:13).

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Table of Contents Chapter

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Abstract .............................................................................................................................. iii Acknowledgements ............................................................................................................ vi Table of Contents ............................................................................................................. viii List of Figures .................................................................................................................... xi List of Tables ................................................................................................................. xxiv 1. Introduction ......................................................................................................................1 1.1. Intercept Receiver Signal Analysis Techniques ........................................................3 1.2. Classical Time-Frequency Analysis Techniques .......................................................4 1.3. The Reassignment Method ........................................................................................9 1.4. The Hough Transform..............................................................................................11 2. Background ....................................................................................................................14 2.1. LPI Radar Overview ................................................................................................14 2.2. LPI Radar Characteristics ........................................................................................15 2.3. LPI Radar Waveforms .............................................................................................16 2.4. LPI Radar Applications and Examples ....................................................................18 2.5. Detection of LPI Radars: Intercept Receiver Overview ..........................................20 2.6. Intercept Receiver Signal Analysis Techniques ......................................................21 2.6.1. Time-Frequency Analysis ..................................................................................21 2.6.2. Wigner-Ville Distribution (WVD) .....................................................................22 2.6.3. Choi-Williams Distribution (CWD) ..................................................................31 2.6.4. Spectrogram (Short-Time Fourier Transform (STFT)) .....................................32 viii

2.6.5. Scalogram (Wavelet Transform)........................................................................34 2.6.6. The Reassignment Method ................................................................................38 2.6.7. The Hough Transform........................................................................................49 2.7. Previous Related Studies..........................................................................................58 2.7.1. ‘Detection of Low Probability of Intercept Radar Signals’ ...............................58 2.7.2. ‘Extraction of Polyphase Radar Modulation Parameters using a Wigner-Ville Distribution Radon Transform’..........................................................................................58 2.7.3. ‘Periodic Wigner-Ville Hough Transform’ .......................................................60 2.7.4. ‘Detection and Parameter Estimation of Chirped Radar Signals’ .....................61 3. Methodology ..................................................................................................................64 4. Detection and Parameter Extraction of Low Probability of Intercept Radar Signals Using the Reassignment Method .......................................................................................85 4.1. Abstract ....................................................................................................................85 4.2. Introduction ..............................................................................................................86 4.3. Methodology ............................................................................................................91 4.4. Results ....................................................................................................................102 4.5. Discussion ..............................................................................................................109 4.6. Conclusions ............................................................................................................119 5. Detection and Parameter Extraction of Low Probability of Intercept Radar Signals Using the Hough Transform ............................................................................................122 5.1. Abstract ..................................................................................................................122 5.2. Introduction ............................................................................................................123 5.3. Methodology ..........................................................................................................128 ix

5.4. Results ....................................................................................................................138 5.5. Discussion ..............................................................................................................148 5.6. Conclusions ............................................................................................................156 6. Joint Sequential Use of the Reassignment Method and the Hough Transform ...........158 7. Overall Conclusions and Future Work ........................................................................161 References ........................................................................................................................165 Appendix A ......................................................................................................................176 Appendix B ......................................................................................................................184

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List of Figures Figure

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Figure 1: WVD of a triangular modulated FMCW signal at an SNR of 0 dB (256 samples) .............................................................................................................................27 Figure 2: Same as Figure 1, but with the SNR level increased to 10 dB. Though some of the noise has disappeared, the cross-terms remain ............................................................27 Figure 3: Same as Figure 1, but with the SNR level increased to 20 dB. Most of the noise has disappeared, but the cross-terms remain, demonstrating that cross-terms are virtually unaffected by the SNR level ..............................................................................................28 Figure 4: Example of cross-term false positives (XFPs). There are 4 true signals (fc1=1KHZ, fc2=1.75KHz, fc3=0.75KHz, fc4=1.25KHz) and 6 XFPs (cross-terms that were wrongly declared as signal detections) (ct1=0.875KHz, ct2=1KHz, ct3=1.125KHz, ct4=1.25KHz, ct5=1.375KHz, ct6=1.5KHz). WVD of a 4-component FSK signal at SNR=10dB (512 samples) .................................................................................................29 Figure 5: RSPWVD of triangular modulated FMCW, 256 samples, SNR=10dB. Threshold of 5% of the maximum intensity (upper left) and 1% threshold (upper right). The SNR has been increased to 25dB (lower-left) and to 50dB (lower-right). The plot on the upper left gives the appearance of perfectly localized distributions for the chirp signals. However, the plot on the upper right (1% threshold) reveals artifacts that are clearly not perfectly localized. In order to determine if these artifacts are due to noise, the SNR is raised to 25dB (lower left). Many of the artifacts disappear, but some remain. The SNR is then raised to 50dB (lower right) to see if the remaining artifacts are also noise related, but most of the artifacts in the 25dB plot are also in the 50dB plot. For time-frequency reassignment, it is known that the localization is slightly degraded at those points where the IF cannot be considered as quasi-linear within a TF smoothing window (i.e. the curved areas of the signal in Figure 5) causing artifacts. In addition, when more than one component is within a TF smoothing window (i.e. the curves where the chirp signals meet in Figure 5), a beating effect occurs, resulting in interference fringes which may manifest as artifacts .............................................................................45 Figure 6: Time-frequency plot on the left and Hough transform plot on the right. A point in the TF plot maps to a sinusoidal curve in the HT plot. A line (signal) in the TF plot maps to a point in the HT plot. The rho and theta values of the point in the HT plot can be used to back-map to the TF plot in order to find the location of the line (signal) (good if time-frequency plot is cluttered with noise and/or cross-term interference and signal is not visible) .........................................................................................................................51 Figure 7: Hough transform of the WVD of a triangular modulated FMCW signal at an SNR of 10 dB (512 samples). Each point has a unique theta and rho value which can be xi

used to back-map to the time-frequency (WVD) representation in order to locate the 4 signals, as depicted in Figure 7 ..........................................................................................53 Figure 8: The unique theta and rho values extracted from Figure 6 are used to back-map to the time-frequency representation (WVD of a triangular modulated FMCW signal (SNR=10dB, #samples=512)) in order to locate the 4 chirp signals that make up the 4 legs of the triangular modulated FMCW signal .................................................................55 Figure 9: Threshold percentage determination. This plot is an amplitude vs. time (x-z view) of CWD of FSK 4-component signal (512 samples, SNR= -2dB). For visually detected low SNR plots (like this one), the percent of max intensity for the peak z-value of each of the signal components was noted (here 82%, 98%, 87%, 72%), and the lowest of these 4 values was recorded (72%). Ten test runs were performed for each timefrequency analysis tool, for each of the 4 waveforms. The average of these recorded low values was determined and then assigned as the threshold for that particular timefrequency analysis tool. Note - the threshold for CWD is 70%........................................68 Figure 10: Percent detection (time-frequency). CWD of 4-component FSK (512 samples, SNR=10dB) with threshold value automatically set to 70%. From this threshold plot, the signal was declared a (visual) detection because at least a portion of each of the 4 FSK signal components was visible ...........................................................................................69 Figure 11: Percent detection (Hough transform). This plot is a theta-intensity (x-z view) of the HT of an RSPWVD (512 samples, SNR=10dB). Signal declared a (visual) detection because at least a portion of each of the signal components exceeded the noise floor threshold ....................................................................................................................70 Figure 12: Example of cross-term false positives (XFPs). WVD of a 4-component FSK signal at SNR=10dB (512 samples). There are 4 true signals (fc1=1KHz, fc2=1.75KHz, fc3=0.75KHz, fc4=1.25KHz) and 6 XFPs (cross-terms that were wrongly declared as signal detections because they passed the signal detection criteria listed in the percent detection section above) (ct1=0.875KHz, ct2=1KHz, ct3=1.125KHz, ct4=1.25KHz, ct5=1.375KHz, ct6=1.5KHz) .............................................................................................71 Figure 13: Determination of carrier frequency. CWD of a triangular modulated FMCW (256 samples, SNR=10dB). From the frequency-intensity (y-z) view, the maximum intensity value is manually determined. The frequency corresponding to the max intensity value is the carrier frequency (here fc=1062.4 Hz).............................................72 Figure 14: Determination of carrier frequency. CWD of a 4-component FSK (512 samples, SNR=10dB). From the frequency-intensity (y-z) view, the 4 maximum intensity values (1 for each carrier frequency) are manually determined. The frequencies corresponding to those 4 max intensity values are the 4 carrier frequency (here fc1=10001 Hz, fc2=1753Hz, fc3=757Hz, fc4=1255Hz) ...................................................73

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Figure 15: Modulation bandwidth determination. CWD of a triangular modulated FMCW (512 samples, SNR=10dB) with threshold value automatically set to 20%. From this threshold plot, the modulation bandwidth was measured manually from the highest frequency value of the signal (top red arrow) to the lowest frequency value of the signal (bottom red arrow) in the y-direction (frequency) .............................................................74 Figure 16: Modulation bandwidth determination. CWD of a 4-component FSK (512 samples, SNR=10dB) with threshold value automatically set to 20%. From this threshold plot, the modulation bandwidth was measured manually from the highest frequency value of the signal (top red arrow) to the lowest frequency value of the signal (bottom red arrow) in the y-direction (frequency) .................................................................................75 Figure 17: Modulation period determination. CWD of a triangular modulated FMCW (512 samples, SNR=10dB) with threshold value automatically set to 20%. From this threshold plot, the modulation period was measured manually from the highest frequency value of the signal (top red arrow) to the lowest frequency value of the signal (bottom red arrow) in the x-direction (time)..........................................................................................77 Figure 18: Modulation period determination. CWD of a 4-component FSK (512 samples, SNR=10dB) with threshold value automatically set to 20%. From this threshold plot, the modulation period was measured manually from the left side of the signal (left red arrow) to the right side of the signal (right red arrow) in the x-direction (time). This was done for all 4 signal components, and the average value was determined .................................78 Figure 19: Time-frequency localization determination. CWD of a triangular modulated FMCW (512 samples, SNR=10dB) with threshold value automatically set to 20%. From this threshold plot, the time-frequency localization was measured manually from the left side of the signal (left red arrow) to the right side of the signal (right red arrow) in both the x-direction (time) and the y-direction (frequency). Measurements were made at the center of each of the 4 ‘legs’, and the average values were determined. Average time and frequency ‘thickness’ values were then converted to: % of entire x-axis and % of entire yaxis .....................................................................................................................................80 Figure 20: Time-frequency localization determination for the CWD of a 4-component FSK (512 samples, SNR=10dB) with threshold value automatically set to 20%. From this threshold plot, the time-frequency localization was measured manually from the top of the signal (top red arrow) to the bottom of the signal (bottom red arrow) in the ydirection (frequency). This frequency ‘thickness’ value was then converted to: % of entire y-axis. .......................................................................................................................81 Figure 21: Lowest detectable SNR (time-frequency). CWD of 4-component FSK (512 samples, SNR=-2dB) with threshold value automatically set to 70%. From this threshold plot, the signal was declared a (visual) detection because at least a portion of each of the 4 FSK signal components was visible. For this case, any lower SNR would have been a non-detect. Compare to Figure 10, which is the same plot, except that it has an SNR level equal to 10dB ............................................................................................................82 xiii

Figure 22: Lowest detectable SNR (Hough transform). This plot is a theta-intensity (x-z view) of the HT of an RSPWVD (512 samples, SNR=-5dB). Signal declared a (visual) detection because at least a portion of each of the signal components exceeded the noise floor threshold. For this case, any lower SNR would have been a non-detect. Compare to Figure 11, which is the same plot, except that it has an SNR level equal to 10dB .......83 Figure 23: Threshold percentage determination. This plot is an amplitude vs. time (x-z view) of CWD of FSK 4-component signal (512 samples, SNR= -2dB). For visually detected low SNR plots (like this one), the percent of max intensity for the peak z-value of each of the signal components was noted (here 82%, 98%, 87%, 72%), and the lowest of these 4 values was recorded (72%). Ten test runs were performed for each timefrequency analysis tool, for each of the 4 waveforms. The average of these recorded low values was determined and then assigned as the threshold for that particular timefrequency analysis tool. Note - the threshold for CWD is 70%......................................95 Figure 24: Percent detection (time-frequency). CWD of 4-component FSK (512 samples, SNR=10dB) with threshold value automatically set to 70%. From this threshold plot, the signal was declared a (visual) detection because at least a portion of each of the 4 FSK signal components was visible ...........................................................................................96 Figure 25: Modulation bandwidth determination. CWD of a triangular modulated FMCW (512 samples, SNR=10dB) with threshold value automatically set to 20%. From this threshold plot, the modulation bandwidth was measured manually from the highest frequency value of the signal (top red arrow) to the lowest frequency value of the signal (bottom red arrow) in the y-direction (frequency) .............................................................98 Figure 26: Modulation bandwidth determination. CWD of a 4-component FSK (512 samples, SNR=10dB) with threshold value automatically set to 20%. From this threshold plot, the modulation bandwidth was measured manually from the highest frequency value of the signal (top red arrow) to the lowest frequency value of the signal (bottom red arrow) in the y-direction (frequency) .................................................................................99 Figure 27: Lowest detectable SNR (time-frequency). CWD of 4-component FSK (512 samples, SNR=-2dB) with threshold value automatically set to 70%. From this threshold plot, the signal was declared a (visual) detection because at least a portion of each of the 4 FSK signal components was visible. For this case, any lower SNR would have been a non-detect. Compare to Figure 24, which is the same plot, except that it has an SNR level equal to 10dB ..........................................................................................................101 Figure 28: Time-frequency localization comparison between the classical time-frequency analysis tools (left) and the reassignment method (right). The uppper two plots are the spectrogram (left) and the reassigned spectrogram (right) for a triangular modulated FMCW signal (256 samples, SNR=10dB), and the lower two plots are the scalogram (left) and the reassigned scalogram (right) for a 4-component FSK signal (512 samples, xiv

SNR=10dB). The reassignment method gives a much more concentrated time-frequency localization than does its classical time-frequency analysis counterparts .......................104 Figure 29: Cross-term interference of the WVD (left) and ability of the RSPWVD to reduce cross-term interference (right). The uppper two plots are the WVD (left) and the RSPWVD (right) for a triangular modulated FMCW signal (512 samples, SNR=10dB), and the lower two plots are the WVD (left) and the RSPWVD (right) for a 8-component FSK signal (512 samples, SNR=10dB). The WVD displays a lot of cross-term interference. There appears to be one additional triangle signal in the middle of the twotriangle signal in the upper-left plot, and there appears to be 9 additional signals to go along with the 8-component signal in the lower-left plot. The RSPWVD has drastically reduced the cross-term interference found in the WVD plots, making for more readable presentations. Notice also that the RSPWVD is more concentrated than its WVD counterpart (as per Table 1) .............................................................................................105 Figure 30: Readability degradation due to reduction in SNR. Spectrogram, triangular modulated FMCW, modulation bandwidth=500Hz, 512 samples. SNR=10dB (left), 0dB (center), -4dB (left). Readability degrades as SNR decreases, negatively affecting the accuracy of the metrics extracted, as per Table 2. ...........................................................107 Figure 31: Comparison between Task 1 (left) and Task 2 (right). The uppper two plots are both spectrogram plots of a triangular modulated FMCW signal (512 samples, SNR=10dB), Task 1 (modulation bandwidth=500Hz) is on the left and Task 2 (modulation bandwidth=2400Hz) is on the right. The lower two plots are CWD plots of the same signals. The Task 2 plots (right) have a larger modulation bandwidth than Task 1 plots, therefore the signals appear taller and more upright than the Task 1 signals .....109 Figure 32: Percent detection (Hough transform). This plot is a theta-intensity (x-z view) of the HT of an RSPWVD (512 samples, SNR=10dB). Signal declared a (visual) detection because at least a portion of each of the signal components exceeded the noise floor threshold ..................................................................................................................132 Figure 33: Threshold percentage determination. This plot is an amplitude vs. time (x-z view) of CWD of FSK 4-component signal (512 samples, SNR= -2dB). For visually detected low SNR plots (like this one), the percent of max intensity for the peak z-value of each of the signal components was noted (here 82%, 98%, 87%, 72%), and the lowest of these 4 values was recorded (72%). Ten test runs were performed for each timefrequency analysis tool, for each of the 4 waveforms. The average of these recorded low values was determined and then assigned as the threshold for that particular timefrequency analysis tool. Note - the threshold for CWD is 70%....................................133 Figure 34: Percent detection (time-frequency). CWD of 4-component FSK (512 samples, SNR=10dB) with threshold value automatically set to 70%. From this threshold plot, the signal was declared a (visual) detection because at least a portion of each of the 4 FSK signal components was visible .........................................................................................134 xv

Figure 35: S Example of cross-term false positives (XFPs). WVD of a 4-component FSK signal at SNR=10dB (512 samples). There are 4 true signals (fc1=1KHz, fc2=1.75KHz, fc3=0.75KHz, fc4=1.25KHz) and 6 XFPs (cross-terms that were wrongly declared as signal detections because they passed the signal detection criteria listed in the percent detection section above) (ct1=0.875KHz, ct2=1KHz, ct3=1.125KHz, ct4=1.25KHz, ct5=1.375KHz, ct6=1.5KHz) ...........................................................................................135 Figure 36: Lowest detectable SNR (Hough transform). This plot is a theta-intensity (x-z view) of the HT of an RSPWVD (512 samples, SNR=-5dB). Signal declared a (visual) detection because at least a portion of each of the signal components exceeded the noise floor threshold. For this case, any lower SNR would have been a non-detect. Compare to Figure 32, which is the same plot, except that it has an SNR level equal to 10dB .....136 Figure 37: Lowest detectable SNR (time-frequency). CWD of 4-component FSK (512 samples, SNR=-2dB) with threshold value automatically set to 70%. From this threshold plot, the signal was declared a (visual) detection because at least a portion of each of the 4 FSK signal components was visible. For this case, any lower SNR would have been a non-detect. Compare to Figure 34, which is the same plot, except that it has an SNR level equal to 10dB ..........................................................................................................137 Figure 38: Cross-term comparison between classical time-frequency analysis techniques (left) and the Hough transform (right). Top left: WVD of a triangular modulated FMCW signal (512 samples, SNR=10dB) (left). Top right: the Hough transform of the WVD of a triangular modulated FMCW signal (512 samples, SNR=10dB). Bottom left: WVD of an FSK (8-component) signal (512 samples, SNR=10dB). Bottom right: the Hough transform of the WVD of an FSK (8-component) signal (512 samples, SNR=10dB). Upper 2 plots – the Hough transform (right) has eliminated the cross-term interference that the WVD (left) displays, making it easier to see the signal (better readability) in the Hough transform plot (the four bright spots which represent the four legs of the triangular modulated FMCW signal). The WVD appears to have another triangle signal between the outer two triangle signals. Lower 2 plots - In the WVD plot (left) there are 8 signal components and 9 cross-term components that appear to be signal components, all melded in together with one another. The 8 signal components are located at 5 distinct frequencies (one at 0.75KHz, two at 1KHz, two at 1.25KHz, two at 1.5KHz, and one at 1.75KHz). In the Hough transform plot (right) there are 8 signal components (located at 5 different frequencies (3 on the left-hand side of the plot and 2 in the middle of the plot)) plus 2 cross-term components, clearly separated from the signal components. The Hough transform plot makes it easier to see the signal components (better readability) as compared to the WVD plot ..............................................................................................141 Figure 39: Low SNR comparison between classical time-frequency analysis techniques (left) and the Hough transform (right). Top left: CWD of a triangular modulated FMCW signal (512 samples, SNR=-6dB) (left). Top right: the Hough transform of the CWD of a triangular modulated FMCW signal (512 samples, SNR=-6dB). Bottom left: CWD of an FSK (8-component) signal (512 samples, SNR=-3dB). Bottom right: the Hough transform of the CWD of an FSK (8-component) signal (512 samples, SNR=-3dB). xvi

Upper 2 plots - Though the signal is not visible in the CWD plot (left) (due to the low SNR (-6dB)), the four bright spots that represent the four legs of the triangular modulated FMCW signal are clearly seen in the Hough transform of the CWD plot (right). Each bright spot has a unique rho and theta value that can be used to back-map to the timefrequency representation (here CWD) and find the location of the 4 (non-visible) chirps that make up the triangular modulated FMCW signal. Lower 2 plots - Though the signal components are not visible in the CWD plot (due to the low SNR (-3dB)), the 5 bright spots (3 on the left and 2 in the middle) corresponding to the 5 different frequencies of the 8 FSK components are clearly visible in the Hough transform of the CWD plot (as are 2 cross-term components). The Hough transform does a good job of detecting the signal components and separating the signal components from the cross-term components, all in a low SNR environment ...................................................................................................142 Figure 40: Readability degradation due to reduction in SNR. Spectrogram, triangular modulated FMCW, modulation bandwidth=500Hz, 512 samples. SNR=10dB (left), 0dB (center), -4dB (left). Readability degrades as SNR decreases, negatively affecting the accuracy of the metrics extracted, as per Table 5 ............................................................144 Figure 41: Comparison between Task 1 (left) and Task 2 (right). The uppper two plots are both spectrogram plots of a triangular modulated FMCW signal (512 samples, SNR=10dB), Task 1 (modulation bandwidth=500Hz) is on the left and Task 2 (modulation bandwidth=2400Hz) is on the right. The lower two plots are CWD plots of the same signals. The Task 2 plots (right) have a larger modulation bandwidth than Task 1 plots, therefore the signals appear taller and more upright than the Task 1 signals .....145 Figure 42: Spectrogram (left) and Hough transform of Spectrogram (right) of area where Task 5 real-world signal was located. Signal was not visible in the spectrogram, but was visible in the Hough transform, due to the Hough transform’s ability to extract signals from low SNR environments. ..........................................................................................146 Figure 43: Hough transform of Spectrogram (zoomed-in on receiver IF bandwidth (~750MHz to 1250MHz)) of area where Task 5 real-world signal was located. Signal is clearly visible at theta=2.872 and rho=228.8 ...................................................................147 Figure 44: Spectrogram (zoomed-in on receiver IF bandwidth (~750MHz to 1250MHz)). Shows how the Hough transform theta and rho values back-map to the time-frequency representation for signal location .....................................................................................148 TASK 1 Figure 45: Test 1 Plot – WVD, 256, 10 ...........................................................................184 Figure 46: Test 2 Plot – WVD, 256, 0 .............................................................................185 Figure 47: Test 3 Plot – WVD, 256, -2 ............................................................................186 xvii

Figure 48: Test 4 Plot – WVD, 512, 10 ...........................................................................187 Figure 49: Test 7 Plot – CWD, 256, 10 ...........................................................................189 Figure 50: Test 8 Plot – CWD, 256, 0 .............................................................................190 Figure 51: Test 9 Plot – CWD, 256, -3 ............................................................................191 Figure 52: Test 10 Plot – CWD, 512, 10 .........................................................................192 Figure 53: Test 11 Plot – CWD, 512, 0 ...........................................................................193 Figure 54: Test 12 Plot – CWD, 512, -3 ..........................................................................194 Figure 55: Test 13 Plot – Spectrogram, 256, 10 ..............................................................195 Figure 56: Test 14 Plot – Spectrogram, 256, 0 ................................................................196 Figure 57: Test 15 Plot – Spectrogram, 256, -3 ...............................................................197 Figure 58: Test 16 Plot – Spectrogram, 512, 10 ..............................................................198 Figure 59: Test 17 Plot – Spectrogram, 512, 0 ................................................................199 Figure 60: Test 18 Plot – Spectrogram, 512, -4 ...............................................................200 Figure 61: Test 19 Plot – Scalogram, 256, 10..................................................................201 Figure 62: Test 20 Plot – Scalogram, 256, 0....................................................................202 Figure 63: Test 21 Plot – Scalogram, 256, -2 ..................................................................203 Figure 64: Test 22 Plot – Scalogram, 512, 10..................................................................204 Figure 65: Test 23 Plot – Scalogram, 512, 0....................................................................205 Figure 66: Test 24 Plot – Scalogram, 512, -3 ..................................................................206 Figure 67: Test 25 Plot – Reassigned Spectrogram, 256, 10 ...........................................207 Figure 68: Test 26 Plot – Reassigned Spectrogram, 256, 0 .............................................208 Figure 69: Test 27 Plot – Reassigned Spectrogram, 256, -2 ............................................209 Figure 70: Test 28 Plot – Reassigned Spectrogram, 512, 10 ...........................................210 xviii

Figure 71: Test 29 Plot – Reassigned Spectrogram, 512, 0 .............................................211 Figure 72: Test 30 Plot – Reassigned Spectrogram, 512, -3 ............................................212 Figure 73: Test 31 Plot – Reassigned Scalogram, 256, 10 ..............................................213 Figure 74: Test 32 Plot – Reassigned Scalogram, 256, 0 ................................................214 Figure 75: Test 33 Plot – Reassigned Scalogram, 256, -2 ...............................................215 Figure 76: Test 34 Plot – Reassigned Scalogram, 512, 10 ..............................................216 Figure 77: Test 35 Plot – Reassigned Scalogram, 512, 0 ................................................217 Figure 78: Test 36 Plot – Reassigned Scalogram, 512, -3 ...............................................218 Figure 79: Test 37 Plot – RSPWVD, 256, 10 ..................................................................219 Figure 80: Test 38 Plot – RSPWVD, 256, 0 ....................................................................220 Figure 81: Test 39 Plot – RSPWVD, 256, -3 ...................................................................221 Figure 82: Test 40 Plot – RSPWVD, 512, 10 ..................................................................222 Figure 83: Test 41 Plot – RSPWVD, 512, 0 ....................................................................223 Figure 84: Test 42 Plot – RSPWVD, 512, -3 ...................................................................224 Figure 85: Test 43 Plot – HT of WVD, 256, 10 ..............................................................225 Figure 86: Test 44 Plot – HT of WVD, 256, 0 ................................................................226 Figure 87: Test 45 Plot – HT of WVD, 256, -3 ...............................................................227 Figure 88: Test 46 Plot – HT of WVD, 512, 10 ..............................................................228 Figure 89: Test 49 Plot – HT of CWD, 256, 10 ...............................................................230 Figure 90: Test 50 Plot – HT of CWD, 256, 0 .................................................................231 Figure 91: Test 51 Plot – HT of CWD, 256, -4 ...............................................................232 Figure 92: Test 52 Plot – HT of CWD, 512, 10 ...............................................................233 Figure 93: Test 53 Plot – HT of CWD, 512, 0 .................................................................234 xix

Figure 94: Test 54 Plot – HT of CWD, 512, -6 ...............................................................235 Figure 95: Test 55 Plot – HT of RSPWVD, 256, 10 .......................................................236 Figure 96: Test 56 Plot – HT of RSPWVD, 256, 0 .........................................................237 Figure 97: Test 57 Plot – HT of RSPWVD, 256, -4 ........................................................238 Figure 98: Test 58 Plot – HT of RSPWVD, 512, 10 .......................................................239 Figure 99: Test 59 Plot – HT of RSPWVD, 512, 0 .........................................................240 Figure 100: Test 60 Plot – HT of RSPWVD, 512, -5 ......................................................241 TASK 2 Figure 101: Test 61 Plot – WVD, 512, 10 .......................................................................242 Figure 102: Test 64 Plot – CWD, 512, 10 .......................................................................244 Figure 103: Test 65 Plot – CWD, 512, 0 .........................................................................245 Figure 104: Test 66 Plot – CWD, 512, -3 ........................................................................246 Figure 105: Test 67 Plot – Spectrogram, 512, 10 ............................................................247 Figure 106: Test 68 Plot – Spectrogram, 512, 0 ..............................................................248 Figure 107: Test 69 Plot – Spectrogram, 512, -4 .............................................................249 Figure 108: Test 70 Plot – Scalogram, 512, 10................................................................250 Figure 109: Test 71 Plot – Scalogram, 512, 0..................................................................251 Figure 110: Test 72 Plot – Scalogram, 512, -3 ................................................................252 Figure 111: Test 73 Plot – Reassigned Spectrogram, 512, 10 .........................................253 Figure 112: Test 74 Plot – Reassigned Spectrogram, 512, 0 ...........................................254 Figure 113: Test 75 Plot – Reassigned Spectrogram, 512, -3 ..........................................255 Figure 114: Test 76 Plot – Reassigned Scalogram, 512, 10 ............................................256 Figure 115: Test 77 Plot – Reassigned Scalogram, 512, 0 ..............................................257 xx

Figure 116: Test 78 Plot – Reassigned Scalogram, 512, -3 .............................................258 Figure 117: Test 79 Plot – RSPWVD, 512, 10 ................................................................259 Figure 118: Test 80 Plot – RSPWVD, 512, 0 ..................................................................260 Figure 119: Test 81 Plot – RSPWVD, 512, -3 .................................................................261 Figure 120: Test 82 Plot – HT of WVD, 512, 10 ............................................................263 Figure 121: Test 85 Plot – HT of CWD, 512, 10 .............................................................264 Figure 122: Test 86 Plot – HT of CWD, 512, 0 ...............................................................265 Figure 123: Test 87 Plot – HT of CWD, 512, -6 .............................................................266 Figure 124: Test 88 Plot – HT of RSPWVD, 512, 10 .....................................................267 Figure 125: Test 89 Plot – HT of RSPWVD, 512, 0 .......................................................268 Figure 126: Test 90 Plot – HT of RSPWVD, 512, -6 ......................................................269 TASK 3 Figure 127: Test 91 Plot – WVD, 512, 10 .......................................................................270 Figure 128: Test 94 Plot – CWD, 512, 10 .......................................................................272 Figure 129: Test 95 Plot – CWD, 512, 0 .........................................................................273 Figure 130: Test 96 Plot – CWD, 512, -2 ........................................................................274 Figure 131: Test 97 Plot – Spectrogram, 512, 10 ............................................................275 Figure 132: Test 98 Plot – Spectrogram, 512, 0 ..............................................................276 Figure 133: Test 99 Plot – Spectrogram, 512, -3 .............................................................277 Figure 134: Test 100 Plot – Scalogram, 512, 10..............................................................278 Figure 135: Test 101 Plot – Scalogram, 512, 0................................................................279 Figure 136: Test 102 Plot – Scalogram, 512, -3 ..............................................................280 Figure 137: Test 103 Plot – Reassigned Spectrogram, 512, 10 .......................................281 xxi

Figure 138: Test 104 Plot – Reassigned Spectrogram, 512, 0 .........................................282 Figure 139: Test 105 Plot – Reassigned Spectrogram, 512, -3 ........................................283 Figure 140: Test 106 Plot – Reassigned Scalogram, 512, 10 ..........................................284 Figure 141: Test 107 Plot – Reassigned Scalogram, 512, 0 ............................................285 Figure 142: Test 108 Plot – Reassigned Scalogram, 512, -4 ...........................................286 Figure 143: Test 109 Plot – RSPWVD, 512, 10 ..............................................................287 Figure 144: Test 110 Plot – RSPWVD, 512, 0 ................................................................288 Figure 145: Test 111 Plot – RSPWVD, 512, -3 ...............................................................289 Figure 146: Test 112 Plot – HT of WVD, 512, 10 ..........................................................290 Figure 147: Test 112 Plot – X-Y View - HT of WVD, 512, 10 ......................................290 Figure 148: Test 113 Plot – HT of CWD, 512, -3 ...........................................................292 Figure 149: Test 113 Plot – X-Y View - HT of CWD, 512, -3 .......................................292 Figure 150: Test 114 Plot – HT of RSPWVD, 512, -4 ....................................................294 Figure 151: Test 114 Plot – X-Y View - HT of RSPWVD, 512, -4 ................................294 TASK 4 Figure 152: Test 115 Plot – WVD, 512, 10 .....................................................................296 Figure 153: Test 118 Plot – CWD, 512, 10 .....................................................................298 Figure 154: Test 119 Plot – CWD, 512, 0 .......................................................................299 Figure 155: Test 120 Plot – CWD, 512, -1 ......................................................................300 Figure 156: Test 121 Plot – Spectrogram, 512, 10 ..........................................................301 Figure 157: Test 122 Plot – Spectrogram, 512, 0 ............................................................302 Figure 158: Test 123 Plot – Spectrogram, 512, -1 ...........................................................303 Figure 159: Test 124 Plot – Scalogram, 512, 10..............................................................304 xxii

Figure 160: Test 125 Plot – Scalogram, 512, 0................................................................305 Figure 161: Test 126 Plot – Scalogram, 512, -3 ..............................................................306 Figure 162: Test 127 Plot – Reassigned Spectrogram, 512, 10 .......................................307 Figure 163: Test 128 Plot – Reassigned Spectrogram, 512, 0 .........................................308 Figure 164: Test 129 Plot – Reassigned Spectrogram, 512, -2 ........................................309 Figure 165: Test 130 Plot – Reassigned Scalogram, 512, 10 ..........................................310 Figure 166: Test 131 Plot – Reassigned Scalogram, 512, 0 ............................................311 Figure 167: Test 132 Plot – Reassigned Scalogram, 512, -2 ...........................................312 Figure 168: Test 133 Plot – RSPWVD, 512, 10 ..............................................................313 Figure 169: Test 134 Plot – RSPWVD, 512, 0 ................................................................314 Figure 170: Test 135 Plot – RSPWVD, 512, -3 ...............................................................315 Figure 171: Test 136 Plot – HT of WVD, 512, 10 ..........................................................316 Figure 172: Test 136 Plot – X-Y View - HT of WVD, 512, 10 ......................................316 Figure 173: Test 137 Plot – HT of CWD, 512, -3 ...........................................................318 Figure 174: Test 137 Plot – X-Y View - HT of CWD, 512, -3 .......................................318 Figure 175: Test 138 Plot – HT of RSPWVD, 512, -4 ....................................................320 Figure 176: Test 138 Plot – X-Y View - HT of RSPWVD, 512, -4 ................................320

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List of Tables Table 1: Signal Processing Tool viewpoint of the overall test metrics (average percent error) for the 4 classical time-frequency analysis techniques (WVD, CWD, spectrogram, scalogram), along with their combined average (TF) and for the 3 reassignment methods (reassigned spectrogram, reassigned scalogram, RSPWVD), along with their combined average (RM). The parameters extracted are listed in the left-hand column: carrier frequency (fc), modulation bandwidth (modbw), modulation period (modper), timefrequency localization in the x-direction (tf res-x), time-frequency localization in the ydirection (tf res-y), chirp rate (cr), percent detection (% det), number of cross-term false positives (#XFP), lowest detectable SNR (low snr), plot time (plottime) .......................103 Table 2: SNR viewpoint of overall test metrics (average percent error) for the classical time-frequency analysis techniques (TF) and for the reassignment method (RM) for SNR=10dB, 0dB, and lowest detectable SNR (low snr). The parameters extracted are listed in the left-hand column: carrier frequency (fc), modulation bandwidth (modbw), modulation period (modper), time-frequency localization in the x-direction (tf res-x), time-frequency localization in the y-direction (tf res-y), chirp rate (cr), percent detection (% det), number of cross-term false positives (#XFP), plot time (plottime) ...................106 Table 3: Task 1, 2, 3, and 4 viewpoint of overall test metrics (average percent error) for the classical time-frequency analysis techniques (TF) and for the reassignment method (RM). Task1=triangular modulated FMCW signal (modulation bandwidth=500Hz), Task2=triangular modulated FMCW signal (modulation bandwidth=2400Hz), Task 3=FSK (4-component) signal, Task 4=FSK (8-component) signal. The parameters extracted are listed in the left-hand column: carrier frequency (fc), modulation bandwidth (modbw), modulation period (modper), time-frequency localization in the x-direction (tf res-x), time-frequency localization in the y-direction (tf res-y), chirp rate (cr), percent detection (% det), number of cross-term false positives (#XFP), lowest detectable SNR (low snr), plot time (plottime) ..........................................................................................108 Table 4: Signal Processing Tool viewpoint of the overall test metrics (average percent error) for the 4 classical time-frequency analysis techniques (WVD, CWD, spectrogram, scalogram) along with their combined average (TF) and for the 2 Hough transform methods (WVD + HT, CWD + HT), along with their combined average (HT). The parameters extracted are listed in the left-hand column: chirp rate (cr), percent detection (% det), # of cross-term false positives (#XFP), lowest detectable SNR (low snr) .........139 Table 5: SNR viewpoint of overall test metrics (average percent error) for the classical time-frequency analysis techniques (TF) and for the Hough transform methods (HT) for SNR=10dB, 0dB, and lowest detectable SNR (low SNR). The parameters extracted are listed in the left-hand column: chirp rate (cr), percent detection (% det), number of crossterm false positives (#XFP). ............................................................................................143 Table 6: Task 1, 2, 3, and 4 viewpoint of overall test metrics (average percent error) for the classical time-frequency analysis techniques (TF) and for the Hough transform (HT). xxiv

Task1=triangular modulated FMCW signal (modulation bandwidth=500Hz), Task2=triangular modulated FMCW signal (modulation bandwidth=2400Hz), Task 3=FSK (4-component) signal, Task 4=FSK (8-component) signal. The parameters extracted are listed in the left-hand column: chirp rate (cr), percent detection (% det), number of cross-term false positives (#XFP), lowest detectable SNR (low snr) ............144 Table 7: Overall test metrics for the classical time-frequency analysis techniques (Classical TF), reassignment method (RM), Hough transform (HT), and reassignment method plus the Hough transform (RM + HT). The parameters extracted are listed in the left-hand column: chirp rate, lowest detectable SNR (Low SNR), and percent detection (% Detection) ...................................................................................................................159 Table 8: Test Matrix for Task 1 through Task 5 ..............................................................176 TASK 1 Table 9: Test 1 Metrics – WVD, 256, 10 .........................................................................184 Table 10: Test 2 Metrics – WVD, 256, 0 .........................................................................185 Table 11: Test 3 Metrics – WVD, 256, -2 .......................................................................186 Table 12: Test 4 Metrics – WVD, 512, 10 .......................................................................187 Table 13: Test 7 Metrics – CWD, 256, 10 .......................................................................189 Table 14: Test 8 Metrics – CWD, 256, 0 .........................................................................190 Table 15: Test 9 Metrics – CWD, 256, -3........................................................................191 Table 16: Test 10 Metrics – CWD, 512, 10 .....................................................................192 Table 17: Test 11 Metrics – CWD, 512, 0 .......................................................................193 Table 18: Test 12 Metrics – CWD, 512, -3......................................................................194 Table 19: Test 13 Metrics – Spectrogram, 256, 10 ..........................................................195 Table 20: Test 14 Metrics – Spectrogram, 256, 0 ............................................................196 Table 21: Test 15 Metrics – Spectrogram, 256, -3 ..........................................................197 Table 22: Test 16 Metrics – Spectrogram, 512, 10 ..........................................................198 Table 23: Test 17 Metrics – Spectrogram, 512, 0 ............................................................199 xxv

Table 24: Test 18 Metrics – Spectrogram, 512, -4 ..........................................................200 Table 25: Test 19 Metrics – Scalogram, 256, 10 .............................................................201 Table 26: Test 20 Metrics – Scalogram, 256, 0 ...............................................................202 Table 27: Test 21 Metrics – Scalogram, 256, -2 ..............................................................203 Table 28: Test 22 Metrics – Scalogram, 512, 10 .............................................................204 Table 29: Test 23 Metrics – Scalogram, 512, 0 ...............................................................205 Table 30: Test 24 Metrics – Scalogram, 512, -3 ..............................................................206 Table 31: Test 25 Metrics – Reassigned Spectrogram, 256, 10.......................................207 Table 32: Test 26 Metrics – Reassigned Spectrogram, 256, 0.........................................208 Table 33: Test 27 Metrics – Reassigned Spectrogram, 256, -2 .......................................209 Table 34: Test 28 Metrics – Reassigned Spectrogram, 512, 10.......................................210 Table 35: Test 29 Metrics – Reassigned Spectrogram, 512, 0.........................................211 Table 36: Test 30 Metrics – Reassigned Spectrogram, 512, -3 .......................................212 Table 37: Test 31 Metrics – Reassigned Scalogram, 256, 10 ..........................................213 Table 38: Test 32 Metrics – Reassigned Scalogram, 256, 0 ............................................214 Table 39: Test 33 Metrics – Reassigned Scalogram, 256, -2...........................................215 Table 40: Test 34 Metrics – Reassigned Scalogram, 512, 10 ..........................................216 Table 41: Test 35 Metrics – Reassigned Scalogram, 512, 0 ............................................217 Table 42: Test 36 Metrics – Reassigned Scalogram, 512, -3...........................................218 Table 43: Test 37 Metrics – RSPWVD, 256, 10..............................................................219 Table 44: Test 38 Metrics – RSPWVD, 256, 0................................................................220 Table 45: Test 39 Metrics – RSPWVD, 256, -3 ..............................................................221 Table 46: Test 40 Metrics – RSPWVD, 512, 10..............................................................222 xxvi

Table 47: Test 41 Metrics – RSPWVD, 512, 0................................................................223 Table 48: Test 42 Metrics – RSPWVD, 512, -3 ..............................................................224 Table 49: Test 43 Metrics – HT of WVD, 256, 10 ..........................................................225 Table 50: Test 44 Metrics – HT of WVD, 256, 0 ............................................................226 Table 51: Test 45 Metrics – HT of WVD, 256, -3 ...........................................................227 Table 52: Test 46 Metrics – HT of WVD, 512, 10 ..........................................................228 Table 53: Test 49 Metrics – HT of CWD, 256, 10 ..........................................................230 Table 54: Test 50 Metrics – HT of CWD, 256, 0 ............................................................231 Table 55: Test 51 Metrics – HT of CWD, 256, -4 ...........................................................232 Table 56: Test 52 Metrics – HT of CWD, 512, 10 ..........................................................233 Table 57: Test 53 Metrics – HT of CWD, 512, 0 ............................................................234 Table 58: Test 54 Metrics – HT of CWD, 512, -6 ...........................................................235 Table 59: Test 55 Metrics – HT of RSPWVD, 256, 10 ...................................................236 Table 60: Test 56 Metrics – HT of RSPWVD, 256, 0 .....................................................237 Table 61: Test 57 Metrics – HT of RSPWVD, 256, -4....................................................238 Table 62: Test 58 Metrics – HT of RSPWVD, 512, 10 ...................................................239 Table 63: Test 59 Metrics – HT of RSPWVD, 512, 0 .....................................................240 Table 64: Test 60 Metrics – HT of RSPWVD, 512, -5....................................................241 TASK 2 Table 65: Test 61 Metrics – WVD, 512, 10 .....................................................................242 Table 66: Test 64 Metrics – CWD, 512, 10 .....................................................................244 Table 67: Test 65 Metrics – CWD, 512, 0 .......................................................................245 Table 68: Test 66 Metrics – CWD, 512, -3......................................................................246 xxvii

Table 69: Test 67 Metrics – Spectrogram, 512, 10 ..........................................................247 Table 70: Test 68 Metrics – Spectrogram, 512, 0 ............................................................248 Table 71: Test 69 Metrics – Spectrogram, 512, -4 ..........................................................249 Table 72: Test 70 Metrics – Scalogram, 512, 10 .............................................................250 Table 73: Test 71 Metrics – Scalogram, 512, 0 ...............................................................251 Table 74: Test 72 Metrics – Scalogram, 512, -3 ..............................................................252 Table 75: Test 73 Metrics – Reassigned Spectrogram, 512, 10.......................................253 Table 76: Test 74 Metrics – Reassigned Spectrogram, 512, 0.........................................254 Table 77: Test 75 Metrics – Reassigned Spectrogram, 512, -3 .......................................255 Table 78: Test 76 Metrics – Reassigned Scalogram, 512, 10 ..........................................256 Table 79: Test 77 Metrics – Reassigned Scalogram, 512, 0 ............................................257 Table 80: Test 78 Metrics – Reassigned Scalogram, 512, -3...........................................258 Table 81: Test 79 Metrics – RSPWVD, 512, 10..............................................................259 Table 82: Test 80 Metrics – RSPWVD, 512, 0................................................................260 Table 83: Test 81 Metrics – RSPWVD, 512, -3 ..............................................................261 Table 84: Test 82 Metrics – HT of WVD, 512, 10 ..........................................................262 Table 85: Test 85 Metrics – HT of CWD, 512, 10 ..........................................................264 Table 86: Test 86 Metrics – HT of CWD, 512, 0 ............................................................265 Table 87: Test 87 Metrics – HT of CWD, 512, -6 ...........................................................266 Table 88: Test 88 Metrics – HT of RSPWVD, 512, 10 ...................................................267 Table 89: Test 89 Metrics – HT of RSPWVD, 512, 0 .....................................................268 Table 90: Test 90 Metrics – HT of RSPWVD, 512, -6....................................................269 TASK 3 xxviii

Table 91: Test 91 Metrics – WVD, 512, 10 .....................................................................270 Table 92: Test 94 Metrics – CWD, 512, 10 .....................................................................272 Table 93: Test 95 Metrics – CWD, 512, 0 .......................................................................273 Table 94: Test 96 Metrics – CWD, 512, -2......................................................................274 Table 95: Test 97 Metrics – Spectrogram, 512, 10 ..........................................................275 Table 96: Test 98 Metrics – Spectrogram, 512, 0 ............................................................276 Table 97: Test 99 Metrics – Spectrogram, 512, -3 ..........................................................277 Table 98: Test 100 Metrics – Scalogram, 512, 10 ...........................................................278 Table 99: Test 101 Metrics – Scalogram, 512, 0 .............................................................279 Table 100: Test 102 Metrics – Scalogram, 512, -3 ..........................................................280 Table 101: Test 103 Metrics – Reassigned Spectrogram, 512, 10...................................281 Table 102: Test 104 Metrics – Reassigned Spectrogram, 512, 0.....................................282 Table 103: Test 105 Metrics – Reassigned Spectrogram, 512, -3 ...................................283 Table 104: Test 106 Metrics – Reassigned Scalogram, 512, 10 ......................................284 Table 105: Test 107 Metrics – Reassigned Scalogram, 512, 0 ........................................285 Table 106: Test 108 Metrics – Reassigned Scalogram, 512, -4.......................................286 Table 107: Test 109 Metrics – RSPWVD, 512, 10..........................................................287 Table 108: Test 110 Metrics – RSPWVD, 512, 0............................................................288 Table 109: Test 111 Metrics – RSPWVD, 512, -3 ..........................................................289 Table 110: Test 112 Metrics – HT of WVD, 512, 10 ......................................................291 Table 111: Test 113 Metrics – HT of CWD, 512, -3 .......................................................293 Table 112: Test 114 Metrics – HT of RSPWVD, 512, -4................................................295 TASK 4 xxix

Table 113: Test 115 Metrics – WVD, 512, 10 .................................................................296 Table 114: Test 118 Metrics – CWD, 512, 10 .................................................................298 Table 115: Test 119 Metrics – CWD, 512, 0 ...................................................................299 Table 116: Test 120 Metrics – CWD, 512, -1..................................................................300 Table 117: Test 121 Metrics – Spectrogram, 512, 10 ......................................................301 Table 118: Test 122 Metrics – Spectrogram, 512, 0 ........................................................302 Table 119: Test 123 Metrics – Spectrogram, 512, -1 ......................................................303 Table 120: Test 124 Metrics – Scalogram, 512, 10 .........................................................304 Table 121: Test 125 Metrics – Scalogram, 512, 0 ...........................................................305 Table 122: Test 126 Metrics – Scalogram, 512, -3 ..........................................................306 Table 123: Test 127 Metrics – Reassigned Spectrogram, 512, 10...................................307 Table 124: Test 128 Metrics – Reassigned Spectrogram, 512, 0.....................................308 Table 125: Test 129 Metrics – Reassigned Spectrogram, 512, -2 ...................................309 Table 126: Test 130 Metrics – Reassigned Scalogram, 512, 10 ......................................310 Table 127: Test 131 Metrics – Reassigned Scalogram, 512, 0 ........................................311 Table 128: Test 132 Metrics – Reassigned Scalogram, 512, -2.......................................312 Table 129: Test 133 Metrics – RSPWVD, 512, 10..........................................................313 Table 130: Test 134 Metrics – RSPWVD, 512, 0............................................................314 Table 131: Test 135 Metrics – RSPWVD, 512, -3 ..........................................................315 Table 132: Test 136 Metrics – HT of WVD, 512, 10 ......................................................317 Table 133: Test 137 Metrics – HT of CWD, 512, -3 .......................................................319 Table 134: Test 138 Metrics – HT of RSPWVD, 512, -4................................................321

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1. Introduction Today, radar systems face serious threats from anti-radiation systems [FAU02] and from electronic attack [GAU02]. In order to be able to perform their functions properly, they must be able to ‘see without being seen’ [PAC09], [WIL06]. This necessitates that they be low probability of intercept (LPI) radars. These radars typically have very low peak power, wide bandwidth, high duty cycle, and power management, making them difficult to be detected and characterized by intercept receivers. In the radar arena, there is a current trend towards radars becoming LPI radars.

On the other side of the fence, the intercept receiver that intercepts these LPI radar signals faces threats as well. It must be able to detect and characterize the signals from these LPI radars in order to provide timely information about threatening systems, in addition to providing information about defensive systems - which is important in maintaining a credible deterrent force to penetrate those defenses. In this case, the knowledge provided by the intercept receiver provides insight for determining the details of the threat so that an effective response can be prepared [WIL06].

LPI radars attempt to detect specific targets at a longer range than the intercept receiver can detect the LPI radar. This means that the success of the LPI radar is measured by how difficult it is for them to be detected by intercept receivers. Consequently, the

1

properties of LPI radar, such as high duty cycle, low power, wide bandwidth, power management, make the LPI radar signals difficult to be detected and characterized by intercept receivers. Examples of intercept receivers are Electronic Support receivers, Electronic Intelligence receivers, and Radar Warning Receivers.

Different from the older pulsed radar systems, which were relatively easy for an intercept receiver to detect due to their high peak power, the LPI radar transmitter makes use of sophisticated phase and frequency modulation to spread the signal bandwidth, making the signal difficult to intercept. The LPI radar receiver is able to make use of the appropriate matched filter so that the radar’s performance is similar to that of traditional pulsed radar radiating the same amount of average power [GAU02].

To make matters even more challenging for intercept receivers, most of the intercept receivers currently in the fleet are analog intercept receivers that were designed to intercept ‘older’ (non-LPI) radar signals, and which perform poorly when faced with LPI radar signals. Digital receivers, which are becoming a growing trend, are seen as the eventual solution to the intercept receiver’s LPI challenge [PAC09].

In this dissertation, the use of the reassignment method and the Hough transform are explored as methods for improving the readability of classical time-frequency analysis representations of LPI radar signals collected by intercept receivers. This improved readability may lead to more accurate signal detection and parameter extraction metrics of these signals. 2

The following sections give a brief overview of the current intercept receiver signal analysis techniques, the classical time-frequency analysis techniques, the reassignment method, and the Hough transform. Each of these will be explained in more detail in the background section of this dissertation.

1.1. Intercept Receiver Signal Analysis Techniques Currently, most of the intercept receivers in the fleet are analog, and since analog intercept receivers perform poorly when faced with LPI radar signals, digital intercept receivers represent the growing trend for LPI signal analysis.

To identify the emitter parameters, Fourier analysis techniques using the FFT have been used as the basic tool of the digital intercept receiver, and make up a majority of the digital intercept receiver techniques that are currently in the fleet [PAC09]. From a Fourier Transform representation, the evolution of the frequency content in time cannot be determined. This is due to the fact that the Fourier Transform is a decomposition of complex exponentials, which are of infinite duration and completely un-localized in time. Time information is in fact encoded in the phase of the Fourier Transform (which is simply ignored by the energy spectrum), but its interpretation is not straightforward and its direct extraction is faced with a number of difficulties such as phase unwrapping [ISI96]. Consequently, when a practical non-stationary signal (such as an LPI radar signal) is processed, the Fourier transform cannot efficiently analyze and process the time-varying characteristics of the signal’s frequency spectrum [XIE08], [STE96]. In 3

order to have a more informative description of such signals, it would be better to directly represent their frequency content while still keeping the time description parameter – which is what time-frequency analysis accomplishes [ISI96], [SOL00].

1.2. Classical Time-Frequency Analysis Techniques From the basic tool of the FFT, more complex signal processing techniques have evolved, such as time-frequency analysis techniques.

In many operational radar

environments, the non-stationary nature of the received radar signal mandates the use of some form of time-frequency analysis technique, which can clearly bring out the nonstationary behavior of the signal. The important virtue of time-frequency analysis is that it provides an identification of the specific times during which certain spectral components of the signal are observed. Time-frequency techniques can also be used to detect and extract the parameters from LPI modulations [MIL02]. For digital intercept receivers, classical time-frequency analysis techniques are currently at the lab phase, being analyzed by industry in an attempt to port the time-frequency analysis algorithms that have been developed to hardware (FPGAs, DSPs) for the purpose of running them in real-time on realistic IF bandwidths [PAC09].

Four of the classical time-frequency analysis techniques (Wigner-Ville Distribution (WVD), Choi-Williams Distribution (CWD), Spectrogram, Scalogram) will be described briefly in this section, and then in more detail in the background section of this dissertation.

4

WVD: One of the most prominent members of the time-frequency analysis techniques family is the WVD. The WVD satisfies a large number of desirable mathematical properties. In particular, it is always real-valued, preserves time and frequency shifts, and satisfies marginal properties [ISI96], [QIA02]. The WVD, which is a transformation of a continuous time signal into the time-frequency domain, is computed by correlating the signal with a time and frequency translated version of itself, making it bilinear. The WVD exhibits the highest signal energy concentration in the time-frequency plane [WIL06].

By using the WVD, an intercept receiver can come close to having a

processing gain near the LPI radar’s matched filter processing gain [PAC09]. The WVD also contains cross term interference between every pair of signal components, which may limit its applications [GUL07], [STE96], and which can make the WVD timefrequency representation very unreadable, especially if the components are numerous or close to each other, and the more so in the presence of noise [BOA03]. This lack of readability can in turn translate into poor signal detection and parameter extraction metrics, potentially placing the intercept receiver signal analyst in harm’s way.

CWD: The CWD is a member of the Cohen’s class of time-frequency distributions which use smoothing kernels [GUL07] to help reduce cross-term interference so prevalent in the WVD [BOA03], [PAC09], [UPP08].

The reduction in cross-term

interference can make the time-frequency representation more readable and can make signal detection and parameter extraction more accurate. The down-side is that the CWD, like all members of Cohen’s class, is faced with an inevitable trade-off between cross-term reduction and time-frequency localization. 5

So the signal detection and

parameter extraction benefits gained by the cross-term reduction may be offset by the decrease in time-frequency localization (smearing or widening of the signal).

Spectrogram: The Short-Time Fourier Transform (STFT) will be examined first, and then connected to the Spectrogram.

The Spectrogram is defined as the magnitude

squared of the STFT [HIP00], [HLA92], [MIT01]. The STFT was the first tool devised for analyzing a signal simultaneously in both time and frequency. The basic idea of the STFT is to analyze a small part of the signal around the time of interest based on Fourier methods in order to determine the frequencies at that time. Since the time interval is short compared to the whole signal, the process is called taking the Short-Time Fourier Transform [STE96]. The observation window allows localization of the spectrum in time, but also smears the spectrum in frequency in accordance with the uncertainty principle, leading to a trade-off between time resolution and frequency resolution. In general, if the window is short, the time resolution is good, but the frequency resolution is poor, and if the window is long, the frequency resolution is good, but the time resolution is poor. For the Spectrogram, its robustness and ease of implementation have made the spectrogram a popular tool for speech analysis [BOA03], [COH95].

Since the

Spectrogram is a quadratic (bilinear) representation, the Spectrogram of the sum of two signals is not the sum of the two Spectrograms (quadratic superposition principle); there is a cross-Spectrogram part and a real part. Therefore, as for every quadratic distribution, the Spectrogram presents interference terms, however, the interference terms are restricted to those regions of the time-frequency plane where the signals overlap. If the

6

signal components are sufficiently far apart so that their spectrograms do not overlap significantly, then the interference terms will be nearly identically zero [ISI96].

Scalogram: The wavelet transform will be examined first, and then connected to the Scalogram. The Scalogram is defined as the magnitude squared of the wavelet transform, and can be used as a time-frequency distribution [COH02], [GAL05], [BOA03]. The wavelet transform is of interest for the analysis of non-stationary signals, because it provides still another alternative to the STFT and to many of the quadratic timefrequency distributions.

The basic difference between the STFT and the wavelet

transform is that the STFT uses a fixed signal analysis window, whereas the wavelet transform uses short windows at high frequencies and long windows at low frequencies. This helps to diffuse the effect of the uncertainty principle by providing good time resolution at high frequencies and good frequency resolution at low frequencies. This approach makes sense especially when the signal at hand has high frequency components for short durations and low frequency components for long durations.

The signals

encountered in practical applications are often of this type. The wavelet transform allows localization in both the time domain via translation of the mother wavelet, and in the scale (frequency) domain via dilations. The wavelet is irregular in shape and compactly supported, thus making it an ideal tool for analyzing signals of a transient nature; the irregularity of the wavelet basis lends itself to analysis of signals with discontinuities or sharp changes, while the compactly supported nature of wavelets enables time localization of a signal’s features [BOA03]. Unlike many of the quadratic functions such as the WVD and CWD, the wavelet transform is a linear transformation; therefore cross7

term interference is not generated. There is another major difference between the STFT and the wavelet transform; the STFT uses sines and cosines as an orthogonal basis set to which the signal of interest is effectively correlated against, whereas the wavelet transform uses special ‘wavelets’ which usually comprise an orthogonal basis set. The wavelet transform then computes coefficients, which represents a measure of the similarities, or correlation, of the signal with respect to the set of wavelets. In other words, the wavelet transform of a signal corresponds to its decomposition with respect to a family of functions obtained by dilations (or contractions) and translations (moving window) of an analyzing wavelet.

A filter bank concept is often used to describe the wavelet transform. The wavelet transform can be interpreted as the result of filtering the signal with a set of bandpass filters, each with a different center frequency.

The Scalogram (the Affine Class counterpart of the Spectrogram) can be considered as an energy distribution of the signal in the time-scale (frequency) plane. As is the case for the wavelet transform, the time and frequency resolutions of the Scalogram are related via the Heisenberg-Gabor principle.

The interference terms of the Scalogram, as for the spectrogram, are also restricted to those regions of the time-frequency plane where the corresponding signals overlap. Therefore, if two signal components are sufficiently far apart in the time-frequency plane, their cross-Scalogram will be essentially zero [ISI96], [HLA92]. 8

Though classical time-frequency analysis techniques, such as those described above, are a great improvement over Fourier analysis techniques, they suffer in general from poor time-frequency localization, cross-term interference, and mediocre performance in low SNR environments. In this dissertation, two signal processing analysis techniques, the reassignment method and the Hough transform are introduced as potential solutions for these deficiencies. The next sections provide a brief overview of these techniques.

1.3. The Reassignment Method A brief overview of the reassignment method is given in this section, while a more detailed description is provided in the background section of this dissertation.

Much work is underway to develop more advanced signal processing techniques which will more effectively and efficiently exploit modern radar signals. These techniques are directed at improving the tasks of detection, classification, and identification, meaning a readability that is based on a good concentration of the signal components and few misleading interference terms. This is what the reassignment method has been devised for.

It’s scheme assumes that the energy distribution in the time-frequency plane

resembles a mass distribution and moves each value of the time-frequency plane located at a point (t, f) to another point (t’, f’), which is the center of gravity of the energy distribution in the area of (t, f). The result is a more focused representation with high intensity, since the value at the point (t’, f’) is the sum of all the neighboring values. Reasoning with the mechanical analogy, the situation is as if the total mass of an object 9

was assigned to its geometrical center, an arbitrary point which except in the very specific case of a homogeneous distribution, has no reason to suit the actual distribution. A much more meaningful choice is to assign the total mass of an object - as well as the time-frequency value – to the center of gravity of their respective distribution. This is what the reassignment method performs [BOA03].

The reassignment method can be applied to most energy distributions as well as to the Spectrogram [HIP00]. One of the most important properties of the reassignment method is that the application of the method theoretically yields perfectly localized distributions for chirp signals, frequency tones and impulses [BOA03] making it a good candidate for analyzing triangular modulated frequency modulated continuous wave (FMCW) and frequency shift keying (FSK) LPI radar signals.

A potential drawback of the

reassignment method is that there is a chance it could generate spurious artifacts due to the reassignment process randomly clustering noise (vice signals). Because of this, its performance may suffer in low SNR scenarios.

The reassignment method can be viewed as a step in the process whose goal is to build a readable time-frequency representation, which consists of: a smoothing, whose main purpose is to rub out oscillatory interferences, but whose drawback is to smear localized components; and a squeezing, whose effect is to refocus the contributions which survived the smoothing [BOA03]. Experimental results have shown for other applications that this method provides, without a drastic increase in computational complexity – enhanced

10

contrast (when compared to smoothed distributions) with a much reduced level of interference (when compared to the WVD) [BOA03], [ISI96].

In this dissertation, the reassignment method, due to its smoothing and squeezing qualities, has been put forth as a potential solution for intercept receivers to the classical time-frequency analysis techniques deficiencies of poor time-frequency localization and cross-term interference.

1.4. The Hough Transform The Hough transform is briefly described in this section, and in more detail in the background section of this dissertation.

The Hough transform (which is very similar to the Radon transform) is used for the detection of straight lines and other curves [BAR92], [BEN05], [ZAI99], [INC07], [BAR95]. The Hough transform of a particular time-frequency representation is found by computing the integral of the time-frequency representation along straight lines at different angles. The presence of a ‘spike’ in the Hough transform representation reveals the presence of high positive values concentrated along a line in the time-frequency representation – whose parameters (such as chirp rate) correspond to the coordinates of the spike (theta and rho values) [BAR92], [YAS06], [BAR95].

Detection can be

achieved by establishing a threshold value for the amplitude of the Hough transform spike. Therefore the Hough transform can be used to convert a difficult global detection

11

problem in the time-frequency representation into a more easily solved local peak detection problem in the Hough transform representation.

As shown in this dissertation, the Hough transform is especially suited for detecting and analyzing low SNR signals, due to its robustness in the presence of noise [SHA07], [DAH08], where the integration carried out by the Hough transform produces an improvement in the SNR. This integration also allows the Hough transform to work well in the presence of multi-component signals which produce cross-terms. This is because the cross-terms have an amplitude modulation which is reduced by the Hough transform integration, while the useful contributions, which are always positive, are correctly integrated [BAR95].

In this dissertation, the Hough transform, because of its ability to suppress cross-term interference, separate signals from cross-term interference, and perform well in a low SNR environment, has been put forth as a potential solution to the classical timefrequency analysis techniques deficiencies of cross-term interference and mediocre performance in low SNR environments.

For digital intercept receivers, the evolution from Fourier-based analysis, to classical time-frequency analysis techniques to the reassignment method and the Hough transform will produce enhanced readability, and as a result, more accurate signal detection and parameter extraction. When the intercept receiver signal analyst correlates these more accurate parameters with existing signals in databases, he will be able to more precisely 12

determine the radar type and threat, which in turn will make for improved response management, resulting in a more informed and safer intercept receiver environment, potentially saving valuable equipment, intelligence, and lives.

13

2. Background The background section of this dissertation elaborates on the introduction section.

2.1. LPI Radar Overview Many users of radar today are specifying LPI [ADA01], [VAN09] as an important tactical requirement [AMS09]. The term LPI (whose meaning is not absolutely precise [WIL06]) is that property of a radar that, because of its low power, high duty cycle, ultralow sidelobes, power management, wide bandwidth [YAV04], frequency/phase modulation [BRO98], or other design attributes, makes it difficult to be detected by means of intercept receivers [WEI04], [ADA04] such as electronic support receivers, electronic intelligence receivers, and radar warning receivers [COR04]. The goal of the LPI radar is to detect targets at longer ranges than the intercept receiver can detect the LPI radar [SHB04]. It is important to note that defining a radar to be LPI necessitates defining the corresponding intercept receiver [SCH06]. That is, the success of an LPI radar is measured by how hard it is for the intercept receiver to detect and intercept the radar emissions.

One formal definition is as follows: A low probability of intercept (LPI) radar is defined as a radar that uses a special emitted waveform intended to prevent a non-cooperative intercept receiver from intercepting and detecting its emission [PAC09].

The LPI emitter has established itself as the premier tactical and strategic radar in the military spectrum.

In addition to surveillance and navigation, the LPI emitter also 14

operates in the time-critical domain for applications such as fire control and missile guidance [WIL06].

2.2. LPI Radar Characteristics Some of the characteristics of the LPI radar are power management, ultra-low side lobes, and pulse compression.

Power management is the radar’s ability to control the power level so that it emits only the necessary power for detection of a target. An intercept receiver is used to seeing an increase in power as the radar approaches. If a power managed LPI radar decreases the power as it approaches the target, an intercept receiver may incorrectly assume that the radar is not approaching, and therefore that no response management is necessary, which could be a deadly decision [WIL06].

Ultra-low side lobes prevent an intercept receiver from detecting radar emissions from the side lobes of the radar. Ultra-low side lobes are required to be -45dB or lower [SCH99].

Pulse compression is another important LPI radar characteristic.

For frequency

modulation LPI radars, the transmitted CW signal is coded with a reference signal that spreads the transmitted energy in frequency, making it more difficult for an intercept receiver to detect and identify the LPI radar.

15

The reference signal can be a linear

frequency modulated signal or an FSK (frequency hopping).

The most popular

implementation has been the FMCW [GUL07].

If the radar uses an FMCW waveform, the processing gain (apart from any noncoherent integration) is the sweep or modulation period bandwidth,

, multiplied by the sweep (input)

(see equation 2.1). That is:

The LPI receiver compresses (correlates) the received signal from the target using the stored reference signal, for the purpose of performing target detection. The correlation receiver is a ‘matched receiver’ if the reference signal is exactly the same duration as the finite duration return signal [WIL06], [PAC09].

2.3. LPI Radar Waveforms This section looks at the FMCW and FSK LPI radar waveforms, which are the waveforms utilized in this dissertation.

FMCW is a signal that is frequently encountered in modern radar systems [WAN08], [WON09], [WAJ08]. The frequency modulation spreads the transmitted energy over a large modulation bandwidth

, providing good range resolution that is critical for

discriminating targets from clutter. The power spectrum of the FMCW signal is nearly rectangular over the modulation bandwidth, so non-cooperative interception is difficult. 16

Since the transmit waveform is deterministic, the form of the return signals can be predicted. This gives it the added advantage of being resistant to interference (such as jamming), since any signal not matching this form can be suppressed [WIL06]. Consequently, it is difficult for an intercept receiver to detect the FMCW waveform and measure the parameters accurately enough to match the jammer waveform to the radar waveform [PAC09].

The most popular linear modulation utilized is the triangular FMCW emitter [LIA09], since it can measure the target’s range and Doppler [MIL02], [LIW08]. Triangular modulated FMCW is one of the waveforms that is employed in this dissertation.

An LPI radar that uses FSK (frequency hopping (FH)) techniques changes the transmitting frequency in time over a wide bandwidth in order to prevent an intercept receiver from intercepting the waveform. The frequency slots used are chosen from an FH sequence, and it is this unknown sequence that gives the radar the advantage over the intercept receiver in terms of processing gain. The frequency sequence appears random to the intercept receiver, and so the possibility of it following the changes in frequency is remote [PAC09]. This prevents a jammer from reactively jamming the transmitted frequency [ADA04].

FH radar performance depends only slightly on the code used, given that certain properties are met. This allows for a larger variety of codes, making it more difficult to intercept. 17

2.4. LPI Radar Applications and Examples This section gives a brief description of some of the applications of LPI radar, including altimeters, landing systems, and surveillance and fire control radar. Also listed are examples of systems for each of these applications.

Altimeters were the earliest FMCW devices and among the most successful [SKO90], [ANS08]. They usually employ triangular modulation, since this yields the most compact and easily amplified spectrum and the Doppler effect balances out [SKO90].

Many radar Altimeters have LPI properties. These radars have special characteristics in that they process only a single target and their antenna’s mainbeam is directed away from potential intercept receivers. Their transmitter signals are generally power managed [SCH06].

For vehicles that fly near the surface, it is necessary to detect and to measure the distance from the surface to the radar, down to almost zero feet.

Frequency modulation

continuous wave radar is the simplest of radar ranging techniques. The transmitter works continuously to produce the CW output, and changes frequency at a constant rate in either a sawtooth pattern or a triangular pattern. The FMCW ranging process occurs by mixing a sample of the linearly varying frequency with the signal reflected from the surface.

The difference produced after mixing is a low-frequency beat signal

proportional to the range of the surface being measured [SKO90], [PAC09]. 18

Some examples of fielded LPI altimeters are: HG-9550 LPI radar altimeter system [HGL02], Collins ALT-55 altimeter [SKO90], AN/APN-232 Combined Altitude Radar Altimeter [CAR03], Bendix ALA-52A altimeter [SKO90].

Landing an aircraft such as an unmanned aerial vehicle presents a number of challenges. Landing involves the air vehicle switching between different modes of operation (e.g. takeoff, landing, and hovering). The air vehicle must also coordinate with the landing site using voice or data links. Automatic and precision landing systems transmit a beacon and can aid in the landing operation, but must be LPI to remain active on the battlefield [WIL06], [PAC09].

Some examples of fielded LPI landing systems are: Tactical Automatic Landing System (TALS) [SNT02], AN/SPN-46

On the battlefield, situational awareness and threat evaluation are achieved using tactical surveillance radar to detect and track targets. In general, the detection and tracking of targets should be as quiet as possible. These systems employ LPI technology to decrease the probability of being passively detected [WIL06], [PAC09].

Some examples of fielded LPI ground-based systems are: Eagle [EAG03], Improved Helicopter and Aircraft/Radar Detection (HARD)-3D [ERI02].

19

Some examples of fielded LPI airborne systems are: AN/APQ-181 [APQ03], AN/APG77 Advanced Multimode Tactical Radar [APG02], AN/APS-147 Multimode Radar [APS03].

2.5. Detection of LPI Radars: Intercept Receiver Overview In this section we switch from the topic of LPI radars, to the topic of those devices that detect and characterize LPI radar signals – intercept receivers.

The three main types of intercept receivers are: electronic support receivers, electronic intelligence receivers, and radar warning receivers.

Electronic intelligence is the result of observing the signals transmitted by radar systems to obtain information about their capabilities: it is the remote sensing of remote sensors. Through electronic intelligence, it is possible to obtain valuable information while remaining remote from the radar itself. Identification is performed by comparing the intercepted signal signature against the signatures contained within its threat library [SCH99], [STE96]. Clearly, the underlying basic function of electronic intelligence is to determine the capabilities of the radar, so that decisions can be made as to what threat it poses [SCH06], [FAU02]. Electronic intelligence receivers are the least time critical of the three intercept receivers. The outputs of the electronic intelligence receiver may be analyzed using off-line analysis tools based on software.

20

Radar warning receivers are designed to give nearly immediate warning if specific threat signals are received (e.g., illumination of an aircraft’s warning receiver by the target tracking radar of a threatening system).

The warning receiver typically has poor

sensitivity and feeds into a near-real-time processor that uses a few parameter measurements to identify a threat. Usually, rough direction (e.g., quadrant or octant) is determined for the threat and the operator has a crude display showing functional radar type, direction, and relative range (strong signals displayed as being nearer than weaker ones). This type of receiver does not provide the kind of output that is analyzed using the methods described later in this dissertation.

Electronic support receivers encompass all actions necessary to provide the information required for immediate decisions involving electronic warfare operations, threat avoidance, targeting, and homing [WIL06].

2.6. Intercept Receiver Signal Analysis Techniques This section describes some of the classical time-frequency analysis techniques as well as the reassignment method and the Hough transform utilized in this dissertation.

2.6.1. Time-Frequency Analysis Time-frequency signal analysis concerns the analysis and processing of signals with time-varying frequency content. Such signals are best represented by a time-frequency distribution [PAP95], [HAN00], which is intended to show how the energy of the signal is distributed over the two-dimensional time-frequency plane [WEI03], [LIX08], 21

[OZD03].

Processing of the signal may then exploit the features produced by the

concentration of signal energy in two dimensions (time and frequency), instead of only one dimension (time or frequency) [BOA03], [LIY03]. Since noise tends to spread out evenly over the time-frequency domain, while signals concentrate their energies within limited time intervals and frequency bands; the local SNR of a noisy signal can be improved simply by using time-frequency analysis [XIA99]. Also, the intercept receiver can increase its processing gain by implementing time-frequency signal analysis [GUL08].

As alluded to previously, time-frequency distributions are useful for the visual interpretation of signal dynamics [RAN01]. An experienced operator can quickly detect a signal and extract the signal parameters by analyzing the time-frequency distribution [ANJ09].

2.6.2. Wigner-Ville Distribution (WVD) The WVD of a signal

is given in equation 2.2 as:

or equivalently in equation 2.3 as:

22

The WVD is the most basic of the quadratic time-frequency representations, first developed in quantum mechanics by Wigner in 1932 [WIG32] and later introduced for signal analysis by Ville [VIL48], [CHE02]. It is a function which gives the distribution of signal energy over time and frequency [WAN98], [YOU04], [GAU02], [DEN06], [GUL07]. It is computed by correlating the signal with a time and frequency translated version of itself, and is said to be bilinear in the signal due to the fact that the signal enters twice into its calculation [COH95]. It has been noted as one of the more useful bilinear time-frequency analysis techniques for signal processing [PAC09]. The WVD representation does not suffer from the inherent trade-off of time resolution versus frequency resolution that limits conventional ‘short-time’ techniques to quasi-stationary signals [KAY85].

It exhibits the highest signal energy concentration in the time-

frequency plane [GUL07], [PAC09] and is totally concentrated along the instantaneous frequency (i.e. well localized) [HIP00] (the WVD is known to be an unbiased estimator of the intermediate frequency (IF) for LFM signals [CIR08], [GUA06]). The WignerVille Distribution, as opposed to the Wigner Distribution, emphasizes the use of the analytic signal and recognizes the contribution of Ville, who, as mentioned above, derived the distribution in a signal-processing context in 1948.

Among all the quadratic time-frequency representations with energetic interpretation, the WVD satisfies an exceptionally large number of desirable properties: the auto-WVD (signal) is always real-valued; the WVD preserves time shifts and frequency shifts of the signal (time-shift invariance (time covariance) – a time shift in the signal causes the same 23

time shift in the WVD; frequency-shift invariance (frequency covariance) - a frequency shift in the signal causes the same frequency shift in the WVD); the WVD satisfies marginal properties [QIA02], [STE96], [ISI96], that is, the time or frequency integrals of the WVD correspond to the signal’s instantaneous power and its spectral energy density, respectively [HLA92]; global energy – integration of the WVD over the entire timefrequency plane yields the signal energy; instantaneous frequency (IF) – for an analytic signal, the first moment (mean) of the WVD with respect to frequency is the instantaneous frequency; time delay – the first moment of the WVD with respect to time is the time delay; convolution invariance – the WVD of the time-convolution of two signals is the time-convolution of the WVDs of the two signals; modulation invariance – the WVD of the frequency-convolution of two signals is the frequency-convolution of the WVDs of the two signals; invertibility – a signal may be recovered from its WVD up to a phase factor; inner-product invariance – the WVD is a unitary transformation – that is, it preserves inner products [BOA03].

Because the WVD is bilinear in the signal rather than linear, it suffers from spurious features called artifacts or cross-terms [GUL07], which appear midway between true signal components in the case of multicomponent signals as well as non-linear mono- and multicomponent FM signals. Stating it another way; cross-terms occur because the WVD of the sum of two signals is not explicitly the sum of the signals’ WVDs, but also of their cross-Wigner-Ville distributions (XWVDs). Cross-terms can make the WVD difficult to interpret (poor readability), especially if the components are numerous or close to each other, and the more so in the presence of noise. Cross-terms between signal components 24

and noise exaggerate the effects of noise and cause rapid degradation of performance as the SNR decreases [DEL02], [GUA06], [WON09]. For such reasons, cross-terms are often regarded as the fundamental limitation on the applicability of quadratic timefrequency methods, and the desire to suppress them has led to several approaches [OZD03].

Cohen’s class of bilinear time-frequency representations is a class of

‘smoothed WVDs’ employing a smoothing kernel that can reduce the sensitivity of the distribution to noise and can suppress cross-terms, at the expense of smearing the distribution in time and frequency. This smearing causes the distribution to be non-zero in regions where the true WVD shows no energy. The general form of Cohen’s class adds a kernel function (a kernel is a weighting function applied to the data – the WVD has a kernel of one [UPP08]) inside of the WVD integral.

The general rules which control the interference geometry of the WVD are that their cross-terms: 1) are located midway between the interacting components; 2) oscillate proportionally to the inter-components’ time-frequency distance; 3) have a direction of oscillation orthogonal to the straight line connecting the components [BOA03]. The cross terms which appear due to the interaction of different signal components (i.e. autocomponents) in a multi-component signal are called outer interference cross terms, whereas the cross terms which appear due to the interaction of a single signal component with itself (i.e. for a non-linear FM signal) is called the inner interference cross terms [HLA97].

25

Unlike the Spectrogram interference terms, the WVD interference terms will be non-zero regardless of the time-frequency distance between the two signal terms [WOO94].

For signal analysis purposes, simple visual inspection of the received signal’s WVD is often sufficient to detect the presence of LPI signals [KAY85] and to extract the parameters of the LPI signal. For example, the carrier frequency is extracted by finding the location of the maximum intensity level within the WVD image [GUL08]. However, the more signals components there are, the more cross-term interference there will be, making it more difficult to detect signals and extract parameters. In general, the more readable the time-frequency representation, the more exact will be the metrics that are extracted.

Another interesting fact pertaining to the WVD is that its cross-terms are virtually unaffected by the SNR level. Attempts have been made to reduce cross-term interference by raising the SNR level. This will eliminate of some of the noise, but the cross-terms will remain virtually unaffected, as displayed in Figures 1, 2, and 3.

26

Figure 1: WVD of a triangular modulated FMCW signal at an SNR of 0dB (256 samples).

Figure 2: Same as Figure 1, but with the SNR level increased to 10dB. Though some of the noise has disappeared, the cross-terms remain. 27

Figure 3: Same as Figure 1, but with the SNR level increased to 20dB. Most of the noise has disappeared, but the cross-terms remain, demonstrating that cross-terms are virtually unaffected by the SNR level. Figure 1 (SNR = 0dB) exhibits a lot of noise and cross-term interference. By raising the SNR to 10 dB (Figure 2), much of the noise has disappeared, but the cross-term interference remains virtually unaffected. Again raising the SNR, this time to 20 dB (Figure 3), nearly all of the noise has disappeared, but again, the cross-term interference remains virtually unaffected. Therefore changing the SNR has very little effect on crossterm interference.

Another negative aspect of the WVD is that a false positive detection can occur when a cross-term is wrongly declared as a signal detection. This is known as a cross-term false positive (XFP). As an example, see Figure 4 below. 28

Figure 4: Example of cross-term false positives (XFPs). There are 4 true signals (fc1=1KHz, fc2=1.75KHz, fc3=0.75KHz, fc4=1.25KHz) and 6 XFPs (cross-terms that were wrongly declared as signal detections) (ct1=0.875KHz, ct2=1KHz, ct3=1.125KHz, ct4=1.25KHz, ct5=1.375KHz, ct6=1.5KHz). WVD of a 4-component FSK signal at SNR=10dB (512 samples). This is a time-frequency representation (WVD) of an FSK 4-component signal. There are 4 signal components: fc1=1KHz (from 0 to .025sec), fc2=1.75KHz (from .025 sec to .05sec), fc3=0.75KHz (from .05sec to .075sec), fc4=1.25KHz (from .075sec to 0.1 sec.).

In general, for the WVD, a cross-term is located mid-way between two signals, and its frequency is one-half the sum of the two signals’ frequencies.

In the case of Figure 4, the cross-terms are located at:

29

½{(fc1 + fc2), (fc1 + fc3), (fc1 + fc4), (fc2 + fc3), (fc2 + fc4), (fc3 + fc4)} KHz

This gives 6 cross-terms located at: ct1=0.875KHz, ct2=1KHz, ct3=1.125KHz, ct4=1.25KHz, ct5=1.375KHz, ct6=1.5KHz. Each of the 4 signals and 6 cross-terms are labeled in Figure 4.

For this test case, all 6 cross-terms were wrongly declared as signal detections (i.e. 6 XFPs). Criteria for signal/XFP detection are given in the methodology chapter of this dissertation.

With all of its benefits, the WVD is very costly with respect to computation time. The computation time can be improved if the algorithms can be coded more efficiently [MIL02].

The WVD has been used in many fields of engineering.

These include optical

implementations of the WVD [LIY88], medical applications [CLA98], [DAR94], [MIL99], image analysis [CRI89], [GON89], target detection [HAY94], [KUM97], and the analysis of non-stationary (LPI) signals [TAB02], [BAR91], [WIL06], [PAC09].

Though the WVD has great time-frequency localization, it is plagued with the worst cross-term interference of the time-frequency techniques, which makes for poor readability of the time-frequency representation, which in turn reduces the accuracy of the metrics extracted, brining potential harm to the intercept receiver environment. As 30

will be addressed in later sections of this dissertation, both the reassignment method and the Hough transform are potential solutions for this deficiency.

2.6.3. Choi-Williams Distribution (CWD) The CWD of a signal

is given by:

As can be seen from equation (2.4), the CWD uses an exponential kernel in the generalized class of bilinear time-frequency distributions. Choi and Williams introduced one of the earliest ‘new’ distributions [CHO89], which they called the Exponential Distribution or ED.

This new distribution overcomes several drawbacks of the

Spectrogram and the WVD, providing decent localization with suppressed interferences [WIL92], [GUL07], [UPP08]. Interference terms tend to lie away from the axes in the ambiguity plane, while autoterms (signals) tend to lie on the axes. The Spectrogram kernel attenuates everything away from the

point, the WVD kernel passes

everything, and the CWD kernel passes everything on the axes and attenuates away from the axes. Thus, the CWD generally attenuates interference terms [PAC09], [HLA92]. This provides its reduced interference characteristic. interference also, but at a cost to the signal concentration.

31

The Spectrogram reduces

Reduced interference may be achieved while maintaining a number of very nice mathematical properties with just a few constraints. The CWD is just one example of the reduced interference distribution class [BOA03], [CHO98], [CUN93], [JEO91], [COH95].

The efficiency of the CWD is strongly dependent on the nature of the analyzed signal. If the signal is composed of synchronized components in time or in frequency, the CWD will present strong interferences.

When the time/frequency supports of the signals

overlap, some ambiguity function interference terms are not completely attenuated (those present around the axes of the ambiguity plane), and the efficiency of the distribution is quite poor [ISI96].

2.6.4. Spectrogram (Short-Time Fourier Transform (STFT)) The Short-Time Fourier Transform (STFT) will be examined first, and then connected to the Spectrogram. As mentioned previously, the Spectrogram is defined as the magnitude squared of the STFT [HIP00], [HLA92], [MIT01], [PAC09], [BOA03]. stationary signals, the STFT is usually in the form of the Spectrogram [GRI08].

The STFT of a signal

is given in equation 2.5 as:

32

For non-

Where

is a short time analysis window localized around

and

. Because

multiplication by the relatively short window

effectively suppresses the signal

outside a neighborhood around the analysis point

, the STFT is a ‘local’ spectrum

of the signal

as sliding along the signal

around . Think of the window

and for each shift function

we compute the usual Fourier transform of the product . The observation window allows localization of the spectrum in

time, but also smears the spectrum in frequency in accordance with the uncertainty principle, leading to a trade-off between time resolution and frequency resolution. In general, if the window is short, the time resolution is good, but the frequency resolution is poor, and if the window is long, the frequency resolution is good, but the time resolution is poor.

The STFT was the first tool devised for analyzing a signal in both time and frequency simultaneously. For analysis of human speech, the main method was, and still is, the STFT. In general, the STFT is still the most widely used method for studying nonstationary signals [COH95].

The Spectrogram (the squared modulus of the STFT) is given by equation 2.6 as:

33

The Spectrogram is a real-valued and non-negative distribution. Since the window h of the STFT is assumed of unit energy, the Spectrogram satisfies the global energy distribution property. Thus we can interpret the Spectrogram as a measure of the energy of the signal contained in the time-frequency domain centered on the point (t, f) and whose shape is independent of this localization.

Here are some properties of the Spectrogram: 1) Time and Frequency covariance - The Spectrogram preserves time and frequency shifts, thus the spectrogram is an element of the class of quadratic time-frequency distributions that are covariant by translation in time and in frequency (i.e. Cohen’s class); 2) Time-Frequency Resolution - The timefrequency resolution of the Spectrogram is limited exactly as it is for the STFT; there is a trade-off between time resolution and frequency resolution. This poor resolution is the main drawback of this representation; 3) Interference Structure - As it is a quadratic (or bilinear) representation, the Spectrogram of the sum of two signals is not the sum of the two Spectrograms (quadratic superposition principle); there is a cross-Spectrogram part and a real part. Thus, as for every quadratic distribution, the Spectrogram presents interference terms; however, those interference terms are restricted to those regions of the time-frequency plane where the signals overlap. Thus if the signal components are sufficiently distant so that their Spectrograms do not overlap significantly, then the interference term will nearly be identically zero [ISI96], [COH95], [HLA92].

2.6.5. Scalogram (Wavelet Transform)

34

The wavelet transform will be examined first, and then connected to the Scalogram. As mentioned previously, the Scalogram is defined as the magnitude squared of the wavelet transform, and can be used as a time-frequency distribution [COH02], [GAL05], [BOA03].

The idea of the wavelet transform (equation (2.7)) is to project a signal

on a family of

zero-mean functions (the wavelets) deduced from an elementary function (the mother wavelet) by translations and dilations:

where

. The variable

sense that taking

dilates the wavelet

corresponds to a scale factor, in the and taking

compresses

. By

definition, the wavelet transform is more a time-scale than a time-frequency representation. frequency

However, for wavelets which are well localized around a non-zero

at a scale

formal identification

, a time-frequency interpretation is possible thanks to the .

The wavelet transform is of interest for the analysis of non-stationary signals, because it provides still another alternative to the STFT and to many of the quadratic timefrequency distributions.

The basic difference between the STFT and the wavelet

transform is that the STFT uses a fixed signal analysis window, whereas the wavelet 35

transform uses short windows at high frequencies and long windows at low frequencies. This helps to diffuse the effect of the uncertainty principle by providing good time resolution at high frequencies and good frequency resolution at low frequencies. This approach makes sense especially when the signal at hand has high frequency components for short durations and low frequency components for long durations.

The signals

encountered in practical applications are often of this type.

The wavelet transform allows localization in both the time domain via translation of the mother wavelet, and in the scale (frequency) domain via dilations. The wavelet is irregular in shape and compactly supported, thus making it an ideal tool for analyzing signals of a transient nature; the irregularity of the wavelet basis lends itself to analysis of signals with discontinuities or sharp changes, while the compactly supported nature of wavelets enables temporal localization of a signal’s features [BOA03]. Unlike many of the quadratic functions such as the WVD and CWD, the wavelet transform is a linear transformation, therefore cross-term interference is not generated. There is another major difference between the STFT and the wavelet transform; the STFT uses sines and cosines as an orthogonal basis set to which the signal of interest is effectively correlated against, whereas the wavelet transform uses special ‘wavelets’ which usually comprise an orthogonal basis set. The wavelet transform then computes coefficients, which represents a measure of the similarities, or correlation, of the signal with respect to the set of wavelets.

In other words, the wavelet transform of a signal corresponds to its

decomposition with respect to a family of functions obtained by dilations (or contractions) and translations (moving window) of an analyzing wavelet. 36

A filter bank concept is often used to describe the wavelet transform. The wavelet transform can be interpreted as the result of filtering the signal with a set of bandpass filters, each with a different center frequency [GRI08], [FAR96], [SAR98], [SAT98]. Like the design of conventional digital filters, the design of a wavelet filter can be accomplished by using a number of methods including weighted least squares [ALN00], [GOH00], orthogonal matrix methods [ZAH99], nonlinear optimization, optimization of a single parameter (e.g. the passband edge) [ZHA00], and a method that minimizes an objective function that bounds the out-of-tile energy [FAR99].

Here are some properties of the wavelet transform: 1) The wavelet transform is covariant by translation in time and scaling. The corresponding group of transforms is called the Affine group (to be compared to the Weyl-Heisenberg group); 2) The signal

can be

recovered from its wavelet transform via the synthesis wavelet; 3) Time and frequency resolutions, like in the STFT case, are related via the Heisenberg-Gabor inequality. However in the wavelet transform case, these two resolutions depend on the frequency: the frequency resolution becomes poorer and the time resolution becomes better as the analysis frequency grows; 4) Because the wavelet transform is a linear transform, it does not contain cross-term interferences [GRI07], [LAR92].

A similar distribution to the Spectrogram can be defined in the wavelet case. Since the wavelet transform behaves like an orthonormal basis decomposition, it can be shown that it preserves energy: 37

where

is the energy of

. This leads us to define the Scalogram (equation (2.8)) of

as the squared modulus of the wavelet transform. It is an energy distribution of the signal in the time-scale plane, associated with the measure

.

As is the case for the wavelet transform, the time and frequency resolutions of the Scalogram are related via the Heisenberg-Gabor principle.

The interference terms of the Scalogram, as for the spectrogram, are also restricted to those regions of the time-frequency plane where the corresponding signals overlap. Therefore, if two signal components are sufficiently far apart in the time-frequency plane, their cross-Scalogram will be essentially zero [ISI96], [HLA92].

For this dissertation, the Morlet Scalogram will be used. The Morlet wavelet is obtained by taking a complex sine wave and by localizing it with a Gaussian envelope. The Mexican hat wavelet isolates a single bump of the Morlet wavelet. The Morlet wavelet has good focusing in both time and frequency [CHE09].

2.6.6. The Reassignment Method

38

Bilinear time-frequency distributions offer a wide range of methods designed for the analysis of non stationary signals. Nevertheless, a critical point of these methods is their readability [STE96], which means both a good concentration of the signal components along with few misleading interference terms. A lack of readability, which is a known deficiency in the classical time-frequency analysis techniques, must be overcome in order to obtain time-frequency distributions that can be both easily read by non-experts and easily included in a signal processing application [BOA03]. Inability to obtain readable time-frequency distributions may lead to inaccurate signal metrics extraction, which in turn can bring about an uninformed and unsafe intercept receiver environment.

Some efforts have been made recently in that direction, and in particular, a general methodology referred to as reassignment.

The original idea of reassignment was introduced in an attempt to improve the Spectrogram [OZD03]. As with any other bilinear energy distribution, the Spectrogram is faced with an unavoidable trade-off between the reduction of misleading interference terms and a sharp localization of the signal components.

We can define the Spectrogram as a two-dimensional convolution of the WVD of the signal by the WVD of the analysis window, as in equation 2.9:

39

Therefore, the distribution reduces the interference terms of the signal’s WVD, but at the expense of time and frequency localization. However, a closer look at equation 2.9 shows that

delimits a time-frequency domain at the vicinity of the

point, inside which a weighted average of the signal’s WVD values is performed. The key point of the reassignment principle is that these values have no reason to be symmetrically distributed around

, which is the geometrical center of this domain.

Therefore, their average should not be assigned at this point, but rather at the center of gravity of this domain, which is much more representative of the local energy distribution of the signal [AUG94].

Reasoning with a mechanical analogy, the local energy

distribution

(as a function of

) can be considered as a

mass distribution, and it is much more accurate to assign the total mass (i.e. the Spectrogram value) to the center of gravity of the domain rather than to its geometrical center. Another way to look at it is this: the total mass of an object is assigned to its geometrical center, an arbitrary point which except in the very specific case of a homogeneous distribution, has no reason to suit the actual distribution. A much more meaningful choice is to assign the total mass of an object, as well as the Spectrogram value, to the center of gravity of their respective distribution [BOA03]

This is exactly how the reassignment method proceeds: it moves each value of the Spectrogram computed at any point

to another point ( , ) which is the center of

gravity of the signal energy distribution around [LIX08]: 40

(see equations 2.10 and 2.11)

and thus leads to a reassigned Spectrogram (equation (2.12)), whose value at any point is the sum of all the Spectrogram values reassigned to this point:

One of the most interesting properties of this new distribution is that it also uses the phase information of the STFT, and not only its squared modulus as in the Spectrogram. It uses this information from the phase spectrum to sharpen the amplitude estimates in time and frequency.

This can be seen from the following expressions of the reassignment

operators:

41

where However,

is the phase of the STFT of

:

(t, f; h)).

these expressions (equations 2.13 and 2.14) do not lead to an efficient

implementation, and have to be replaced by equations 2.15 (local group delay) and 2.16 (local instantaneous frequency):

where

and

. This leads to an efficient implementation

for the Reassigned Spectrogram without explicitly computing the partial derivatives of phase. The Reassigned Spectrogram may thus be computed by using 3 STFTs, each having a different window (the window function h; the same window with a weighted time ramp t*h; the derivative of the window function h with respect to time (dh/dt)). Reassigned Spectrograms are therefore very easy to implement, and do not require a drastic increase in computational complexity.

42

Since time-frequency reassignment is not a bilinear operation, it does not permit a stable reconstruction of the signal. In addition, once the phase information has been used to reassign the amplitude coefficients, it is no longer available for use in reconstruction. For this reason, the reassignment method has received limited attention from engineers, and its greatest potential seems to be where reconstruction is not necessary, that is, where signal analysis is an end unto itself.

One of the most important properties of the reassignment method is that the application of the reassignment process to any distribution of Cohen’s class theoretically yields perfectly localized distributions for chirp signals, frequency tones, and impulses, since the WVD does so also.

As mentioned earlier, this is one of the reasons that the

reassignment method was chosen for this dissertation as a signal process analysis tool for analyzing LPI radar waveforms such as triangular modulated FMCW waveforms (which can be viewed as back-to-back chirps) and FSK waveforms (which can be viewed as frequency tones).

In Figure 5, the reassignment method (RSPWVD) is used for a

triangular modulated FMCW signal (back-to-back chirps). From the threshold of 5% of the maximum intensity plot (the threshold value used for all of the plots in Appendix B of this dissertation) on the upper left, it appears as though the chirp signals are indeed perfectly localized, but in the 1% threshold plot on the upper right, artifacts are seen near the chirp signals which are clearly not perfectly localized. In order to determine if these artifacts are due to noise, the SNR is raised to 25dB (lower left). Many of the artifacts (apparently due to noise) disappear, but some remain. The SNR is then raised to 50dB (lower right), to see if the remaining artifacts are also noise related, but most of the 43

artifacts in the 25dB plot are also in the 50dB plot. For time-frequency reassignment it is known that the localization is slightly degraded at those points where the IF cannot be considered as quasi-linear within a TF smoothing window (i.e. the curved areas of the signal in Figure 5) causing artifacts [FLA03].

In addition, when more than one

component is within a TF smoothing window (i.e. the curves where the chirp signals meet in Figure 5), a beating effect occurs, resulting in interference fringes which may manifest as artifacts [FLA03].

44

Figure 5: RSPWVD of triangular modulated FMCW, 256 samples, SNR=10dB. Threshold of 5% of the maximum intensity (upper left) and 1% threshold (upper right). The SNR has been increased to 25dB (lower-left) and to 50dB (lower-right). The plot on the upper left gives the appearance of perfectly localized distributions for the chirp signals. However, the plot on the upper right (1% threshold) reveals artifacts that are clearly not perfectly localized. In order to determine if these artifacts are due to noise, the SNR is raised to 25dB (lower left). Many of the artifacts disappear, but some remain. The SNR is then raised to 50dB (lower right) to see if the remaining artifacts are also noise related, but most of the artifacts in the 25dB plot are also in the 50dB plot. For time-frequency reassignment, it is known that the localization is slightly degraded at those points where the IF cannot be considered as quasi-linear within a TF smoothing window (i.e. the curved areas of the signal in Figure 5) causing artifacts. In addition, when more than one component is within a TF smoothing window (i.e. the curves where the chirp signals meet in Figure 5), a beating effect occurs, resulting in interference fringes which may manifest as artifacts. The reassignment method provides readability improvement. The components are much better localized and very concentrated, and there are very few cross-terms. This is yet another reason why I chose the reassignment method as a signal process analysis tool. In 45

order to rectify the classical time-frequency analysis deficiencies of poor time-frequency localization and cross-term interference, there needs to be a method that produces more concentrated distributions and reduces cross-terms, which the reassignment method does. This improves readability, which makes for more accurate metrics extraction and consequently a more effective intercept receiver environment.

The reassignment principle for the Spectrogram allows for a straight-forward extension of its use to other distributions as well [HIP00]. If we consider the general expression of a distribution of the Cohen’s class as a two-dimensional convolution of the WVD, as in equation 2.17:

replacing the particular smoothing kernel

by an arbitrary kernel

simply

defines the reassignment of any member of Cohen’s class (equations 2.18 through 2.20):

46

The resulting reassigned distributions efficiently combine a reduction of the interference terms provided by a well adapted smoothing kernel and an increased concentration of the signal components achieved by the reassignment. In addition, the reassignment operators and

are almost as easy to compute as for the Spectrogram [AUG95].

Similarly, the reassignment method can also be applied to the time-scale energy distributions [RIO92]. Starting from the general expression in equation 2.21:

we can see that the representation value at any point weighted WVD values on the points

is the average of the

located in a domain centered on

and

bounded by the essential support of . In order to avoid the resultant signal components broadening while preserving the cross-terms attenuation, it seems once again appropriate to assign this average to the center of gravity of these energy measures, whose coordinates are shown in equations 2.22 and 2.23:

47

rather than to the point

where it is computed. The value of the resulting

modified time-scale representation on any point

is then the sum of all the

representation values moved to this point, and is known as the reassigned Scalogram (equation 2.24):

As for Cohen’s class, it can be shown that these modified distributions are also theoretically perfectly localized for chirps and impulses.

As alluded to above, a potential drawback of the reassignment method is that there is a chance it could generate spurious artifacts due to the reassignment process randomly clustering noise (vice signals). Because of this, its performance may suffer in low SNR scenarios.

Recapping, it is noted that the smoothing and squeezing qualities of the reassignment method lead to improved readability, which leads to more accurate metrics extraction, which creates a more informed and safer intercept receiver environment. 48

The following three reassignments are utilized in this dissertation: 

Reassigned Spectrogram



Reassigned Scalogram



Reassigned Smoothed Pseudo WVD (RSPWVD)

2.6.7. The Hough Transform P.V.C. Hough patented the Hough transform in 1962 [HOU62], and it was later used in work accomplished by Duda and Hart [DUD72].

Consider the case where we have straight lines in an image. For every point the image, all the straight lines pass through that point satisfy

in

for varying

values of line slope and intercept.

Now if we reverse our variables and look instead at the values of the image point coordinates

, then

as a function of

becomes

which

also describes a straight line.

Consider two points

and

, which lie on the same line in the

space. For each

point, we can represent all possible lines through it by a single line in the Therefore a line in the

space.

space that passes through both points must lie on the

intersection of the two lines in the

space representing the two points. This means

49

that all points which lie on the same line in the which all pass through a single point in the

To avoid the problem of infinite

space are represented by lines

space.

values which occurs when vertical lines exist in the

image, an alternative formulation,

(the parametric representation of

a line) can be used to describe a line [CAR94], [DAH08]. This means that a point in the space (image space) is now represented by a sinusoid in

space (parameter

space) rather than by a straight line. Points lying on the same line in the

space

define sinusoids in the parameter space which all intersect at the same point. The more points that exist on that particular line in image space; the more sinusoids will intercept at that particular point in parameter space, and consequently, the more the accumulator value at this point (parameter space) will increase, forming a ‘spike’ in the parameter space. Therefore, ‘spikes’ (peak values) in the parameter space correspond to lines in the image space.

The coordinates of the point of intersection of the sinusoids in the

parameter space define the parameters of the line in the

space (image space). For

example, if we apply the Hough transform to the WVD of a chirp (line), we obtain a peak in the parameter space located in a position which depends on the parameter values (such as chirp rate) of the chirp (line) in the image space (the WVD plot) [SHA07] [XUL93].

This can best be shown by Figure 6 below:

50

RAS

Mapping from T-F plane (image space) to Hough plane (parameter space)

Each point on the line in the Image Space is transformed to a sinusoidal curve in the Parameter Space (Hough Plane) Y-axis (freq)

1

rho

2

theta(x) is the angle between the vertical axis above the ctr.pt.and the normal of the signal line passing through the center

3

2

1

3 rho(x) is the perpendicular distance from the center to the signal line rho(x) center of the image Parameter Space (Hough Plane) (Birdseye view of 3-D Param Space)

Image Space X-axis (time)

theta(x)

theta

Figure 6: Time-frequency plot on the left and Hough transform plot on the right. A point • Themaps intersection the sinusoidal curves 1, 2,in3 (parameter the line point rho(x), theta(x) in the TF plot to aof sinusoidal curve the HT space) plot. at A (signal) in the TF plot corresponds line connecting 2, 3and (imagetheta space)values that has rho, rho(x), maps to a pointtointhethe HT plot.the points The 1,rho of theta the values pointofin thetheta(x) HT plot can • The more sinusoidal curves (parameter space) that pass through a point – the higher the accumulator value is and the higher be used to back-map to the TF plot in order to find the location of the line (signal) (good the 3-D Hough Peak (2-D plane is birdseye view of 3-D plane, therefore pt of intersect in 2-D plane =s Spike in 3-D plane) is if time-frequency plot is cluttered with noise and/or cross-term interference and signal • Note that each line in the Image Space can be represented by a unique rho, theta value which maps to the unique point rho, not visible). theta in the Parameter Space

In Figure 6, the image space (time-frequency plot) is on the left and the parameter space (two-dimensional Hough transform plot) is on the right. Each point in the image space maps to a sinusoidal curve in the parameter space. The points 1, 2, and 3 in the image space map to the sinusoidal curves 1, 2, and 3 in the parameter space. In the parameter space, the intersection of the sinusoidal curves 1, 2, 3 at the point rho(x), theta(x) corresponds to the line connecting the points 1, 2, and 3 in the image space (same rho(x) and theta(x) values) [ISI96]. The more sinusoidal curves in the parameter space that pass through a particular point, the higher the accumulator value of that point will be and the higher the three-dimensional Hough Transform ‘spike’ will be [OLM01]. The presence of a peak in the parameter space reveals the presence of high positive values concentrated 51

along a line in the image space – whose parameters are exactly the coordinates of the peak. The peak in the parameter space is located in a position which depends on the chirp rate of the line in the image space [BAR95].

The two-dimensional Hough

transform plot is simply a birds-eye view of the three-dimensional plot, therefore a ‘point’ (or ‘bright spot’) in two-dimensional Hough transform plot is equivalent to a ‘spike’ in the three-dimensional Hough transform plot. The Hough transform converts a difficult global detection problem in the image space into a more easily solved local peak detection problem in the parameter space [THU04].

The Hough Transform of a given function

Where point

is defined in equation 2.25 as:

is the Dirac delta function. With

(as noted in the figure above), each

in the original image , is transformed into a sinusoid

where, in the image, line at an angle

,

is the perpendicular distance from the center of the image to the

from the vertical axis passing through the center of the image. Again,

points that lie on the same line in the image will produce sinusoids that all cross at a single point in the Hough plot.

52

The expression above gives the projection (line integral) of

along an arbitrary line

in the xy plane. By definition, the Hough transform computes the integration of the values of an image over all its lines.

From the signal location (rho and theta values) of the Hough transform plot, it is possible to back-map back to the signal location in the time-frequency representation, using the same exact rho and theta values.

Let’s give an example of back-mapping with two of the test plots from this dissertation, starting with the Hough Transform plot in Figure 7:

Figure 7: Hough transform of the WVD of a triangular modulated FMCW signal at an SNR of 10dB (512 samples). Each point has a unique theta and rho value which can be used to back-map to the time-frequency (WVD) representation in order to locate the 4 signals, as depicted in Figure 7.

53

4 signals are clearly seen in the Hough transform plot (Figure 7); from left to right they are (theta, rho, intensity):

Signal 1: .7854, 136, 8994 Signal 2: 2.381, 43.22, 11430 Signal 3: 3.902, 43.93, 11540 Signal 4: 5.498, 136.7, 9543

The values of rho and theta allow for back-mapping to the time-frequency distribution in order to determine the location of these 4 signals in the time-frequency distribution.

Theta is in radians, therefore we multiply by 57.3 to obtain degrees.

Rho is the number of samples, therefore we divide by 512 (the number of samples of the Y-Axis of the time-frequency distribution) to obtain rho (length) in terms of percent of the length of the entire Y-Axis of the time-frequency distribution.

Signal 1: 45.0 degrees, 26% of Y-Axis Signal 2: 136.4 degrees, 8% of Y-Axis Signal 3: 223.6 degrees, 8% of Y-Axis Signal 4: 315.0 degrees, 26% of Y-Axis

54

With these values, we can now back-map from the Hough transform plot (Figure 7) to the time-frequency distribution to find where the 4 signals are located in the time-frequency distribution (Figure 8).

Figure 8: The unique theta and rho values extracted from Figure 6 are used to back-map to the time-frequency representation (WVD of a triangular modulated FMCW signal (SNR=10dB, #samples=512)) in order to locate the 4 chirp signals that make up the 4 legs of the triangular modulated FMCW signal. Figure 8 shows how the unique theta and rho values from the Hough transform plot can be used to back-map to the time-frequency distribution in order to find the location of the signals in the time-frequency distribution. This would be beneficial in the case where the signals in the time-frequency distribution were unable to be seen, due to cross-term interference and/or noise, but the signals were seen in the Hough transform plot. We could then back-map from the Hough transform plot, using the theta and rho values of the

55

signals, to find the location of the signals in the time-frequency representation. These types of cases will be demonstrated later in this dissertation.

It is important to note that the Hough transform method works well in the presence of multi-component signals, in spite of the cross-terms produced by time-frequency distributions such as the WVD [TOR07]. Since the cross-terms have an amplitude modulation, the integration implicit in the Hough transform reduces them, while the useful contributions, which are always positive, are correctly integrated [BAR92], [BAR95]. Likewise, in the presence of noise, the integration carried out by the Hough transform produces an improvement in the SNR [INC07], [YAS06], [NIK08].

The Hough transform is very similar to the Radon transform. The Hough transform, like the Radon transform is a mapping from image space to parameter space. The Radon transform is usually treated as a reading paradigm (how a data point in the destination space is obtained from the data in the source space). The Hough transform is usually treated as a writing paradigm (how a data point in the source space maps onto data points in the destination space) [GIN04], [ZAI99].

Some additional advantages of the Hough transform are its ability to discard features belonging to other objects and its robustness against incomplete data [CAR06], [BEN05].

56

The Hough transform finds many uses today, from signal processing (such as low SNR signal extraction and chirp rate determination) to image processing (such as locating iris features in frontal face images [TOE02]).

The ability of the Hough Transform to perform well in low SNR environments as well as in heavy cross-term environments makes it an ideal signal analysis tool to offset the classical time-frequency analysis deficiencies of cross-term interference and mediocre performance in low SNR environments. This will make for better readability, leading to more accurate parameter extractions for the intercept receiver signal analyst.

The following three Hough transforms are used in this dissertation: 

Hough transform of the WVD



Hough transform of the CWD



Hough transform of the reassigned smoothed pseudo WVD (RSPWVD)

In summary, this section has described some of the classical time-frequency analysis techniques as well as the reassignment method and the Hough transform.

The

reassignment method addresses the classical time-frequency analysis deficiencies of poor time-frequency localization and cross-term interference, and the Hough transform addresses the classical time-frequency analysis deficiencies of cross-term interference and mediocre performance in low SNR environments. Addressing these deficiencies of the current cutting edge, state-of-the art signal analysis technology for digital intercept receivers, and eventually implementing the reassignment method and the Hough 57

transform as the new cutting edge, state-of-the-art signal analysis techniques for digital receivers will result in an improvement in readability, leading to more accurate parameter extractions, which the intercept receiver signal analyst requires.

2.7. Previous Related Studies This section presents findings from several studies as related to the research done in this dissertation.

2.7.1. ‘Detection of Low Probability of Intercept Radar Signals’ In [WON09] the researchers used a form of the WVD followed by the Hough transform as one of their detection and parameter estimation algorithms, but, for the FMCW case, it was used for just a single chirp signal, therefore there was no cross-term interference as was seen in this dissertation in the plots of the WVD of a triangular modulated FMCW signal or an FSK signal. Without cross-term interference to reduce readability, there’s no consequent degradation of the detection and parameter metrics.

2.7.2. ‘Extraction of Polyphase Radar Modulation Parameters Using a Wigner-Ville Distribution Radon Transform’ In [GUL08] the researchers use the pseudo Wigner-Ville Distribution (PWVD) + Radon transform. As mentioned earlier in this dissertation, the Hough transform and the Radon transform are nearly identical, therefore PWVD + Radon transform is virtually identical to PWVD + Hough transform.

58

Their results indicate that the overall errors are reasonably small for 0 dB, but at the low SNR value (-6 dB), the largest errors occurred. This is consistent with the results of the testing of this dissertation, which showed that the percent error for the metrics generally increased as the SNR decreased.

Their results also indicate that the extraction algorithm that combines the PWVD with the Radon transform is not affected by the cross-terms present within the PWVD images. The reason stated is that the integration of the cross-term projections is small compared to the modulation projections. This is consistent with what was brought out earlier in this dissertation, that since the cross-terms have an amplitude modulation, the integration implicit in the Hough transform reduces them, while the useful contributions (the signals), which are always positive, are correctly integrated

Their algorithm attempts to efficiently autonomously extract signal parameters, which is a future goal of this dissertation work as well. For those types of intercept receivers where speed (of the signal analysis process) is essential, autonomous extraction of signal parameters using time-frequency transform and the Radon (Hough) transform is particularly beneficial.

They used the PWVD + Radon transform only for parameter extraction. Consideration should be given towards using the PWVD + Radon for signal detection as well (particularly in low SNR environments). The results of this dissertation bring out the fact

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that signal detection in a low SNR environment is one of the strong points of the Hough (in their case Radon) transform.

2.7.3. ‘Periodic Wigner-Ville Hough Transform’ In [GER09] the researchers’ algorithm is similar to the WVD + Hough transform, except that it searches for patterns such as the sawtooth FMCW waveform in the timefrequency image, vice searching for lines. It uses a matching function that compensates exactly for the time-frequency characteristic of the sawtooth FMCW waveform (a realistic LPI waveform).

Their results state that the periodic Wigner-Ville Hough transform is optimal for the detection and estimation of sawtooth FMCW waveforms and is superior to the WVD + HT in the area of detection performance.

They assume that the phase is coherent from one LFM ramp to the next, which for a simple VCO type transmitter, the phase of one LFM segment is not related to the phase of the next LFM segment. It could be coherent in this way if the transmitter used digital phase modulation to generate the whole signal, but for present altimeters, this is not the case.

In addition, for their research, one needs to search for the right repetition period, the right starting frequency and the right slope.

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2.7.4 ‘Detection and Parameter Estimation of Chirped Radar Signals’ In [HIP00] the researchers utilize eleven different time-frequency/scale transformations as signal analysis tools (Wavelet packet, Cosine packet, Wavelet, Cosine pursuit, WVD, PWVD, RPWVD, SPWVD, RSPWVD, Spectrogram, Reassigned Spectrogram).

In

addition, they use the Radon transform (similar to the Hough transform), but it appears only to be used in combination with the Spectrogram.

Their results indicate that time-frequency transformations lead to better focused images when dealing with noisy chirp signals, and to better estimation of the modulation parameters than do wavelet based decompositions, which agrees with the results of the testing in this dissertation.

Their results indicate that the reassignment method improved the performances at higher SNRs, which agrees with the statement made earlier in this dissertation that the performance of the reassignment method may degrade in low SNR environments, due to the reassignment process clustering noise instead of signals.

Their results indicate that the combination of time-frequency/scale transformations plus the Radon (Hough in our case) transform lead to small detection errors. This agrees with the results of the testing of this dissertation that show that one of the strong points of the Hough transform, is signal detection, particularly in low SNR and/or high cross-term environments.

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Their analysis is done predominantly on a signal chirp signal (and occasionally for two parallel chirp signals). As mentioned previously, a single chirp signal will not produce any cross-term interference, and consequently will not degrade readability. In addition, the Radon transform is only performed on the Spectrogram.

Due to the fact that the previous related studies used different waveforms, signal parameters, SNR levels, and analysis tools than were used for this dissertation, it is difficult to make a direct comparison between the results of the previous related studies and the results of this dissertation.

Based on background research and literature reviews performed in preparation for this dissertation research, it appears that little research has been done in the area of analyzing triangular modulated FMCW LPI radar signals and FSK LPI radar signals using the reassignment method and the Hough transform.

The background section of this dissertation has presented an overview of LPI radars as well as those devices that intercept and analyze LPI radar signals, known as intercept receivers. Since the trend in radar is towards LPI radars, the trend in intercept receivers is towards intercepting and analyzing LPI radar signals, which requires that intercept receivers be digital. Most of the digital intercept receivers currently in the fleet use Fourier-based analysis, which is less than adequate for signals that change in frequency over time, such as LPI radar signals. Because of this inadequacy, there is a push towards using classical time-frequency analysis techniques (currently in the lab phase) for 62

analyzing these non-stationary signals. However, the classical time-frequency analysis techniques have deficiencies, such as poor time-frequency localization, cross-term interference, and mediocre performance in low SNR environments, all of which inhibit the readability of the time-frequency representations. Limited readability translates to poor metrics extraction, which for the intercept receiver signal analyst, can lead to an ineffective intercept receiver environment. We hypothesize that the reassignment method will address the deficiencies of poor time-frequency localization and cross-term interference, and that the Hough transform will address the deficiencies of mediocre performance in low SNR environments and cross-term interference. This will make for a more readable representation and therefore more accurate metrics.

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3. Methodology The methodologies detailed in this section describe the processes involved in obtaining and comparing metrics between the classical time-frequency analysis techniques and the reassignment method and the Hough transform for the detection and parameter extraction of LPI radar signals.

The tools used for this testing were: MATLAB (version 7.7), Signal Processing Toolbox (version 6.10), Wavelet Toolbox (version 4.3), Image Processing Toolbox (version 6.2), Time-Frequency Toolbox (version 1.0) (http://tftb.nongnu.org/).

All the testing was accomplished on a desktop computer (HP Compaq, 2.5GHz processor, AMD Athlon 64X2 Dual Core Processor 4800+, 2.00GB Memory (RAM), 32 Bit Operating System).

Testing was performed for 4 different waveforms (2 triangular modulated FMCWs and 2 FSKs), each waveform representing a different task (Task 1 through Task 4). For each waveform, parameters were chosen for academic validation of signal processing techniques. Due to computer processing resources they were not meant to represent realworld values. The number of samples for each test was chosen to be either 256 or 512, which seemed to be the optimum size for the desktop computer. Testing was performed at three different SNR levels: 10dB, 0dB, and the lowest SNR at which the signal could be detected. The noise added was white Gaussian noise, which best reflected the thermal noise present in the IF section of an intercept receiver [PAC09]. Kaiser windowing was 64

used, when windowing was applicable.

25 runs were performed for each test, for

statistical purposes. The plots included in this dissertation were done at a threshold of 5% of the maximum intensity and were linear scale (not dB) of analytic (complex) signals; the color bar represented intensity. The signal processing tools used for each task were:

Classical time-frequency analysis techniques: WVD, CWD, Spectrogram, Scalogram

Reassignment method: reassigned Spectrogram, reassigned Scalogram, reassigned smoothed pseudo Wigner Ville Distribution (RSPWVD)

Hough transform method: Hough transform of WVD, Hough transform of CWD, Hough transform of RSPWVD

The test matrix for all of the testing is included in Appendix A. The test plots and test metrics tables for each of the tests that were run are included in Appendix B.

Task 1 consisted of analyzing a triangular modulated FMCW signal (most prevalent LPI radar waveform [LIA09]) whose parameters were: sampling frequency=4KHz; carrier frequency=1KHz; modulation bandwidth=500Hz; modulation period=.02sec.

Task 2 was similar to Task 1, but with different parameters: sampling frequency=6KHz; carrier frequency=1.5KHz; modulation bandwidth=2400Hz; modulation period=.15sec. 65

The different parameters were chosen to see how the different shapes/heights of the triangles of the triangular modulated FMCW would affect the cross-term interference and the metrics.

Task 3 consisted of analyzing an FSK (prevalent in the LPI arena [AMS09]) 4component signal whose parameters were: sampling frequency=5KHz; carrier frequencies=1KHz, 1.75KHz, 0.75KHz, 1.25KHz; modulation bandwidth=1000Hz; modulation period=.025sec.

Task 4 was similar to Task 3, but for an FSK 8-component signal whose parameters were: sampling frequency=5KHz; carrier frequencies=1.5KHz, 1KHz, 1.25KHz, 1.5KHz, 1.75KHz, 1.25KHz, 0.75KHz, 1KHz; modulation bandwidth=1000Hz; modulation period=.0125sec. The different number of components and different parameters between Task 3 and Task 4 were chosen to see how the different number/lengths of FSK components would affect the cross-term interference and the metrics.

Because of computational complexity, the WVD tests and the Hough transform of WVD tests for 512 samples, SNR=0dB and 512 samples, SNR= low SNR - were not able to be performed for any of the 4 waveforms. It was noted that a single run was still processing after more than 8 hours. The WVD is known to be very computationally complex [MIL02].

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After each particular run of each test, metrics were extracted from the time-frequency representations and/or the Hough transform. The metrics extracted were as follows (both=applies to both the time-frequency representation (includes classical timefrequency analysis techniques and the reassignment method) and the Hough transform; TF=time-frequency representation; HT=Hough transform):

1) Plot (processing) time (both): time required for plot to be displayed. 2) Percent detection TF: percent of time signal was detected - signal was declared a detection if any portion of each of the signal components (4 chirp components for triangular modulated FMCW, and 4 or 8 signal components for FSK) exceeded a set threshold (a certain percentage of the maximum intensity of the time-frequency representation).

Threshold percentages were determined based on visual detections of low SNR signals (lowest SNR at which the signal could be visually detected in the time-frequency representation) (see Figure 9).

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Figure 9: Threshold percentage determination. This plot is an amplitude vs. time (x-z view) of CWD of FSK 4-component signal (512 samples, SNR= -2dB). For visually detected low SNR plots (like this one), the percent of max intensity for the peak z-value of each of the signal components was noted (here 82%, 98%, 87%, 72%), and the lowest of these 4 values was recorded (72%). Ten test runs were performed for each timefrequency analysis tool, for each of the 4 waveforms. The average of these recorded low values was determined and then assigned as the threshold for that particular timefrequency analysis tool. Note - the threshold for CWD is 70%. Thresholds were assigned as follows: CWD (70%); Spectrogram (60%); Scalogram, Reassigned Spectrogram, Reassigned Scalogram, RSPWVD, WVD (4-component FSK) (50%); WVD (triangular modulated FMCW) (35%); WVD (8-component FSK) (20%).

For percent detection determination, these threshold values were included in the timefrequency plot algorithms so that the thresholds could be applied automatically during the plotting process. From the threshold plot, the signal was declared a detection if any portion of each of the signal components was visible (see Figure 10).

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Figure 10: Percent detection (time-frequency). CWD of 4-component FSK (512 samples, SNR=10dB) with threshold value automatically set to 70%. From this threshold plot, the signal was declared a (visual) detection because at least a portion of each of the 4 FSK signal components was visible. Automatically applying a threshold value to the time-frequency plot algorithms for percent detection determination can be seen as a first step towards the future work of automating the metrics extraction process.

HT: percent of time signal was detected – signal was declared a detection if any portion of each of the signal components exceeded the noise floor threshold (see Figure 11).

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Figure 11: Percent detection (Hough transform). This plot is a theta-intensity (x-z view) of the HT of an RSPWVD (512 samples, SNR=10dB). Signal declared a (visual) detection because at least a portion of each of the signal components exceeded the noise floor threshold. 3) Cross-term false positives (XFPs): The number of cross-terms that were wrongly declared as signal detections. For the time-frequency representation, the XFP detection criteria is the same as the time-frequency signal detection criteria listed in the percent detection section above. For the HT, the XFP detection criteria is the same as the HT signal detection criteria listed in the percent detection section above. Figure 12, repeated from the WVD section of this dissertation, shows 4 true signals and also shows 6 crossterms, all of which were XFPs.

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Figure 12: Example of cross-term false positives (XFPs). WVD of a 4-component FSK signal at SNR=10dB (512 samples). There are 4 true signals (fc1=1KHz, fc2=1.75KHz, fc3=0.75KHz, fc4=1.25KHz) and 6 XFPs (cross-terms that were wrongly declared as signal detections because they passed the signal detection criteria listed in the percent detection section above) (ct1=0.875KHz, ct2=1KHz, ct3=1.125KHz, ct4=1.25KHz, ct5=1.375KHz, ct6=1.5KHz). 4) Carrier frequency (TF): the frequency corresponding to the maximum intensity of the time-frequency representation (there are multiple carrier frequencies for FSK waveforms) (see Figure 13 (for triangular modulated FMCW) and Figure 14 (for FSK)).

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Figure 13: Determination of carrier frequency. CWD of a triangular modulated FMCW (256 samples, SNR=10dB). From the frequency-intensity (y-z) view, the maximum intensity value is manually determined. The frequency corresponding to the max intensity value is the carrier frequency (here fc=1062.4 Hz).

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Figure 14: Determination of carrier frequency. CWD of a 4-component FSK (512 samples, SNR=10dB). From the frequency-intensity (y-z) view, the 4 maximum intensity values (1 for each carrier frequency) are manually determined. The frequencies corresponding to those 4 max intensity values are the 4 carrier frequency (here fc1=10001 Hz, fc2=1753Hz, fc3=757Hz, fc4=1255Hz). 5) Modulation bandwidth (TF): distance from highest frequency value of signal (at a threshold of 20% maximum intensity) to lowest frequency value of signal (at same threshold) in Y-direction (frequency).

The threshold percentage was determined based on manual measurement of the modulation bandwidth of the signal in the time-frequency representation. This was accomplished for ten test runs of each time-frequency analysis tool, for each of the 4 waveforms. During each manual measurement, the percent of max intensity of the high and low measuring points was recorded. The average of the percent of max intensity

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values for these test runs was 20%. This was adopted as the threshold value, and is representative of what is obtained when performing manual measurements.

For modulation bandwidth determination, the 20% threshold value was included in the time-frequency plot algorithms so that the threshold could be applied automatically during the plotting process. From the threshold plot, the modulation bandwidth was manually measured (see Figure 15 (for triangular modulated FMCW) and Figure 16 (for FSK)).

Figure 15: Modulation bandwidth determination. CWD of a triangular modulated FMCW (512 samples, SNR=10dB) with threshold value automatically set to 20%. From this threshold plot, the modulation bandwidth was measured manually from the highest frequency value of the signal (top red arrow) to the lowest frequency value of the signal (bottom red arrow) in the y-direction (frequency).

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Figure 16: Modulation bandwidth determination. CWD of a 4-component FSK (512 samples, SNR=10dB) with threshold value automatically set to 20%. From this threshold plot, the modulation bandwidth was measured manually from the highest frequency value of the signal (top red arrow) to the lowest frequency value of the signal (bottom red arrow) in the y-direction (frequency). Automatically applying a threshold value to the time-frequency plot algorithms for modulation bandwidth determination can be seen as a first step towards the future work of automating the metrics extraction process.

6) Modulation period (TF): for triangular modulated FMCW - distance from highest frequency value of signal (at a threshold of 20% maximum intensity) to lowest frequency value of signal (at same threshold) in X-direction (time) – for FSK - width of FSK component (at a threshold of 20% maximum intensity) in X-direction (time).

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The threshold percentage was determined based on manual measurement of the modulation period of the signal in the time-frequency representation. This was accomplished for ten test runs of each time-frequency analysis tool, for each of the 4 waveforms. During each manual measurement, the max intensity of the two measuring points was recorded. The average of the max intensity values for these test runs was 20%. This was adopted as the threshold value, and is representative of what is obtained when performing manual measurements.

For modulation period determination, the 20% threshold value was included in the timefrequency plot algorithms so that the threshold could be applied automatically during the plotting process. From the threshold plot, the modulation period was manually measured (see Figure 17 (for triangular modulated FMCW) and Figure 18 (for FSK)).

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Figure 17: Modulation period determination. CWD of a triangular modulated FMCW (512 samples, SNR=10dB) with threshold value automatically set to 20%. From this threshold plot, the modulation period was measured manually from the highest frequency value of the signal (top red arrow) to the lowest frequency value of the signal (bottom red arrow) in the x-direction (time).

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Figure 18: Modulation period determination. CWD of a 4-component FSK (512 samples, SNR=10dB) with threshold value automatically set to 20%. From this threshold plot, the modulation period was measured manually from the left side of the signal (left red arrow) to the right side of the signal (right red arrow) in the x-direction (time). This was done for all 4 signal components, and the average value was determined.

Automatically applying a threshold value to the time-frequency plot algorithms for modulation period determination can be seen as a first step towards the future work of automating the metrics extraction process.

7) Time-frequency localization (TF): a measure of the thickness of a signal component (at a threshold of 20% maximum intensity on each side of the component) – converted to % of entire X-Axis, and % of entire Y-Axis.

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The threshold percentage was determined based on manual measurement of the timefrequency localization of the signal in the time-frequency representation. This was accomplished for ten test runs of each time-frequency analysis tool, for each of the 4 waveforms. During each manual measurement, the max intensity of the two measuring points was recorded. The average of the max intensity values for these test runs was 20%. This was adopted as the threshold value, and is representative of what is obtained when performing manual measurements.

For time-frequency localization determination, the 20% threshold value was included in the time-frequency plot algorithms so that the threshold could be applied automatically during the plotting process. From the threshold plot, the time-frequency localization was manually measured (see Figure 19 (for triangular modulated FMCW) and Figure 20 (for FSK)).

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Figure 19: Time-frequency localization determination. CWD of a triangular modulated FMCW (512 samples, SNR=10dB) with threshold value automatically set to 20%. From this threshold plot, the time-frequency localization was measured manually from the left side of the signal (left red arrow) to the right side of the signal (right red arrow) in both the x-direction (time) and the y-direction (frequency). Measurements were made at the center of each of the 4 ‘legs’, and the average values were determined. Average time and frequency ‘thickness’ values were then converted to: % of entire x-axis and % of entire yaxis.

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Figure 20: Time-frequency localization determination for the CWD of a 4-component FSK (512 samples, SNR=10dB) with threshold value automatically set to 20%. From this threshold plot, the time-frequency localization was measured manually from the top of the signal (top red arrow) to the bottom of the signal (bottom red arrow) in the ydirection (frequency). This frequency ‘thickness’ value was then converted to: % of entire y-axis. Automatically applying a threshold value to the time-frequency plot algorithms for timefrequency localization determination can be seen as a first step towards the future work of automating the metrics extraction process.

8) Chirp rate TF: (modulation bandwidth)/(modulation period) – (for Task 1 and Task2 only). HT: chirp rate

– (for Task 1 and

Task 2 only).

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9) Lowest detectable SNR TF: the lowest SNR level at which at least a portion of each of the signal components exceeded the set threshold listed in the percent detection section above.

For lowest detectable SNR determination, these threshold values were included in the time-frequency plot algorithms so that the thresholds could be applied automatically during the plotting process. From the threshold plot, the signal was declared a detection if any portion of each of the signal components was visible. The lowest SNR level for which the signal was declared a detection is the lowest detectable SNR (see Figure 21).

Figure 21: Lowest detectable SNR (time-frequency). CWD of 4-component FSK (512 samples, SNR=-2dB) with threshold value automatically set to 70%. From this threshold plot, the signal was declared a (visual) detection because at least a portion of each of the 4 FSK signal components was visible. For this case, any lower SNR would have been a non-detect. Compare to Figure 10, which is the same plot, except that it has an SNR level equal to 10dB. 82

HT: the lowest SNR level for which each signal component exceeded the noise floor threshold (see Figure 22).

Figure 22: Lowest detectable SNR (Hough transform). This plot is a theta-intensity (x-z view) of the HT of an RSPWVD (512 samples, SNR=-5dB). Signal declared a (visual) detection because at least a portion of each of the signal components exceeded the noise floor threshold. For this case, any lower SNR would have been a non-detect. Compare to Figure 11, which is the same plot, except that it has an SNR level equal to 10dB. The data from all 25 runs for each test was used to produce the mean, standard deviation, variance, actual, error, and percent error for each of the metrics listed above.

The metrics from the classical time-frequency analysis techniques were then compared to the metrics from the reassignment method and the Hough transform. By and large, the

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reassignment method and the Hough transform outperformed the classical time-frequency analysis techniques, as will be shown later in this dissertation

For Task 5, data from a CD was analyzed, with the only a priori knowledge being that the data contained an LPI radar signal in a low SNR environment (between -5dB and -10dB), and that the data was collected at a sampling frequency of 4GHz.

The data (19

megabytes) was first processed using the Spectrogram (because it is the fastest timefrequency analysis tool). The signal was not visible in the Spectrogram time-frequency representation, due to low SNR. The data was then processed with the Hough transform of the Spectrogram, and the signal was detected, but had almost zero slope (like an FSK (tonal) signal). A MATLAB script was written which allowed for ‘zooming-in’ on the receiver IF bandwidth (~ 750MHz to 1250MHz) (Y-axis zoom-in only), and then the data was re-processed with the Hough transform of the Spectrogram, this time for the purpose of determining if the signal had slope or if it was tonal. From the Hough transform plot it was observed that the signal had slope (i.e. was a chirp signal). The Hough transform plot allowed for not only detection of the chirp signal, but also for extraction of the chirp rate. A back-mapping from the Hough plot to the time-frequency representation was then performed.

The signal was located in the time-frequency representation, and the

modulation bandwidth and modulation period (and consequently the chirp rate) were extracted from the time-frequency representation. From these metrics, the type/source of the signal was identified. Additional details/results of Task 5 testing are addressed later in this dissertation.

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4. Detection and Parameter Extraction of Low Probability of Intercept Radar Signals Using the Reassignment Method

Material from this chapter will be published as a journal paper and is presented here in the corresponding form.

The author of this disseertation was the primary student

contributor of the work presented in this chapter.

4.1. Abstract Digital intercept receivers are currently moving away from Fourier-based analysis and towards classical time-frequency analysis techniques, based on the Wigner-Ville distribution, Choi-Williams distribution, spectrogram, and scalogram, for the purpose of analyzing low probability of intercept radar signals (e.g. triangular modulated frequency modulated continuous wave and frequency shift keying). Although these classical timefrequency analysis techniques are an improvement over the Fourier-based analysis, they still suffer from a lack of readability, due to poor time-frequency localization and crossterm interference.

This lack of readability may lead to inaccurate detection and

parameter extraction of these radar signals, making for a less informed (and therefore less safe) intercept receiver environment.

In this dissertation, the reassignment method,

because of its ability to smooth out cross-term interference and improve time-frequency localization, is proposed as an improved signal analysis technique. With these qualities, the reassignment method has the potential to produce better readability and consequently, more accurate signal detection and parameter extraction metrics.

Simulations are

presented that compare time-frequency representations of the classical time-frequency 85

techniques with those of the reassignment method. Two different triangular modulated frequency modulated continuous wave low probability of intercept radar signals and two different frequency shift keying low probability of intercept radar signals (4-component and 8-component) were analyzed. The following metrics were used for evaluation of the analysis: percent error of: carrier frequency, modulation bandwidth, modulation period, time-frequency localization, and chirp rate. Also used were: percent detection, number of cross-term false positives, lowest signal-to-noise ratio for signal detection, and plot (processing) time.

Experimental results demonstrate that the ‘squeezing’ and

‘smoothing’ qualities of the reassignment method did lead to improved readability over the classical time-frequency analysis techniques, and consequently, provided more accurate signal detection and parameter extraction metrics (smaller percent error from true value) than the classical time-frequency analysis techniques.

In summary, this

dissertation provides evidence that the reassignment method has the potential to outperform the classical time-frequency analysis techniques, which are the current stateof-the-art, cutting-edge techniques for this arena. An improvement in performance can easily translate into saved equipment and lives. Future work will include automation of the metrics extraction process, analysis of additional low probability of intercept radar waveforms of interest, and analysis of other real-world low probability of intercept radar signals utilizing more powerful computing platforms.

4.2. Introduction Today, radar systems face serious threats from anti-radiation systems [FAU02], electronic attack [GAU02] and other areas. In order to be able to perform their functions 86

properly, they must be able to ‘see without being seen’ [PAC09], [WIL06].

This

necessitates that they be low probability of intercept (LPI) radars. These radars typically have very low peak power, wide bandwidth, high duty cycle, and power management, making them difficult to be detected and characterized by intercept receivers.

On the other side of the fence, the intercept receiver that intercepts these LPI radar signals faces threats as well. It must be able to detect and characterize the signals from these LPI radars for the purpose of providing timely information about threatening systems, in addition to providing information about defensive systems - which is important in maintaining a credible deterrent force to penetrate those defenses. In this case, knowledge provided by the intercept receiver provides insight for determining the details of the threat so that an effective response can be prepared [WIL06].

To make matters even more challenging for intercept receivers, most of the intercept receivers currently in the fleet are analog and were designed to intercept ‘older’ (i.e. nonLPI) radar signals, and which perform poorly when faced with LPI radar signals. Digital receivers, which are becoming a growing trend, are seen as the eventual solution to the intercept receiver’s LPI challenge.

Fourier analysis techniques using the FFT have been employed as the basic tool of the digital intercept receiver for detecting and extracting parameters of LPI radar signals, and make up a majority of the digital intercept receiver techniques currently in the fleet [PAC09].

When a practical non-stationary signal (such as an LPI radar signal) is 87

processed, the Fourier transform cannot efficiently analyze and process the time-varying characteristics of the signal’s frequency spectrum, because time and frequency information cannot be combined to tell how the frequency content is changing in time [XIE08], [STE96]. The non-stationary nature of the received radar signal mandates the use of some form of time-frequency analysis for signal detection and parameter extraction [MIL02].

Some of the more common classical time-frequency analysis techniques include the Wigner-Ville distribution (WVD), Choi-Williams distribution (CWD), spectrogram, and scalogram. The WVD exhibits the highest signal energy concentration [WIL06], but has the worse cross-term interference, which can severely limit the readability of a timefrequency representation [GUL07], [STE96], [BOA03]. The CWD is a member of Cohen’s class, which adds a smoothing kernel to help reduce cross-term interference [BOA03], [UPP08]. The CWD, as with all members of Cohen’s class, is faced with a trade-off between cross-term reduction and time-frequency localization.

The

Spectrogram is the magnitude squared of the short-time Fourier transform [HLA92], [MIT01]. It has poorer time-frequency localization but less cross-term interference than the WVD or CWD, and its cross-terms are limited to regions where the signals overlap [ISI96]. The Scalogram is the magnitude squared of the wavelet transform, and can be used as a time-frequency distribution [COH02], [GAL05], [BOA03].

Like the

Spectrogram, the Scalogram has cross-terms that are limited to regions where the signals overlap [ISI96], [HLA92]. Currently, for digital intercept receivers, these classical time-

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frequency analysis techniques are primarily at the lab phase [PAC09], [WIL06], [GAU02], [GUL07], [MIL02].

Though classical time-frequency analysis techniques, such as those described above, are a great improvement over Fourier analysis techniques, they suffer in general from poor time-frequency localization and cross-term interference, as described above. This may result in degraded readability of time-frequency representations, potentially leading to inaccurate LPI radar signal detection and parameter extraction. This, in turn, can place the intercept receiver environment in harm’s way.

A promising avenue for overcoming these deficiencies is the utilization of the reassignment method. The reassignment method, which can be applied to most energy distributions [HIP00], has, in theory, a perfectly localized distribution for chirps, tones, and impulses [ISI96], [BOA03], making it a good candidate for the analysis of certain LPI radar signals, such as the triangular modulated frequency modulated continuous wave (FMCW) (which can be viewed as back-to-back chirps) and the frequency shift keying (FSK) (which can be viewed as tones).

The reassignment method’s scheme assumes that the energy distribution in the timefrequency plane resembles a mass distribution and moves each value of the timefrequency plane located at a point (t, f) to another point (t’, f’), which is the center of gravity of the energy distribution in the area of (t, f).

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The result is a focused

representation with high intensity, since the value at the point (t’, f’) is the sum of all the neighboring values [LIX08], [AUG94].

A potential drawback of the reassignment method is that there is a chance it could generate spurious artifacts due to the reassignment process randomly clustering noise (vice signals).

Because of this, its performance may suffer in low SNR scenarios

[BOA93].

The reassignment method can be viewed as helping to build a more readable timefrequency representation. The first step in the process is to reduce (smooth) the crossterm interference.

An unfortunate side-effect of this smoothing is that the signal

components become smeared. The second step in the process is then to refocus (squeeze) the components which were smeared during the smoothing process [ISI96], [BOA03]. This ‘smoothing’ and ‘squeezing’ helps to create a more readable time-frequency representation.

In related work, [HIP00] performed some comparisons between the reassignment method and the WVD and spectrogram, but this was done predominantly for a single chirp signal, which presents no cross-term interference. It appears that there has been little or no research done in the area of using the reassignment method for the analysis of triangular modulated FMCW LPI radar signals and FSK LPI radar signals.

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In this dissertation, the reassignment method is evaluated as a technique for improving the readability of the classical time-frequency analysis representations by improving time-frequency localization and reducing cross-term interference.

This approach is

assessed using 2 triangular modulated FMCW LPI radar signals and 2 FSK LPI radar signals (4-component and 8-component). Metrics designed include percent error of: carrier frequency, modulation bandwidth, modulation period, time-frequency localization and chirp rate. Additional metrics include: percent detection, number of cross-term false positives, lowest SNR for signal detection, and plot time.

The rest of this chapter is organized as follows: methodology is presented in section 4.3.

Description of the proposed

Experimental results comparing the

reassignment method and classical time-frequency analysis techniques are presented in section 4.4, followed by discussion and conclusions.

4.3. Methodology The methodologies detailed in this dissertation describe the processes involved in obtaining and comparing metrics between the classical time-frequency analysis techniques and the reassignment method for the detection and parameter extraction of LPI radar signals.

The tools used for this testing were: MATLAB (version 7.7), Signal Processing Toolbox (version 6.10), Wavelet Toolbox (version 4.3), Image Processing Toolbox (version 6.2), Time-Frequency Toolbox (version 1.0) (http://tftb.nongnu.org/). 91

All the testing was accomplished on a desktop computer (HP Compaq, 2.5GHz processor, AMD Athlon 64X2 Dual Core Processor 4800+, 2.00GB Memory (RAM), 32 Bit Operating System).

Testing was performed for 4 different waveforms (2 triangular modulated FMCWs and 2 FSKs), each waveform representing a different task (Task 1 through Task 4). For each waveform, parameters were chosen for academic validation of signal processing techniques. Due to computer processing limitations they were not meant to represent real-world values. The number of samples for each test was chosen to be either 256 or 512, which seemed to be the optimum size for the desktop computer. Testing was performed at three different SNR levels: 10dB, 0dB, and low SNR (the lowest SNR at which the signal could be detected). The noise added was white Gaussian noise, which best reflected the thermal noise present in the IF section of an intercept receiver [PAC09]. Kaiser windowing was used, when windowing was applicable. 25 runs were performed for each test, for statistical purposes. The plots included in this dissertation were done at a threshold of 5% of the maximum intensity and were linear scale (not dB) of analytic (complex) signals; the color bar represented intensity. The signal processing tools used for each task were:

Classical time-frequency analysis techniques: WVD, CWD, spectrogram, scalogram

92

Reassignment method: reassigned spectrogram, reassigned scalogram, reassigned smoothed pseudo Wigner Ville distribution (RSPWVD)

Task 1 consisted of analyzing a triangular modulated FMCW signal (most prevalent LPI radar waveform [LIA09]) whose parameters were: sampling frequency=4KHz; carrier frequency=1KHz; modulation bandwidth=500Hz; modulation period=.02sec.

Task 2 was similar to Task 1, but with different parameters: sampling frequency=6KHz; carrier frequency=1.5KHz; modulation bandwidth=2400Hz; modulation period=.15sec. The different parameters were chosen to see how the different shapes/heights of the triangles of the triangular modulated FMCW would affect the cross-term interference and the metrics.

Task 3 consisted of analyzing an FSK (prevalent in the LPI arena [AMS09]) 4component signal whose parameters were: sampling frequency=5KHz; carrier frequencies=1KHz, 1.75KHz, 0.75KHz, 1.25KHz; modulation bandwidth=1000Hz; modulation period=.025sec.

Task 4 was similar to Task 3, but for an FSK 8-component signal whose parameters were: sampling frequency=5KHz; carrier frequencies=1.5KHz, 1KHz, 1.25KHz, 1.5KHz, 1.75KHz, 1.25KHz, 0.75KHz, 1KHz; modulation bandwidth=1000Hz; modulation period=.0125sec. The different number of components and different parameters between

93

Task 3 and Task 4 were chosen to see how the different number/lengths of FSK components would affect the cross-term interference and the metrics.

Because of computational complexity, the WVD tests for 512 samples, SNR=0dB and 512 samples, SNR= low SNR were not able to be performed for any of the 4 waveforms. It was noted that a single run was still processing after more than 8 hours. The WVD is known to be very computationally complex [MIL02].

After each particular run of each test, metrics were extracted from the time-frequency representation. The metrics extracted were as follows:

1) Plot (processing) time: time required for plot to be displayed.

2) Percent detection:

percent of time signal was detected - signal was declared a

detection if any portion of each of the signal components (4 chirp components for triangular modulated FMCW, and 4 or 8 signal components for FSK) exceeded a set threshold (a certain percentage of the maximum intensity of the time-frequency representation).

Threshold percentages were determined based on visual detections of low SNR signals (lowest SNR at which the signal could be visually detected in the time-frequency representation) (see Figure 23).

94

Figure 23: Threshold percentage determination. This plot is an amplitude vs. time (x-z view) of CWD of FSK 4-component signal (512 samples, SNR= -2dB). For visually detected low SNR plots (like this one), the percent of max intensity for the peak z-value of each of the signal components was noted (here 82%, 98%, 87%, 72%), and the lowest of these 4 values was recorded (72%). Ten test runs were performed for each timefrequency analysis tool, for each of the 4 waveforms. The average of these recorded low values was determined and then assigned as the threshold for that particular timefrequency analysis tool. Note - the threshold for CWD is 70%. Thresholds were assigned as follows: CWD (70%); Spectrogram (60%); Scalogram, Reassigned Spectrogram, Reassigned Scalogram, RSPWVD, WVD (4-component FSK) (50%); WVD (triangular modulated FMCW) (35%); WVD (8-component FSK) (20%).

For percent detection determination, these threshold values were included in the timefrequency plot algorithms so that the thresholds could be applied automatically during the plotting process. From the threshold plot, the signal was declared a detection if any portion of each of the signal components was visible (see Figure 24).

95

Figure 24: Percent detection (time-frequency). CWD of 4-component FSK (512 samples, SNR=10dB) with threshold value automatically set to 70%. From this threshold plot, the signal was declared a (visual) detection because at least a portion of each of the 4 FSK signal components was visible. 3) Cross-term false positives (XFPs): The number of cross-terms that were wrongly declared as signal detections. For the time-frequency representation, the XFP detection criteria is the same as the time-frequency signal detection criteria listed in the percent detection section above.

4) Carrier frequency: the frequency corresponding to the maximum intensity of the timefrequency representation (there are multiple carrier frequencies for FSK waveforms).

5) Modulation bandwidth: distance from highest frequency value of signal (at a threshold of 20% maximum intensity) to lowest frequency value of signal (at same threshold) in Ydirection (frequency). 96

The threshold percentage was determined based on manual measurement of the modulation bandwidth of the signal in the time-frequency representation. This was accomplished for ten test runs of each time-frequency analysis tool, for each of the 4 waveforms. During each manual measurement, the max intensity of the high and low measuring points was recorded. The average of the max intensity values for these test runs was 20%. This was adopted as the threshold value, and is representative of what is obtained when performing manual measurements.

For modulation bandwidth determination, the 20% threshold value was included in the time-frequency plot algorithms so that the threshold could be applied automatically during the plotting process. From the threshold plot, the modulation bandwidth was manually measured (see Figure 25 (for triangular modulated FMCW) and Figure 26 (for FSK)).

97

Figure 25: Modulation bandwidth determination. CWD of a triangular modulated FMCW (512 samples, SNR=10dB) with threshold value automatically set to 20%. From this threshold plot, the modulation bandwidth was measured manually from the highest frequency value of the signal (top red arrow) to the lowest frequency value of the signal (bottom red arrow) in the y-direction (frequency).

98

Figure 26: Modulation bandwidth determination. CWD of a 4-component FSK (512 samples, SNR=10dB) with threshold value automatically set to 20%. From this threshold plot, the modulation bandwidth was measured manually from the highest frequency value of the signal (top red arrow) to the lowest frequency value of the signal (bottom red arrow) in the y-direction (frequency). 6) Modulation period:

for triangular modulated FMCW - distance from highest

frequency value of signal (at a threshold of 20% maximum intensity (same threshold as modulation bandwidth)) to lowest frequency value of signal (at same threshold) in Xdirection (time) – for FSK - width of FSK component (at a threshold of 20% maximum intensity) in X-direction (time).

From Figure 25 (for triangular modulated FMCW

signal), the modulation period is measured manually from the top red arrow to the bottom red arrow, but this time in the x-direction (time). From Figure 26 (for FSK signal), the modulation period is the manual measurement of the width of each of the 4 signals in the x-direction (time), and then the average of the 4 signals is calculated.

99

7) Time-frequency localization: a measure of the thickness of a signal component (from one side of the signal component to the other) (at a threshold of 20% maximum intensity (same threshold as modulation bandwidth)) on each side of the component – converted to % of entire X-Axis, and % of entire Y-Axis. From Figure 25 (for triangular modulated FMCW signal), the time-frequency localization is a manual measurement of the ‘thickness’ of the signal component at the center of each of the 4 ‘legs’, and then the average of the 4 values is calculated. Average time and frequency ‘thickness’ values are then converted to: percent of entire x-axis and percent of entire y-axis. From Figure 26 (for FSK signal), the time-frequency localization is a manual measurement of the thickness of the center of each of the 4 signal components (y-direction only), and then the average of the 4 values are determined. The average frequency ‘thickness’ is then converted to: percent of the entire y-axis.

8) Chirp rate: (modulation bandwidth)/(modulation period) – for Task 1 and Task2 only.

9) Lowest detectable SNR: the lowest SNR level at which at least a portion of each of the signal components exceeded the set threshold listed in the percent detection section above.

For lowest detectable SNR determination, these threshold values were included in the time-frequency plot algorithms so that the thresholds could be applied automatically during the plotting process. From the threshold plot, the signal was declared a detection

100

if any portion of each of the signal components was visible. The lowest SNR level for which the signal was declared a detection is the lowest detectable SNR (see Figure 27).

Figure 27: Lowest detectable SNR (time-frequency). CWD of 4-component FSK (512 samples, SNR=-2dB) with threshold value automatically set to 70%. From this threshold plot, the signal was declared a (visual) detection because at least a portion of each of the 4 FSK signal components was visible. For this case, any lower SNR would have been a non-detect. Compare to Figure 24, which is the same plot, except that it has an SNR level equal to 10dB. Automatically applying a threshold value to the time-frequency plot algorithms for the determination of percent detection, modulation bandwidth, modulation period, timefrequency localization, and lowest detectable SNR can be seen as a first step towards the future work of automating the metrics extraction process.

The data from all 25 runs for each test was used to produce the mean, standard deviation, variance, actual, error, and percent error for each of these metrics listed above. 101

The metrics from the classical time-frequency analysis techniques were then compared to the metrics from the reassignment method. By and large, the reassignment method outperformed the classical time-frequency analysis techniques, as will be shown in the results section.

4.4. Results Some of the graphical and statistical results of the testing are presented in this section.

Table 1 presents the overall test metrics (signal processing tool viewpoint) for the 4 timefrequency analysis techniques and the 3 reassignment methods used in this testing.

102

Table 1: Signal Processing Tool viewpoint of the overall test metrics (average percent error) for the 4 classical time-frequency analysis techniques (WVD, CWD, spectrogram, scalogram), along with their combined average (TF) and for the 3 reassignment methods (reassigned spectrogram, reassigned scalogram, RSPWVD), along with their combined average (RM). The parameters extracted are listed in the left-hand column: carrier frequency (fc), modulation bandwidth (modbw), modulation period (modper), timefrequency localization in the x-direction (tf loc-x), time-frequency localization in the ydirection (tf loc-y), chirp rate (cr), percent detection (% det), number of cross-term false positives (#XFP), lowest detectable SNR (low snr), plot time (plottime). params

wvd

cwd

spectro

scalo

fc

1.61%

3.38%

4.36%

modbw

respect

rescalo rspwvd RM

5.14% 3.62%

2.31%

4.88%

3.76%

3.65%

5.3%

12.95% 20.23% 25.6% 16.0%

6.51%

5.83%

4.91%

5.75%

modper

7.5%

7.1%

4.95%

4.52% 6.02%

4.59%

3.11%

4.72%

4.14%

tf loc-x

0.63%

1.88%

2.88%

4.4%

2.45%

1.18%

1.46%

0.66%

1.1%

tf loc-y

1.45%

4.83%

7.32%

8.85% 5.61%

2.79%

1.83%

1.18%

1.93%

cr

5.29%

11.49% 16.25% 28.5% 15.4%

3.84%

4.94%

4.54%

4.44%

% det

94.6%

92.5%

96.4%

90.4% 93.4%

100%

98.8%

98.0%

98.9%

# XFP

25

0

0

0

25

0

0

0

0

low snr

-2db

-2.4db

-3db

-2.8db

-2.5db

-2.6db

-2.8db

-3db

-2.8db

4.25s

4.95s

4m:31s 49.1s

13.63s

33.97s

32.2s

plottime 17m:44s 10.34s

TF

From Table 1, the WVD has much better time-frequency localization percent error (x: 0.63%/y: 1.45%) than its classical time-frequency analysis techniques counterparts, the CWD (x: 1.88%/y: 4.83%), the spectrogram (x: 2.88%/y: 7.32%), and the scalogram (x: 4.4%/y: 8.85%), and only slightly better time-frequency localization percent error than the reassigned spectrogram (x: 1.18%/y: 2.79%) and the reassigned scalogram (x: 1.46%/y: 1.83%). The RSPWVD (x: 0.66%/y: 1.18%) was nearly identical to the WVD in the x-direction and slightly outperformed the WVD in the y-direction. Figure 28 103

shows that the reassigned spectrogram gives a more concentrated time-frequency localization than the spectrogram, and that the reassigned scalogram gives a more concentrated time-frequency localization for each signal component than does the scalogram.

Figure 28: Time-frequency localization comparison between the classical time-frequency analysis tools (left) and the reassignment method (right). The uppper two plots are the spectrogram (left) and the reassigned spectrogram (right) for a triangular modulated FMCW signal (256 samples, SNR=10dB), and the lower two plots are the scalogram (left) and the reassigned scalogram (right) for a 4-component FSK signal (512 samples, SNR=10dB). The reassignment method gives a much more concentrated time-frequency localization than does its classical time-frequency analysis counterpart. Table 1 also shows that the WVD has a large number of cross-term false positives (25). Figure 29 shows WVD cross-term interference, along with the ability of the RSPWVD to reduce these cross-terms, making for a more readable presentation.

104

Figure 29: Cross-term interference of the WVD (left) and ability of the RSPWVD to reduce cross-term interference (right). The uppper two plots are the WVD (left) and the RSPWVD (right) for a triangular modulated FMCW signal (512 samples, SNR=10dB), and the lower two plots are the WVD (left) and the RSPWVD (right) for a 8-component FSK signal (512 samples, SNR=10dB). The WVD displays a lot of cross-term interference. There appears to be one additional triangle signal in the middle of the twotriangle signal in the upper-left plot, and there appears to be 9 additional signals to go along with the 8-component signal in the lower-left plot. The RSPWVD has drastically reduced the cross-term interference found in the WVD plots, making for more readable presentations. Notice also that the RSPWVD is more concentrated than its WVD counterpart (as per Table 1). Additionally, Table 1 shows that the WVD has the slowest plot time (17min 44sec), and that the plot times of the other 3 classical time-frequency analysis techniques, the CWD (10.34sec), the spectrogram (4.25sec) and the scalogram (4.95sec) are all faster than each of the 3 reassignment methods; the reassigned spectrogram (49.1sec),the reassigned scalogram (13.63sec), and the RSPWVD (33.97sec). Also, Table 1 showed that the WVD had the worst low SNR value (-2dB). Overall, the reassignment method 105

outperforms the classical time-frequency analysis techniques in the areas of percent detection (98.9% to 93.4%) and low SNR (-2.8dB to -2.5dB).

Table 2 presents the overall test metrics (SNR viewpoint) for the testing performed in this dissertation. Table 2: SNR viewpoint of overall test metrics (average percent error) for the classical time-frequency analysis techniques (TF) and for the reassignment method (RM) for SNR=10dB, 0dB, and lowest detectable SNR (low snr). The parameters extracted are listed in the left-hand column: carrier frequency (fc), modulation bandwidth (modbw), modulation period (modper), time-frequency localization in the x-direction (tf loc-x), time-frequency localization in the y-direction (tf loc-y), chirp rate (cr), percent detection (% det), number of cross-term false positives (#XFP), plot time (plottime). params

TF 10dB

TF 0dB

TF low snr

RM 10dB

RM 0dB

RM low snr

fc

4.89%

2.00%

5.31%

4.12%

2.54%

2.30%

modbw

14.78%

15.4%

18.81%

4.44%

5.35%

7.46%

modper

5.0%

3.93%

6.52%

4.63%

4.25%

3.58%

tf loc-x

2.34%

2.51%

2.61%

0.78%

1.11%

1.41%

tf loc-y

4.87%

5.47%

6.64%

1.25%

2.08%

2.61%

cr

13.73%

14.51%

17.04%

4.82%

3.35%

5.14%

% det

100%

82.4%

N/A

100%

97.87%

N/A

# XFP

21

2

2

0

0

0

plottime

4m:49sec

3m:54sec

7m:01sec

21.0sec

23.65sec

52.11sec

Table 2 shows that the percent error of the modulation bandwidth, time-frequency localization, chirp rate, percent detection, and plot time tend to worsen with lowering SNR values, for both the classical time-frequency analysis techniques and the reassignment method.

Figure 30 gives a visual representation of the readability 106

degradation that accompanies a reduction in SNR. The XFP numbers in Table 2 are representative of the fact that, due to computational complexity, there was no WVD testing accomplished at lower than 10dB (except for the 256 sample cases).

Figure 30: Readability degradation due to reduction in SNR. Spectrogram, triangular modulated FMCW, modulation bandwidth=500Hz, 512 samples. SNR=10dB (left), 0dB (center), -4dB (left). Readability degrades as SNR decreases, negatively affecting the accuracy of the metrics extracted, as per Table 2. Table 3 presents the overall test metrics (Task 1, 2, 3, and 4 viewpoint) for the testing performed in this dissertation.

107

Table 3: Task 1, 2, 3, and 4 viewpoint of overall test metrics (average percent error) for the classical time-frequency analysis techniques (TF) and for the reassignment method (RM). Task1=triangular modulated FMCW signal (modulation bandwidth=500Hz), Task2=triangular modulated FMCW signal (modulation bandwidth=2400Hz), Task 3=FSK (4-component) signal, Task 4=FSK (8-component) signal. The parameters extracted are listed in the left-hand column: carrier frequency (fc), modulation bandwidth (modbw), modulation period (modper), time-frequency localization in the x-direction (tf loc-x), time-frequency localization in the y-direction (tf loc-y), chirp rate (cr), percent detection (% det), number of cross-term false positives (#XFP), lowest detectable SNR (low snr), plot time (plottime). TF

TF

TF

TF

RM

RM

RM

RM

Task1

Task2

Task3

Task4

Task1

Task2

Task3

Task4

fc

4.02%

8.15%

0.65%

0.40%

4.15%

4.48%

0.45%

0.37%

modbw

20.97%

5.60%

19.37%

20.67% 4.59%

3.36%

7.65%

8.57%

modper

0.70%

0.41%

10.7%

17.47% 0.78%

0.56%

6.55%

12.0%

tf loc-x

2.78%

2.55%

7.50%

8.50%

1.00%

1.28%

2.47%

2.63%

tf loc-y

6.42%

2.35%

N/A

N/A

1.94%

0.92%

N/A

N/A

cr

12.09%

5.47%

N/A

N/A

4.89%

2.86%

N/A

N/A

% det

88%

93.7%

100%

100%

97.33% 100%

100%

100.0%

# XFP

8

2

6

9

0

0

0

low snr

-2.8db

-3.3db

-2.67db

-1.67db -2.67db -3.0db

-3.33db

-2.33db

plottime

2m:25s

7m:19s

2m:23s

1m:44s

33.6sec

22.8sec

params

31.1sec

0

35.3sec

Table 3 shows that the percent error of the modulation bandwidth, chirp rate, and timefrequency localization (y-direction) are lower for Task 2 (triangular modulated FMCW, modulation bandwidth=2400Hz) than for Task 1 (triangular modulated FMCW, modulation bandwidth=500Hz) for both the classical time-frequency analysis techniques and the reassignment method.

Figure 31 gives visual insight into this, as will be

discussed in the discussion section. 108

Figure 31: Comparison between Task 1 (left) and Task 2 (right). The uppper two plots are both spectrogram plots of a triangular modulated FMCW signal (512 samples, SNR=10dB), Task 1 (modulation bandwidth=500Hz) is on the left and Task 2 (modulation bandwidth=2400Hz) is on the right. The lower two plots are CWD plots of the same signals. The Task 2 plots (right) have a larger modulation bandwidth than Task 1 plots, therefore the signals appear taller and more upright than the Task 1 signals. Table 3 also shows that the percent error of the carrier frequency for Tasks 3 and 4 (FSK signals) is much lower than for Tasks 1 and 2 (triangular modulated FMCW signals). Also, the percent error of the modulation period for Tasks 1 and 2 (triangular modulated FMCW signals) is much lower than for Tasks 3 and 4 (FSK signals). In addition, the lowest detectable SNR is lower for Task 3 (4-component FSK signals) than for Task 4 (8component FSK signal).

4.5. Discussion 109

This section of the dissertation will elaborate on the results from the previous section.

From Table 1 (signal processing tool viewpoint of overall test metrics), the performance of each of the 7 signal processing analysis tools will be summarized, including strengths, weaknesses, and generic scenarios in which a particular tool might be used.

WVD: The WVD had excellent time-frequency localization percent error (0.63%/1.45%), but had the worst cross-term interference (25 XFPs), plot time (17min 44sec), and low SNR (-2dB). Its excellent time-frequency localization can be attributed to the fact that the WVD exhibits the highest signal energy concentration in the time-frequency plane [GUL07], [PAC09] and is totally concentrated along the instantaneous frequency [CIR08], [GUA06]. The WVD’s cross-term interference problem, which is well-known [GUL07], makes it very difficult to see the actual signal [DEL02], [GUA06], [WON09], reducing the readability of the time-frequency distribution. Figure 29 clearly shows the cross-term interference problem with the WVD.

The cross-terms produced a false

positive (XFP) triangle in the middle of the two-triangle signal (upper-left plot), and also produced 9 XFPs in the FSK 8-component signal (lower-left plot). Cross-terms are located half-way between signal components [DEL02], [WON09]. The long plot time is indicative of the fact that the WVD is known to be computationally intensive [MIL02], due in part to its cross-term interference. Though the WVD is highly concentrated in time and frequency, it is also highly non-linear and non-local, and is therefore very sensitive to noise [AUG95], [FLA03], which accounts for its poor low SNR performance (-2dB) [DEL02], [GUA06]. The WVD might be a good tool to use if excellent time110

frequency localization is a requirement, but readability, speed, and low SNR environments are not an issue, such as in a scenario where off-line analysis is performed, without any time constraints.

The readability issue can be alleviated if a single-

component signal is used, which would eliminate the cross-term interference, but which is unrealistic for LPI radar signals.

CWD: The CWD did not perform the best or the worst in any one category. Two of its strongest areas were time-frequency localization percent error (1.88%/4.8%), where it performed better than the spectrogram and the scalogram, and plot time (10.34sec) where it performed better than the WVD, reassigned spectrogram, reassigned scalogram, and RSPWVD. The better than average performance in both time-frequency localization percent error and plot time can be attributed to the fact that the CWD is part of the Cohen’s class of time-frequency distributions, which use a smoothing kernel to smooth out cross-term interference, but at the expense of time-frequency localization [CHO89], [WIL92].

In this sense, the CWD is seen as a mid-point between the WVD (good

localization, poor cross-term interference) and the spectrogram (poor localization, good cross-term interference). This reduction in cross-term interference speeds up the CWD. The fact that it doesn’t smooth out all of the cross-term interference allows for decent localization. ‘Middle-of-the-road’ would be the best way to describe the performance of the CWD. The CWD might be used in a scenario where above average localization is required (i.e. somewhere between the WVD and the spectrogram) and where a fairly short plot time is required. The goal of such a scenario would be to obtain above average signal metrics in a short amount of time. 111

Spectrogram: The spectrogram had the best plot time (4.25sec), best low SNR (-3dB) (tied with RSPWVD), and the best percent detection (96.4%) of the classical timefrequency analysis techniques, but suffered in time-frequency localization percent error (2.88%/7.32%) (only the scalogram (4.4%/8.85%) was worse) [ISI96], [COH95], [HLA92]. Each of these metrics (the good and the bad) can be accounted for by the spectrogram’s extreme reduction of cross-term interference, which accounts for good plot time, low SNR, and percent detection, but at the expense of poor time-frequency localization percent error. The spectrogram might be used in a scenario where a short plot time is necessary, in a fairly low SNR environment, and where time-frequency localization is not an issue. Such a scenario might be a ‘quick and dirty’ check to see if a signal is present, without precise extraction of its parameters.

Scalogram: The scalogram had good plot time (4.95sec), good low SNR (-2.8dB), but had the worst percent detection (90.4%) and time-frequency localization percent error (4.4%/8.85%). The scalogram suppresses almost all cross-terms [GRI07], [LAR92], accounting for its good plot time and good low SNR performance.

Because of this

cross-term reduction, it is surprising that the scalogram did not perform better in the area of percent detection. This could be due to its bad time-frequency localization percent error, or to the fact that a wavelet/scalogram performs better on signals that change rapidly in frequency over time, vice the triangular modulated FMCW and FSK signals used in this dissertation. Like the spectrogram, the scalogram might be used in a scenario where short plot time is necessary, in a fairly low SNR environment, and where time112

frequency localization is not an issue, or, as mentioned previously, in a scenario that detects/analyzes signals that change rapidly in frequency over time.

Reassignment method: The three reassignment methods performed fairly similarly, so they will be lumped together, with any specific differences being brought out separately. The reassignment methods performed very well in the time-frequency localization percent error category, coming close to the performance attained by the WVD (as mentioned earlier, the WVD exhibits the highest signal energy concentration in the timefrequency plane [GUL07], [PAC09]), and even surpassing the WVD in one area (RSPWVD – time-frequency localization percent error in the y-direction (1.18% vs. 1.45% for the WVD)). Figure 28 displays this ‘squeezing’ quality of the reassignment method [BOA03]. This quality is due to the fact that the reassignment method reassigns the signal energy value from the center of the analysis window, to the center of gravity of the analysis window, giving a much more concentrated signal representation [LIX08]. This supports the earlier hypothesis that, since the reassignment method is, in theory, a perfectly localized distribution for chirps, tones, and impulses [ISI96], it will work well for the triangular modulated FMCW (which can be viewed as back-to-back chirps) and the FSK (which can be viewed as tones), which can be seen in Figure 28. Another presupposition was that the reassignment method produces a more concentrated signal [ISI96], therefore the time-frequency distribution becomes more readable, and a more readable time-frequency distribution makes for more accurate signal detection and parameter extraction. The metrics in Table 1 confirm this, showing that the reassignment method by-and-large had more accurate signal detection and parameter extraction metrics 113

than the classical time-frequency analysis techniques in the areas of percent error of the modulation bandwidth (5.75% to 16.9%), percent error of time-frequency localization (both x (1.1% to 2.45%) and y (1.93% to 5.61%) direction), percent error of the chirp rate (4.44% to 15.4%), percent detection (98.9% to 93.4%), number XFPs (0 to 25), and plot time (32.2 sec to 4min 31sec).

Figure 29 is a prime example of the reassignment

method’s ability to smooth out cross-term interference [BOA03]. This figure shows how the readability of the time-frequency representation can be drastically increased through the cross-term reduction provided by the reassignment method. It was proposed that this increase in readability due to cross-term interference smoothing would bear an improvement over the classical time-frequency analysis techniques in the detection and parameter extraction metrics, which, as noted, is seen to be the case in Table 1. The 3 reassignment methods had slightly slower plot times than the classical time-frequency analysis techniques (except the WVD). This is due to the reassignment process being slightly more computationally complex than the CWD, spectrogram, and scalogram. As far as the differences between the 3 reassignment methods; the RSPWVD had the best time-frequency localization percent error and best low SNR; the reassigned scalogram had the fastest plot time, and the reassigned spectrogram had the best percent detection. Since the spectrogram has the worst localization (and the best cross-term interference) of any of Cohen’s class (which would include the smoothed-pseudo WVD (SPWVD)), then since the SPWVD yields better localization of signal components than the spectrogram, it follows that it’s reassigned version the RSPWVD would be more localized (x=0.66%, y=1.18%) than the reassigned spectrogram (x=1.18%, y=2.79%), and also less sensitive to noise (-3dB) than the reassigned spectrogram (-2.6dB) [FLA03]. 114

Any of the 3

reassignment methods might be used in a scenario where excellent time-frequency localization and low cross-term interference is required, perhaps due to a crucial need for very accurate metrics extraction.

From Table 2, it was seen that in general, the percent error of the modulation bandwidth, time-frequency localization (x and y), percent error of the chirp rate, percent detection, and plot time all tended to worsen as the SNR decreased, for both the classical timefrequency analysis techniques and the reassignment method. A visual representation of the degradation of the readability of the time-frequency representation as the SNR decreases is shown in Figure 30.

This confirms an initial hypothesis that as the

readability of the time-frequency representation degrades, the accuracy of the metrics will degrade as well, as validated in Table 2. It seems plausible that as SNR is lowered, the accuracy of the metrics will decrease. However, it was noted that for the modulation bandwidth, time-frequency localization (x and y) and chirp rate percent error metrics, the classical time-frequency analysis techniques experienced a greater percent degradation of metrics going from 0dB to low SNR than from 10dB to 0dB (119% greater), whereas for the reassignment method, it was just the opposite; it experienced less of a percent degradation of metrics going from 0 dB to low SNR than from 10dB to 0dB (38.5% less). This highlights the classical time-frequency analysis techniques mediocre performance in a low SNR environment, and also highlights the reassignment method’s smoothing and squeezing capabilities, which improve readability, and therefore help to offset readability degradation at low SNR levels. As noted above, the XFP numbers in Table 2 are representative of the fact that, due to computational complexity, there was no WVD 115

testing accomplished at lower than 10dB (except for the 256 sample cases). Had testing been able to be accomplished at lower than 10dB SNR levels for the 512 sample cases, the XFP numbers would have likely increased as the SNR level decreased. Table 2 shows by-and-large that the reassignment method’s metrics were more accurate than the classical time-frequency analysis techniques’ metrics at every SNR level.

From Table 3 it was seen that Task 2 (triangular modulated FMCW, modulation bandwidth=2400 Hz) had a much lower percent error for time-frequency localization (ydirection) than Task 1 (triangular modulated FMCW, modulation bandwidth=500Hz), for both the classical time-frequency analysis techniques (2.35% to 6.42%) and the reassignment method (0.92% to 1.94%). As per Figure 31, this is due to the fact that the Task 2 signal is more upright than the Task 1 signal, and therefore more of Task 2 signal’s ‘thickness’ is in the x-direction than that of the Task 1 signal. It was also noted that percent error of the modulation bandwidth was lower for Task 2 than for Task 1, for both the classical time-frequency analysis techniques (3.36% to 4.59%) and the reassignment method (5.60% to 20.97%). The modulation bandwidth is a measure from the highest frequency point of a signal to the lowest frequency point of a signal. Therefore, the ‘thickness’ of a signal will affect the modulation bandwidth measurement of a ‘shorter’ signal (Task 1 – left-hand side of Figure 31)) more than that of a ‘taller’ signal (Task 2 – right-hand side of Figure 31)).

Because of this, the modulation

bandwidth percent error will be lower for Task 2 (the ‘taller’ signal) than for Task 1 (the ‘shorter’ signal). It should be noted that though the definition of modulation bandwidth is the measure in frequency from the highest frequency value (at a 20% threshold of the 116

max intensity for this testing) of a signal to the lowest frequency value (at a 20% threshold of the max intensity) of a signal, an experienced intercept receiver signal analyst may choose to manually ‘override’ the 20% threshold value, and manually measure the modulation bandwidth of a ‘thick’ signal at points which he believes would give a more accurate modulation bandwidth measurement. This method may work for time-frequency representations with good readability, but becomes more difficult to accomplish as the signal readability degrades due to low SNR and cross-terms, and the more so for a less experienced intercept receiver signal analyst. Table 3 also shows that the chirp rate percent error is lower for Task 2 than for Task 1 for both the classical timefrequency analysis techniques (5.47% to 12.09%) and the reassignment method (2.86% to 4.89%). This follows from the fact that the modulation bandwidth percent error is lower for Task 2 than Task 1, and that chirp rate, which equals modulation bandwidth/modulation period, is directly proportional to the modulation bandwidth.

From Table 3 it is also noted that the carrier frequency percent error for the FSK waveforms (Task 3 and Task 4) is lower than the carrier frequency percent error for the triangular modulated FMCW waveforms (Task 1 and Task 2) for both the classical timefrequency analysis techniques and the reassignment method. This is due to the fact that the FSK waveforms are basically frequency tones, with each frequency tone representing a carrier frequency, making for a more accurate carrier frequency (since it’s a tone). Also noted was the fact that the modulation period percent error for the FSK waveforms (Task 3 and Task 4) was greater than the modulation period percent error for the triangular modulated FMCW waveforms (Task 1 and Task 2), for both the classical time-frequency 117

analysis techniques and the reassignment method. This is due to the fact that for a triangular modulated FMCW signal, the modulation period is the measure of the point where the first and second chirp legs of the signal meet to the point where the second and third chirp legs of the signal meet (measured in the x-direction), and the location of these two points changes very little based on time-frequency localization (signal ‘thickness’) or the SNR level (see Figure 28, the top two plots). For the FSK waveform, the modulation bandwidth is a measure of the width of the entire signal component, and can vary drastically based on time-frequency localization and SNR level. From Table 3, it was also noted that the modulation period percent error for the 4-component FSK signal (Task 3) was smaller than the modulation period percent error for the 8-component FSK signal (Task 4). This may be due to the fact that the size of the gap between signal components in the x-direction for the 4-component signal is roughly the same as the size of the gap between signal components for the 8-component signal, leaving less signal space in the xdirection for the 8-component signal (i.e. shorter components in the x-direction) and therefore a larger percent error of the modulation bandwidth (measure of the x-direction width of the signal). Another observation from Table 3 was that the low SNR value for the 4-component FSK signal (Task 3) was 1dB lower than the 8-component FSK signal (Task 4). This may be due to the fact that the Task 3 signal has only 4 components, each of which is twice as long as the Task 4 signal’s 8-components. This means that at low SNR levels, the Task 3 signal has a better chance of at least a portion of each of its 4 (longer) signal components exceeding the low SNR threshold than does the Task 4 signal with its 8 (shorter) components.

118

Recapping, the metrics data backs up the following introductory assumptions:

The classical time-frequency analysis techniques are deficient in the areas of timefrequency localization and cross-term interference, making for poor readability and consequently, for inaccurate detection and parameter extraction of LPI radar signals.

Since the reassignment method is, in theory, a perfectly localized distribution for chirps, tones, and impulses, it should (and does) work well with LPI radar signals such as the triangular modulated FMCW and the FSK.

The smoothing and squeezing qualities of the reassignment method make for better readability and consequently more accurate signal detection and parameter extraction of LPI radar signals, making for a more informed and effective intercept receiver environment.

4.6. Conclusions Digital intercept receivers, whose main job is to detect and extract parameters from LPI radar signals, are currently moving away from Fourier-based analysis and towards classical time-frequency analysis techniques, such as the Wigner-Ville distribution, ChoiWilliams distribution, spectrogram, and scalogram, for the purpose of analyzing LPI radar signals. Though these classical time-frequency techniques are an improvement over the Fourier-based techniques, it was shown through testing plots and results metrics that these techniques suffer from a lack of readability, due to poor time-frequency 119

localization and cross-term interference. It was observed that this lack of readability led to inaccurate detection and parameter extraction of the LPI radar signals, which would undoubtedly make for a less informed (and therefore less safe) intercept receiver environment. The reassignment method, in theory being a perfectly localized distribution for chirps, tones, and impulses, worked well for the triangular modulated FMCW (backto-back chirps) and the FSK (tones) LPI radar waveforms. Simulations were presented that compared time-frequency representations of the classical time-frequency techniques with those of the reassignment method. Two different triangular modulated frequency modulated continuous wave low probability of intercept radar signals and two different frequency shift keying low probability of intercept radar signals (4-component and 8component) were analyzed. The following metrics were used for evaluation of the analysis: percent error of: carrier frequency, modulation bandwidth, modulation period, time-frequency localization, and chirp rate. Also used were: percent detection, number of cross-term false positives, lowest signal-to-noise ratio for signal detection, and plot time. Experimental results demonstrated that the ‘squeezing’ and ‘smoothing’ qualities of the reassignment method did indeed lead to improved readability over the classical timefrequency analysis techniques, and consequently, provided more accurate signal detection and parameter extraction metrics (smaller percent error from true value) than the classical time-frequency analysis techniques. In summary, this dissertation provided evidence that the reassignment method has the potential to outperform the classical time-frequency analysis techniques, which are the current state-of-the-art, cutting-edge techniques for this arena. An improvement in performance can easily translate into saved equipment and lives. Future plans include automation of the metrics extraction process, analysis of 120

additional low probability of intercept radar waveforms of interest, and analysis of other real-world low probability of intercept radar signals utilizing more powerful computing platforms.

121

5. Detection and Parameter Extraction of Low Probability of Intercept Radar Signals Using the Hough Transform

Material from this chapter will be published as a journal paper and is presented here in the corresponding form.

The author of this dissertation was the primary student

contributor of the work presented in this chapter.

5.1. Abstract Digital intercept receivers are currently moving away from Fourier-based analysis and towards classical time-frequency analysis techniques, such as the Wigner-Ville distribution, Choi-Williams distribution, spectrogram, and scalogram, for the purpose of analyzing low probability of intercept radar signals (e.g. triangular modulated frequency modulated continuous wave and frequency shift keying). Although these classical timefrequency techniques are an improvement over the Fourier-based analysis, they still suffer from a lack of readability, due to cross-term interference, and a mediocre performance in low SNR environments. This lack of readability may lead to inaccurate detection and parameter extraction of these radar signals, making for a less informed (and therefore less safe) intercept receiver environment. In this dissertation, the use of the Hough transform, because of its ability to suppress cross-term interference, separate signals from cross-terms, and perform well in the presence of noise, is proposed as an improved signal analysis technique. With these qualities, the Hough transform has the potential to produce better readability and consequently, more accurate signal detection 122

and parameter extraction metrics.

Two different triangular modulated frequency

modulated continuous wave low probability of intercept radar signals and two different frequency shift keying low probability of intercept radar signals (4-component and 8component) were analyzed. The following metrics were used for evaluation of the analysis: percent error of chirp rate, percent detection, number of cross-term false positives, and lowest signal-to-noise ratio for signal detection. Experimental results demonstrate that the qualities of suppressing cross-term interference, separating signals from cross-terms, and performing well in a low SNR environment did lead to improved readability over the classical time-frequency analysis techniques, and consequently, provided more accurate signal detection and parameter extraction metrics (smaller percent error from true value) than the classical time-frequency analysis techniques. In addition, the Hough transform was utilized to detect, extract parameters, and properly identify a real-world low probability of intercept radar signal in a low signal-to-noise ratio environment, where the classical time-frequency analysis failed. In summary, this dissertation provides evidence that the Hough transform has the potential to outperform the classical time-frequency analysis techniques, which are the current state-of-the-art, cutting-edge techniques for this arena.

An improvement in performance can easily

translate into saved equipment and lives. Future work will include automation of the metrics extraction process, analysis of additional low probability of intercept radar waveforms of interest, and analysis of other real-world low probability of intercept radar signals utilizing more powerful computing platforms.

5.2. Introduction 123

Today, radar systems face serious threats from anti-radiation systems [FAU02], electronic attack [GAU02] and other areas. In order to be able to perform their functions properly, they must be able to ‘see without being seen’ [PAC09], [WIL06].

This

necessitates that they be low probability of intercept (LPI) radars. These radars typically have very low peak power, wide bandwidth, high duty cycle, and power management, making them difficult to be detected and characterized by intercept receivers.

On the other side of the fence, the intercept receiver that intercepts these LPI radar signals faces threats as well. It must be able to detect and characterize the signals from these LPI radars for the purpose of providing timely information about threatening systems, in addition to providing information about defensive systems - which is important in maintaining a credible deterrent force to penetrate those defenses. In this case, knowledge provided by the intercept receiver provides insight for determining the details of the threat so that an effective response can be prepared [WIL06].

To make matters even more challenging for intercept receivers, most of the intercept receivers currently in the fleet are analog and were designed to intercept ‘older’ (i.e. nonLPI) radar signals, which perform poorly when faced with LPI radar signals. Digital receivers, which are becoming a growing trend, are seen as the eventual solution to the intercept receiver’s LPI challenge.

Fourier analysis techniques using the FFT have been employed as the basic tool of the digital intercept receiver for detecting and extracting parameters of LPI radar signals, and 124

make up a majority of the digital intercept receiver techniques currently in the fleet [PAC09].

When a practical non-stationary signal (such as an LPI radar signal) is

processed, the Fourier transform cannot efficiently analyze and process the time-varying characteristics of the signal’s frequency spectrum, because time and frequency information cannot be combined to tell how the frequency content is changing in time [XIE08], [STE96]. The non-stationary nature of the received radar signal mandates the use of some form of time-frequency analysis for signal detection and parameter extraction [MIL02].

Some of the more common classical time-frequency analysis techniques include the Wigner-Ville distribution (WVD), Choi-Williams distribution (CWD), spectrogram, and scalogram. The WVD exhibits the highest signal energy concentration [WIL06], but has the worse cross-term interference, which can severely limit the readability of a timefrequency representation [GUL07], [STE96], [BOA03]. The CWD is a member of Cohen’s class, which adds a smoothing kernel to help reduce cross-term interference [BOA03], [UPP08]. The CWD, as with all members of Cohen’s class, is faced with a trade-off between cross-term reduction and time-frequency localization.

The

Spectrogram is the magnitude squared of the short-time Fourier transform [HLA92], [MIT01]. It has poorer time-frequency localization but less cross-term interference than the WVD or CWD, and its cross-terms are limited to regions where the signals overlap [ISI96]. The Scalogram is the magnitude squared of the wavelet transform, and can be used as a time-frequency distribution [COH02], [GAL05], [BOA03].

Like the

Spectrogram, the Scalogram has cross-terms that are limited to regions where the signals 125

overlap [ISI96], [HLA92]. Currently, for digital intercept receivers, these classical timefrequency analysis techniques are primarily at the lab phase [GAU02], [GUL07], [MIL02], [PAC09], [WIL06].

Though classical time-frequency analysis techniques, such as those described above, are a great improvement over Fourier analysis techniques, they suffer in general from crossterm interference and a mediocre performance in low SNR environments, as described above.

This may result in degraded readability of time-frequency representations,

potentially leading to inaccurate LPI radar signal detection and parameter extraction metrics.

This, in turn, can lead to an ineffective and unsafe intercept receiver

environment.

A promising avenue for overcoming these shortfalls is the utilization of the Hough Transform, which is very similar to the Radon transform, and is used, for the detection of straight lines and other curves [BAR95], [BEN05], [ZAI99], [INC07].

The Hough

transform of a particular time-frequency representation is found by computing the integral of the time-frequency representation along straight lines at different angles. The presence of a ‘spike’ in the Hough transform representation reveals the presence of high positive values concentrated along a line in the time-frequency representation – whose parameters (such as chirp rate) correspond to the coordinates of the spike (theta and rho values) [BAR92], [YAS06], [BAR95]. Detection can be achieved by establishing a threshold value for the amplitude of the Hough transform spike. Therefore the Hough transform can be used to convert a difficult global detection problem in the time126

frequency representation into a more easily solved local peak detection problem in the Hough transform representation.

Since cross-terms have amplitude modulation, the integration implicit in the Hough transform reduces the cross-terms, while the useful contributions, which are always positive, are correctly integrated [TOR07], [BAR92], [BAR95].

Likewise, in the

presence of noise, the integration carried out by the Hough transform produces an improvement in SNR [INC07], [YAS06], [NIK08]. These qualities make the Hough transform a viable candidate for analyzing LPI radar signals.

In related work, [WON09] performed research using the WVD followed by the Hough transform as one of their detection and parameter estimation algorithms, but this was utilized for a single chirp signal, which presents no cross-term interference like the triangular modulated FMCW and FSK waveforms that are examined in this dissertation. [GUL08] utilized the pseudo Wigner-Ville distribution followed by the Radon transform, but used it only for parameter extraction, and not for signal detection. This dissertation utilized the Hough transform for both parameter extraction and signal detection. [GER09] used an algorithm similar to the WVD followed by the Hough transform, called the periodic Wigner-Ville Hough transform. This algorithm was used on a sawtooth FMCW waveform, which is a viable LPI radar waveform. Their research assumed that phase is coherent from on LFM ramp to the next which, for present altimeters, is not the case. Also, for their research, one needed to search for the right repetition period, the right starting frequency and the right slope. Overall, it appears that little research has 127

been done in the area of using the Hough transform for the analysis of triangular modulated FMCW LPI radar signals and FSK LPI radar signals.

In this dissertation, the Hough transform is evaluated as a technique for improving the readability of the classical time-frequency analysis representations by suppressing crossterm interference, separating signals from cross-terms, and performing well in a low SNR environment. This approach is assessed using 2 triangular modulated FMCW LPI radar signals and 2 FSK LPI radar signals (4-component and 8-component). Metrics designed include: percent error of chirp rate, percent detection, number of cross-term false positives, and lowest SNR for signal detection.

The rest of this chapter is organized as follows: methodology is presented in section 5.3.

Description of the proposed

Experimental results comparing the

reassignment method and classical time-frequency analysis techniques are presented in section 5.4, followed by discussion and conclusions.

5.3. Methodology The methodologies detailed in this dissertation describe the processes involved in obtaining and comparing metrics between the classical time-frequency analysis techniques and the Hough transform for the detection and parameter extraction of LPI radar signals.

128

The tools used for this testing were: MATLAB (version 7.7), Signal Processing Toolbox (version 6.10), Wavelet Toolbox (version 4.3), Image Processing Toolbox (version 6.2), Time-Frequency Toolbox (version 1.0) (http://tftb.nongnu.org/).

All the testing was accomplished on a desktop computer (HP Compaq, 2.5GHz processor, AMD Athlon 64X2 Dual Core Processor 4800+, 2.00GB Memory (RAM), 32 Bit Operating System).

Testing was performed for 4 different waveforms (2 triangular modulated FMCWs and 2 FSKs), each waveform representing a different task (Task 1 through Task 4). For each waveform, parameters were chosen for academic validation of signal processing techniques. Due to computer processing limitations they were not meant to represent real-world values. The number of samples for each test was chosen to be either 256 or 512, which seemed to be the optimum size for the desktop computer. Testing was performed at three different SNR levels: 10dB, 0dB, and low SNR (the lowest SNR at which the signal could be detected). The noise added was white Gaussian noise, which best reflected the thermal noise present in the IF section of an intercept receiver [PAC09]. Kaiser windowing was used, when windowing was applicable. 25 runs were performed for each test, for statistical purposes. The plots included in this dissertation were done at a threshold of 5% of the maximum intensity and were linear scale (not dB) of analytic (complex) signals; the color bar represented intensity. The signal processing tools used for each task were:

129

Classical time-frequency analysis techniques: WVD, CWD, spectrogram, scalogram

Hough transform method: Hough transform of WVD and Hough transform of CWD

Task 1 consisted of analyzing a triangular modulated FMCW signal (most prevalent LPI radar waveform [LIA09]) whose parameters were: sampling frequency=4KHz; carrier frequency=1KHz; modulation bandwidth=500Hz; modulation period=.02sec.

Task 2 was similar to Task 1, but with different parameters: sampling frequency=6KHz; carrier frequency=1.5KHz; modulation bandwidth=2400Hz; modulation period=.15sec. The different parameters were chosen to see how the different shapes/heights of the triangles of the triangular modulated FMCW would affect the cross-term interference and the metrics.

Task 3 consisted of analyzing an FSK (prevalent in the LPI arena [AMS09]) 4component signal whose parameters were:

sampling frequency=5KHz; carrier

frequencies=1KHz, 1.75KHz, 0.75KHz, 1.25KHz; modulation bandwidth=1000Hz; modulation period=.025sec.

Task 4 was similar to Task 3, but for an FSK 8-component signal whose parameters were:

sampling frequency=5KHz; carrier frequencies=1.5KHz, 1KHz, 1.25KHz,

1.5KHz, 1.75KHz, 1.25KHz, 0.75KHz, 1KHz; modulation bandwidth=1000Hz; modulation period=.0125sec.

The different number of components and different 130

parameters between Task 3 and Task 4 were chosen to see how the different number/lengths of FSK components would affect the cross-term interference and the metrics.

Because of computational complexity, the WVD tests and the Hough transform of WVD tests for 512 samples, SNR=0dB and 512 samples, SNR=low SNR – were not able to be performed for any of the 4 waveforms. It was noted that a single run was still processing after more than 8 hours. The WVD is known to be very computationally complex [MIL02].

After each particular run of each test, metrics were extracted from the time-frequency representation and the Hough transform plot. The metrics extracted were as follows (TF=time-frequency representation; HT=Hough transform):

1) Percent detection HT: percent of time signal was detected – signal was declared a detection if any portion of each of the signal components exceeded the noise floor threshold (see Figure 32).

131

Figure 32: Percent detection (Hough transform). This plot is a theta-intensity (x-z view) of the HT of an RSPWVD (512 samples, SNR=10dB). Signal declared a (visual) detection because at least a portion of each of the signal components exceeded the noise floor threshold.

TF: percent of time signal was detected - signal was declared a detection if any portion of each of the signal components (4 chirp components for triangular modulated FMCW, and 4 or 8 signal components for FSK) exceeded a set threshold (a certain percentage of the maximum intensity of the time-frequency representation).

Threshold percentages were determined based on visual detections of low SNR signals (lowest SNR at which the signal could be visually detected in the time-frequency representation) (see Figure 33).

132

Figure 33: Threshold percentage determination. This plot is an amplitude vs. time (x-z view) of CWD of FSK 4-component signal (512 samples, SNR= -2dB). For visually detected low SNR plots (like this one), the percent of max intensity for the peak z-value of each of the signal components was noted (here 82%, 98%, 87%, 72%), and the lowest of these 4 values was recorded (72%). Ten test runs were performed for each timefrequency analysis tool, for each of the 4 waveforms. The average of these recorded low values was determined and then assigned as the threshold for that particular timefrequency analysis tool. Note - the threshold for CWD is 70%. Thresholds were assigned as follows: CWD (70%); spectrogram (60%); scalogram, WVD (4-component FSK) (50%); WVD (triangular modulated FMCW) (35%); WVD (8component FSK) (20%).

For percent detection determination, these threshold values were included in the timefrequency plot algorithms so that the thresholds could be applied automatically during the plotting process. From the threshold plot, the signal was declared a detection if any portion of each of the signal components was visible (see Figure 34).

133

Figure 34: Percent detection (time-frequency). CWD of 4-component FSK (512 samples, SNR=10dB) with threshold value automatically set to 70%. From this threshold plot, the signal was declared a (visual) detection because at least a portion of each of the 4 FSK signal components was visible. Automatically applying a threshold value to the time-frequency plot algorithms for percent detection determination can be seen as a first step towards the future work of automating the metrics extraction process.

2) Cross-term false positives (XFPs): The number of cross-terms that were wrongly declared as signal detections. For the time-frequency representation, the XFP detection criteria is the same as the time-frequency signal detection criteria listed in the percent detection section above. For the HT, the XFP detection criteria is the same as the HT signal detection criteria listed in the percent detection section above. Figure 35 shows a WVD plot with 4 true signals and 6 cross-terms, all 6 of which were XFPs.

134

Figure 35: Example of cross-term false positives (XFPs). WVD of a 4-component FSK signal at SNR=10dB (512 samples). There are 4 true signals (fc1=1KHz, fc2=1.75KHz, fc3=0.75KHz, fc4=1.25KHz) and 6 XFPs (cross-terms that were wrongly declared as signal detections because they passed the signal detection criteria listed in the percent detection section above) (ct1=0.875KHz, ct2=1KHz, ct3=1.125KHz, ct4=1.25KHz, ct5=1.375KHz, ct6=1.5KHz). 3) Chirp rate HT:

– (for Task 1 and Task2

only).

TF: (modulation bandwidth/modulation period) – (for Task 1 and Task 2 only).

4) Lowest detectable SNR HT: the lowest SNR level for which each signal component exceeded the noise floor threshold (see Figure 36). 135

Figure 36: Lowest detectable SNR (Hough transform). This plot is a theta-intensity (x-z view) of the HT of an RSPWVD (512 samples, SNR=-5dB). Signal declared a (visual) detection because at least a portion of each of the signal components exceeded the noise floor threshold. For this case, any lower SNR would have been a non-detect. Compare to Figure 32, which is the same plot, except that it has an SNR level equal to 10dB. TF: the lowest SNR level at which at least a portion of each of the signal components exceeded the set threshold listed in the percent detection section above.

For lowest detectable SNR determination, these threshold values were included in the time-frequency plot algorithms so that the thresholds could be applied automatically during the plotting process. From the threshold plot, the signal was declared a detection if any portion of each of the signal components was visible. The lowest SNR level for which the signal was declared a detection is the lowest detectable SNR (see Figure 37).

136

Figure 37: Lowest detectable SNR (time-frequency). CWD of 4-component FSK (512 samples, SNR=-2dB) with threshold value automatically set to 70%. From this threshold plot, the signal was declared a (visual) detection because at least a portion of each of the 4 FSK signal components was visible. For this case, any lower SNR would have been a non-detect. Compare to Figure 34, which is the same plot, except that it has an SNR level equal to 10dB. Automatically applying a threshold value to the time-frequency plot algorithms for the determination of the lowest detectable SNR can be seen as a first step towards the future work of automating the metrics extraction process.

The data from all 25 runs for each test was used to produce the mean, standard deviation, variance, actual, error, and percent error for each of these metrics listed above.

The metrics from the classical time-frequency analysis techniques were then compared to the metrics from the Hough transform. By and large, the Hough transform outperformed the classical time-frequency analysis techniques, as will be shown in the results section. 137

For Task 5, data from a CD was analyzed, with the only a priori knowledge being that the data contained an LPI radar signal in a low SNR environment (between -5dB and -10dB), and that the data was collected at a sampling frequency of 4GHz.

The data (19

megabytes) was first processed using the Spectrogram (because it is the fastest timefrequency analysis tool). The signal was not visible in the Spectrogram time-frequency representation, due to low SNR. The data was then processed with the Hough transform of the Spectrogram, and the signal was detected, but had almost zero slope (like an FSK (tonal) signal). A MATLAB script was written which allowed for decimation of the Yaxis for the receiver IF bandwidth (~ 750MHz to 1250MHz), and then the data was reprocessed with the Hough transform of the Spectrogram, this time for the purpose of determining if the signal had slope or if it was tonal. From the Hough transform plot it was observed that the signal had slope (i.e. was a chirp signal). The Hough transform plot allowed for not only detection of the chirp signal, but also for extraction of the chirp rate. A back-mapping from the Hough plot to the time-frequency representation was then performed.

The signal was located in the time-frequency representation, and the

modulation bandwidth and modulation period (and consequently the chirp rate) were extracted from the time-frequency representation. From these metrics, the type/source of the signal was identified. Additional details/results of Task 5 testing are addressed later in this chapter.

5.4. Results Some of the graphical and statistical results of the testing are presented in this section. 138

Table 4 presents the overall test metrics (signal processing tool viewpoint) for the 4 timefrequency analysis techniques and the 2 Hough transform methods used in this testing. Table 4: Signal Processing Tool viewpoint of the overall test metrics (average percent error) for the 4 classical time-frequency analysis techniques (WVD, CWD, spectrogram, scalogram) along with their combined average (TF) and for the 2 Hough transform methods (WVD + HT, CWD + HT), along with their combined average (HT). The parameters extracted are listed in the left-hand column: chirp rate (cr), percent detection (% det), # of cross-term false positives (#XFP), lowest detectable SNR (low snr). params

wvd

cwd

spectro

scalo

TF

wvd+ht

cwd+ht

HT

cr

5.29%

11.49%

16.25%

28.5%

15.4%

5.40%

2.74%

4.07%

% det

94.6%

92.5%

96.4%

90.4%

93.4%

100%

98.7%

99.4%

# XFP

25

0

0

0

25

4

4

8

low snr

-2db

-2.4db

-3db

-2.8db

-2.5db

-3db

-4.4db

-3.7db

From Table 4, the WVD had the best percent error of chirp rate (5.29%) of any of the classical time-frequency analysis tools, but performed the poorest out of all of the 6 signal processing techniques in the areas of number of cross-term false positives (25) and low SNR (-2dB). Figure 38 (left-hand side) shows the cross-term interference problem that the WVD has.

The CWD performed ‘middle-of-the-road’ in every category (cr=11.49%; % det=92.5%; #XFP=0; low snr= -2.4dB) , as compared to the other classical time-frequency analysis techniques.

139

The spectrogram had the best low SNR (-3dB) and percent detection (96.4%) of the classical time-frequency analysis techniques, but had a poor percent error of chirp rate (16.25%).

The scalogram had the worst percent detection (90.4%) and percent error of chirp rate (28.5%) of the classical time-frequency analysis techniques, but did well in low SNR (2.6dB).

The WVD + HT had a good percent error of chirp rate (5.40%) that was on par with that of the WVD (5.29%), and also had the best percent detection (100%) of all the 6 signal processing techniques. In addition, the WVD + HT had a low SNR value (-3dB) that equaled the best low SNR value of the classical time-frequency analysis techniques (spectrogram), and its number of cross-term false positives (4) was lower than that of the WVD (25) (see Figure 38 for comparison).

140

Figure 38: Cross-term comparison between classical time-frequency analysis techniques (left) and the Hough transform (right). Top left: WVD of a triangular modulated FMCW signal (512 samples, SNR=10dB) (left). Top right: the Hough transform of the WVD of a triangular modulated FMCW signal (512 samples, SNR=10dB). Bottom left: WVD of an FSK (8-component) signal (512 samples, SNR=10dB). Bottom right: the Hough transform of the WVD of an FSK (8-component) signal (512 samples, SNR=10dB). Upper 2 plots – the Hough transform (right) has eliminated the cross-term interference that the WVD (left) displays, making it easier to see the signal (better readability) in the Hough transform plot (the four bright spots which represent the four legs of the triangular modulated FMCW signal). The WVD appears to have another triangle signal between the outer two triangle signals. Lower 2 plots - In the WVD plot (left) there are 8 signal components and 9 cross-term components that appear to be signal components, all melded in together with one another. The 8 signal components are located at 5 distinct frequencies (one at 0.75KHz, two at 1KHz, two at 1.25KHz, two at 1.5KHz, and one at 1.75KHz). In the Hough transform plot (right) there are 8 signal components (located at 5 different frequencies (3 on the left-hand side of the plot and 2 in the middle of the plot)) plus 2 cross-term components, clearly separated from the signal components. The Hough transform plot makes it easier to see the signal components (better readability) as compared to the WVD plot.

141

The CWD + HT performed the best of all the 6 signal processing techniques in the areas of percent error of chirp rate (2.74%) and low SNR (-4.4dB) (see figure 39 (right-hand side)). It also performed very well for percent detection (98.7%).

Figure 39: Low SNR comparison between classical time-frequency analysis techniques (left) and the Hough transform (right). Top left: CWD of a triangular modulated FMCW signal (512 samples, SNR=-6dB) (left). Top right: the Hough transform of the CWD of a triangular modulated FMCW signal (512 samples, SNR=-6dB). Bottom left: CWD of an FSK (8-component) signal (512 samples, SNR=-3dB). Bottom right: the Hough transform of the CWD of an FSK (8-component) signal (512 samples, SNR=-3dB). Upper 2 plots - Though the signal is not visible in the CWD plot (left) (due to the low SNR (-6dB)), the four bright spots that represent the four legs of the triangular modulated FMCW signal are clearly seen in the Hough transform of the CWD plot (right). Each bright spot has a unique rho and theta value that can be used to back-map to the timefrequency representation (here CWD) and find the location of the 4 (non-visible) chirps that make up the triangular modulated FMCW signal. Lower 2 plots - Though the signal components are not visible in the CWD plot (due to the low SNR (-3dB)), the 5 bright spots (3 on the left and 2 in the middle) corresponding to the 5 different frequencies of the 8 FSK components are clearly visible in the Hough transform of the CWD plot (as are 2 cross-term components). The Hough transform does a good job of detecting the signal components and separating the signal components from the cross-term components, all in a low SNR environment. 142

Overall from Table 4, the Hough transform methods outperformed the classical timefrequency analysis techniques in percent error of chirp rate (4.07% to 15.4%), percent detection (99.4% to 93.4%), number of cross-term false positives (8 to 25), and low SNR (-3.7dB to -2.5dB).

Table 5 presents the overall test metrics (SNR viewpoint) for the testing performed in this dissertation.

Table 5: SNR viewpoint of overall test metrics (average percent error) for the classical time-frequency analysis techniques (TF) and for the Hough transform methods (HT) for SNR=10dB, 0dB, and lowest detectable SNR (low SNR). The parameters extracted are listed in the left-hand column: chirp rate (cr), percent detection (% det), number of crossterm false positives (#XFP). params

TF 10dB

TF 0dB

TF low snr

HT 10dB

HT 0dB

HT low snr

cr

13.73%

14.51%

17.04%

4.0%

6.23%

3.24%

% det

100%

82.4%

N/A

100%

98.7%

N/A

# XFP

21

2

2

2

0

4

Table 5 shows that the percent error of chirp rate and percent detection tended to worsen with lowering SNR values (see Figure 40) for both the classical time-frequency analysis techniques and the Hough transform (except for HT low SNR). The XFP numbers in Table 5 are representative of the fact that, due to computational complexity, there was no WVD testing accomplished at lower than 10dB (except for the 256 sample cases).

143

Figure 40: Readability degradation due to reduction in SNR. Spectrogram, triangular modulated FMCW, modulation bandwidth=500Hz, 512 samples. SNR=10dB (left), 0dB (center), -4dB (left). Readability degrades as SNR decreases, negatively affecting the accuracy of the metrics extracted, as per Table 5. Table 6 presents the overall test metrics (Task 1, 2, 3, and 4 viewpoint) for the testing performed in this dissertation.

Table 6: Task 1, 2, 3, and 4 viewpoint of overall test metrics (average percent error) for the classical time-frequency analysis techniques (TF) and for the Hough transform (HT). Task1=triangular modulated FMCW signal (modulation bandwidth=500Hz), Task2=triangular modulated FMCW signal (modulation bandwidth=2400Hz), Task 3=FSK (4-component) signal, Task 4=FSK (8-component) signal. The parameters extracted are listed in the left-hand column: chirp rate (cr), percent detection (% det), number of cross-term false positives (#XFP), lowest detectable SNR (low snr). TF

TF

TF

TF

HT

HT

HT

HT

Task1

Task2

Task3

Task4

Task1

Task2

Task3

Task4

cr

12.09%

5.47%

N/A

N/A

3.68%

0.31%

N/A

N/A

% det

88%

93.7%

100%

100%

99.27% 100%

99.27%

100.0%

# XFP

8

2

6

9

0

0

4

4

low snr

-2.8db

-3.3db

-2.67db

-1.67db

-4.4db

-6.0db

-3.5db

-3.5db

params

Table 6 shows that the percent error of chirp rate, percent detection, and low SNR were all better for Task 2 (triangular modulated FMCW, modulation bandwidth=2400Hz) than for Task 1 (triangular modulated FMCW, modulation bandwidth=500Hz) for both the classical time-frequency analysis techniques and the Hough transform (see Figure 41). 144

Also, the low SNR was lower for Task 3 (4-component FSK signal) than for Task 4 (8component FSK signal).

Figure 41: Comparison between Task 1 (left) and Task 2 (right). The uppper two plots are both spectrogram plots of a triangular modulated FMCW signal (512 samples, SNR=10dB), Task 1 (modulation bandwidth=500Hz) is on the left and Task 2 (modulation bandwidth=2400Hz) is on the right. The lower two plots are CWD plots of the same signals. The Task 2 plots (right) have a larger modulation bandwidth than Task 1 plots, therefore the signals appear taller and more upright than the Task 1 signals. Figure 42 shows a Spectrogram plot (left) of the area where the Task 5 real-world signal was located, though the signal was not visible in the Spectrogram plot (yellow area is inband portion, and orange areas are out-of-band portion of band pass filter). The Hough transform of the Spectrogram (right) was performed, and the signal became visible, despite the low SNR environment (-5dB to -10dB). The Hough transform plot showed 145

that the signal values were near theta=0 (or pi) and rho=0, which back-mapped to a nearly flat signal (i.e. tone) which was located near the center (frequency-wise) of the time-frequency representation. For this case, either the signal was a tonal (perhaps an FSK component), or the signal was a chirp; but because the bandwidth of the Spectrogram was so wide (2GHz) compared to the modulation bandwidth of the chirp, the chirp signal appeared ‘flat’. For this case, the bandwidth of the Spectrogram needed to be reduced so that the slope of the signal (given that it is a chirp) would become apparent.

Figure 42: Spectrogram (left) and Hough transform of Spectrogram (right) of area where Task 5 real-world signal was located. Signal was not visible in the spectrogram, but was visible in the Hough transform, due to the Hough transform’s ability to extract signals from low SNR environments. To investigate this further, a MATLAB script was utilized that that allowed for ‘zooming-in’ on the receiver IF bandwidth (representing the yellow portion of the Spectrogram in Figure 42 (~750MHz to 1250MHz)) (Y-axis zoom-in only). The data was processed again using the Hough transform of the Spectrogram (Figure 43). The signal appeared at theta=2.872 and rho=228.8, which indicated that the signal was indeed a chirp signal, with a chirp rate of 146

or (-tan (2.872*57.3)(500MHz/471usec) = 0.28MHz/usec.

Figure 43: Hough transform of Spectrogram (zoomed-in on receiver IF bandwidth (~750MHz to 1250MHz)) of area where Task 5 real-world signal was located. Signal is clearly visible at theta=2.872 and rho=228.8 Using these theta and rho values, back-mapping was performed from the Hough transform to the time-frequency representation (Spectrogram), which allowed the signal to be located (see Figure 44).

147

Figure 44: Spectrogram (zoomed-in on receiver IF bandwidth (~750MHz to 1250MHz)). Shows how the Hough transform theta and rho values back-map to the time-frequency representation for signal location. Once the chirp signal was detected in the Spectrogram, the modulation bandwidth (103MHz) and modulation period (389.34usec) metrics were extracted. The chirp rate (modulation bandwidth/modulation period) was then calculated to be 0.264 MHz/usec (very close to the chirp rate of 0.28MHz/usec obtained from the Hough transform plot).

Reading through LPI radar related literature, it was noted that the modulation bandwidth of a certain LPI radar device was listed as 100MHz, and its chirp rate was listed as 0.26MHz/usec [CAR03], [PAC09]. Based on these parameters, it was determined that the signal was the front-end chirp of this particular LPI radar device, which was confirmed by the personnel who supplied the CD for testing.

5.5. Discussion 148

This section of the dissertation will elaborate on the results from the previous section.

From Table 4 (signal processing tool viewpoint of overall test metrics), the performance of each of the 6 signal processing tools will be summarized, including strengths, weaknesses, and generic scenarios in which a particular tool might be used.

The WVD had the best percent error of chirp rate (5.29%) of all of the classical timefrequency analysis tools, but performed the poorest of all the 6 signal processing techniques in the areas of number of cross-term false positives (25) and low SNR (-2dB). Chirp rate can be seen as directly related to time-frequency localization. As per the methodology section, chirp rate is proportional to the modulation bandwidth. Since modulation bandwidth is a measure from the highest frequency value of a signal to the lowest frequency value of a signal, then the ‘thinner’ the signal (i.e. good time-frequency localization) the more accurate the modulation bandwidth (and therefore the more accurate the chirp rate), and the ‘thicker’ the signal (i.e. poor time-frequency localization) the less accurate the modulation bandwidth (and therefore the less accurate the chirp rate). Based on this, it can be said that the WVD’s excellent chirp rate is due to its excellent time-frequency localization. This can be attributed to the fact that the WVD exhibits the highest signal energy concentration in the time-frequency plane [GUL07], [PAC09] and is totally concentrated along the instantaneous frequency [CIR08], [GUA06]. The WVD’s cross-term interference problem, which is well-known [GUL07], makes it very difficult to see the actual signal [DEL02], [GUA06], [WON09], reducing the readability of the time-frequency distribution. Figure 38 clearly shows the cross-term 149

interference problem that the WVD has. The cross-terms produced a false positive (XFP) triangle in the middle of the two-triangle signal (upper-left plot), and also produced 9 XFPs in the FSK 8-component signal (lower-left plot). Cross-terms are located half-way between signal components [DEL02], [WON09].

Though the WVD is highly

concentrated in time and frequency, it is also highly non-linear and non-local, and is therefore very sensitive to noise [AUG95], [FLA03], which accounts for its poor low SNR performance (-2dB). The WVD might be a good tool to use if excellent chirp rate (time-frequency localization) is a requirement, but readability and low SNR environments are not an issue, such as in a scenario where off-line analysis is performed, without any time constraints (because of the WVD’s slow plot time). The readability issue can be alleviated if a single-component signal is used, which would eliminate the cross-term interference, but which is unrealistic for LPI radar signals.

The CWD performed ‘middle-of-the-road’ in every category, as compared to the other classical time-frequency analysis techniques. Its decent performance for percent error of chirp rate (11.49%) (time-frequency localization) can be attributed to the fact that the CWD is part of the Cohen’s class of time-frequency distributions, which use a smoothing kernel to smooth out cross-term interference, but at the expense of time-frequency localization [CHO89], [WIL92]. In this sense, the CWD is seen as a mid-point between the WVD (good localization, poor cross-term interference) and the spectrogram (poor localization, good cross-term interference). The fact that it doesn’t smooth out all of the cross-term interference allows for decent localization (and therefore decent chirp rate). The CWD might be used in a scenario where above average localization (chirp rate) is 150

required (i.e. somewhere between the WVD and the spectrogram). The goal of such a scenario would be to obtain above average signal metrics in a short amount of time (due to the CWD’s fairly quick plot time).

The spectrogram had the best low SNR (-3dB) and percent detection (96.4%) of the classical time-frequency analysis techniques, but had a poor percent error of chirp rate (16.25%). It is known that the spectrogram suffers in time-frequency localization (and therefore chirp rate) [ISI96], [COH95], [HLA92]. The results of these 3 metrics can be attributed to the spectrogram’s extreme reduction of cross-term interference, which accounts for good low SNR, and percent detection, but at the expense of poor timefrequency localization (chirp rate). The spectrogram might be used in a scenario where a short plot time is necessary (since it is the fastest time-frequency technique), in a fairly low SNR environment, and where time-frequency localization (chirp rate) is not an issue. Such a scenario might be a ‘quick and dirty’ check to see if a signal is present, without precise extraction of its parameters.

The scalogram had the worst percent detection (90.4%) and percent error of chirp rate (28.5%) of the classical time-frequency analysis techniques, but did well in low SNR (2.8dB). The scalogram suppresses almost all cross-terms [GRI07], [LAR92], accounting for its good low SNR performance. Because of this cross-term reduction, it is surprising that the scalogram did not perform better in the area of percent detection. This could be due to its bad time-frequency localization (chirp rate), or to the fact that a wavelet/scalogram performs better on signals that change rapidly in frequency over time, 151

vice the triangular modulated FMCW and FSK signals used in this dissertation. Like the spectrogram, the scalogram might be used in a scenario where short plot time is necessary (the scalogram has a very fast plot time), in a fairly low SNR environment, and where time-frequency localization (chirp rate) is not an issue, or in a scenario that detects/analyzes signals that change rapidly in frequency over time.

The WVD + HT and the CWD + HT both had good percent error of chirp rates (5.40%/2.74%), that were on par with or better than that of the WVD (5.29%). In addition, the WVD + HT and the CWD + HT both had good percent detections (100%/98.7%) and low SNRs (-3dB/-4.4dB) (see Figure 39). In the presence of noise, the integration carried out by the Hough transform produces an improvement in SNR [INC07], [YAS06], [NIK08]. The WVD + HT and the CWD + HT both had a lower number of cross-term false positives (4/4) than did the WVD (25) (see Figures 38 and 39 for comparison). Since cross-terms have amplitude modulation, the integration implicit in the Hough transform reduces the cross-terms, while the useful contributions, which are always positive, are correctly integrated [TOR07], [BAR95].

Overall from Table 4, the Hough transform methods outperformed the classical timefrequency analysis techniques in percent error of chirp rate (4.07% to 15.4%), percent detection (99.4% to 93.4%), number of cross-term false positives (8 to 25), and low SNR (-3.7dB to -2.5dB).

152

From Table 5, it was seen that in general, the percent error of chirp rate and percent detection metrics tended to worsen with lowering SNR values, for both the classical timefrequency analysis techniques and the Hough transform (except for HT low SNR). Figure 40 shows how readability is degraded as the SNR level is lowered. It is noted that the for the chirp rate metrics, the classical time-frequency analysis techniques experienced a 17.4% degradation of metrics while going from 0dB to low SNR, while the HT experienced a 48% improvement in metrics while going from 0dB to low SNR. This highlights the classical time-frequency analysis techniques mediocre performance in a low SNR environment, and also highlights the Hough transform’s robust performance in a low SNR environment. This translates to improved readability of the Hough transform plot over the classical time-frequency analysis representation in low SNR environments. As noted previously, the XFPs in Table 5 are representative of the fact that, due to computational complexity, there was no WVD testing accomplished at lower than 10dB (except for the 256 sample cases). Had testing been able to be accomplished at lower than 10dB SNR levels for the 512 sample cases, the XFP numbers would have likely increased as the SNR level decreased. Table 5 shows by-and-large that the Hough transform’s metrics were more accurate than the classical time-frequency analysis techniques’ metrics at every SNR level.

From Table 6, it was seen that the percent error of chirp rate, percent detection, and low SNR are all better for Task 2 (triangular modulated FMCW, modulation bandwidth=2400Hz) than for Task 1 (triangular modulated FMCW, modulation bandwidth=500Hz) for both the classical time-frequency analysis techniques and the 153

Hough transform. As per Figure 41, the modulation bandwidth is a measure from the highest frequency point of a signal to the lowest frequency point of a signal. Therefore, the ‘thickness’ of a signal will affect the modulation bandwidth measurement of a ‘shorter’ signal (Task 1 – left-hand side of Figure 41)) more than that of a ‘taller’ signal (Task 2 – right-hand side of Figure 41)). Because of this, the modulation bandwidth percent error will be lower for Task 2 (the ‘taller’ signal) than for Task 1 (the ‘shorter’ signal). Since chirp rate is proportional to modulation bandwidth, then chirp rate percent error will be lower for Task 2 (the ‘taller’ signal) than for Task 1 (the ‘shorter’ signal). As mentioned previously, chirp rate is proportional to time-frequency localization; therefore time-frequency localization will be better for Task 2 than for Task1. Better time-frequency localization translates to better readability, which in turn makes for better percent detection and low SNR values, which accounts for Task 2 having better metrics for these 2 parameters than Task 1. For the Hough transform, the ‘longer’, ‘tighter’ signal of Task 2 translates to a ‘higher’ (greater accumulator value) and ‘tighter’ spike in the Hough transform plot. The ‘higher’ spike accounts for the better percent detection and low SNR values of Task 2, because the signal (spike) is that much higher than the noise floor. The ‘tighter’ signal makes for a more accurate theta value extraction, which in turn makes for a more accurate chirp rate (since chirp rate is proportional to theta per the methodology section). It was also noted that for the classical time-frequency analysis techniques, the low SNR value was lower for Task 3 (4-component FSK signal) than for Task 4 (8-component FSK signal). This may be due to the fact that the Task 3 signal has only 4 components, each of which is twice as long as the Task 4 signal’s 8-components. This means that at low SNR levels, the Task 3 signal has a better chance of at least a 154

portion of each of its 4 (longer) signal components exceeding the low SNR threshold than does the Task 4 signal with its 8 (shorter) components. Table 6 shows by-and-large that the Hough transform’s metrics were more accurate than the classical time-frequency analysis techniques’ metrics for each of the 4 Tasks.

Figure 42 shows that the Spectrogram, though the fastest classical time-frequency analysis technique [HLA92], did not have the ability to detect the signal in a low SNR environment (-5dB to -10dB). However, in the presence of noise, the integration carried out by the Hough transform produces an improvement in SNR [INC07], [YAS06], [NIK08], and therefore the Hough transform was able to ‘dig’ the signal out of the noise. The ability to back-map from the Hough transform to the time-frequency representation, based on theta and rho values of the signal, allowed for a quick deduction that the signal (in the time-frequency representation) was nearly ‘flat’ and that it was located near the middle of the plot (frequency-wise). The ability to zoom-in (Y direction only) on the receiver IF bandwidth (the yellow portion of the Spectrogram of Figure 42) and then perform a Hough transform of the Spectrogram (Figure 43) made possible the detection and chirp rate extraction (0.28MHz/usec) of the signal, and also the determination that the signal was a chirp and not a tone. Back-mapping once again allowed for the signal location to be found in the time-frequency representation.

Once the modulation

bandwidth (103MHz) and modulation period (389.34usec) values were extracted, and the chirp rate (modulation bandwidth/modulation period) was calculated (0.264MHz/usec, very close to the Hough transform calculated value of 0.28MHz/usec), then identification of the LPI radar device that emitted the signal was straight-forward. 155

Recapping, the metrics data backs up the following introductory assumptions:

The classical time-frequency analysis techniques are deficient in the areas of cross-term interference and mediocre performance in low SNR environments, making for poor readability and consequently, inaccurate detection and parameter extraction of LPI radar signals.

The Hough transform’s qualities of cross-term reduction, separating signals from crossterms, and good performance in low SNR environments make for better readability and consequently, for more accurate signal detection and parameter extraction of LPI radar signals, making for a more effective intercept receiver environment.

5.6. Conclusions It was noted that digital intercept receivers are currently moving away from Fourierbased analysis and towards classical time-frequency analysis techniques, such as the Wigner-Ville distribution, Choi-Williams distribution, spectrogram, and scalogram, for the purpose of analyzing low probability of intercept radar signals (e.g. triangular modulated FMCW and FSK). Though these classical time-frequency techniques are an improvement over the Fourier-based analysis, it was shown through the testing plots that they suffer from a lack of readability, due to cross-term interference, and a mediocre performance in low SNR environments, as brought out in the discussion section of this chapter.

It was shown through testing metrics that this lack of readability led to 156

inaccurate detection and parameter extraction of the LPI radar signals, which would undoubtedly make for a less informed (and therefore less safe) intercept receiver environment. Simulations were presented that compared time-frequency representations of the classical time-frequency techniques with those of the Hough transform. Two different triangular modulated FMCW LPI radar signals and two different FKS LPI radar signals (4-component and 8-component) were analyzed. The following metrics were used for evaluation of the analysis: percent error of chirp rate, percent detection, number of cross-term false positives, and lowest signal-to-noise ratio for signal detection. Experimental results demonstrated that the Hough transform’s ability to suppress crossterm interference, separate signals from cross-terms, and perform well in the presence of noise did indeed lead to improved readability over the classical time-frequency analysis techniques, and consequently, provided more accurate signal detection and parameter extraction metrics (smaller percent error from true value) than the classical timefrequency analysis techniques. In summary, this dissertation provided evidence that the Hough transform has the potential to outperform the classical time-frequency analysis techniques, which are the current state-of-the-art, cutting-edge techniques for this arena. An improvement in performance can easily translate into saved equipment and lives. In addition, the Hough transform was utilized to detect, extract parameters, and properly identify a real-world LPI radar signal in a low signal-to-noise ratio environment where classical time-frequency analysis failed. Future plans include automation of the metrics extraction process, analysis of additional low probability of intercept radar waveforms of interest, and analysis of other real-world low probability of intercept radar signals utilizing more powerful computing platforms. 157

6. Joint Sequential Use of the Reassignment Method and the Hough Transform

Chapter 4 of this dissertation addressed using the Reassignment method for the analysis and LPI radar signals, and chapter 5 addressed using the Hough transform for the analysis of LPI radar signals.

This chapter takes a brief look at the joint sequential use of the reassignment method and the Hough transform for the analysis of LPI radar signals.

In chapters 4 and 5 of this dissertation, it was shown that both the reassignment method and the Hough transform (combined with time-frequency analysis) clearly outperformed the classical time-frequency analysis techniques for the detection and parameter extraction of LPI radar signals.

This chapter shows that using the reassignment method together with the Hough transform will result in more accurate metrics than the reassignment method by itself or the Hough transform by itself.

Table 7 shows the overall test metrics for the classical time-frequency analysis techniques, reassignment method, Hough transform, and reassignment method plus the Hough transform, for percent error of chirp rate, lowest detectable SNR and percent detection. 158

Table 7: Overall test metrics for the classical time-frequency analysis techniques (Classical TF), reassignment method (RM), Hough transform (HT), and reassignment method plus the Hough transform (RM + HT). The parameters extracted are listed in the left-hand column: chirp rate, lowest detectable SNR (Low SNR), and percent detection (% Detection). Parameters Classical RM HT RM + HT Extracted TF Chirp rate 10.0%

4.44%

3.94%

0.65%

Low SNR

-2.6dB

-2.8dB

-4.1dB

-4.6dB

% Detection

93.7%

98.9%

100%

100%

(% error)

For the percent error of chirp rate, low SNR, and percent detection metrics listed in Table 7, a progression is seen: classical time-frequency analysis techniques (worst) (10.0%, -2.6dB, 93.7%); reassignment method (better) (4.44%, -2.8dB, 98.9%); Hough transform (better yet) (3.94%, -4.1dB, 100%); joint use of the reassignment method and the Hough transform (best) (0.65%, -4.6dB, 100%).

These metrics are representative of the classical time-frequency analysis techniques deficiencies (poor time-frequency localization, cross-term interference, mediocre performance in low SNR environments – leading to less accurate signal detection and parameter extraction of LPI radar signals), the reassignment method qualities (smoothing and squeezing), and the Hough transform qualities (cross-term reduction, separation of signals from cross-terms, good performance in low SNR environments) – all of which were discussed in chapters 4 and 5 of this dissertation. 159

For the RM + HT combination, ‘tighter’ signals in the time-frequency plane (due to the squeezing quality of the reassignment method), make for ‘tighter’ (equals more accurate theta value extraction and therefore more accurate chirp rate extraction), ‘higher’ (equals detecting the signal at lower SNR values and better percent detection due to the signal being that much higher than the noise floor) spikes in the Hough plane. Therefore the joint use of the reassignment method and the Hough transform allows for more accurate signal detection and parameter extraction of LPI radar signals than either the RM or the HT individually, making for an even more informed, effective, and safer intercept receiver environment, which could potentially save valuable equipment, intelligence, and lives.

160

7. Overall Conclusions and Future Work This chapter summarizes the unique contributions presented in this dissertation and an overview of future work.

The first unique contribution is a novel application of a signal analysis technique for use in digital intercept receivers for the detection and parameter extraction of an important class of LPI radar signals. The applied signal analysis technique is the time-frequency reassignment method. This novel application has been experimentally validated using simulated signal data. The simulated class of signals includes triangular modulated FMCW LPI radar signals and FSK LPI radar signals, with varying modulation parameters.

Currently, the cutting-edge, state-of-the-art signal analysis technique in this arena, the classical time-frequency analysis technique, suffers from lack of readability, due to deficiencies such as poor time-frequency localization and cross-term interference. This lack of readability leads to inaccurate detection and parameter extraction of LPI radar signals.

This can lead to a compromise in the safety of the intercept receiver

environment, endangering valuable equipment, intelligence, and lives.

The reassignment method, with its squeezing and smoothing qualities, addresses these deficiencies, making for improved readability, and consequently the extraction of more accurate metrics.

161

Based on the results of this dissertation, the reassignment method clearly outperformed the classical time-frequency analysis techniques based on the following metrics that were extracted: modulation bandwidth (percent error from actual), chirp rate (percent error from actual), time-frequency localization, percent detection, number of cross-term false positives, and lowest SNR at which signal was detected.

The second unique contribution is the novel application of another signal analysis technique for use in digital intercept receivers for the detection and parameter extraction of the previously identified class of LPI radar signals. The applied analysis technique is the Hough transform, which together with time-frequency analysis and/or reassignment method pre-processing, greatly improves receiver detection and parameter estimation performance.

This novel application has been experimentally validated using both

simulated and collected data.

Again, the current cutting-edge, state-of-the-art signal analysis technique, the classical time-frequency analysis technique, suffers from lack of readability, due to deficiencies such as cross-term interference and mediocre performance in low SNR environments. This lack of readability equates to inaccurate detection and parameter extraction of LPI radar signals.

The Hough transform addresses these deficiencies through its ability to suppress crossterm interference, separate signals from cross-terms, and perform well in the presence of noise, making for better readability and consequently. 162

Based on the results of this dissertation, the Hough transform clearly outperformed the classical time-frequency analysis techniques based on the following metrics that were extracted: chirp rate (percent error from actual), percent detection, number of cross-term false positives, and lowest SNR at which signal was detected.

Metrics results obtained for the reassignment method and the Hough transform in this dissertation suggest a need for more analysis to be accomplished with additional realworld LPI radar signals. There are a number of research facilities that emit/collect LPI radar signals, which may be a potential resource for such data.

In addition to the

simulations conducted, included in this dissertation are initial results regarding the processing of actual collection data, demonstrating the utility of the Hough transform when used in conjunction with the spectrogram.

To efficiently analyze LPI radar signals at more realistic bandwidths at faster speeds requires a computer more powerful than that used for the testing in this dissertation. As more powerful computing resources become available, they will be utilized for future testing.

After additional analysis has been accomplished, and automation of the metrics extraction process has been achieved; the intent is to implement these algorithms into an actual digital intercept receiver in order to evaluate their performance in a real-world setting. If

163

the testing goes well, the goal will be to expand the LPI radar waveform base to include additional LPI radar waveforms of interest.

Other future research can include a more extensive detection performance analysis using for example, receiver operating characteristic (ROC) curves. This method inherently takes into account both probabilities of detection and probabilities of false alarms (false positives). By plotting false alarm probabilities versus detection probabilities, with SNR as a parameter, one can better assess the performance of the various processing methods described in this dissertation. With automated detection and false alarm metrics, a Monte-Carlo type experimental analysis can be used to generate the ROC curves. These could also then be compared with theoretically derived performance results.

Overall, the direction of the future work outlined in this chapter should allow for continued improvement in the application of the reassignment method and Hough transform signal analysis techniques, equating to more accurate metrics, and a more effective intercept receiver environment.

164

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175

Appendix A Table 8: Test Matrix for Task 1 through Task 5 TASK 1 – Analysis of Triangular Modulated FMCW 2Tri fs=4KHz fc=1KHz modBW=500Hz modperiod=20ms Test #

SP Tool

WF

fs (KHz)

fc (KHz)

Mod BW (Hz)

Mod Per. sec

# samp

SNR (dB)

Time Smooth Window ‘g’

Freq Smooth Window ‘h’

# Runs

1

WVD

FMCW

4

1

500

.02

256

10

N/A

N/A

25

2

WVD

FMCW

4

1

500

.02

256

0

N/A

N/A

25

3

WVD

FMCW

4

1

500

.02

256

low

N/A

N/A

25

4

WVD

FMCW

4

1

500

.02

512

10

N/A

N/A

25

5

WVD

FMCW

4

1

500

.02

512

0

N/A

N/A

25

6

WVD

FMCW

4

1

500

.02

512

low

N/A

N/A

25

7

CWD

FMCW

4

1

500

.02

256

10

Kaiser, 9

Kaiser, 27

25

8

CWD

FMCW

4

1

500

.02

256

0

Kaiser, 9

Kaiser, 27

25

9

CWD

FMCW

4

1

500

.02

256

low

Kaiser, 9

Kaiser, 27

25

10

CWD

FMCW

4

1

500

.02

512

10

Kaiser, 9

Kaiser, 27

25

11

CWD

FMCW

4

1

500

.02

512

0

Kaiser, 9

Kaiser, 27

25

12

CWD

FMCW

4

1

500

.02

512

low

Kaiser, 9

Kaiser, 27

25

13

Spectro

FMCW

4

1

500

.02

256

10

N/A

Kaiser, 27

25

14

Spectro

FMCW

4

1

500

.02

256

0

N/A

Kaiser, 27

25

15

Spectro

FMCW

4

1

500

.02

256

low

N/A

Kaiser, 27

25

16

Spectro

FMCW

4

1

500

.02

512

10

N/A

Kaiser, 27

25

17

Spectro

FMCW

4

1

500

.02

512

0

N/A

Kaiser, 27

25

18

Spectro

FMCW

4

1

500

.02

512

low

N/A

Kaiser, 27

25

19

Scalo

FMCW

4

1

500

.02

256

10

N/A

N/A

25

20

Scalo

FMCW

4

1

500

.02

256

0

N/A

N/A

25

21

Scalo

FMCW

4

1

500

.02

256

low

N/A

N/A

25

22

Scalo

FMCW

4

1

500

.02

512

10

N/A

N/A

25

23

Scalo

FMCW

4

1

500

.02

512

0

N/A

N/A

25

24

Scalo

FMCW

4

1

500

.02

512

low

N/A

N/A

25

FMCW

4

1

500

.02

256

10

N/A

Kaiser, 27

25

FMCW

4

1

500

.02

256

0

N/A

Kaiser, 27

25

FMCW

4

1

500

.02

256

low

N/A

Kaiser, 27

25

FMCW

4

1

500

.02

512

10

N/A

Kaiser, 27

25

FMCW

4

1

500

.02

512

0

N/A

Kaiser, 27

25

FMCW

4

1

500

.02

512

low

N/A

Kaiser, 27

25

FMCW

4

1

500

.02

256

10

N/A

N/A

25

FMCW

4

1

500

.02

256

0

N/A

N/A

25

25 26 27 28 29 30 31 32

Re Spectro Re Spectro Re Spectro Re Spectro Re Spectro Re Spectro Re Scalo Re Scalo

176

33 34 35 36

Re Scalo Re Scalo Re Scalo Re Scalo

FMCW

4

1

500

.02

256

low

N/A

N/A

25

FMCW

4

1

500

.02

512

10

N/A

N/A

25

FMCW

4

1

500

.02

512

0

N/A

N/A

25

FMCW

4

1

500

.02

512

low

N/A

N/A

25

Kaiser, 17

25

Kaiser, 17

25

Kaiser, 17

25

Kaiser, 17

25

Kaiser, 17

25

Kaiser, 17

25

Kaiser, 15 Kaiser, 15 Kaiser, 15 Kaiser, 15 Kaiser, 15 Kaiser, 15

37

RSPWVD

FMCW

4

1

500

.02

256

10

38

RSPWVD

FMCW

4

1

500

.02

256

0

39

RSPWVD

FMCW

4

1

500

.02

256

low

40

RSPWVD

FMCW

4

1

500

.02

512

10

41

RSPWVD

FMCW

4

1

500

.02

512

0

42

RSPWVD

FMCW

4

1

500

.02

512

low

FMCW

4

1

500

.02

256

10

N/A

N/A

25

FMCW

4

1

500

.02

256

0

N/A

N/A

25

FMCW

4

1

500

.02

256

low

N/A

N/A

25

FMCW

4

1

500

.02

512

10

N/A

N/A

25

FMCW

4

1

500

.02

512

0

N/A

N/A

25

FMCW

4

1

500

.02

512

low

N/A

N/A

25

FMCW

4

1

500

.02

256

10

Kaiser, 9

Kaiser, 27

25

FMCW

4

1

500

.02

256

0

Kaiser, 9

Kaiser, 27

25

FMCW

4

1

500

.02

256

low

Kaiser, 9

Kaiser, 27

25

FMCW

4

1

500

.02

512

10

Kaiser, 9

Kaiser, 27

25

FMCW

4

1

500

.02

512

0

Kaiser, 9

Kaiser, 27

25

FMCW

4

1

500

.02

512

low

Kaiser, 9

Kaiser, 27

25

Kaiser, 17

25

Kaiser, 17

25

Kaiser, 17

25

Kaiser, 17

25

Kaiser, 17

25

Kaiser, 17

25

43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

WVD + HT WVD + HT WVD + HT WVD + HT WVD + HT WVD + HT CWD + HT CWD + HT CWD + HT CWD + HT CWD + HT CWD + HT RSPWVD + HT RSPWVD + HT RSPWVD + HT RSPWVD + HT RSPWVD + HT RSPWVD + HT

FMCW

4

1

500

.02

256

10

FMCW

4

1

500

.02

256

0

FMCW

4

1

500

.02

256

low

FMCW

4

1

500

.02

512

10

FMCW

4

1

500

.02

512

0

FMCW

4

1

500

.02

512

low

Kaiser, 15 Kaiser, 15 Kaiser, 15 Kaiser, 15 Kaiser, 15 Kaiser, 15

TASK 2 – Analysis of Triangular Modulated FMCW 2Tri fs=6KHz fc=1.5KHz modBW=2400Hz modperiod=15ms Test #

SP Tool

WF

fs (KHz)

fc (KHz)

Mod BW (Hz)

Mod Per. sec

# samp

SNR (dB)

Time Smooth Window ‘g’

Freq Smooth Window ‘h’

# Runs

61

WVD

FMCW

6

1.5

2400

.015

512

10

N/A

N/A

25

62

WVD

FMCW

6

1.5

2400

.015

512

0

N/A

N/A

25

63

WVD

FMCW

6

1.5

2400

.015

512

low

N/A

N/A

25

177

64

CWD

FMCW

6

1.5

2400

.015

512

10

Kaiser, 9

Kaiser, 27

25

65

CWD

FMCW

6

1.5

2400

.015

512

0

Kaiser, 9

Kaiser, 27

25

66

CWD

FMCW

6

1.5

2400

.015

512

low

Kaiser, 9

Kaiser, 27

25

67

Spectro

FMCW

6

1.5

2400

.015

512

10

N/A

Kaiser, 27

25

68

Spectro

FMCW

6

1.5

2400

.015

512

0

N/A

Kaiser, 27

25

69

Spectro

FMCW

6

1.5

2400

.015

512

low

N/A

Kaiser, 27

25

70

Scalo

FMCW

6

1.5

2400

.015

512

10

N/A

N/A

25

71

Scalo

FMCW

6

1.5

2400

.015

512

0

N/A

N/A

25

72

Scalo

FMCW

6

1.5

2400

.015

512

low

N/A

N/A

25

FMCW

6

1.5

2400

.015

512

10

N/A

Kaiser, 27

25

FMCW

6

1.5

2400

.015

512

0

N/A

Kaiser, 27

25

FMCW

6

1.5

2400

.015

512

low

N/A

Kaiser, 27

25

FMCW

6

1.5

2400

.015

512

10

N/A

N/A

25

FMCW

6

1.5

2400

.015

512

0

N/A

N/A

25

FMCW

6

1.5

2400

.015

512

low

N/A

N/A

25

73 74 75 76 77 78

Re Spectro Re Spectro Re Spectro Re Scalo Re Scalo Re Scalo

79

RSPWVD

FMCW

6

1.5

2400

.015

512

10

Kaiser, 15

Kaiser, 17

25

80

RSPWVD

FMCW

6

1.5

2400

.015

512

0

Kaiser, 15

Kaiser, 17

25

81

RSPWVD

FMCW

6

1.5

2400

.015

512

low

Kaiser, 15

Kaiser, 17

25

WVD 82 FMCW 6 1.5 2400 .015 512 10 N/A N/A + HT WVD 83 FMCW 6 1.5 2400 .015 512 0 N/A N/A + HT WVD 84 FMCW 6 1.5 2400 .015 512 low N/A N/A + HT CWD 85 FMCW 6 1.5 2400 .015 512 10 Kaiser, 9 Kaiser, 27 + HT CWD 86 FMCW 6 1.5 2400 .015 512 0 Kaiser, 9 Kaiser, 27 + HT CWD 87 FMCW 6 1.5 2400 .015 512 low Kaiser, 9 Kaiser, 27 + HT RSPWVD 88 FMCW 6 1.5 2400 .015 512 10 Kaiser, 15 Kaiser, 17 + HT RSPWVD 89 FMCW 6 1.5 2400 .015 512 0 Kaiser, 15 Kaiser, 17 + HT RSPWVD 90 FMCW 6 1.5 2400 .015 512 low Kaiser, 15 Kaiser, 17 + HT TASK 3 – Analysis of FSK 4-component signal fs=5KHz fc=1KHz, 1.75KHz, .75KHz, 1.25KHz modBW=1KHz modper=25msec Time Freq Mod Mod Test fs fc # SNR Smooth Smooth SP Tool WF BW Per. # (KHz) (KHz) samp (dB) Window Window (Hz) msec ‘g’ ‘h’ 1 1.75 91 WVD FSK 5 1000 25 512 10 N/A N/A .75 1.25 1 1.75 92 WVD FSK 5 1000 25 512 0 N/A N/A .75 1.25 1 1.75 93 WVD FSK 5 1000 25 512 low N/A N/A .75 1.25

178

25 25 25 25 25 25 25 25 25

# Runs

25

25

25

94

CWD

FSK

5

95

CWD

FSK

5

96

CWD

FSK

5

97

Spectro

FSK

5

98

Spectro

FSK

5

99

Spectro

FSK

5

100

Scalo

FSK

5

101

Scalo

FSK

5

102

Scalo

FSK

5

103

Re Spectro

FSK

5

104

Re Spectro

FSK

5

105

Re Spectro

FSK

5

106

Re Scalo

FSK

5

107

Re Scalo

FSK

5

108

Re Scalo

FSK

5

109

RSPWVD

FSK

5

110

RSPWVD

FSK

5

1 1.75 .75 1.25 1 1.75 .75 1.25 1 1.75 .75 1.25 1 1.75 .75 1.25 1 1.75 .75 1.25 1 1.75 .75 1.25 1 1.75 .75 1.25 1 1.75 .75 1.25 1 1.75 .75 1.25 1 1.75 .75 1.25 1 1.75 .75 1.25 1 1.75 .75 1.25 1 1.75 .75 1.25 1 1.75 .75 1.25 1 1.75 .75 1.25 1 1.75 .75 1.25 1 1.75 .75 1.25

1000

25

512

10

Kaiser, 9

Kaiser, 27

25

1000

25

512

0

Kaiser, 9

Kaiser, 27

25

1000

25

512

low

Kaiser, 9

Kaiser, 27

25

1000

25

512

10

N/A

Kaiser, 27

25

1000

25

512

0

N/A

Kaiser, 27

25

1000

25

512

low

N/A

Kaiser, 27

25

1000

25

512

10

N/A

N/A

25

1000

25

512

0

N/A

N/A

25

1000

25

512

low

N/A

N/A

25

1000

25

512

10

N/A

Kaiser, 27

25

1000

25

512

0

N/A

Kaiser, 27

25

1000

25

512

low

N/A

Kaiser, 27

25

1000

25

512

10

N/A

N/A

25

1000

25

512

0

N/A

N/A

25

1000

25

512

low

N/A

N/A

25

1000

25

512

10

Kaiser, 15

Kaiser, 17

25

1000

25

512

0

Kaiser, 15

Kaiser, 17

25

179

1 1.75 111 RSPWVD FSK 5 1000 25 512 low Kaiser, 15 Kaiser, 17 25 .75 1.25 1 WVD 1.75 112 FSK 5 1000 25 512 10 N/A N/A 25 + HT .75 1.25 1 CWD 1.75 113 FSK 5 1000 25 512 low Kaiser, 9 Kaiser, 27 25 + HT .75 1.25 1 RSPWVD 1.75 114 FSK 5 1000 25 512 low Kaiser, 15 Kaiser, 17 25 + HT .75 1.25 TASK 4 – Analysis of FSK 8-component signal fs=5KHz fc=1.5KHz, 1KHz, 1.25KHz, 1.5KHz, 1.75KHz, 1.25KHz, .75KHz, 1KHz modBW=1KHz modperiod=12.5ms Time Freq Mod Mod Test fs fc # SNR Smooth Smooth # SP Tool WF BW Per. # (KHz) (KHz) samp (dB) Window Window Runs (Hz) msec ‘g’ ‘h’ 1.5 1 1.25 1.5 115 WVD FSK 5 1000 12.5 512 10 N/A N/A 25 1.75 1.25 .75 1 1.5 1 1.25 1.5 116 WVD FSK 5 1000 12.5 512 0 N/A N/A 25 1.75 1.25 .75 1 1.5 1 1.25 1.5 117 WVD FSK 5 1000 12.5 512 low N/A N/A 25 1.75 1.25 .75 1 1.5 1 1.25 1.5 118 CWD FSK 5 1000 12.5 512 10 Kaiser, 9 Kaiser, 27 25 1.75 1.25 .75 1 1.5 1 1.25 1.5 119 CWD FSK 5 1000 12.5 512 0 Kaiser, 9 Kaiser, 27 25 1.75 1.25 .75 1 1.5 1 1.25 120 CWD FSK 5 1000 12.5 512 low Kaiser, 9 Kaiser, 27 25 1.5 1.75 1.25

180

121

Spectro

FSK

5

122

Spectro

FSK

5

123

Spectro

FSK

5

124

Scalo

FSK

5

125

Scalo

FSK

5

126

Scalo

FSK

5

127

Re Spectro

FSK

5

128

Re Spectro

FSK

5

129

Re Spectro

FSK

5

.75 1 1.5 1 1.25 1.5 1.75 1.25 .75 1 1.5 1 1.25 1.5 1.75 1.25 .75 1 1.5 1 1.25 1.5 1.75 1.25 .75 1 1.5 1 1.25 1.5 1.75 1.25 .75 1 1.5 1 1.25 1.5 1.75 1.25 .75 1 1.5 1 1.25 1.5 1.75 1.25 .75 1. 1.5 1 1.25 1.5 1.75 1.25 .75 1 1.5 1 1.25 1.5 1.75 1.25 .75 1 1.5 1

1000

12.5

512

10

N/A

Kaiser, 27

25

1000

12.5

512

0

N/A

Kaiser, 27

25

1000

12.5

512

low

N/A

Kaiser, 27

25

1000

12.5

512

10

N/A

N/A

25

1000

12.5

512

0

N/A

N/A

25

1000

12.5

512

low

N/A

N/A

25

1000

12.5

512

10

N/A

Kaiser, 27

25

1000

12.5

512

0

N/A

Kaiser, 27

25

1000

12.5

512

low

N/A

Kaiser, 27

25

181

130

Re Scalo

FSK

5

131

Re Scalo

FSK

5

132

Re Scalo

FSK

5

133

RSPWVD

FSK

5

134

RSPWVD

FSK

5

135

RSPWVD

FSK

5

136

WVD + HT

FSK

5

137

CWD + HT

FSK

5

1.25 1.5 1.75 1.25 .75 1 1.5 1 1.25 1.5 1.75 1.25 .75 1 1.5 1 1.25 1.5 1.75 1.25 .75 1 1.5 1 1.25 1.5 1.75 1.25 .75 1 1.5 1 1.25 1.5 1.75 1.25 .75 1 1.5 1 1.25 1.5 1.75 1.25 .75 1. 1.5 1 1.25 1.5 1.75 1.25 .75 1k. 1.5 1 1.25 1.5 1.75 1.25 .75 1k 1.5 1 1.25 1.5 1.75 1.25

1000

12.5

512

10

N/A

N/A

25

1000

12.5

512

0

N/A

N/A

25

1000

12.5

512

low

N/A

N/A

25

1000 .

12.5

512

10

Kaiser, 15

Kaiser, 17

25

1000

12.5

512

0

Kaiser, 15

Kaiser, 17

25

1000

12.5

512

low

Kaiser, 15

Kaiser, 17

25

1000

12.5

512

10

N/A

N/A

25

1000

12.5

512

low

Kaiser, 9

Kaiser, 27

25

182

138

RSPWVD + HT

FSK

.75 1k 1.5 1 1.25 1.5 1.75 1.25 .75 1k.

5

1000

12.5

512

low

Kaiser, 15

Kaiser, 17

25

TASK 5 – Analysis of Real-World LPI Signal Test #

SP Tool

WF

fs (KHz)

fc (KHz)

Mod BW (Hz)

Mod Per. sec

# samp

SNR (dB)

Time Smooth Window ‘g’

Freq Smooth Window ‘h’

# Runs

139

Spectro + HT

Unknown

Unk.

Unk.

Unk.

Unk.

Unk.

Unk.

N/A

N/A

N/A

183

Appendix B Test Plots and the Test Metrics Tables Test 1

Figure 45: Test 1 Plot – WVD, 256, 10

Table 9: Test 1 Metrics – WVD, 256, 10 TEST 1 WVD TriModFMCW 2 Tri fs=4KHz fc=1KHz modBW=500 modperiod=20ms #samples=256 SNR=10dB (25 runs)

Plot Time (sec)

Carrier Freq (Hz)

Mod BW (Hz)

mean std. dev. variance actual error % error

40.42 3.93 15.47 N/A N/A N/A

1001.64 41.63 1733.14 1000 1.64 0.16%

475.8 12.53 156.88 500 24.30 4.84%

Mod Period (sec)

Time (sec)-Frequency (Hz) Localization

.01997 .0005 .00043 .00016 0 0 .02 N/A .00003 N/A 0.15% (0.63% of X-Axis) % detection = 100% # x-term false positives=2 % false positive=100%

184

28.08 5.39 29.08 N/A N/A (1.40% of Y-Axis)

Chirp Rate (Hz/sec) 23842.64 968.18 9.37e5 25000 1157.36 4.63%

Test 2

Figure 46: Test 2 Plot – WVD, 256, 0

Table 10: Test 2 Metrics – WVD, 256, 0 TEST 2 WVD TriModFMCW 2 Tri fs=4KHz fc=1KHz modBW=500 modperiod=20ms #samples=256 SNR=0dB (25 runs)

Plot Time min:sec

Carrier Freq (Hz)

Mod BW (Hz)

mean std. dev. variance actual error % error

15:17.52 2.06 4.23 N/A N/A N/A

990.18 74.01 5477.86 1000 9.82 0.98%

480.52 38.06 1448.86 500 19.48 3.90%

Mod Period (sec)

Time (sec)-Frequency (Hz) Localization

.02006 .00053 .00069 .00015 0 0 .02 N/A .00006 N/A 0.30% (0.66% of X-Axis) % detection = 68% # x-term false positives=2 % false positive=100%

185

29.68 8.86 78.57 N/A N/A (1.48% of Y-Axis)

Chirp Rate (Hz/sec) 24003.87 2408.80 5.80e6 25000 996.13 3.98%

Test 3

Figure 47: Test 3 Plot – WVD, 256, -2

Table 11: Test 3 Metrics – WVD, 256, -2 TEST 3 WVD TriModFMCW 2 Tri fs=4KHz fc=1KHz modBW=500 modperiod=20ms #samples=256 SNR=lowest detectable (-2dB) (25 runs)

Plot Time min:sec

Carrier Freq (Hz)

Mod BW (Hz)

mean std. dev. variance actual error % error

27:29.82 5.21 27.10 N/A N/A N/A

1089.04 135.90 18469.81 1000 89.04 8.90%

442.04 28.78 828.57 500 57.96 11.59%

Mod Period (sec)

Time (sec)-Frequency (Hz) Localization

.01966 .00075 .00043 .00033 0 0 .02 N/A .00034 N/A 1.70% (0.94% of X-Axis) % detection = 20% # x-term false positives=2 % false positive=100%

186

42.34 13.45 180.93 N/A N/A (2.12% of Y-Axis)

Chirp Rate (Hz/sec) 22576.02 1752.15 3.07e6 25000 2423.98 9.70%

Test 4

Figure 48: Test 4 Plot – WVD, 512, 10

Table 12: Test 4 Metrics – WVD, 512, 10 TEST 4 WVD TriModFMCW 2 Tri fs=4KHz fc=1KHz modBW=500 modperiod=20ms #samples=512 SNR=10dB (25 runs)

Plot Time min:sec

Carrier Freq (Hz)

Mod BW (Hz)

mean std. dev. variance actual error % error

8:09.08 0.63 0.40 N/A N/A N/A

1004.33 33.89 1148.63 1000 4.33 0.43%

468.28 9.54 91.13 500 31.72 6.34%

Mod Period (sec)

Time (sec)-Frequency (Hz) Localization

.02008 .00028 .00015 .00014 0 0 .02 N/A .00008 N/A 0.40% (0.35% of X-Axis) % detection = 100% # x-term false positives=2 % false positive=100%

187

23.0 9.28 86.12 N/A N/A (1.15% of Y-Axis)

Chirp Rate (Hz/sec) 23323.72 485.28 2.35e5 25000 1676.28 6.71%

Test 5 Did not run Test 5 due to inadequate computational resources (1 run was still processing after 8 hours)

Test 6 Did not run Test 6 due to inadequate computational resources (1 run was still processing after 8 hours)

188

Test 7

Figure 49: Test 7 Plot – CWD, 256, 10

Table 13: Test 7 Metrics – CWD, 256, 10 TEST 7 CWD TriModFMCW 2 Tri fs=4KHz fc=1KHz modBW=500 modperiod=20ms #samples=256 SNR=10dB (25 runs)

Plot Time (sec)

Carrier Freq (Hz)

Mod BW (Hz)

mean std. dev. variance actual error % error

2.09 0.20 0.04 N/A N/A N/A

1105.52 115.9 13435.23 1000 105.52 10.55%

560.17 18.69 349.50 500 60.17 12.03%

Mod Period (sec)

Time (sec)-Frequency (Hz) Localization

.02013 .0016 .00051 .00036 0 0 .02 N/A .00013 N/A 0.65% (2.08% of X-Axis) % detection = 100%

189

93.0 13.51 182.52 N/A N/A (4.65% of Y-Axis)

Chirp Rate (Hz/sec) 27845.97 1025.42 1.05e6 25000 2845.97 11.38%

Test 8

Figure 50: Test 8 Plot – CWD, 256, 0

Table 14: Test 8 Metrics – CWD, 256, 0 TEST 8 CWD TriModFMCW 2 Tri fs=4KHz fc=1KHz modBW=500 modperiod=20ms #samples=256 SNR=0dB (25 runs)

Plot Time (sec)

Carrier Freq (Hz)

Mod BW (Hz)

mean std. dev. variance actual error % error

3.98 0.28 0.08 N/A N/A N/A

1030.48 173.43 30075.44 1000 30.48 3.05%

553.96 38.38 1472.76 500 53.96 10.79%

Mod Period (sec)

Time (sec)-Frequency (Hz) Localization

.01994 .00175 .00096 .00049 0 0 .02 N/A .00006 N/A 0.30% (2.19% of X-Axis) % detection = 68%

190

90.72 6.55 42.87 N/A N/A (4.54% of Y-Axis)

Chirp Rate (Hz/sec) 27858.74 2535.28 6.43e6 25000 2858.74 11.43%

Test 9

Figure 51: Test 9 Plot – CWD, 256, -3

Table 15: Test 9 Metrics – CWD, 256, -3 TEST 9 CWD TriModFMCW 2 Tri fs=4KHz fc=1KHz modBW=500 modperiod=20ms #samples=256 SNR=lowest detectable (-3dB) (25 runs)

Plot Time (sec)

Carrier Freq (Hz)

Mod BW (Hz)

mean std. dev. variance actual error % error

5.61 0.37 0.14 N/A N/A N/A

935.20 137.02 18775.18 1000 64.8 6.48%

572.76 35.45 1256.90 500 72.76 14.55%

Mod Period (sec)

Time (sec)-Frequency (Hz) Localization

.01994 .00155 .0008 .00068 0 0 .02 N/A .00006 N/A 0.30% (1.94% of X-Axis) % detection = 100%

191

117.92 54.69 2991.35 N/A N/A (5.90% of Y-Axis)

Chirp Rate (Hz/sec) 28779.23 2162.65 4.68e6 25000 3779.23 15.12%

Test 10

Figure 52: Test 10 Plot – CWD, 512, 10

Table 16: Test 10 Metrics – CWD, 512, 10 TEST 10 CWD TriModFMCW 2 Tri fs=4KHz fc=1KHz modBW=500 modperiod=20ms #samples=512 SNR=10dB (25 runs)

Plot Time (sec)

Carrier Freq (Hz)

Mod BW (Hz)

mean std. dev. variance actual error % error

3.17 0.09 0.009 N/A N/A N/A

970.76 145.31 21114.61 1000 29.24 2.92%

563.28 7.68 58.98 500 63.28 12.66%

Mod Period (sec)

Time (sec)-Frequency (Hz) Localization

.02008 .0016 .00049 .00039 0 0 .02 N/A .00008 N/A 0.40% (2.0% of X-Axis) % detection = 100%

192

83.16 8.22 67.57 N/A N/A (4.16% of Y-Axis)

Chirp Rate (Hz/sec) 28069.92 798.10 6.37e5 25000 3069.92 12.28%

Test 11

Figure 53: Test 11 Plot – CWD, 512, 0

Table 17: Test 11 Metrics – CWD, 512, 0 TEST 11 CWD TriModFMCW 2 Tri fs=4KHz fc=1KHz modBW=500 modperiod=20ms #samples=512 SNR=0dB (25 runs)

Plot Time (sec)

Carrier Freq (Hz)

Mod BW (Hz)

mean std. dev. variance actual error % error

10.25 1.31 1.73 N/A N/A N/A

1018.76 147.78 21840.33 1000 18.76 1.88%

576.48 22.47 505.06 500 76.48 15.30%

Mod Period (sec)

Time (sec)-Frequency (Hz) Localization

.01981 .0011 .00048 .00029 0 0 .02 N/A .00019 N/A 0.95% (1.38% of X-Axis) % detection = 80%

193

78.48 15.62 243.87 N/A N/A (3.92% of Y-Axis)

Chirp Rate (Hz/sec) 29120.28 1540.84 2.37e6 25000 4120.28 16.48%

Test 12

Figure 54: Test 12 Plot – CWD, 512, -3

Table 18: Test 12 Metrics – CWD, 512, -3 TEST 12 CWD TriModFMCW 2 Tri fs=4KHz fc=1KHz modBW=500 modperiod=20ms #samples=512 SNR=lowest detectable (-3dB) (25 runs)

Plot Time (sec)

Carrier Freq (Hz)

Mod BW (Hz)

mean std. dev. variance actual error % error

27.62 2.47 6.09 N/A N/A N/A

974.97 137.41 18881.52 1000 25.03 2.50%

559.37 24.84 616.88 500 59.37 11.87%

Mod Period (sec)

Time (sec)-Frequency (Hz) Localization

.01994 .0017 .00062 .00047 0 0 .02 N/A .00006 N/A 0.30% (2.13% of X-Axis) % detection = 8%

194

101.56 29.76 885.78 N/A N/A (5.08% of Y-Axis)

Chirp Rate (Hz/sec) 28086.19 1649.08 2.72e6 25000 3086.19 12.34%

Test 13

Figure 55: Test 13 Plot – Spectrogram, 256, 10

Table 19: Test 13 Metrics – Spectrogram, 256, 10 TEST 13 Spectrogram TriModFMCW 2 Tri fs=4KHz fc=1KHz modBW=500 modperiod=20ms #samples=256 SNR=10 (25 runs)

Plot Time (sec)

Carrier Freq (Hz)

Mod BW (Hz)

mean std. dev. variance actual error % error

1.10 0.06 0.0034 N/A N/A N/A

962.48 151.22 22866.23 1000 37.52 3.75%

606.08 9.87 97.34 500 106.08 21.22%

Mod Period (sec)

Time (sec)-Frequency (Hz) Localization

.02009 .00228 .00036 .00026 0 0 .02 N/A .00009 N/A 0.45% (2.85% of X-Axis) % detection = 100%

195

115.56 16.89 285.14 N/A N/A (5.78% of Y-Axis)

Chirp Rate (Hz/sec) 30169.44 636.09 4.05e5 25000 5169.44 20.68%

Test 14

Figure 56: Test 14 Plot – Spectrogram, 256, 0

Table 20: Test 14 Metrics – Spectrogram, 256, 0 TEST 14 Spectrogram TriModFMCW 2 Tri fs=4KHz fc=1KHz modBW=500 modperiod=20ms #samples=256 SNR=0 (25 runs)

Plot Time (sec)

Carrier Freq (Hz)

Mod BW (Hz)

mean std. dev. variance actual error % error

1.51 0.05 0.003 N/A N/A N/A

1013.16 132.81 17639.77 1000 13.16 1.32%

588.92 22.12 489.42 500 88.92 17.78%

Mod Period (sec)

Time (sec)-Frequency (Hz) Localization

.02019 .0025 .00072 .00047 0 0 .02 N/A .00019 N/A 0.95% (3.13% of X-Axis) % detection = 80%

196

137.56 23.06 531.96 N/A N/A (6.88% of Y-Axis)

Chirp Rate (Hz/sec) 29199.73 1346.06 1.81e6 25000 4199.73 16.80%

Test 15

Figure 57: Test 15 Plot – Spectrogram, 256, -3

Table 21: Test 15 Metrics – Spectrogram, 256, -3 TEST 15 Spectrogram TriModFMCW 2 Tri fs=4KHz fc=1KHz modBW=500 modperiod=20ms #samples=256 SNR=lowest detectable (-3dB) (25 runs)

Plot Time (sec)

Carrier Freq (Hz)

Mod BW (Hz)

mean std. dev. variance actual error % error

3.20 0.21 0.044 N/A N/A N/A

1131.28 65.20 4250.73 1000 131.28 13.13%

592.0 23.80 566.4 500 92.0 18.40%

Mod Period (sec)

Time (sec)-Frequency (Hz) Localization

.01984 .00272 .00086 .00079 0 0 .02 N/A .00016 N/A 0.80% (3.40% of X-Axis) % detection = 4%

197

140.32 41.73 1741.22 N/A N/A (7.02% of Y-Axis)

Chirp Rate (Hz/sec) 29890.33 1902.54 3.62e6 25000 4890.33 19.56%

Test 16

Figure 58: Test 16 Plot – Spectrogram, 512, 10

Table 22: Test 16 Metrics – Spectrogram, 512, 10 TEST 16 Spectrogram TriModFMCW 2 Tri fs=4KHz fc=1KHz modBW=500 modperiod=20ms #samples=512 SNR=10dB (25 runs)

Plot Time (sec)

Carrier Freq (Hz)

Mod BW (Hz)

mean std. dev. variance actual error % error

1.88 0.08 0.0072 N/A N/A N/A

963.32 138.32 19132.70 1000 36.68 3.67%

620.36 13.32 177.51 500 120.36 24.07%

Mod Period (sec)

Time (sec)-Frequency (Hz) Localization

.01989 .00217 .00029 .00039 0 0 .02 N/A .00011 N/A 0.55% (2.71% of X-Axis) % detection = 100%

198

114.08 9.94 98.76 N/A N/A (5.70% of Y-Axis)

Chirp Rate (Hz/sec) 31165.50 983.97 9.68e5 25000 6165.50 24.66%

Test 17

Figure 59: Test 17 Plot – Spectrogram, 512, 0

Table 23: Test 17 Metrics – Spectrogram, 512, 0 TEST 17 Spectrogram TriModFMCW 2 Tri fs=4KHz fc=1KHz modBW=500 modperiod=20ms #samples=512 SNR=0dB (25 runs)

Plot Time (sec)

Carrier Freq (Hz)

Mod BW (Hz)

mean std. dev. variance actual error % error

3.27 0.19 0.036 N/A N/A N/A

954.68 135.76 18430.94 1000 45.32 4.53%

628.96 16.18 261.84 500 128.96 25.79%

Mod Period (sec)

Time (sec)-Frequency (Hz) Localization

.01983 .00186 .00033 .00057 0 0 .02 N/A .00017 N/A 0.85% (2.33% of X-Axis) % detection = 92%

199

146.96 25.16 633.18 N/A N/A (7.35% of Y-Axis)

Chirp Rate (Hz/sec) 31733.23 1117.82 1.25e6 25000 6733.23 26.93%

Test 18

Figure 60: Test 18 Plot – Spectrogram, 512, -4

Table 24: Test 18 Metrics – Spectrogram, 512, -4 TEST 18 Spectrogram TriModFMCW 2 Tri fs=4KHz fc=1KHz modBW=500 modperiod=20ms #samples=512 SNR=lowest detectable (-4dB) (25 runs)

Plot Time (sec)

Carrier Freq (Hz)

Mod BW (Hz)

mean std. dev. variance actual error % error

12.73 1.15 1.33 N/A N/A N/A

1011.0 179.91 32368.76 1000 11.0 1.10%

623.4 19.33 373.66 500 123.4 24.68%

Mod Period (sec)

Time (sec)-Frequency (Hz) Localization

.0203 .00264 .0006 .00081 0 0 .02 N/A .0003 N/A 1.50% (3.30% of X-Axis) % detection = 8%

200

193.8 56.47 3188.95 N/A N/A (9.69% of Y-Axis)

Chirp Rate (Hz/sec) 30286.60 1374.70 1.89e6 25000 5286.60 21.15%

Test 19

Figure 61: Test 19 Plot – Scalogram, 256, 10

Table 25: Test 19 Metrics – Scalogram, 256, 10 TEST 19 Scalogram TriModFMCW 2 Tri fs=4KHz fc=1KHz modBW=500 modperiod=20ms #samples=256 SNR=10dB (25 runs)

Plot Time (sec)

Carrier Freq (Hz)

Mod BW (Hz)

mean std. dev. variance actual error % error

2.22 0.08 0.006 N/A N/A N/A

814.0 4.93 24.36 1000 186.0 18.60%

668.6 16.27 264.61 500 168.6 33.72%

Mod Period (sec)

Time (sec)-Frequency (Hz) Localization

.02025 .00363 .00053 .00071 0 0 .02 N/A .00025 N/A 1.25% (4.54% of X-Axis) % detection = 100%

201

217.2 21.34 455.47 N/A N/A (10.86% of Y-Axis)

Chirp Rate (Hz/sec) 33025.65 737.26 5.44e5 25000 8025.65 32.10%

Test 20

Figure 62: Test 20 Plot – Scalogram, 256, 0

Table 26: Test 20 Metrics – Scalogram, 256, 0 TEST 20 Scalogram TriModFMCW 2 Tri fs=4KHz fc=1KHz modBW=500 modperiod=20ms #samples=256 SNR=0dB (25 runs)

Plot Time (sec)

Carrier Freq (Hz)

Mod BW (Hz)

mean std. dev. variance actual error % error

2.38 0.08 0.006 N/A N/A N/A

958.92 116.71 13622.15 1000 41.08 4.11%

674.92 38.22 1460.48 500 174.92 34.98%

Mod Period (sec)

Time (sec)-Frequency (Hz) Localization

.02 .00463 .00083 .00084 0 0 .02 N/A 0.00 N/A 0.00% (5.79% of X-Axis) % detection = 60%

202

210.96 25.81 665.96 N/A N/A (10.55% of Y-Axis)

Chirp Rate (Hz/sec) 33829.28 2853.17 8.14e6 25000 8829.28 35.32%

Test 21

Figure 63: Test 21 Plot – Scalogram, 256, -2

Table 27: Test 21 Metrics – Scalogram, 256, -2 TEST 21 Scalogram TriModFMCW 2 Tri fs=4KHz fc=1KHz modBW=500 modperiod=20ms #samples=256 SNR=lowest detectable (-2dB) (25 runs)

Plot Time (sec)

Carrier Freq (Hz)

Mod BW (Hz)

mean std. dev. variance actual error % error

2.33 0.09 0.009 N/A N/A N/A

1003.18 149.01 22202.58 1000 3.18 0.32%

657.68 58.72 3447.64 500 157.68 31.54%

Mod Period (sec)

Time (sec)-Frequency (Hz) Localization

.02 .00472 .00118 .00184 0 0 .02 N/A 0.00 N/A 0.00% (5.90% of X-Axis) % detection = 16%

203

232.76 54.86 3009.58 N/A N/A (11.64% of Y-Axis)

Chirp Rate (Hz/sec) 32981.60 3473.98 1.20e7 25000 7981.60 31.93%

Test 22

Figure 64: Test 22 Plot – Scalogram, 512, 10

Table 28: Test 22 Metrics – Scalogram, 512, 10 TEST 22 Scalogram TriModFMCW 2 Tri fs=4KHz fc=1KHz modBW=500 modperiod=20ms #samples=512 SNR=10dB (25 runs)

Plot Time (sec)

Carrier Freq (Hz)

Mod BW (Hz)

mean std. dev. variance actual error % error

5.17 0.11 0.011 N/A N/A N/A

836.76 69.49 4829.31 1000 163.24 16.32%

667.20 33.55 1125.72 500 167.20 33.44%

Mod Period (sec)

Time (sec)-Frequency (Hz) Localization

.02006 .00319 .00069 .00046 0 0 .02 N/A 0.00006 N/A 0.30% (3.99% of X-Axis) % detection = 100%

204

160.12 17.84 318.37 N/A N/A (8.00% of Y-Axis)

Chirp Rate (Hz/sec) 33332.87 1901.58 3.62e6 25000 8332.87 33.33%

Test 23

Figure 65: Test 23 Plot – Scalogram, 512, 0

Table 29: Test 23 Metrics – Scalogram, 512, 0 TEST 23 Scalogram TriModFMCW 2 Tri fs=4KHz fc=1KHz modBW=500 modperiod=20ms #samples=512 SNR=0dB (25 runs)

Plot Time (sec)

Carrier Freq (Hz)

Mod BW (Hz)

mean std. dev. variance actual error % error

5.44 0.08 0.007 N/A N/A N/A

966.4 163.18 26628.66 1000 33.6 3.36%

709.92 59.16 3499.66 500 209.92 41.98%

Mod Period (sec)

Time (sec)-Frequency (Hz) Localization

.02012 .00331 .00062 .00051 0 0 .02 N/A 0.00012 N/A 0.60% (4.14% of X-Axis) % detection = 72%

205

197.68 35.16 1236.19 N/A N/A (9.88% of Y-Axis)

Chirp Rate (Hz/sec) 35293.58 2967.41 8.80e6 25000 10293.58 41.17%

Test 24

Figure 66: Test 24 Plot – Scalogram, 512, -3

Table 30: Test 24 Metrics – Scalogram, 512, -3 TEST 24 Scalogram TriModFMCW 2 Tri fs=4KHz fc=1KHz modBW=500 modperiod=20ms #samples=512 SNR=lowest detectable (-3dB) (25 runs)

Plot Time (sec)

Carrier Freq (Hz)

Mod BW (Hz)

mean std. dev. variance actual error % error

5.98 0.12 0.014 N/A N/A N/A

955.96 182.08 33152.76 1000 44.04 4.40%

749.24 61.08 3731.11 500 249.24 49.85%

Mod Period (sec)

Time (sec)-Frequency (Hz) Localization

.0195 .00383 .00065 .00114 0 0 .02 N/A 0.0005 N/A 2.50% (4.79% of X-Axis) % detection = 16%

206

271.04 60.24 3628.81 N/A N/A (13.55% of Y-Axis)

Chirp Rate (Hz/sec) 38430.43 2985.99 8.92e6 25000 13430.43 53.72%

Test 25

Figure 67: Test 25 Plot – Reassigned Spectrogram, 256, 10

Table 31: Test 25 Metrics – Reassigned Spectrogram, 256, 10 TEST 25 Reassigned Spectrogram TriModFMCW 2 Tri fs=4KHz fc=1KHz modBW=500 modperiod=20ms #samples=256 SNR=10dB (25 runs)

Plot Time (sec)

Carrier Freq (Hz)

Mod BW (Hz)

mean std. dev. variance actual error % error

4.19 0.20 0.039 N/A N/A N/A

1023.48 160.79 25854.15 1000 23.48 2.35%

465.68 14.34 205.50 500 34.32 6.86%

Mod Period (sec)

Time (sec)-Frequency (Hz) Localization

.02006 .00081 .00062 .0003 0 0 .02 N/A 0.00006 N/A 0.30% (1.01% of X-Axis) % detection = 100%

207

39.0 13.26 175.76 N/A N/A (1.95% of Y-Axis)

Chirp Rate (Hz/sec) 23228.38 934.94 8.74e5 25000 1771.62 7.09%

Test 26

Figure 68: Test 26 Plot – Reassigned Spectrogram, 256, 0

Table 32: Test 26 Metrics – Reassigned Spectrogram, 256, 0 TEST 26 Reassigned Spectrogram TriModFMCW 2 Tri fs=4KHz fc=1KHz modBW=500 modperiod=20ms #samples=256 SNR=0dB (25 runs)

Plot Time (sec)

Carrier Freq (Hz)

Mod BW (Hz)

mean std. dev. variance actual error % error

6.00 0.58 0.33 N/A N/A N/A

982.72 110.59 12230.78 1000 17.28 1.73%

500.00 27.52 757.12 500 0.00 0.00%

Mod Period (sec)

Time (sec)-Frequency (Hz) Localization

.02019 .00112 .00059 .00057 0 0 .02 N/A 0.00019 N/A 0.95% (1.40% of X-Axis) % detection = 100%

208

50.08 9.95 99.05 N/A N/A (2.50% of Y-Axis)

Chirp Rate (Hz/sec) 24789.92 1618.40 2.62e6 25000 210.08 0.84%

Test 27

Figure 69: Test 27 Plot – Reassigned Spectrogram, 256, -2

Table 33: Test 27 Metrics – Reassigned Spectrogram, 256, -2 TEST 27 Reassigned Spectrogram TriModFMCW 2 Tri fs=4KHz fc=1KHz modBW=500 modper=20ms #samp=256 SNR=low detect (-2dB) (25 runs)

Plot Time (sec)

Carrier Freq (Hz)

Mod BW (Hz)

mean std. dev. variance actual error % error

12.98 3.74 13.96 N/A N/A N/A

987.52 163.74 26810.20 1000 12.48 1.25%

531.20 32.89 1081.60 500 31.20 6.24%

Mod Period (sec)

Time (sec)-Frequency (Hz) Localization

.02006 .0015 .00062 .00067 0 0 .02 N/A 0.00006 N/A 0.30% (1.88% of X-Axis) % detection = 20%

209

65.64 14.31 204.71 N/A N/A (3.28% of Y-Axis)

Chirp Rate (Hz/sec) 26493.61 1734.04 3.01e6 25000 1493.61 5.97%

Test 28

Figure 70: Test 28 Plot – Reassigned Spectrogram, 512, 10

Table 34: Test 28 Metrics – Reassigned Spectrogram, 512, 10 TEST 28 Reassigned Spectrogram TriModFMCW 2 Tri fs=4KHz fc=1KHz modBW=500 modperiod=20ms #samples=512 SNR=10dB (25 runs)

Plot Time (sec)

Carrier Freq (Hz)

Mod BW (Hz)

mean std. dev. variance actual error % error

41.75 3.27 10.70 N/A N/A N/A

1035.24 150.80 22740.78 1000 35.24 3.52%

488.12 10.03 100.66 500 11.88 2.38%

Mod Period (sec)

Time (sec)-Frequency (Hz) Localization

.02013 .0005 .00026 .00022 0 0 .02 N/A 0.00013 N/A 0.65% (0.63% of X-Axis) % detection = 100%

210

17.88 5.20 27.02 N/A N/A (0.89% of Y-Axis)

Chirp Rate (Hz/sec) 24258.58 693.08 4.80e5 25000 741.42 2.97%

Test 29

Figure 71: Test 29 Plot – Reassigned Spectrogram, 512, 0

Table 35: Test 29 Metrics – Reassigned Spectrogram, 512, 0 TEST 29 Reassigned Spectrogram TriModFMCW 2 Tri fs=4KHz fc=1KHz modBW=500 modperiod=20ms #samples=512 SNR=0dB (25 runs)

Plot Time (sec)

Carrier Freq (Hz)

Mod BW (Hz)

mean std. dev. variance actual error % error

46.75 6.63 43.93 N/A N/A N/A

1122.64 123.49 15249.11 1000 122.64 12.26%

506.32 13.80 190.36 500 6.32 1.26%

Mod Period (sec)

Time (sec)-Frequency (Hz) Localization

.02003 .00081 .00034 .00037 0 0 .02 N/A 0.00003 N/A 0.15% (1.01% of X-Axis) % detection = 100%

211

50.00 11.19 125.23 N/A N/A (2.50% of Y-Axis)

Chirp Rate (Hz/sec) 25284.21 851.23 7.25e5 25000 284.21 1.14%

Test 30

Figure 72: Test 30 Plot – Reassigned Spectrogram, 512, -3

Table 36: Test 30 Metrics – Reassigned Spectrogram, 512, -3 TEST 30 Reassigned Spectrogram TriModFMCW 2 Tri fs=4KHz fc=1KHz modBW=500 modper=20ms #samp=512 SNR=low detect (-3dB) (25 runs)

Plot Time min:sec

Carrier Freq (Hz)

Mod BW (Hz)

mean std. dev. variance actual error % error

2:35.55 22.45 503.94 N/A N/A N/A

988.58 143.95 20722.79 1000 11.42 1.14%

540.6 19.07 363.6 500 40.6 8.12%

Mod Period (sec)

Time (sec)-Frequency (Hz) Localization

.02009 .00103 .00036 .00021 0 0 .02 N/A 0.00009 N/A 0.45% (1.29% of X-Axis) % detection = 8%

212

57.08 13.33 177.82 N/A N/A (2.85% of Y-Axis)

Chirp Rate (Hz/sec) 26906.23 899.37 8.09e5 25000 1906.23 7.62%

Test 31

Figure 73: Test 31 Plot – Reassigned Scalogram, 256, 10

Table 37: Test 31 Metrics – Reassigned Scalogram, 256, 10 TEST 31 Reassigned Scalogram TriModFMCW 2 Tri fs=4KHz fc=1KHz modBW=500 modperiod=20ms #samples=256 SNR=10 (25 runs)

Plot Time (sec)

Carrier Freq (Hz)

Mod BW (Hz)

mean std. dev. variance actual error % error

3.36 0.27 0.07 N/A N/A N/A

971.84 174.66 30507.63 1000 28.16 2.82%

492.2 24.67 608.4 500 7.8 1.56%

Mod Period (sec)

Time (sec)-Frequency (Hz) Localization

.02025 .00113 .00053 .00037 0 0 .02 N/A 0.00025 N/A 1.25% (1.41% of X-Axis) % detection = 100%

213

45.32 15.48 239.58 N/A N/A (2.27% of Y-Axis)

Chirp Rate (Hz/sec) 24329.65 1523.66 2.32e6 25000 670.35 2.68%

Test 32

Figure 74: Test 32 Plot – Reassigned Scalogram, 256, 0

Table 38: Test 32 Metrics – Reassigned Scalogram, 256, 0 TEST 32 Reassigned Scalogram TriModFMCW 2 Tri fs=4KHz fc=1KHz modBW=500 modperiod=20ms #samples=256 SNR=0 (25 runs)

Plot Time (sec)

Carrier Freq (Hz)

Mod BW (Hz)

mean std. dev. variance actual error % error

3.38 0.20 0.04 N/A N/A N/A

1031.28 134.31 18037.86 1000 31.28 3.13%

437.6 16.44 270.4 500 62.4 12.48%

Mod Period (sec)

Time (sec)-Frequency (Hz) Localization

.01962 .00128 .00084 .00056 0 0 .02 N/A 0.00038 N/A 1.90% (1.60% of X-Axis) % detection = 88%

214

62.56 22.13 489.54 N/A N/A (3.13% of Y-Axis)

Chirp Rate (Hz/sec) 22335.37 1286.57 1.66e6 25000 2664.63 10.66%

Test 33

Figure 75: Test 33 Plot – Reassigned Scalogram, 256, -2

Table 39: Test 33 Metrics – Reassigned Scalogram, 256, -2 TEST 33 Reassigned Scalogram TriModFMCW 2 Tri fs=4KHz fc=1KHz modBW=500 modper=20ms #samp=256 SNR=low detect (-2dB) (25 runs)

Plot Time (sec)

Carrier Freq (Hz)

Mod BW (Hz)

mean std. dev. variance actual error % error

4.51 0.48 0.23 N/A N/A N/A

1026.6 186.71 34858.92 1000 26.60 2.66%

490.64 42.37 1795.46 500 9.36 1.87%

Mod Period (sec)

Time (sec)-Frequency (Hz) Localization

.01987 .00116 .00071 .00026 0 0 .02 N/A 0.00013 N/A 0.65% (1.45% of X-Axis) % detection = 12%

215

57.88 22.22 493.90 N/A N/A (2.89% of Y-Axis)

Chirp Rate (Hz/sec) 24697.31 2127.30 4.53e6 25000 302.69 1.21%

Test 34

Figure 76: Test 34 Plot – Reassigned Scalogram, 512, 10

Table 40: Test 34 Metrics – Reassigned Scalogram, 512, 10 TEST 34 Reassigned Scalogram TriModFMCW 2 Tri fs=4KHz fc=1KHz modBW=500 modperiod=20ms #samples=512 SNR=10 (25 runs)

Plot Time (sec)

Carrier Freq (Hz)

Mod BW (Hz)

mean std. dev. variance actual error % error

15.52 1.08 1.17 N/A N/A N/A

1132.8 143,44 20576.46 1000 132.8 13.28%

471.04 7.41 54.94 500 28.96 5.79%

Mod Period (sec)

Time (sec)-Frequency (Hz) Localization

.02025 .00061 .00044 .00016 0 0 .02 N/A 0.00025 N/A 1.25% (0.76% of X-Axis) % detection = 100%

216

21.93 4.59 21.11 N/A N/A (1.10% of Y-Axis)

Chirp Rate (Hz/sec) 23269.26 704.66 4.97e5 25000 1730.74 6.92%

Test 35

Figure 77: Test 35 Plot – Reassigned Scalogram, 512, 0

Table 41: Test 35 Metrics – Reassigned Scalogram, 512, 0 TEST 35 Reassigned Scalogram TriModFMCW 2 Tri fs=4KHz fc=1KHz modBW=500 modperiod=20ms #samples=512 SNR=0 (25 runs)

Plot Time (sec)

Carrier Freq (Hz)

Mod BW (Hz)

mean std. dev. variance actual error % error

15.13 2.31 5.32 N/A N/A N/A

1032.8 159.33 25385.71 1000 32.8 3.28%

478.08 18.49 341.86 500 21.92 4.38%

Mod Period (sec)

Time (sec)-Frequency (Hz) Localization

.02006 .0005 .00055 .00016 0 0 .02 N/A 0.00006 N/A 0.30% (0.63% of X-Axis) % detection = 100%

217

29.76 9.65 93.06 N/A N/A (1.49% of Y-Axis)

Chirp Rate (Hz/sec) 23856.23 1356.43 1.84e6 25000 1143.77 4.58%

Test 36

Figure 78: Test 36 Plot – Reassigned Scalogram, 512, -3

Table 42: Test 36 Metrics – Reassigned Scalogram, 512, -3 TEST 36 Reassigned Scalogram TriModFMCW 2 Tri fs=4KHz fc=1KHz modBW=500 modper=20ms #samp=512 SNR=low detect (-3dB) (25 runs)

Plot Time (sec)

Carrier Freq (Hz)

Mod BW (Hz)

mean std. dev. variance actual error % error

36.18 5.89 34.74 N/A N/A N/A

1057.84 123.27 15194.57 1000 57.84 5.78%

503.0 28.97 839.40 500 3.0 0.60%

Mod Period (sec)

Time (sec)-Frequency (Hz) Localization

.01987 .00067 .0004 .00033 0 0 .02 N/A 0.00013 N/A 0.65% (0.84% of X-Axis) % detection = 8%

218

34.32 20.28 411.12 N/A N/A (1.22% of Y-Axis)

Chirp Rate (Hz/sec) 25311.39 1427.51 2.04e6 25000 311.39 1.25%

Test 37

Figure 79: Test 37 Plot – RSPWVD, 256, 10

Table 43: Test 37 Metrics – RSPWVD, 256, 10 TEST 37 RSPWVD TriModFMCW 2 Tri fs=4KHz fc=1KHz modBW=500 modperiod=20ms #samples=256 SNR=10dB (25 runs)

Plot Time (sec)

Carrier Freq (Hz)

Mod BW (Hz)

mean std. dev. variance actual error % error

5.17 0.26 0.067 N/A N/A N/A

899.20 163.55 26748.34 1000 100.80 10.08%

444.44 12.43 154.50 500 55.56 11.11%

Mod Period (sec)

Time (sec)-Frequency (Hz) Localization

.02031 .00047 .00079 .00016 0 0 .02 N/A 0.00031 N/A 1.55% (0.59% of X-Axis) % detection = 100%

219

25.0 6.26 39.16 N/A N/A (1.25% of Y-Axis)

Chirp Rate (Hz/sec) 21922.90 1311.08 1.72e6 25000 3077.10 12.31%

Test 38

Figure 80: Test 38 Plot –RSPWVD, 256, 0

Table 44: Test 38 Metrics – RSPWVD, 256, 0 TEST 38 RSPWVD TriModFMCW 2 Tri fs=4KHz fc=1KHz modBW=500 modperiod=20ms #samples=256 SNR=0dB (25 runs)

Plot Time (sec)

Carrier Freq (Hz)

Mod BW (Hz)

mean std. dev. variance actual error % error

5.16 0.40 0.16 N/A N/A N/A

1003.12 171.07 29263.89 1000 3.12 0.31%

486.28 43.40 1883.59 500 13.72 2.74%

Mod Period (sec)

Time (sec)-Frequency (Hz) Localization

.02028 .00075 .00089 .00026 0 0 .02 N/A 0.00028 N/A 1.40% (0.94% of X-Axis) % detection = 80%

220

34.44 12.33 152.02 N/A N/A (1.72% of Y-Axis)

Chirp Rate (Hz/sec) 23989.94 1995.50 3.98e6 25000 1010.06 4.04%

Test 39

Figure 81: Test 39 Plot – RSPWVD, 256, -3

Table 45: Test 39 Metrics – RSPWVD, 256, -3 TEST 39 RSPWVD TriModFMCW 2 Tri fs=4KHz fc=1KHz modBW=500 modperiod=20ms #samples=256 SNR=low detect (-3dB) (25 runs)

Plot Time (sec)

Carrier Freq (Hz)

Mod BW (Hz)

mean std. dev. variance actual error % error

6.91 1.33 1.76 N/A N/A N/A

898.4 89.57 8023.54 1000 101.6 10.16%

453.12 44.88 2014.50 500 46.88 9.38%

Mod Period (sec)

Time (sec)-Frequency (Hz) Localization

.02019 .00112 .00122 .00056 0 0 .02 N/A 0.00019 N/A 0.95% (1.40% of X-Axis) % detection = 12%

221

52.44 14.74 217.26 N/A N/A (2.62% of Y-Axis)

Chirp Rate (Hz/sec) 22507.00 2454.31 6.02e6 25000 2493 9.97%

Test 40

Figure 82: Test 40 Plot – RSPWVD, 512, 10

Table 46: Test 40 Metrics – RSPWVD, 512, 10 TEST 40 RSPWVD TriModFMCW 2 Tri fs=4KHz fc=1KHz modBW=500 modperiod=20ms #samples=512 SNR=10dB (25 runs)

Plot Time (sec)

Carrier Freq (Hz)

Mod BW (Hz)

mean std. dev. variance actual error % error

58.88 3.05 9.33 N/A N/A N/A

897.68 171.04 29256.35 1000 102.32 10.23%

474.88 5.90 34.80 500 25.12 5.02%

Mod Period (sec)

Time (sec)-Frequency (Hz) Localization

.02016 .00017 .0003 .0001 0 0 .02 N/A 0.00016 N/A 0.80% (0.21% of X-Axis) % detection = 100%

222

10.00 3.37 11.38 N/A N/A (0.50% of Y-Axis)

Chirp Rate (Hz/sec) 23565.00 345.27 1.19e5 25000 1435 5.74%

Test 41

Figure 83: Test 41 Plot – RSPWVD, 512, 0

Table 47: Test 41 Metrics – RSPWVD, 512, 0 TEST 41 RSPWVD TriModFMCW 2 Tri fs=4KHz fc=1KHz modBW=500 modperiod=20ms #samples=512 SNR=0dB (25 runs)

Plot Time (sec)

Carrier Freq (Hz)

Mod BW (Hz)

mean std. dev. variance actual error % error

36.24 2.91 8.46 N/A N/A N/A

1007.22 111.30 12387.32 1000 7.22 0.72%

496.56 9.24 85.41 500 3.44 0.69%

Mod Period (sec)

Time (sec)-Frequency (Hz) Localization

.01987 .00038 .00051 .00013 0 0 .02 N/A 0.00013 N/A 0.65% (0.48% of X-Axis) % detection = 100%

223

25.04 8.51 72.43 N/A N/A (1.25% of Y-Axis)

Chirp Rate (Hz/sec) 24920.69 547.27 2.99e5 25000 79.31 0.32%

Test 42

Figure 84: Test 42 Plot – RSPWVD, 512, -3

Table 48: Test 42 Metrics – RSPWVD, 512, -3 TEST 42 RSPWVD TriModFMCW 2 Tri fs=4KHz fc=1KHz modBW=500 modperiod=20ms #samples=512 SNR=low detect (-3dB) (25 runs)

Plot Time min:sec

Carrier Freq (Hz)

Mod BW (Hz)

mean std. dev. variance actual error % error

1:42:22 4.05 16.43 N/A N/A N/A

1021.92 152.15 23150.46 1000 21.92 2.19%

510.68 21.58 465.76 500 10.68 2.14%

Mod Period (sec)

Time (sec)-Frequency (Hz) Localization

.01988 .0005 .0003 .00016 0 0 .02 N/A 0.00012 N/A 0.60% (0.63% of X-Axis) % detection = 8%

224

31.37 9.26 85.69 N/A N/A (1.57% of Y-Axis)

Chirp Rate (Hz/sec) 25706.06 1293.66 1.67e6 25000 706.06 2.82%

Test 43

Figure 85: Test 43 Plot – HT of WVD, 256, 10

Table 49: Test 43 Metrics – HT of WVD, 256, 10 TEST 43 HT of WVD TriModFMCW 2 Tri fs=4KHz fc=1KHz modBW=500 modperiod=20ms #samples=256 SNR=10dB (25 runs)

Plot Time (sec)

θ1 (rad.)

mean

38.53

0.7697

66.96

std dev

1.84

0.019

var.

3.39

actual

ρ1 (samp.)

chirp rate1

θ2 (rad.)

ρ2 (samp.)

chirp rate2

θ3 (rad.)

ρ3 (samp.)

chirp rate3

θ4 (rad.)

ρ4 (samp.)

-24244

2.43

19.71

21565

3.849

1.43

935.55

0.025

0.71

1081.65

0.00037

2.06

8.75e5

0.0006

0.50

N/A

0.7853

N/A

-25000

2.356

error

N/A

0.0156

N/A

-756

% error

N/A

1.99%

N/A

3.02%

20.26

-21422

5.547

65.61

22692

0.022

0.63

945.84

0.032

2.59

1466.59

1.17e6

0.0005

0.40

8.95e5

0.001

6.74

2.15e6

N/A

25000

3.927

N/A

-25000

5.497

N/A

25000

0.074

N/A

3435

0.078

N/A

-3578

0.05

N/A

2308

3.14%

N/A

13.74%

1.99%

N/A

14.31%

0.91%

N/A

9.23%

% detection = 100%

225

chirp rate4

Test 44

Figure 86: Test 44 Plot – HT of WVD, 256, 0

Table 50: Test 44 Metrics – HT of WVD, 256, 0 TEST 44 HT of WVD TriModFMCW 2 Tri fs=4KHz fc=1KHz modBW=500 modperiod=20ms #samples=256 SNR=0dB (25 runs)

Plot Time min:sec

θ1 (rad.)

ρ1 (samp.)

mean

23:37.4

0.7510

65.10

std dev

3.03

0.048

var.

9.16

actual

chirp rate1

θ2 (rad.)

ρ2 (samp.)

chirp rate2

θ3 (rad.)

ρ3 (samp.)

chirp rate3

θ4 (rad.)

ρ4 (samp.)

-23431

2.42

19.12

22014

3.863

2.58

2371.22

0.026

1.63

1243.49

0.0023

6.67

5.62e6

0.0007

2.66

N/A

0.7853

N/A

-25000

2.356

error

N/A

0.0343

N/A

-1569

% error

N/A

4.37%

N/A

6.28%

21.02

-22016

5.532

65.71

23421

0.026

1.08

1237.53

0.051

4.54

2436.0

1.55e6

0.0007

1.17

1.53e6

0.0026

20.64

5.93e6

N/A

25000

3.927

N/A

-25000

5.497

N/A

25000

0.064

N/A

2986

0.064

N/A

-2984

0.035

N/A

1579

2.72%

N/A

11.94%

1.63%

N/A

11.94%

0.64%

N/A

6.32%

% detection = 100%

226

chirp rate4

Test 45

Figure 87: Test 45 Plot – HT of WVD, 256, -3

Table 51: Test 45 Metrics – HT of WVD, 256, -3 TEST 45 HT of WVD TriModFMCW 2 Tri fs=4KHz fc=1KHz modBW=500 modperiod=20ms #samples=256 SNR= low detect (-3dB) (25 runs)

Plot Time min:sec

θ1 (rad.)

ρ1 (samp.)

mean

46:17.5

0.7854

std dev

5.25

var.

chirp rate1

θ2 (rad.)

ρ2 (samp.)

chirp rate2

θ3 (rad.)

ρ3 (samp.)

chirp rate3

θ4 (rad.)

ρ4 (samp.)

65.45

-25546

2.435

20.31

21418

3.932

0.112

9.21

5937.14

0.05

2.64

2249.75

27.60

0.0127

84.79

3.52e7

0.0025

6.96

actual

N/A

0.7853

N/A

-25000

2.356

error

N/A

0.0001

N/A

-546

% error

N/A

0.01%

N/A

2.18%

22.01

-25650

5.532

65.60

23586

0.099

3.82

5004.28

0.09

5.65

4162.13

5.06e6

0.0098

14.62

2.50e7

0.008

31.91

1.73e7

N/A

25000

3.927

N/A

-25000

5.497

N/A

25000

0.079

N/A

3582

0.005

N/A

-650

0.035

N/A

1414

3.35%

N/A

14.33%

0.13%

N/A

2.60%

0.64%

N/A

5.66%

% detection = 16%

227

chirp rate4

Test 46

Figure 88: Test 46 Plot – HT of WVD, 512, 10

Table 52: Test 46 Metrics – HT of WVD, 512, 10 TEST 46 HT of WVD TriModFMCW 2 Tri fs=4KHz fc=1KHz modBW=500 modperiod=20ms #samples=512 SNR=10dB (25 runs)

Plot Time min:sec

θ1 (rad.)

ρ1 (samp.)

chirp rate1

θ2 (rad.)

ρ2 (samp.)

mean

8:13.08

0.7793

135.23

-24696

2.388

std dev

0.199

0.013

1.69

667.81

var.

0.039

0.0002

2.88

actual

N/A

0.7853

error

N/A

% error

N/A

chirp rate2

θ3 (rad.)

ρ3 (samp.)

chirp rate3

θ4 (rad.)

ρ4 (samp.)

chirp rate4

42.65

23460

3.894

43.43

-23400

5.509

135.03

24881

0.017

0.873

786.5

0.0099

0.479

458.93

0.0158

2.234

1147.21

4.46e5

0.0003

0.762

6.19e5

0.0001

0.23

2.11e5

0.0003

4.99

1.32e6

N/A

-25000

2.356

N/A

25000

3.927

N/A

-25000

5.497

N/A

25000

0.006

N/A

-304

0.032

N/A

1540

0.033

N/A

-1600

0.012

N/A

119

0.76%

N/A

1.22%

1.36%

N/A

6.16%

0.84%

N/A

6.40%

0.22%

N/A

0.48%

% detection = 100%

228

Test 47 Did not run Test 47 due to inadequate computational resources (1 run was still processing after 8 hours)

Test 48 Did not run Test 48 due to inadequate computational resources (1 run was still processing after 8 hours)

229

Test 49

Figure 89: Test 49 Plot – HT of CWD, 256, 10

Table 53: Test 49 Metrics – HT of CWD, 256, 10 TEST 49 HT of CWD TriModFMCW 2 Tri fs=4KHz fc=1KHz modBW=500 modperiod=20ms #samples=256 SNR=10dB (25 runs)

Plot Time (sec)

θ1 (rad.)

ρ1 (samp.)

chirp rate1

θ2 (rad.)

ρ2 (samp.)

chirp rate2

θ3 (rad.)

ρ3 (samp.)

chirp rate3

θ4 (rad.)

ρ4 (samp.)

chirp rate4

mean

4.65

0.7584

65.88

-23692

2.403

20.31

22779

3.868

21.02

-22249

5.527

67.23

23567

std dev

0.28

0.0078

0.299

362.4

0.018

0.599

823.56

0.024

0.599

1044.65

0.0105

0.674

489.94

var.

0.079

0.0001

0.0896

1.31e5

0.0003

0.358

6.78e5

0.0006

0.358

1.09e6

0.0001

0.454

2.40e5

actual

N/A

0.7853

N/A

-25000

2.356

N/A

25000

3.927

N/A

-25000

5.497

N/A

25000

error

N/A

0.027

N/A

-1308.0

0.047

N/A

2221

0.059

N/A

-2751

0.03

N/A

1433

% error

N/A

3.43%

N/A

5.23%

1.99%

N/A

8.88%

1.50%

N/A

11.00%

0.55%

N/A

5.73%

% detection = 100%

230

Test 50

Figure 90: Test 50 Plot – HT of CWD, 256, 0

Table 54: Test 50 Metrics – HT of CWD, 256, 0 TEST 50 HT of CWD TriModFMCW 2 Tri fs=4KHz fc=1KHz modBW=500 modperiod=20ms #samples=256 SNR=0dB (25 runs)

Plot Time (sec)

θ1 (rad.)

ρ1 (samp.)

mean

6.09

0.7584

std dev

0.34

var.

chirp rate1

θ2 (rad.)

ρ2 (samp.)

chirp rate2

θ3 (rad.)

ρ3 (samp.)

chirp rate3

θ4 (rad.)

ρ4 (samp.)

66.38

-23764

2.398

20.59

23064

3.846

0.042

3.23

2039.44

0.043

1.16

2076.97

0.115

0.0018

10.45

4.16e6

0.0019

1.34

actual

N/A

0.7853

N/A

-25000

2.356

error

N/A

0.027

N/A

-1236

% error

N/A

3.43%

N/A

4.94%

19.95

-22187

5.520

67.16

23986

0.053

2.75

1840.11

0.042

3.18

2106.02

4.31e6

0.0028

7.56

3.39e6

0.0018

10.10

4.44e6

N/A

25000

3.927

N/A

-25000

5.497

N/A

25000

0.042

N/A

1936

0.081

N/A

-2813

0.023

N/A

1014

1.78%

N/A

7.74%

2.06%

N/A

11.25%

0.42%

N/A

4.06%

% detection = 92%

231

chirp rate4

Test 51

Figure 91: Test 51 Plot – HT of CWD, 256, -4

Table 55: Test 51 Metrics – HT of CWD, 256, -4 TEST 51 HT of CWD TriModFMCW 2 Tri fs=4KHz fc=1KHz modBW=500 modperiod=20ms #samples=256 SNR=low detect (-4dB) (25 runs)

Plot Time (sec)

θ1 (rad.)

ρ1 (samp.)

mean

9.78

0.7805

std dev

0.60

var.

chirp rate1

θ2 (rad.)

ρ2 (samp.)

chirp rate2

θ3 (rad.)

ρ3 (samp.)

chirp rate3

θ4 (rad.)

ρ4 (samp.)

66.16

-24954

2.398

21.21

23220

3.844

0.062

3.21

3337.55

0.078

2.00

3565.65

0.362

0.004

10.28

1.11e7

0.006

4.00

actual

N/A

0.7853

N/A

-25000

2.356

error

N/A

0.0048

N/A

-46

% error

N/A

0.61%

N/A

0.18%

22.44

-21537

5.557

67.51

24345

0.114

9.56

4626.49

0.169

6.019

4295.16

1.27e7

0.013

91.49

2.14e7

0.029

36.23

1.84e7

N/A

25000

3.927

N/A

-25000

5.497

N/A

25000

0.042

N/A

1780

0.083

N/A

-3463

0.06

N/A

655

1.78%

N/A

7.12%

2.11%

N/A

13.85%

1.09%

N/A

2.62%

% detection = 16%

232

chirp rate4

Test 52

Figure 92: Test 52 Plot – HT of CWD, 512, 10

Table 56: Test 52 Metrics – HT of CWD, 512, 10 TEST 52 HT of CWD TriModFMCW 2 Tri fs=4KHz fc=1KHz modBW=500 modperiod=20ms #samples=512 SNR=10dB (25 runs)

Plot Time (sec)

θ1 (rad.)

ρ1 (samp.)

chirp rate1

mean

13.11

0.7805

134.98

-24766

2.373

43.65

24167

3.905

43.65

-23752

5.493

136.4

25214

std dev

0.487

0.013

2.38

666.03

0.011

0.686

501.75

0.013

0.367

864.45

0.016

3.02

806.84

var.

0.237

0.00017

5.68

4.44e5

0.00011

0.47

2.52e5

0.00017

0.134

7.47e5

0.00027

9.12

6.51e5

actual

N/A

0.7853

N/A

-25000

2.356

N/A

25000

3.927

N/A

-25000

5.497

N/A

25000

error

N/A

0.0048

N/A

-234

0.017

N/A

833

0.022

N/A

-1248

0.004

N/A

214

% error

N/A

0.61%

N/A

0.94%

0.72%

N/A

3.33%

0.56%

N/A

4.99%

0.07%

N/A

0.86%

θ2 (rad.)

ρ2 (samp.)

chirp rate2

% detection = 100%

233

θ3 (rad.)

ρ3 (samp.)

chirp rate3

θ4 (rad.)

ρ4 (samp.)

chirp rate4

Test 53

Figure 93: Test 53 Plot – HT of CWD, 512, 0

Table 57: Test 53 Metrics – HT of CWD, 512, 0 TEST 53 HT of CWD TriModFMCW 2 Tri fs=4KHz fc=1KHz modBW=500 modperiod=20ms #samples=512 SNR=0dB (25 runs)

Plot Time (sec)

θ1 (rad.)

mean

25.99

0.7719

133.4

-24384

2.375

std dev

2.64

0.031

4.42

1575.40

var.

6.94

0.00099

19.57

actual

N/A

0.7853

error

N/A

% error

N/A

ρ1 (samp.)

chirp rate1

θ2 (rad.)

ρ2 (samp.)

chirp rate2

θ3 (rad.)

ρ3 (samp.)

chirp rate3

θ4 (rad.)

ρ4 (samp.)

43.43

24128

3.892

0.030

1.74

1523.07

2.48e6

0.00094

3.02

N/A

-25000

2.356

0.0134

N/A

-616

1.71%

N/A

2.46%

43.29

-23340

5.494

136.8

25271

0.06

3.07

2792.90

0.048

6.30

2556.3

2.32e6

0.0035

9.41

7.80e6

0.0023

39.65

6.53e6

N/A

25000

3.927

N/A

-25000

5.497

N/A

25000

0.019

N/A

872

0.035

N/A

-1660

0.003

N/A

271

0.81%

N/A

3.49%

0.89%

N/A

6.64%

0.05%

N/A

1.08%

% detection = 100%

234

chirp rate4

Test 54

Figure 94: Test 54 Plot – HT of CWD, 512, -6

Table 58: Test 54 Metrics – HT of CWD, 512, -6 TEST 54 HT of CWD TriModFMCW 2 Tri fs=4KHz fc=1KHz modBW=500 modperiod=20ms #samples=512 SNR=low detect (-6dB) (25 runs)

Plot Time min:sec

θ1 (rad.)

ρ1 (samp.)

chirp rate1

θ2 (rad.)

ρ2 (samp.)

mean

1:42.8

0.7895

135.08

-25217

2.412

std dev

9.87

0.014

3.33

710.72

var.

97.5

0.0002

11.06

actual

N/A

0.7853

error

N/A

% error

N/A

chirp rate2

θ3 (rad.)

ρ3 (samp.)

chirp rate3

θ4 (rad.)

ρ4 (samp.)

chirp rate4

40.21

22490

3.917

42.25

-25060

5.482

137.45

26017

0.065

1.77

2948.88

0.116

3.13

5872.98

0.078

11.57

4085.72

5.05e5

0.0042

3.13

8.70e6

0.013

9.82

3.45e7

0.006

133.95

1.67e7

N/A

-25000

2.356

N/A

25000

3.927

N/A

-25000

5.497

N/A

25000

0.0042

N/A

-217

0.056

N/A

2510

0.010

N/A

-60

0.015

N/A

1017

0.53%

N/A

0.87%

2.38%

N/A

10.04%

0.25%

N/A

0.24%

0.27%

N/A

4.07%

% detection = 8%

235

Test 55

Figure 95: Test 55 Plot – HT of RSPWVD, 256, 10

Table 59: Test 55 Metrics – HT of RSPWVD, 256, 10 TEST 55 HT of RSPWVD TriModFMCW 2 Tri fs=4KHz fc=1KHz modBW=500 modperiod=20ms #samples=256 SNR=10dB (25 runs)

Plot Time (sec)

θ1 (rad.)

ρ1 (samp.)

mean

5.89

0.7857

std dev

0.11

var.

chirp rate1

θ2 (rad.)

ρ2 (samp.)

chirp rate2

θ3 (rad.)

ρ3 (samp.)

chirp rate3

θ4 (rad.)

ρ4 (samp.)

64.6

-25016

2.348

19.53

25411

3.926

0.007

1.80

241.48

0.014

1.08

712.4

0.012

0

3.22

5.83e4

0.0002

1.16

actual

N/A

0.7853

N/A

-25000

2.356

error

N/A

0.0004

N/A

-16

% error

N/A

0.05%

N/A

0.06%

20.95

-24971

5.498

67.8

25009

0.015

0.389

197.44

0.05

0.87

1222.9

5.08e5

0.0002

0.151

3.89e4

0.0006

0.76

1.49e6

N/A

25000

3.927

N/A

-25000

5.497

N/A

25000

0.008

N/A

411

0.001

N/A

-29

0.001

N/A

0.34%

N/A

1.64%

0.03%

N/A

0.12%

0.02%

N/A

% detection = 100%

236

chirp rate4

9 0.04%

Test 56

Figure 96: Test 56 Plot – HT of RSPWVD, 256, 0

Table 60: Test 56 Metrics – HT of RSPWVD, 256, 0 TEST 56 HT of RSPWVD TriModFMCW 2 Tri fs=4KHz fc=1KHz modBW=500 modperiod=20ms #samples=256 SNR=0dB (25 runs)

Plot Time (sec)

θ1 (rad.)

ρ1 (samp.)

mean

6.48

0.7915

66.3

std dev

0.09

0.046

var.

0.008

actual

chirp rate1

θ2 (rad.)

ρ2 (samp.)

chirp rate2

θ3 (rad.)

ρ3 (samp.)

chirp rate3

θ4 (rad.)

ρ4 (samp.)

-25394

2.37

19.31

24267

3.919

7.15

2444.2

0.051

2.033

2436.1

0.002

51.07

5.97e6

0.0026

4.133

N/A

0.7853

N/A

-25000

2.356

error

N/A

0.0062

N/A

-394

% error

N/A

0.79%

N/A

1.58%

20.59

-24627

5.506

65.88

24632

0.029

1.67

1447.9

0.038

4.80

1816.3

5.93e6

0.0008

2.77

2.09e6

0.0014

23.04

3.29e6

N/A

25000

3.927

N/A

-25000

5.497

N/A

25000

0.014

N/A

733

0.008

N/A

-373

0.009

N/A

368

0.59%

N/A

2.93%

0.20%

N/A

1.49%

0.16%

N/A

1.47%

% detection = 100%

237

chirp rate4

Test 57

Figure 97: Test 57 Plot – HT of RSPWVD, 256, -4

Table 61: Test 57 Metrics – HT of RSPWVD, 256, -4 TEST 57 HT of RSPWVD TriModFMCW 2 Tri fs=4KHz fc=1KHz modBW=500 modperiod=20ms #samples=256 SNR=low detect (-4dB) (25 runs)

Plot Time (sec)

θ1 (rad.)

ρ1 (samp.)

mean

9.00

0.7854

std dev

1.32

var.

chirp rate1

θ2 (rad.)

ρ2 (samp.)

chirp rate2

θ3 (rad.)

ρ3 (samp.)

chirp rate3

θ4 (rad.)

ρ4 (samp.)

64.39

-25072

2.352

22.01

25309

3.912

0.0448

2.08

2239.8

0.052

2.19

2571.9

1.73

0.002

4.33

5.01e6

0.0028

4.79

actual

N/A

0.7853

N/A

-25000

2.356

error

N/A

0.0001

N/A

-72

% error

N/A

0.01%

N/A

0.29%

21.16

-24412

5.507

62.47

24678

0.054

3.46

2764.3

0.066

9.31

3450.53

6.61e6

0.0029

11.95

7.64e6

0.0044

86.71

1.19e7

N/A

25000

3.927

N/A

-25000

5.497

N/A

25000

0.004

N/A

309

0.015

N/A

-588

0.01

N/A

322

0.17%

N/A

1.24%

0.38%

N/A

2.35%

0.18%

N/A

1.29%

% detection = 20%

238

chirp rate4

Test 58

Figure 98: Test 58 Plot – HT of RSPWVD, 512, 10

Table 62: Test 58 Metrics – HT of RSPWVD, 512, 10 TEST 58 HT of RSPWVD TriModFMCW 2 Tri fs=4KHz fc=1KHz modBW=500 modperiod=20ms #samples=512 SNR=10dB (25 runs)

Plot Time min:sec

θ1 (rad.)

ρ1 (samp.)

chirp rate1

θ2 (rad.)

ρ2 (samp.)

mean

1:01.08

0.78537

137.18

-25002

2.3563

44.78

24983.7

3.931

std dev

0.88

.00006

2.69

1.732

.00058

1.16

28.29

var.

0.774

0

7.26

3.0

0

1.35

actual

N/A

0.7853

N/A

-25000

2.356

error

N/A

0.00007

N/A

-2

% error

N/A

0.01%

N/A

0.01%

chirp rate2

chirp rate3

θ4 (rad.)

ρ4 (samp.)

chirp rate4

44.92

-25206

5.502

139.16

24772

0.007

1.28

356.8

0.0069

2.68

341.79

800.3

.00005

1.65

1.27e5

.00005

7.16

1.17e5

N/A

25000

3.927

N/A

-25000

5.497

N/A

25000

0.0003

N/A

16.3

0.004

N/A

-206

0.005

N/A

228

0.01%

N/A

0.06%

0.10%

N/A

0.82%

0.09%

N/A

0.91%

% detection = 100%

239

θ3 (rad.)

ρ3 (samp.)

Test 59

Figure 99: Test 59 Plot – HT of RSPWVD, 512, 0

Table 63: Test 59 Metrics – HT of RSPWVD, 512, 0 TEST 59 HT of RSPWVD TriModFMCW 2 Tri fs=4KHz fc=1KHz modBW=500 modperiod=20ms #samples=512 SNR=0dB (25 runs)

Plot Time (sec)

θ1 (rad.)

ρ1 (samp.)

chirp rate1

mean

39.86

0.7846

136.62

std dev

0.878

0.0013

var.

0.771

actual

θ2 (rad.)

ρ2 (samp.)

chirp rate2

θ3 (rad.)

ρ3 (samp.)

chirp rate3

θ4 (rad.)

ρ4 (samp.)

chirp rate4

-24964

2.36

43.77

24813

3.923

42.51

-24821

5.498

140.72

24973

3.24

66.97

0.014

2.22

690.5

0.013

2.50

703.7

0.012

5.59

598.5

0

10.48

4485.3

0.0002

4.95

4.77e5

0.0002

6.27

4.95e5

0.0001

31.23

3.58e5

N/A

0.7853

N/A

-25000

2.356

N/A

25000

3.927

N/A

-25000

5.497

N/A

25000

error

N/A

0.0007

N/A

-36

0.004

N/A

187

0.004

N/A

-179

0.001

N/A

% error

N/A

0.09%

N/A

0.14%

0.17%

N/A

0.75%

0.10%

N/A

0.72%

0.02%

N/A

% detection = 100%

240

27 0.11%

Test 60

Figure 100: Test 60 Plot – HT of RSPWVD, 512, -5

Table 64: Test 60 Metrics – HT of RSPWVD, 512, -5 TEST 60 HT of RSPWVD TriModFMCW 2 Tri fs=4KHz fc=1KHz modBW=500 modperiod=20ms #samples=512 SNR=low detect (-5dB) (25 runs)

Plot Time min:sec

θ1 (rad.)

ρ1 (samp.)

chirp rate1

θ2 (rad.)

ρ2 (samp.)

mean

2:18.36

0.7854

128.74

-25009

2.360

std dev

20.69

0.0123

5.91

615.52

var.

428.2

0.0002

34.91

actual

N/A

0.7853

error

N/A

% error

N/A

chirp rate2

θ3 (rad.)

ρ3 (samp.)

chirp rate3

θ4 (rad.)

ρ4 (samp.)

chirp rate4

44.21

24808

3.935

42.11

-25530

5.498

138.14

25220

0.025

5.38

1274.0

0.048

6.76

2516.2

0.086

10.33

4147.4

3.78e5

.0006

29.0

1.62e6

0.0023

45.67

6.33e6

0.007

106.77

1.72e7

N/A

-25000

2.356

N/A

25000

3.927

N/A

-25000

5.497

N/A

25000

0.001

N/A

-9

0.004

N/A

192

0.008

N/A

-530

0.001

N/A

220

0.13%

N/A

0.04%

0.16%

N/A

0.77%

0.20%

N/A

2.12%

0.02%

N/A

0.88%

% detection = 16%

241

Test 61

Figure 101: Test 61 Plot – WVD, 512, 10

Table 65: Test 61 Metrics – WVD, 512, 10 TEST 61 WVD TriModFMCW 2Tri fs=6KHz fc=1.5KHz modBW=2400 modperiod=15ms #samples=512 SNR=10dB (25 runs)

Plot Time h:m:s

Carrier Freq (Hz)

Mod BW (Hz)

mean std. dev. variance actual error % error

1:11:53 3.51 12.31 N/A N/A N/A

1510.56 92.49 8554.25 1500 10.56 0.70%

2425.84 51.00 2601.71 2400 25.84 1.08%

Mod Period (sec)

Time (sec)-Frequency (Hz) Localization

.01495 .00032 .00013 .0001 0 0 .015 N/A 0.00005 N/A 0.33% (0.53% of X-Axis) % detection = 100% # x-term false positives=2 % false positive=100%

242

8.04 3.08 9.47 N/A N/A (0.27% of Y-Axis)

Chirp Rate (Hz/sec) 162234 3344.38 1.12e7 160000 2234 1.39%

Test 62 Did not run Test 62 due to inadequate computational resources (1 run was still processing after 8 hours)

Test 63 Did not run Test 63 due to inadequate computational resources (1 run was still processing after 8 hours)

243

Test 64

Figure 102: Test 64 Plot – CWD, 512, 10

Table 66: Test 64 Metrics – CWD, 512, 10 TEST 64 CWD TriModFMCW 2Tri fs=6KHz fc=1.5KHz modBW=2400 modperiod=15ms #samples=512 SNR=10dB (25 runs)

Plot Time (sec)

Carrier Freq (Hz)

Mod BW (Hz)

mean std. dev. variance actual error % error

2.99 0.186 0.034 N/A N/A N/A

1362.8 312.73 97797 1500 137.2 9.15%

2467.9 21.24 451.23 2400 67.9 2.83%

Mod Period (sec)

Time (sec)-Frequency (Hz) Localization

.01502 .00105 .00015 .00008 0 0 .015 N/A 0.00002 N/A 0.13% (1.75% of X-Axis) % detection = 100%

244

28.12 7.44 55.29 N/A N/A (0.94% of Y-Axis)

Chirp Rate (Hz/sec) 164283 2150.1 4.62e6 160000 4283 2.68%

Test 65

Figure 103: Test 65 Plot – CWD, 512, 0

Table 67: Test 65 Metrics – CWD, 512, 0 TEST 65 CWD TriModFMCW 2Tri fs=6KHz fc=1.5KHz modBW=2400 modperiod=15ms #samples=512 SNR=0dB (25 runs)

Plot Time (sec)

Carrier Freq (Hz)

Mod BW (Hz)

mean std. dev. variance actual error % error

11.18 1.75 1.38 N/A N/A N/A

1483.68 320.27 1.03e5 1500 16.32 1.09%

2434.02 81.92 6711.16 2400 34.02 1.42%

Mod Period (sec)

Time (sec)-Frequency (Hz) Localization

.01502 .00105 .00031 .00039 0 0 .015 N/A 0.00002 N/A 0.13% (1.75% of X-Axis) % detection = 92%

245

30.48 8.92 79.63 N/A N/A (1.02% of Y-Axis)

Chirp Rate (Hz/sec) 162073 6365.85 4.05e7 160000 2073 1.29%

Test 66

Figure 104: Test 66 Plot – CWD, 512, -3

Table 68: Test 66 Metrics – CWD, 512, -3 TEST 66 CWD TriModFMCW 2Tri fs=6KHz fc=1.5KHz modBW=2400 modperiod=15ms #samples=512 SNR=low detect (-3dB) (25 runs)

Plot Time (sec)

Carrier Freq (Hz)

Mod BW (Hz)

mean std. dev. variance actual error % error

27.19 2.17 4.70 N/A N/A N/A

1359.4 439.81 1.93e5 1500 140.6 9.37%

2523.0 85.88 7376.2 2400 123.0 5.13%

Mod Period (sec)

Time (sec)-Frequency (Hz) Localization

.01514 .00105 .00033 .00038 0 0 .015 N/A 0.00014 N/A 0.93% (1.75% of X-Axis) % detection = 32%

246

46.8 11.22 126 N/A N/A (1.56% of Y-Axis)

Chirp Rate (Hz/sec) 166667 5446.6 2.97e7 160000 6667 4.17%

Test 67

Figure 105: Test 67 Plot – Spectrogram, 512, 10

Table 69: Test 67 Metrics – Spectrogram, 512, 10 TEST 67 Spectrogram TriModFMCW 2Tri fs=6KHz fc=1.5KHz modBW=2400 modperiod=15ms #samples=512 SNR=10dB (25 runs)

Plot Time (sec)

Carrier Freq (Hz)

Mod BW (Hz)

mean std. dev. variance actual error % error

2.14 0.215 0.046 N/A N/A N/A

1272.7 423.1 1.79e5 1500 227.3 15.2%

2573.2 6.38 40.74 2400 173.2 7.22%

Mod Period (sec)

Time (sec)-Frequency (Hz) Localization

.01514 .00162 .0001 .0001 0 0 .015 N/A 0.00014 N/A 0.93% (2.7% of X-Axis) % detection = 100%

247

67.8 12.88 165.8 N/A N/A (2.26% of Y-Axis)

Chirp Rate (Hz/sec) 169960 819.9 6.72e5 160000 9960 6.23%

Test 68

Figure 106: Test 68 Plot – Spectrogram, 512, 0

Table 70: Test 68 Metrics – Spectrogram, 512, 0 TEST 68 Spectrogram TriModFMCW 2Tri fs=6KHz fc=1.5KHz modBW=2400 modperiod=15ms #samples=512 SNR=0dB (25 runs)

Plot Time (sec)

Carrier Freq (Hz)

Mod BW (Hz)

mean std. dev. variance actual error % error

3.60 0.233 0.054 N/A N/A N/A

1387.4 387.16 1.50e5 1500 112.56 7.50%

2519.5 35.92 1289.95 2400 119.5 4.98%

Mod Period (sec)

Time (sec)-Frequency (Hz) Localization

.01502 .00176 .00027 .00048 0 0 .015 N/A 0.00002 N/A 0.13% (2.93% of X-Axis) % detection = 92%

248

96.0 28.09 788.9 N/A N/A (3.20% of Y-Axis)

Chirp Rate (Hz/sec) 167736 3215.0 1.03e7 160000 7736 4.84%

Test 69

Figure 107: Test 69 Plot – Spectrogram, 512, -4

Table 71: Test 69 Metrics – Spectrogram, 512, -4 TEST 69 Spectrogram TriModFMCW 2Tri fs=6KHz fc=1.5KHz modBW=2400 modperiod=15ms #samples=512 SNR=low detect (-4dB) (25 runs)

Plot Time (sec)

Carrier Freq (Hz)

Mod BW (Hz)

mean std. dev. variance actual error % error

13.73 2.02 4.08 N/A N/A N/A

1331.2 232.8 5.42e4 1500 168.8 11.25%

2528.8 47.81 2286.1 2400 128.8 5.37%

Mod Period (sec)

Time (sec)-Frequency (Hz) Localization

.015 .00157 .0003 .00081 0 0 .015 N/A 0.0 N/A 0.0% (2.62% of X-Axis) % detection = 16%

249

117.04 24.94 621.91 N/A N/A (3.90% of Y-Axis)

Chirp Rate (Hz/sec) 168672 5979.98 3.58e7 160000 8672 5.42%

Test 70

Figure 108: Test 70 Plot – Scalogram, 512, 10

Table 72: Test 70 Metrics – Scalogram, 512, 10 TEST 70 Scalogram TriModFMCW 2Tri fs=6KHz fc=1.5KHz modBW=2400 modperiod=15ms #samples=512 SNR=10dB (25 runs)

Plot Time (sec)

Carrier Freq (Hz)

Mod BW (Hz)

mean std. dev. variance actual error % error

5.22 0.098 0.01 N/A N/A N/A

1725 259.3 6.72e4 1500 225 15.0%

2582.9 42.64 1818.2 2400 182.9 7.62%

Mod Period (sec)

Time (sec)-Frequency (Hz) Localization

.01484 .00239 .00023 .0001 0 0 .015 N/A 0.00016 N/A 1.07% (3.98% of X-Axis) % detection = 100%

250

82.4 11.75 138.1 N/A N/A (2.75% of Y-Axis)

Chirp Rate (Hz/sec) 174126 3494.6 1.22e7 160000 14126 8.83%

Test 71

Figure 109: Test 71 Plot – Scalogram, 512, 0

Table 73: Test 71 Metrics – Scalogram, 512, 0 TEST 71 Scalogram TriModFMCW 2Tri fs=6KHz fc=1.5KHz modBW=2400 modperiod=15ms #samples=512 SNR=0dB (25 runs)

Plot Time (sec)

Carrier Freq (Hz)

Mod BW (Hz)

mean std. dev. variance actual error % error

5.92 0.207 0.043 N/A N/A N/A

1584.4 241.16 5.82e4 1500 84.4 5.63%

2601.4 6.22 38.72 2400 201.4 8.39%

Mod Period (sec)

Time (sec)-Frequency (Hz) Localization

.01507 .00272 .00023 .00039 0 0 .015 N/A 0.00007 N/A 0.47% (4.53% of X-Axis) % detection = 72%

251

100.56 26.73 717.35 N/A N/A (3.35% of Y-Axis)

Chirp Rate (Hz/sec) 172652 2878.2 8.28e6 160000 12652 7.91%

Test 72

Figure 110: Test 72 Plot – Scalogram, 512, -3

Table 74: Test 72 Metrics – Scalogram, 512, -3 TEST 72 Scalogram TriModFMCW 2Tri fs=6KHz fc=1.5KHz modBW=2400 modperiod=15ms #samples=512 SNR=low detect (-3dB) (25 runs)

Plot Time (sec)

Carrier Freq (Hz)

Mod BW (Hz)

mean std. dev. variance actual error % error

5.82 0.111 0.012 N/A N/A N/A

1400.36 235.27 5.54e4 1500 99.64 6.64%

2686.1 12.93 167.1 2400 286.1 11.92%

Mod Period (sec)

Time (sec)-Frequency (Hz) Localization

.015 .00178 .00033 .00021 0 0 .015 N/A 0.0 N/A 0.0% (2.97% of X-Axis) % detection = 24%

252

128.8 24.61 605.6 N/A N/A (4.29% of Y-Axis)

Chirp Rate (Hz/sec) 179140 4013.8 1.61e7 160000 19140 11.96%

Test 73

Figure 111: Test 73 Plot – Reassigned Spectrogram, 512, 10

Table 75: Test 73 Metrics – Reassigned Spectrogram, 512, 10 TEST 73 Reassigned Spectrogram TriModFMCW 2Tri fs=6KHz fc=1.5KHz modBW=2400 modperiod=15ms #samples=512 SNR=10dB (25 runs)

Plot Time (sec)

Carrier Freq (Hz)

Mod BW (Hz)

mean std. dev. variance actual error % error

21.57 0.845 0.713 N/A N/A N/A

1603.1 257.1 6.61e4 1500 103.1 6.87%

2325.3 6.83 46.6 2400 74.7 3.11%

Mod Period (sec)

Time (sec)-Frequency (Hz) Localization

.01505 .00047 .0002 .00008 0 0 .015 N/A 0.00005 N/A 0.33% (0.78% of X-Axis) % detection = 100%

253

13.1 2.18 4.75 N/A N/A (0.44% of Y-Axis)

Chirp Rate (Hz/sec) 154542 1920.3 3.69e6 160000 5458 3.41%

Test 74

Figure 112: Test 74 Plot – Reassigned Spectrogram, 512, 0

Table 76: Test 74 Metrics – Reassigned Spectrogram, 512, 0 TEST 74 Reassigned Spectrogram TriModFMCW 2Tri fs=6KHz fc=1.5KHz modBW=2400 modperiod=15ms #samples=512 SNR=0dB (25 runs)

Plot Time (sec)

Carrier Freq (Hz)

Mod BW (Hz)

mean std. dev. variance actual error % error

35.18 1.54 2.37 N/A N/A N/A

1493.0 304.83 9.29e4 1500 7.0 0.47%

2423.4 29.23 854.1 2400 23.4 0.98%

Mod Period (sec)

Time (sec)-Frequency (Hz) Localization

.015 .00063 .00033 .0001 0 0 .015 N/A 0.0 N/A 0.0% (1.05% of X-Axis) % detection = 100%

254

34.64 16.81 282.73 N/A N/A (1.15% of Y-Axis)

Chirp Rate (Hz/sec) 161592 2204.8 4.86e6 160000 1592 1.0%

Test 75

Figure 113: Test 75 Plot – Reassigned Spectrogram, 512, -3

Table 77: Test 75 Metrics – Reassigned Spectrogram, 512, -3 TEST 75 Reassigned Spectrogram TriModFMCW 2Tri fs=6KHz fc=1.5KHz modBW=2400 modper=15ms #samp=512 SNR=low detect (-3dB) (25 runs)

Plot Time min:sec

Carrier Freq (Hz)

Mod BW (Hz)

mean std. dev. variance actual error % error

1:39.2 5.18 26.8 N/A N/A N/A

1481.3 325.9 1.06e5 1500 18.7 1.25%

2477.5 44.34 1965.9 2400 77.5 3.23%

Mod Period (sec)

Time (sec)-Frequency (Hz) Localization

.01481 .00098 .00026 .00031 0 0 .015 N/A 0.00019 N/A 1.27% (1.63% of X-Axis) % detection = 16%

255

44.0 23.36 545.5 N/A N/A (1.47% of Y-Axis)

Chirp Rate (Hz/sec) 167316 5166.0 2.67e7 160000 7316 4.57%

Test 76

Figure 114: Test 76 Plot – Reassigned Scalogram, 512, 10

Table 78: Test 76 Metrics – Reassigned Scalogram, 512, 10 TEST 76 Reassigned Scalogram TriModFMCW 2Tri fs=6KHz fc=1.5KHz modBW=2400 modperiod=15ms #samples=512 SNR=10dB (25 runs)

Plot Time (sec)

Carrier Freq (Hz)

Mod BW (Hz)

mean std. dev. variance actual error % error

10.11 0.34 0.12 N/A N/A N/A

1550.0 386.7 1.46e5 1500 50.0 3.33%

2446.8 20.83 433.86 2400 46.8 1.95%

Mod Period (sec)

Time (sec)-Frequency (Hz) Localization

.01519 .0008 .00026 .0001 0 0 .015 N/A 0.00019 N/A 1.27% (1.33% of X-Axis) % detection = 100%

256

38.5 5.69 32.42 N/A N/A (1.28% of Y-Axis)

Chirp Rate (Hz/sec) 161137 4036.3 1.63e7 160000 1137 0.71%

Test 77

Figure 115: Test 77 Plot – Reassigned Scalogram, 512, 0

Table 79: Test 77 Metrics – Reassigned Scalogram, 512, 0 TEST 77 Reassigned Scalogram TriModFMCW 2Tri fs=6KHz fc=1.5KHz modBW=2400 modperiod=15ms #samples=512 SNR=0dB (25 runs)

Plot Time (sec)

Carrier Freq (Hz)

Mod BW (Hz)

mean std. dev. variance actual error % error

14.05 0.85 0.72 N/A N/A N/A

1535.2 180.24 3.25e4 1500 35.2 2.35%

2561.8 29.38 863.14 2400 161.8 6.74%

Mod Period (sec)

Time (sec)-Frequency (Hz) Localization

.015 .00136 .00033 .00048 0 0 .015 N/A 0.0 N/A 0.0% (2.27% of X-Axis) % detection = 100%

257

38.5 17.31 299.78 N/A N/A (1.28% of Y-Axis)

Chirp Rate (Hz/sec) 170823 2355.34 5.54e6 160000 10823 6.76%

Test 78

Figure 116: Test 78 Plot – Reassigned Scalogram, 512, -3

Table 80: Test 78 Metrics – Reassigned Scalogram, 512, -3 TEST 78 Reassigned Scalogram TriModFMCW 2Tri fs=6KHz fc=1.5KHz modBW=2400 modper=15ms #samp=512 SNR=low detect (-3dB) (25 runs)

Plot Time (sec)

Carrier Freq (Hz)

Mod BW (Hz)

mean std. dev. variance actual error % error

31.48 1.95 3.81 N/A N/A N/A

1430.1 275.5 7.59e4 1500 69.9 4.66%

2598.4 9.14 38.36 2400 198.4 8.27%

Mod Period (sec)

Time (sec)-Frequency (Hz) Localization

.01481 .00169 .00026 .00056 0 0 .015 N/A 0.00019 N/A 1.27% (2.82% of X-Axis) % detection = 16%

258

38.64 19.25 370.73 N/A N/A (1.29% of Y-Axis)

Chirp Rate (Hz/sec) 175462 3186.8 1.02e7 160000 15462 9.66%

Test 79

Figure 117: Test 79 Plot – RSPWVD, 512, 10

Table 81: Test 79 Metrics – RSPWVD, 512, 10 TEST 79 RSPWVD TriModFMCW 2Tri fs=6KHz fc=1.5KHz modBW=2400 modperiod=15ms #samples=512 SNR=10dB (25 runs)

Plot Time (sec)

Carrier Freq (Hz)

Mod BW (Hz)

mean std. dev. variance actual error % error

20.74 1.21 1.47 N/A N/A N/A

1371.6 202.53 4.10e4 1500 128.4 8.56%

2345.8 12.33 152.11 2400 54.2 2.26%

Mod Period (sec)

Time (sec)-Frequency (Hz) Localization

.01491 .0002 .00013 .00003 0 0 .015 N/A 0.00009 N/A 0.6% (0.33% of X-Axis) % detection = 100%

259

8.4 3.29 10.8 N/A N/A (0.28% of Y-Axis)

Chirp Rate (Hz/sec) 157378 1482.2 2.20e6 160000 2622 1.64%

Test 80

Figure 118: Test 80 Plot – RSPWVD, 512, 0

Table 82: Test 80 Metrics – RSPWVD, 512, 0 TEST 80 RSPWVD TriModFMCW 2Tri fs=6KHz fc=1.5KHz modBW=2400 modperiod=15ms #samples=512 SNR=0dB (25 runs)

Plot Time (sec)

Carrier Freq (Hz)

Mod BW (Hz)

mean std. dev. variance actual error % error

27.43 6.97 48.61 N/A N/A N/A

1660.6 287.76 8.28e4 1500 160.6 10.71%

2419.5 26.31 692.47 2400 19.5 0.81%

Mod Period (sec)

Time (sec)-Frequency (Hz) Localization

.015 .00035 .00008 .00009 0 0 .015 N/A 0.0 N/A 0.0% (0.58% of X-Axis) % detection = 100%

260

16.0 4.58 20.94 N/A N/A (0.53% of Y-Axis)

Chirp Rate (Hz/sec) 161305 1961.79 3.85e6 160000 1305 0.82%

Test 81

Figure 119: Test 81 Plot – RSPWVD, 512, -3

Table 83: Test 81 Metrics – RSPWVD, 512, -3 TEST 81 RSPWVD TriModFMCW 2Tri fs=6KHz fc=1.5KHz modBW=2400 modperiod=15ms #samples=512 SNR=-3dB (25 runs)

Plot Time (sec)

Carrier Freq (Hz)

Mod BW (Hz)

mean std. dev. variance actual error % error

58.0 0.97 0.95 N/A N/A N/A

1531.9 291.11 8.47e4 1500 31.9 2.13%

2470.3 12.82 164.27 2400 70.3 2.93%

Mod Period (sec)

Time (sec)-Frequency (Hz) Localization

.01495 .00048 .00006 .0001 0 0 .015 N/A 0.00005 N/A 0.33% (0.80% of X-Axis) % detection = 12%

261

18.48 2.76 7.63 N/A N/A (0.62% of Y-Axis)

Chirp Rate (Hz/sec) 165190 1177.3 1.39e6 160000 5190 3.24%

Test 82

Figure 120: Test 82 Plot – HT of WVD, 512, 10

Table 84: Test 82 Metrics – HT of WVD, 512, 10 TEST 82 HT of WVD TriModFMCW 2 Tri fs=6KHz fc=1.5KHz modBW=2400 modperiod=15ms #samples=512 SNR=10dB (25 runs)

Plot Time h:m

θ1 (rad.)

ρ1 (samp.)

chirp rate1 (neg)

mean

1:12.0

1.276

183.5

164697

1.875

std dev

3.51

0.001

0.495

841.73

var.

12.33

0.0

0.245

actual

N/A

1.268

error

N/A

% error

N/A

θ2 (rad.)

ρ2 (samp.)

chirp rate3 (neg)

chirp rate2

θ3 (rad.)

ρ3 (samp.)

60.22

159017

4.404

60.91

157322

0.0058

0.046

3335.32

0.0022

0.027

7.09e5

0.00003

0.002

1.11e7

0.0

N/A

160000

1.873

N/A

160000

0.008

N/A

4697

0.002

N/A

0.63%

N/A

2.94%

0.11%

N/A

ρ4 (samp.)

chirp rate4

5.012

183.92

161827

1195.68

0.0066

0.383

3754.09

0.0008

1.43e6

0.00004

0.147

1.41e7

4.409

N/A

160000

5.015

N/A

160000

983

0.005

N/A

2678

0.003

N/A

1827

0.61%

0.11%

N/A

1.67%

0.06%

N/A

1.14%

% detection = 100%

262

θ4 (rad.)

Test 83 Did not run Test 83 due to inadequate computational resources (1 run was still processing after 8 hours)

Test 84 Did not run Test 84 due to inadequate computational resources (1 run was still processing after 8 hours)

263

Test 85

Figure 121: Test 85 Plot – HT of CWD, 512, 10

Table 85: Test 85 Metrics – HT of CWD, 512, 10 TEST 85 HT of CWD TriModFMCW 2 Tri fs=6KHz fc=1.5KHz modBW=2400 modperiod=15ms #samples=512 SNR=10dB (25 runs)

Plot Time (sec)

θ1 (rad.)

ρ1 (samp.)

chirp rate1 (neg)

mean

11.99

1.266

182.94

159260

1.868

std dev

0.073

0.005

0.313

3025.85

var.

0.005

0.0

0.098

actual

N/A

1.268

error

N/A

% error

N/A

θ2 (rad.)

ρ2 (samp.)

chirp rate3 (neg)

chirp rate2

θ3 (rad.)

ρ3 (samp.)

60.21

163491

4.411

61.64

160952

5.019

184.1

157506

0.006

0.013

3335.3

0.0066

0.009

3774.9

0.0009

0.313

466.89

9.16e6

0.0

0.0002

1.11e7

0.0

0.0

1.42e7

0.0

0.098

2.18e5

N/A

160000

1.873

N/A

160000

4.409

N/A

160000

5.015

N/A

160000

0.002

N/A

740

0.005

N/A

3491

0.002

N/A

952

0.004

N/A

2494

0.16%

N/A

0.46%

0.27%

N/A

2.18%

0.05%

N/A

0.59%

0.08%

N/A

1.56%

% detection = 100%

264

θ4 (rad.)

ρ4 (samp.)

chirp rate4

Test 86

Figure 122: Test 86 Plot – HT of CWD, 512, 0

Table 86: Test 86 Metrics – HT of CWD, 512, 0 TEST 86 HT of CWD TriModFMCW 2 Tri fs=6KHz fc=1.5KHz modBW=2400 modperiod=15ms #samples=512 SNR=0dB (25 runs)

Plot Time (sec)

θ1 (rad.)

ρ1 (samp.)

chirp rate1 (neg)

mean

28.95

1.266

183.2

159260

1.873

std dev

1.88

0.005

0.626

3025.85

var.

3.53

0.0

0.3922

actual

N/A

1.268

error

N/A

% error

N/A

θ2 (rad.)

ρ2 (samp.)

chirp rate3 (neg)

chirp rate2

θ3 (rad.)

ρ3 (samp.)

60.36

160508

4.413

61.50

162330

5.017

184.1

159086

0.007

0.318

4084.9

0.007

0.318

3774.9

0.005

0.313

3065.2

9.16e6

0.0

0.101

1.67e7

0.0

0.101

1.42e7

0.0

0.098

9.39e6

N/A

160000

1.873

N/A

160000

4.409

N/A

160000

5.015

N/A

160000

0.002

N/A

740

0.0

N/A

508

0.004

N/A

2330

0.002

N/A

914

0.16%

N/A

0.46%

0.0%

N/A

0.32%

0.09%

N/A

1.46%

0.04%

N/A

0.57%

% detection = 100%

265

θ4 (rad.)

ρ4 (samp.)

chirp rate4

Test 87

Figure 123: Test 87 Plot – HT of CWD, 512, -6

Table 87: Test 87 Metrics – HT of CWD, 512, -6 TEST 87 HT of CWD TriModFMCW 2 Tri fs=6KHz fc=1.5KHz modBW=2400 modperiod=15ms #samples=512 SNR=low detect (-6dB) (25 runs)

Plot Time min:sec

θ1 (rad.)

ρ1 (samp.)

chirp rate1 (neg)

mean

1:32.4

1.269

183.5

160613

1.875

θ2 (rad.)

ρ2 (samp.)

chirp rate3 (neg)

chirp rate2

θ3 (rad.)

ρ3 (samp.)

θ4 (rad.)

ρ4 (samp.)

chirp rate4

60.24

159240

4.418

62.98

165316

5.014

184.6

160457

std dev

7.44

0.007

0.86

3705.89

0.01

0.027

5846.23

0.012

2.63

7179.1

0.007

0.383

3754.1

var.

55.37

0.0

0.74

1.37e7

0.0001

0.0008

3.42e7

0.0001

6.90

5.15e7

0.0

0.147

1.41e7

actual

N/A

1.268

N/A

160000

1.873

N/A

160000

4.409

N/A

160000

5.015

N/A

160000

error

N/A

0.001

N/A

613

0.002

N/A

760

0.009

N/A

5316

0.001

N/A

457

% error

N/A

0.08%

N/A

0.38%

0.11%

N/A

0.48%

0.20%

N/A

3.32%

0.02%

N/A

0.29%

% detection = 28%

266

Test 88

Figure 124: Test 88 Plot – HT of RSPWVD, 512, 10

Table 88: Test 88 Metrics – HT of RSPWVD, 512, 10 TEST 88 HT of RSPWVD TriModFMCW 2 Tri fs=6KHz fc=1.5KHz modBW=2400 modperiod=15ms #samples=512 SNR=10dB (25 runs)

Plot Time (sec)

θ1 (rad.)

ρ1 (samp.)

chirp rate1 (neg)

θ2 (rad.)

mean

22.70

1.268

183.1

160178

1.8737

std dev

0.81

0.006

0.33

3934.6

var.

0.65

.00005

0.11

actual

N/A

1.268

error

N/A

0.0

% error

N/A

0.0%

ρ2 (samp.)

chirp rate3 (neg)

chirp rate2

θ3 (rad.)

ρ3 (samp.)

59.92

160011

4.410

61.75

160492

5.015

184.6

159996

0.0075

0.346

4305.8

0.0069

0.22

3979.1

0.007

0.85

4852.7

1.54e7

.00006

0.12

1.85e7

.00005

0.048

1.58e7

.00005

0.72

2.35e7

N/A

160000

1.873

N/A

160000

4.409

N/A

160000

5.015

N/A

160000

N/A

178

0.0007

N/A

11

0.001

N/A

492

0.0

N/A

4

N/A

0.11%

0.04%

N/A

0.007%

0.02%

N/A

0.30%

0.0%

N/A

0.002%

% detection = 100%

267

θ4 (rad.)

ρ4 (samp.)

chirp rate4

Test 89

Figure 125: Test 89 Plot – HT of RSPWVD, 512, 0

Table 89: Test 89 Metrics – HT of RSPWVD, 512, 0 TEST 89 HT of RSPWVD TriModFMCW 2 Tri fs=6KHz fc=1.5KHz modBW=2400 modperiod=15ms #samples=512 SNR=0dB (25 runs)

Plot Time (sec)

θ1 (rad.)

ρ1 (samp.)

chirp rate1 (neg)

mean

28.55

1.264

182.9

158007

1.872

std dev

0.34

0.009

1.05

5362.4

var.

0.12

0.0001

1.10

actual

N/A

1.268

error

N/A

% error

N/A

θ2 (rad.)

ρ2 (samp.)

chirp rate3 (neg)

chirp rate2

θ3 (rad.)

ρ3 (samp.)

60.58

161254

4.409

61.66

159918

5.016

184.2

159429

0.0075

0.71

4305.88

0.006

0.04

3446

0.006

0.57

3427

2.88e7

0.0

0.50

1.85e7

0.0

0.0016

1.19e7

0.0

0.33

1.17e7

N/A

160000

1.873

N/A

160000

4.409

N/A

160000

5.015

N/A

160000

0.004

N/A

1993

0.001

N/A

1254

0.0

N/A

82

0.001

N/A

571

0.32%

N/A

1.25%

0.05%

N/A

0.78%

0.0%

N/A

0.05%

0.02%

N/A

0.36%

% detection = 100%

268

θ4 (rad.)

ρ4 (samp.)

chirp rate4

Test 90

Figure 126: Test 90 Plot – HT of RSPWVD, 512, -6

Table 90: Test 90 Metrics – HT of RSPWVD, 512, -6 TEST 90 HT of RSPWVD TriModFMCW 2 Tri fs=6KHz fc=1.5KHz modBW=2400 modperiod=15ms #samples=512 SNR=low detect (-6dB) (25 runs)

Plot Time min:sec

θ1 (rad.)

ρ1 (samp.)

chirp rate1 (neg)

θ2 (rad.)

mean

1:19.21

1.271

184.2

161813

1.8745

std dev

2.45

0.006

1.28

3410.9

var.

6.01

0.0

1.63

actual

N/A

1.268

error

N/A

% error

N/A

ρ2 (samp.)

chirp rate3 (neg)

chirp rate2

θ3 (rad.)

ρ3 (samp.)

60.39

159676

4.412

61.29

161641

5.019

183.7

157983

0.012

0.36

6647.7

0.007

0.71

3979.1

0.014

1.33

7605.4

1.16e7

0.00014

0.126

4.42e7

0.0

0.50

1.58e7

0.00019

1.76

5.78e7

N/A

160000

1.873

N/A

160000

4.409

N/A

160000

5.015

N/A

160000

0.003

N/A

1813

0.0015

N/A

324

0.003

N/A

1641

0.004

N/A

2017

0.24%

N/A

1.13%

0.08%

N/A

0.20%

0.07%

N/A

1.03%

0.08%

N/A

1.26%

% detection = 28%

269

θ4 (rad.)

ρ4 (samp.)

chirp rate4

Test 91

Figure 127: Test 91 Plot – WVD, 512, 10

Table 91: Test 91 Metrics – WVD, 512, 10 TEST 91 WVD FSK 4-component fs=5KHz fc=1KHz, 1.75KHz, .75KHz, 1.25KHz modBW=1KHz modperiod=25ms #samples=512 SNR=10dB (25 runs)

Plot Time min:sec

fc1 (Hz)

fc2 (Hz)

mean std. dev. variance actual error % error

22:49.0 1.09 1.19 N/A N/A N/A

1000 2.24 5 1000 0.0 0.0%

1750.6 2.30 5.30 1750 0.6 0.03%

fc3 (Hz)

fc4 (Hz)

Mod BW (Hz)

Mod Period (sec)

750.8 1250.4 1032.0 .01934 1.64 1.82 5.71 .00069 2.7 3.3 32.5 0 750 1250 1000 .025 0.8 0.4 32.0 0.00566 0.11% 0.03% 3.20% 22.64% % detection = 100% # x-term false positives=6 % false positive=100% (for each of the 6 false positives)

270

T-F localization 35.6 2.30 5.30 N/A N/A 1.42% of Y-Axis

Test 92 Did not run Test 92 due to inadequate computational resources (1 run was still processing after 8 hours)

Test 93 Did not run Test 93 due to inadequate computational resources (1 run was still processing after 8 hours)

271

Test 94

Figure 128: Test 94 Plot – CWD, 512, 10

Table 92: Test 94 Metrics – CWD, 512, 10 TEST 94 CWD FSK 4-component fs=5KHz fc=1KHz, 1.75KHz, .75KHz, 1.25KHz modBW=1KHz modperiod=25ms #samples=512 SNR=10dB (25 runs)

Plot Time (sec)

fc1 (Hz)

fc2 (Hz)

fc3 (Hz)

fc4 (Hz)

mean std. dev. variance actual error % error

3.26 0.105 0.011 N/A N/A N/A

998.5 4.18 17.5 1000 1.5 0.15%

1748.3 5.0 25.0 1750 2.3 0.13%

748.0 1251.1 2.24 8.22 5.0 67.5 750 1250 2.0 1.1 0.27% 0.08% % detection = 100%

272

Mod BW (Hz)

Mod Period (sec)

T-F localization

1157.44 3.39 11.49 1000 157.44 15.74%

.02288 .00057 0 .025 0.00212 8.48%

140.8 6.29 39.58 N/A N/A 5.63% of Y-Axis

Test 95

Figure 129: Test 95 Plot – CWD, 512, 0

Table 93: Test 95 Metrics – CWD, 512, 0 TEST 95 CWD FSK 4-component fs=5KHz fc=1KHz, 1.75KHz, .75KHz, 1.25KHz modBW=1KHz modperiod=25ms #samples=512 SNR=0dB (25 runs)

Plot Time (sec)

fc1 (Hz)

fc2 (Hz)

fc3 (Hz)

fc4 (Hz)

mean std. dev. variance actual error % error

11.51 0.455 0.209 N/A N/A N/A

997.9 10.01 102.0 1000 2.1 0.21%

1758.3 11.66 135.95 1750 8.3 0.47%

753.5 1244.1 17.2 4.24 295.9 17.95 750 1250 3.5 5.9 0.46% 0.47% % detection = 100%

273

Mod BW (Hz)

Mod Period (sec)

T-F localization

1173.9 21.14 447.04 1000 173.9 17.39%

.02218 .0007 0 .025 0.00282 11.28%

161.44 22.25 495.24 N/A N/A 6.46% of Y-Axis

Test 96

Figure 130: Test 96 Plot – CWD, 512, -2

Table 94: Test 96 Metrics – CWD, 512, -2 TEST 96 CWD FSK 4-component fs=5KHz fc=1KHz, 1.75KHz, .75KHz, 1.25KHz modBW=1KHz modperiod=25ms #samples=512 SNR=low detect (-2dB) (25 runs)

Plot Time (sec)

fc1 (Hz)

fc2 (Hz)

fc3 (Hz)

fc4 (Hz)

mean std. dev. variance actual error % error

17.12 2.08 4.33 N/A N/A N/A

1018.4 13.26 175.93 1000 18.4 1.84%

1745.6 8.38 70.18 1750 4.4 0.25%

724.24 1220.4 13.45 24.77 180.79 613.36 750 1250 25.76 29.6 3.43% 2.37% % detection = 100%

274

Mod BW (Hz)

Mod Period (sec)

T-F localization

1210.7 3.79 14.38 1000 210.7 21.07%

.0191 .00025 0 .025 0.0059 23.6%

179.9 32.43 1051.68 N/A N/A 7.20% of Y-Axis

Test 97

Figure 131: Test 97 Plot – Spectrogram, 512, 10

Table 95: Test 97 Metrics – Spectrogram, 512, 10 TEST 97 Spectrogram FSK 4-component fs=5KHz fc=1KHz, 1.75KHz, .75KHz, 1.25KHz modBW=1KHz modperiod=25ms #samples=512 SNR=10dB (25 runs)

Plot Time (sec)

fc1 (Hz)

fc2 (Hz)

fc3 (Hz)

fc4 (Hz)

mean std. dev. variance actual error % error

1.76 .017 .0003 N/A N/A N/A

1002.7 6.24 38.89 1000 2.74 0.27%

1754.3 8.4 70.57 1750 4.3 0.25%

751.0 1252.34 7.09 5.33 50.37 28.43 750 1250 1.0 2.34 0.13% 0.19% % detection = 100%

275

Mod BW (Hz)

Mod Period (sec)

T-F localization

1236.68 9.21 84.89 1000 236.68 23.67%

.02456 .0003 0 .025 0.00044 1.76%

216.6 16.04 257.18 N/A N/A 8.66% of Y-Axis

Test 98

Figure 132: Test 98 Plot – Spectrogram, 512, 0

Table 96: Test 98 Metrics – Spectrogram, 512, 0 TEST 98 Spectrogram FSK 4-component fs=5KHz fc=1KHz, 1.75KHz, .75KHz, 1.25KHz modBW=1KHz modperiod=25ms #samples=512 SNR=0dB (25 runs)

Plot Time (sec)

fc1 (Hz)

fc2 (Hz)

fc3 (Hz)

fc4 (Hz)

mean std. dev. variance actual error % error

3.36 0.099 0.0097 N/A N/A N/A

998.6 14.45 208.89 1000 1.4 0.14%

1723.6 24.07 579.3 1750 26.4 1.51%

751.5 1242.9 21.99 18.81 483.38 353.93 750 1250 1.5 7.1 0.20% 0.57% % detection = 100%

276

Mod BW (Hz)

Mod Period (sec)

T-F localization

1252.0 23.67 560.25 1000 252.0 25.2%

.02324 .00081 0 .025 0.00176 7.04%

236.3 19.97 398.83 N/A N/A 9.45% of Y-Axis

Test 99

Figure 133: Test 99 Plot – Spectrogram, 512, -3

Table 97: Test 99 Metrics – Spectrogram, 512, -3 TEST 99 Spectrogram FSK 4-component fs=5KHz fc=1KHz, 1.75KHz, .75KHz, 1.25KHz modBW=1KHz modperiod=25ms #samples=512 SNR=low detect (-3dB) (25 runs)

Plot Time (sec)

fc1 (Hz)

fc2 (Hz)

fc3 (Hz)

mean std. dev. variance actual error % error

6.05 0.471 0.222 N/A N/A N/A

1024.74 28.38 805.19 1000 24.74 2.47%

1729.6 5.01 25.09 1750 20.4 1.17%

732.5 30.21 912.88 750 17.5 2.33%

fc4 (Hz) 1273.1 27.35 748.0 1250 23.1 1.85% % detection = 24%

277

Mod BW (Hz)

Mod Period (sec)

T-F localization

1279.4 22.66 513.68 1000 279.4 27.94%

.02194 .00147 0 .025 0.00306 12.24%

259.7 42.46 1802.83 N/A N/A 10.39% of Y-Axis

Test 100

Figure 134: Test 100 Plot – Scalogram, 512, 10

Table 98: Test 100 Metrics – Scalogram, 512, 10 TEST 100 Scalogram FSK 4-component fs=5KHz fc=1KHz, 1.75KHz, .75KHz, 1.25KHz modBW=1KHz modperiod=25ms #samples=512 SNR=10dB (25 runs)

Plot Time (sec)

fc1 (Hz)

fc2 (Hz)

fc3 (Hz)

fc4 (Hz)

mean std. dev. variance actual error % error

5.49 0.045 0.002 N/A N/A N/A

996.9 1.75 3.05 1000 3.1 0.31%

1747.2 4.60 21.2 1750 2.8 0.16%

749.6 1252.4 4.34 4.33 18.8 18.77 750 1250 0.4 2.4 0.05% 0.19% % detection = 100%

278

Mod BW (Hz)

Mod Period (sec)

T-F localization

1177.7 9.75 95.08 1000 177.7 17.77%

.024 .00066 0 .025 0.001 4.0%

170 26.23 688.13 N/A N/A 6.8% of Y-Axis

Test 101

Figure 135: Test 101 Plot – Scalogram, 512, 0

Table 99: Test 101 Metrics – Scalogram, 512, 0 TEST 101 Scalogram FSK 4-component fs=5KHz fc=1KHz, 1.75KHz, .75KHz, 1.25KHz modBW=1KHz modperiod=25ms #samples=512 SNR=0dB (25 runs)

Plot Time (sec)

fc1 (Hz)

fc2 (Hz)

fc3 (Hz)

fc4 (Hz)

mean std. dev. variance actual error % error

5.63 0.067 0.004 N/A N/A N/A

994.9 11.07 122.55 1000 5.1 0.51%

1755.1 17.54 307.0 1750 5.1 0.29%

747.6 1259.7 5.18 25.05 26.8 627.7 750 1250 2.4 9.7 0.32% 0.78% % detection = 100%

279

Mod BW (Hz)

Mod Period (sec)

T-F localization

1194.9 24.92 621.18 1000 194.9 19.49%

.0232 .00067 0 .025 0.0018 7.2%

213 47.03 2211.38 N/A N/A 8.52% of Y-Axis

Test 102

Figure 136: Test 102 Plot – Scalogram, 512, -3

Table 100: Test 102 Metrics – Scalogram, 512, -3 TEST 102 Scalogram FSK 4-component fs=5KHz fc=1KHz, 1.75KHz, .75KHz, 1.25KHz modBW=1KHz modperiod=25ms #samples=512 SNR=low detect (-3dB) (25 runs)

Plot Time (sec)

fc1 (Hz)

fc2 (Hz)

fc3 (Hz)

mean std. dev. variance actual error % error

6.0 0.054 0.0029 N/A N/A N/A

991.4 4.77 22.74 1000 8.6 0.86%

1764.8 21.75 472.97 1750 14.8 0.85%

747.6 10.56 111.43 750 0.4 0.05%

fc4 (Hz) 1252.1 8.38 70.28 1250 2.1 0.17% % detection = 20%

280

Mod BW (Hz)

Mod Period (sec)

T-F localization

1222.7 17.74 314.7 1000 222.7 22.27%

.02282 .00115 0 .025 0.00218 8.72%

269.7 95.29 9079.95 N/A N/A 10.79% of Y-Axis

Test 103

Figure 137: Test 103 Plot – Reassigned Spectrogram, 512, 10

Table 101: Test 103 Metrics – Reassigned Spectrogram, 512, 10 TEST 103 Reassigned Spectrogram FSK 4-component fs=5KHz fc=1KHz, 1.75KHz, .75KHz, 1.25KHz modBW=1KHz modperiod=25ms #samples=512 SNR=10dB (25 runs)

Plot Time (sec)

fc1 (Hz)

fc2 (Hz)

fc3 (Hz)

fc4 (Hz)

mean std. dev. variance actual error % error

39.55 0.538 0.29 N/A N/A N/A

1002.8 4.58 20.96 1000 2.8 0.28%

1752.4 2.17 4.72 1750 2.4 0.14%

747.6 1252.1 5.19 4.46 26.95 19.93 750 1250 2.4 2.1 0.32% 0.17% % detection = 100%

281

Mod BW (Hz)

Mod Period (sec)

T-F localization

1042.7 19.34 374.2 1000 42.7 4.27%

.02266 .00049 0 .025 0.00234 9.36%

40.4 6.32 39.93 N/A N/A 1.62% of Y-Axis

Test 104

Figure 138: Test 104 Plot – Reassigned Spectrogram, 512, 0

Table 102: Test 104 Metrics – Reassigned Spectrogram, 512, 0 TEST 104 Reassigned Spectrogram FSK 4-component fs=5KHz fc=1KHz, 1.75KHz, .75KHz, 1.25KHz modBW=1KHz modperiod=25ms #samples=512 SNR=0dB (25 runs)

Plot Time (sec)

fc1 (Hz)

fc2 (Hz)

fc3 (Hz)

fc4 (Hz)

mean std. dev. variance actual error % error

46.5 1.28 1.64 N/A N/A N/A

998.8 29.54 872.33 1000 1.2 0.12%

1759.0 13.71 187.88 1750 9.0 0.51%

765.4 1238.28 34.53 21.14 1192.18 446.72 750 1250 15.4 11.72 2.05% 0.94% % detection = 100%

282

Mod BW (Hz)

Mod Period (sec)

T-F localization

1109.4 24.86 617.93 1000 109.4 10.94%

.0229 .00087 0 .025 0.0021 8.40%

101.4 36.23 1312.3 N/A N/A 4.06% of Y-Axis

Test 105

Figure 139: Test 105 Plot – Reassigned Spectrogram, 512, -3

Table 103: Test 105 Metrics – Reassigned Spectrogram, 512, -3 TEST 105 Reassigned Spectrogram FSK 4-component fs=5KHz fc=1KHz, 1.75KHz, .75KHz, 1.25KHz modBW=1KHz modperiod=25ms #samples=512 SNR=low detect (-3dB) (25 runs)

Plot Time (min:sec)

fc1 (Hz)

fc2 (Hz)

fc3 (Hz)

fc4 (Hz)

mean std. dev. variance actual error % error

1:19.9 8.94 79.92 N/A N/A N/A

1004.7 28.63 819.83 1000 4.7 0.47%

1739.5 30.16 909.38 1750 10.5 0.6%

765.7 1234.3 31.63 11.16 1000.45 124.58 750 1250 15.7 15.7 2.09% 1.26% % detection = 28%

283

Mod BW (Hz)

Mod Period (sec)

T-F localization

1172.6 25.4 645.3 1000 172.6 17.26%

.02246 .001 0 .025 0.00254 10.16%

161.7 29.22 853.81 N/A N/A 6.47% of Y-Axis

Test 106

Figure 140: Test 106 Plot – Reassigned Scalogram, 512, 10

Table 104: Test 106 Metrics – Reassigned Scalogram, 512, 10 TEST 106 Reassigned Scalogram FSK 4-component fs=5KHz fc=1KHz, 1.75KHz, .75KHz, 1.25KHz modBW=1KHz modperiod=25ms #samples=512 SNR=10dB (25 runs)

Plot Time (sec)

fc1 (Hz)

fc2 (Hz)

fc3 (Hz)

fc4 (Hz)

mean std. dev. variance actual error % error

6.64 0.07 0.005 N/A N/A N/A

998.1 2.83 8.0 1000 1.9 0.19%

1745.4 10.08 101.68 1750 4.6 0.26%

750.8 1254.1 1.8 5.46 3.24 29.86 750 1250 0.8 4.1 0.11% 0.33% % detection = 100%

284

Mod BW (Hz)

Mod Period (sec)

T-F localization

1046.8 10.98 120.5 1000 46.8 4.68%

.02344 .00105 0 .025 0.00156 6.24%

46.9 10.59 112.18 N/A N/A 1.88% of Y-Axis

Test 107

Figure 141: Test 107 Plot – Reassigned Scalogram, 512, 0

Table 105: Test 107 Metrics – Reassigned Scalogram, 512, 0 TEST 107 Reassigned Scalogram FSK 4-component fs=5KHz fc=1KHz, 1.75KHz, .75KHz, 1.25KHz modBW=1KHz modperiod=25ms #samples=512 SNR=0dB (25 runs)

Plot Time (sec)

fc1 (Hz)

fc2 (Hz)

fc3 (Hz)

fc4 (Hz)

mean std. dev. variance actual error % error

9.56 0.853 0.727 N/A N/A N/A

989.5 14.47 209.26 1000 10.46 1.05%

1748.1 12.05 145.08 1750 1.9 0.11%

757.4 1243.1 9.07 8.17 82.18 66.71 750 1250 7.4 6.9 0.99% 0.55% % detection = 100%

285

Mod BW (Hz)

Mod Period (sec)

T-F localization

1068.5 11.07 122.63 1000 68.5 6.85%

.02324 .00122 0 .025 0.00176 7.04%

54.7 20.36 414.7 N/A N/A 2.19% of Y-Axis

Test 108

Figure 142: Test 108 Plot – Reassigned Scalogram, 512, -4

Table 106: Test 108 Metrics – Reassigned Scalogram, 512, -4 TEST 108 Reassigned Scalogram FSK 4-component fs=5KHz fc=1KHz, 1.75KHz, .75KHz, 1.25KHz modBW=1KHz modperiod=25ms #samples=512 SNR=low detect (-4dB) (25 runs)

Plot Time (sec)

fc1 (Hz)

fc2 (Hz)

fc3 (Hz)

fc4 (Hz)

mean std. dev. variance actual error % error

22.73 2.59 6.73 N/A N/A N/A

1000.8 10.85 117.83 1000 0.8 0.08%

1740.72 17.58 309.2 1750 9.28 0.53%

749.6 4.35 18.88 750 0.4 0.05%

1249.8 30.38 922.83 1250 0.2 0.02% % detection = 12%

286

Mod BW (Hz)

Mod Period (sec)

T-F localization

1095.7 12.62 159.2 1000 95.7 9.57%

.0252 .0027 0 .025 0.0002 0.80%

62.4 14.78 218.68 N/A N/A 2.50% of Y-Axis

Test 109

Figure 143: Test 109 Plot – RSPWVD, 512, 10

Table 107: Test 109 Metrics – RSPWVD, 512, 10 TEST 109 RSPWVD FSK 4-component fs=5KHz fc=1KHz, 1.75KHz, .75KHz, 1.25KHz modBW=1KHz modperiod=25ms #samples=512 SNR=10dB (25 runs)

Plot Time (sec)

fc1 (Hz)

fc2 (Hz)

fc3 (Hz)

fc4 (Hz)

mean std. dev. variance actual error % error

23.57 0.927 0.859 N/A N/A N/A

1000.1 1.60 2.57 1000 0.1 0.01%

1749.6 4.11 16.87 1750 0.4 0.02%

749.9 1245.8 2.16 10.71 4.67 114.6 750 1250 0.1 4.2 0.01% 0.34% % detection = 100%

287

Mod BW (Hz)

Mod Period (sec)

T-F localization

1029.3 11.71 137.08 1000 29.3 2.93%

.0243 .00061 0 .025 0.0007 2.80%

14.8 5.85 34.2 N/A N/A 0.59% of Y-Axis

Test 110

Figure 144: Test 110 Plot – RSPWVD, 512, 0

Table 108: Test 110 Metrics – RSPWVD, 512, 0 TEST 110 RSPWVD FSK 4-component fs=5KHz fc=1KHz, 1.75KHz, .75KHz, 1.25KHz modBW=1KHz modperiod=25ms #samples=512 SNR=0dB (25 runs)

Plot Time (sec)

fc1 (Hz)

fc2 (Hz)

fc3 (Hz)

fc4 (Hz)

mean std. dev. variance actual error % error

27.44 4.56 20.8 N/A N/A N/A

1002.2 22.91 524.5 1000 2.2 0.22%

1755.8 28.86 832.73 1750 5.8 0.33%

744.1 1248.75 8.87 6.29 78.73 39.58 750 1250 5.9 1.25 0.79% 0.10% % detection = 100%

288

Mod BW (Hz)

Mod Period (sec)

T-F localization

1055.8 22.06 486.56 1000 55.8 5.58%

.02226 .00075 0 .025 0.00274 10.96%

30.1 16.81 282.68 N/A N/A 1.20% of Y-Axis

Test 111

Figure 145: Test 111 Plot – RSPWVD, 512, -3

Table 109: Test 111 Metrics – RSPWVD, 512, -3 TEST 111 RSPWVD FSK 4-component fs=5KHz fc=1KHz, 1.75KHz, .75KHz, 1.25KHz modBW=1KHz modperiod=25ms #samples=512 SNR=low detect (-3dB) (25 runs)

Plot Time (sec)

fc1 (Hz)

fc2 (Hz)

fc3 (Hz)

mean std. dev. variance actual error % error

47.13 9.18 84.28 N/A N/A N/A

1001.9 36.22 1311.73 1000 1.9 0.19%

1751.0 18.01 324.5 1750 1.0 0.06%

747.8 31.59 997.73 750 2.2 0.29%

fc4 (Hz) 1244.1 22.38 500.67 1250 5.9 0.47% % detection = 8%

289

Mod BW (Hz)

Mod Period (sec)

T-F localization

1068.4 7.67 58.8 1000 68.4 6.84%

.0242 .003 0 .025 0.0008 3.20%

43.8 26.72 714.2 N/A N/A 1.75% of Y-Axis

Test 112

Figure 146: Test 112 Plot – HT of WVD, 512, 10

Figure 147: Test 112 Plot – X-Y View – HT of WVD, 512, 10

290

Table 110: Test 112 Metrics – HT of WVD, 512, 10 TEST 112 HT of WVD FSK 4-component fs=5KHz fc=1KHz, 1.75KHz, .75KHz, 1.25KHz modBW=1KHz modper=.025sec #samples=512 SNR=10dB (25 runs)

Plot Time (m:s)

θ1 (rad.)

mean

ρ1 (samp.)

z1 (int.)

θ2 (rad.)

ρ2 (samp.)

z2 (int.)

θ3 (rad.)

ρ3 (samp.)

z3 (int.)

θ4 (rad.)

ρ4 (samp.)

z4 (int.)

22:57

3.1419

51.0

8755.5

0.006

103.4

8133.0

3.143

102.0

7654.2

0.012

0.472

8622

std dev

1.59

0.013

0.714

173.25

0.007

1.404

123.87

0.024

0.984

108.4

0.022

0.512

163.6

var.

2.53

0.0002

0.509

3.0e4

0.0

1.97

1.53e4

0.0006

0.968

1.17e4

0.0005

0.262

2.67e4

actual

N/A

3.142

51.2

N/A

0.0

102.4

N/A

3.142

102.4

N/A

0.0

0.0

N/A

error

N/A

0.0001

0.2

N/A

0.006

1.0

N/A

0.001

0.4

N/A

0.012

0.472

N/A

% error

N/A

0.003%

0.39%

N/A

N/A

0.97%

N/A

0.03%

0.39%

N/A

N/A

N/A

N/A

% detection = 100% # x-term false positives=2

% false positive=100% (for each of the 2 false positives)

291

Test 113

Figure 148: Test 113 Plot – HT of CWD, 512, -3

Figure 149: Test 113 Plot – X-Y View – HT of CWD, 512, -3

292

Table 111: Test 113 Metrics – HT of CWD, 512, -3 TEST 113 HT of CWD FSK 4-component fs=5KHz fc=1KHz, 1.75KHz, .75KHz, 1.25KHz modBW=1KHz modper=.025sec #samples=512 SNR=-3dB (25 runs)

Plot Time (sec)

θ1 (rad.)

ρ1 (samp.)

mean

29.2

3.144

51.0

4195

0.011

std dev

0.142

0.036

0.692

82.7

var.

0.02

0.001

0.478

actual

N/A

3.142

error

N/A

% error

N/A

z1 (int.)

θ2 (rad.)

ρ2 (samp.)

z2 (int.)

θ3 (rad.)

ρ3 (samp.)

z3 (int.)

θ4 (rad.)

102.7

3433.2

3.1434

101.3

3735

0.0122

0.393

3374

0.019

1.288

72.9

0.025

1.237

81.3

0.023

0.422

92.4

6839.3

0.0004

1.659

5314.4

0.0006

1.53

6609.7

0.0005

0.178

8537.7

51.2

N/A

0.0

102.4

N/A

3.142

102.4

N/A

0.0

0.0

N/A

0.002

0.2

N/A

0.011

0.3

N/A

0.0014

1.1

N/A

0.0122

0.393

N/A

0.06%

0.39%

N/A

N/A

0.29%

N/A

0.04%

1.07%

N/A

N/A

N/A

N/A

% detection = 32% # x-term false positives=2

% false positive=100% (for each of the 2 false positives)

293

ρ4 (samp.)

z4 (int.)

Test 114

Figure 150: Test 114 Plot – HT of RSPWVD, 512, -4

Figure 151: Test 114 Plot – X-Y View – HT of RSPWVD, 512, -4

294

Table 112: Test 114 Metrics – HT of RSPWVD, 512, -4 TEST 114 HT of RSPWVD FSK 4-component fs=5KHz fc=1KHz, 1.75KHz, .75KHz, 1.25KHz modBW=1KHz modper=.025sec #samples=512 SNR=-4dB (25 runs)

Plot Time (sec)

θ1 (rad.)

ρ1 (samp.)

mean

55.1

3.1425

50.5

5228.5

0.004

std dev

0.24

0.076

0.67

154.7

var.

0.058

0.006

0.449

actual

N/A

3.142

error

N/A

% error

N/A

z1 (int.)

θ2 (rad.)

ρ2 (samp.)

z2 (int.)

θ3 (rad.)

ρ3 (samp.)

z3 (int.)

θ4 (rad.)

102.7

5187.0

3.166

103.1

5781

0.01

0.44

5192

0.008

1.08

132.3

0.023

1.03

101.9

0.02

0.57

123.1

2.39e4

0.0

1.17

1.75e4

0.0005

1.06

1.03e4

0.0004

0.325

1.52e4

51.2

N/A

0.0

102.4

N/A

3.142

102.4

N/A

0.0

0.0

N/A

0.0005

0.7

N/A

0.004

0.3

N/A

0.024

0.7

N/A

0.01

0.44

N/A

0.01%

1.37%

N/A

N/A

0.29%

N/A

0.76%

0.68%

N/A

N/A

N/A

N/A

% detection = 28% # x-term false positives=2

% false positive=100% (for each of the 2 false positives)

295

ρ4 (samp.)

z4 (int.)

Test 115

Figure 152: Test 115 Plot – WVD, 512, 10

Table 113: Test 115 Metrics – WVD, 512, 10 TEST 115 WVD FSK 8-component fs=5KHz fc=1.5KHz, 1KHz, 1.25KHz, 1.5KHz, 1.75KHz, 1.25KHz, .75KHz, 1KHz modBW=1KHz modperiod=12.5ms #samples=512 SNR=10dB 25 runs

Plot Time (min: sec)

fc1 (Hz)

fc2 (Hz)

fc3 (Hz)

fc4 (Hz)

fc5 (Hz)

fc6 (Hz)

fc7 (Hz)

fc8 (Hz)

Mod BW (Hz)

Mod Period (sec)

T-F localization

mean std dev var. actual error % error

16:24 0.452 0.205 N/A N/A N/A

1500.0 4.18 17.5 1500 0.0 0.0%

1000.0 2.24 5.0 1000 0.0 0.0%

1251.1 6.52 42.47 1250 1.1 .09%

1498.9 6.02 36.19 1500 1.1 0.07%

1749.1 7.23 52.3 1750 0.9 0.05%

1254.6 5.77 33.3 1250 0.4 0.03%

750.8 6.62 43.82 750 0.8 0.1%

999.8 3.19 10.17 1000 0.2 0.02%

1061.4 6.56 43.05 1000 61.4 6.14%

.00913 .00095 0 .0125 0.003 26.96%

55.66 7.39 54.75 N/A N/A 2.27% of Y-Axis

# x-term false positives=9

% detection = 100% % false positive=100% (for each of the 9 false positives)

296

Test 116 Did not run Test 116 due to inadequate computational resources (1 run was still processing after 8 hours)

Test 117 Did not run Test 117 due to inadequate computational resources (1 run was still processing after 8 hours)

297

Test 118

Figure 153: Test 118 Plot – CWD, 512, 10

Table 114: Test 118 Metrics – CWD, 512, 10 TEST 118 CWD FSK 8-component fs=5KHz fc=1.5KHz, 1KHz, 1.25KHz, 1.5KHz, 1.75KHz, 1.25KHz, .75KHz, 1KHz modBW=1KHz modperiod=12.5ms #samples=512 SNR=10dB 25 runs

Plot Time (sec)

fc1 (Hz)

fc2 (Hz)

fc3 (Hz)

fc4 (Hz)

fc5 (Hz)

fc6 (Hz)

fc7 (Hz)

fc8 (Hz)

Mod BW (Hz)

Mod Period (sec)

T-F localization

mean std dev var. actual error % error

3.15 0.054 0.003 N/A N/A N/A

1504.0 5.77 33.33 1500 4.0 0.27%

997.9 7.47 55.73 1000 2.1 0.21%

1250.1 4.08 16.67 1250 0.1 .01%

1500.9 5.43 29.49 1500 0.9 0.06%

1750.5 6.46 41.78 1750 0.5 0.03%

1246.8 2.36 5.59 1250 3.2 0.26%

747.3 2.49 6.30 750 2.7 0.36%

1002.5 4.38 19.16 1000 2.5 0.25%

1163.3 8.54 72.92 1000 163.3 16.3%

.011 .00053 0 .0125 0.0015 12.0%

163.5 9.23 85.17 N/A N/A 6.54% of Y-Axis

% detection = 100%

298

Test 119

Figure 154: Test 119 Plot – CWD, 512, 0

Table 115: Test 119 Metrics – CWD, 512, 0 TEST 119 CWD FSK 8-component fs=5KHz fc=1.5KHz, 1KHz, 1.25KHz, 1.5KHz, 1.75KHz, 1.25KHz, .75KHz, 1KHz modBW=1KHz modperiod=12.5ms #samples=512 SNR=0dB 25 runs

Plot Time (sec)

fc1 (Hz)

fc2 (Hz)

fc3 (Hz)

fc4 (Hz)

fc5 (Hz)

fc6 (Hz)

fc7 (Hz)

fc8 (Hz)

Mod BW (Hz)

Mod Period (sec)

T-F localization

mean std dev var. actual error % error

11.49 0.567 0.321 N/A N/A N/A

1500.3 12.77 163.08 1500 0.3 0.02%

1003.4 15.09 227.73 1000 3.4 0.34%

1247.6 9.36 87.56 1250 0.4 .03%

1509.1 14.28 203.9 1500 9.1 0.61%

1744.4 11.59 134.23 1750 5.6 0.32%

1250.0 14.34 205.5 1250 0.0 0.0%

743.4 24.33 586.4 750 6.6 0.88%

1004.3 7.38 54.56 1000 4.3 0.43%

1175.6 9.00 81.06 1000 175.6 17.6%

.0102 .00147 0 .0125 0.0023 18.4%

171.0 15.92 253.5 N/A N/A 6.84% of Y-Axis

% detection = 100%

299

Test 120

Figure 155: Test 120 Plot – CWD, 512, -1

Table 116: Test 120 Metrics – CWD, 512, -1 TEST 120 CWD FSK 8-component fs=5KHz fc=1.5KHz, 1KHz, 1.25KHz, 1.5KHz, 1.75KHz, 1.25KHz, .75KHz, 1KHz modBW=1KHz modperiod=12.5ms #samples=512 SNR=low SNR (-1dB) 25 runs

Plot Time (sec)

fc1 (Hz)

fc2 (Hz)

fc3 (Hz)

fc4 (Hz)

fc5 (Hz)

fc6 (Hz)

fc7 (Hz)

fc8 (Hz)

Mod BW (Hz)

Mod Period (sec)

T-F localization

mean std dev var. actual error % error

14.43 0.557 0.311 N/A N/A N/A

1496.8 14.5 210.25 1500 3.2 0.21%

991.6 10.58 111.91 1000 8.4 0.84%

1247.6 16.13 260.21 1250 2.4 .19%

1504.4 20.24 409.9 1500 4.4 0.29%

1736.0 25.28 638.83 1750 14.0 0.80%

1236.7 10.42 108.61 1250 13.3 1.06%

764.3 24.5 600.25 750 14.3 1.91%

1002.3 29.73 884.06 1000 2.3 0.23%

1194.4 22.59 510.21 1000 194.4 19.4%

.00893 .002 0 .0125 0.00357 28.56%

200.13 23.75 563.89 N/A N/A 8.01% of Y-Axis

% detection = 32%

300

Test 121

Figure 156: Test 121 Plot – Spectrogram, 512, 10

Table 117: Test 121 Metrics – Spectrogram, 512, 10 TEST 121 Spectrogram FSK 8-component fs=5KHz fc=1.5KHz, 1KHz, 1.25KHz, 1.5KHz, 1.75KHz, 1.25KHz, .75KHz, 1KHz modBW=1KHz modperiod=12.5ms #samples=512 SNR=10dB 25 runs

Plot Time (sec)

fc1 (Hz)

fc2 (Hz)

fc3 (Hz)

fc4 (Hz)

fc5 (Hz)

fc6 (Hz)

fc7 (Hz)

fc8 (Hz)

Mod BW (Hz)

Mod Period (sec)

T-F localization

mean std dev var. actual error % error

2.05 0.056 0.003 N/A N/A N/A

1499.2 5.76 33.17 1500 0.8 0.05%

1000.9 5.83 34.01 1000 0.9 0.09%

1252.8 5.75 33.06 1250 2.8 .22%

1499.4 9.74 94.8 1500 0.6 0.04%

1750.5 5.04 25.35 1750 0.5 0.03%

1246.1 5.91 34.95 1250 3.9 0.31%

749.4 9.36 87.56 750 0.6 0.08%

999.5 5.98 35.78 1000 0.5 0.05%

1242.6 9.36 87.6 1000 242.6 24.3%

.0113 .0005 0 .0125 0.0012 9.6%

232.2 5.01 25.1 N/A N/A 9.29% of Y-Axis

% detection = 100%

301

Test 122

Figure 157: Test 122 Plot – Spectrogram, 512, 0

Table 118: Test 122 Metrics – Spectrogram, 512, 0 TEST 122 Spectrogram FSK 8-component fs=5KHz fc=1.5KHz, 1KHz, 1.25KHz, 1.5KHz, 1.75KHz, 1.25KHz, .75KHz, 1KHz modBW=1KHz modperiod=12.5ms #samples=512 SNR=0dB 25 runs

Plot Time (sec)

fc1 (Hz)

fc2 (Hz)

fc3 (Hz)

fc4 (Hz)

fc5 (Hz)

fc6 (Hz)

fc7 (Hz)

fc8 (Hz)

Mod BW (Hz)

Mod Period (sec)

T-F localization

mean std dev var. actual error % error

3.49 0.122 0.015 N/A N/A N/A

1491.3 26.9 723.75 1500 8.7 0.58%

995.6 13.66 186.56 1000 4.4 0.44%

1252.4 12.35 152.56 1250 2.4 .19%

1515.1 15.73 247.39 1500 15.1 1.01%

1753.2 12.52 156.83 1750 3.2 0.18%

1259.4 13.92 193.79 1250 9.4 0.74%

759.3 9.31 86.75 750 9.3 1.24%

1002.9 8.77 76.96 1000 2.9 0.29%

1253.1 9.44 89.21 1000 253.1 25.3%

.0108 .00126 0 .0125 0.0017 13.6%

249.1 12.51 156.6 N/A N/A 9.96% of Y-Axis

% detection = 100%

302

Test 123

Figure 158: Test 123 Plot – Spectrogram, 512, -1

Table 119: Test 123 Metrics – Spectrogram, 512, -1 TEST 123 Spectrogram FSK 8-component fs=5KHz fc=1.5KHz, 1KHz, 1.25KHz, 1.5KHz, 1.75KHz, 1.25KHz, .75KHz, 1KHz modBW=1KHz modperiod=12.5ms #samples=512 SNR=low SNR (-1dB) 25 runs

Plot Time (sec)

fc1 (Hz)

fc2 (Hz)

fc3 (Hz)

fc4 (Hz)

fc5 (Hz)

fc6 (Hz)

fc7 (Hz)

fc8 (Hz)

Mod BW (Hz)

Mod Period (sec)

T-F localization

mean std dev var. actual error % error

3.85 0.064 0.004 N/A N/A N/A

1493.1 17.16 294.56 1500 6.9 0.46%

1004.5 2.97 8.79 1000 4.5 0.45%

1261.1 13.97 195.2 1250 11.1 .89%

1513.3 11.76 138.24 1500 13.3 0.88%

1746.1 16.74 280.23 1750 3.9 0.23%

1250.3 1.19 1.43 1250 0.3 0.02%

742.1 26.46 700.39 750 7.9 1.05%

997.3 8.09 65.46 1000 2.7 0.27%

1274.6 18.67 348.56 1000 274.6 27.5%

.0095 .00129 0 .0125 0.003 24.0%

256.3 29.22 853.75 N/A N/A 10.25% of Y-Axis

% detection = 36%

303

Test 124

Figure 159: Test 124 Plot – Scalogram, 512, 10

Table 120: Test 124 Metrics – Scalogram, 512, 10 TEST 124 Scalogram FSK 8-component fs=5KHz fc=1.5KHz, 1KHz, 1.25KHz, 1.5KHz, 1.75KHz, 1.25KHz, .75KHz, 1KHz modBW=1KHz modperiod=12.5ms #samples=512 SNR=10dB 25 runs

Plot Time (sec)

fc1 (Hz)

fc2 (Hz)

fc3 (Hz)

fc4 (Hz)

fc5 (Hz)

fc6 (Hz)

fc7 (Hz)

fc8 (Hz)

Mod BW (Hz)

Mod Period (sec)

T-F localization

mean std dev var. actual error % error

4.95 0.083 0.007 N/A N/A N/A

1498.0 5.88 34.66 1500 2.0 0.13%

998.5 5.10 26.01 1000 1.5 0.15%

1245.2 5.67 32.14 1250 4.80 .38%

1500.1 7.75 60.06 1500 0.1 0.01%

1740.8 4.75 22.56 1750 9.2 0.52%

1245.5 5.26 27.7 1250 4.5 0.36%

749.5 5.20 27.0 750 0.5 0.07%

999.2 7.36 54.29 1000 0.8 0.08%

1213.5 9.57 91.7 1000 213.5 21.4%

.0115 .00058 0 .0125 0.001 8.0%

219.63 18.41 339.06 N/A N/A 8.78% of Y-Axis

% detection = 100%

304

Test 125

Figure 160: Test 125 Plot – Scalogram, 512, 0

Table 121: Test 125 Metrics – Scalogram, 512, 0 TEST 125 Scalogram FSK 8-component fs=5KHz fc=1.5KHz, 1KHz, 1.25KHz, 1.5KHz, 1.75KHz, 1.25KHz, .75KHz, 1KHz modBW=1KHz modperiod=12.5ms #samples=512 SNR=0dB 25 runs

Plot Time (sec)

fc1 (Hz)

fc2 (Hz)

fc3 (Hz)

fc4 (Hz)

fc5 (Hz)

fc6 (Hz)

fc7 (Hz)

fc8 (Hz)

Mod BW (Hz)

Mod Period (sec)

T-F localization

mean std dev var. actual error % error

5.64 0.104 0.011 N/A N/A N/A

1505.8 5.17 26.73 1500 5.8 0.38%

1001.1 16.94 287.1 1000 1.1 0.11%

1243.3 15.08 227.42 1250 6.70 .54%

1494.3 12.68 160.75 1500 5.7 0.38%

1743.3 23.3 543.6 1750 6.7 0.38%

1245.6 11.97 143.23 1250 4.4 0.35%

747.0 12.52 156.83 750 3.0 0.40%

1001.2 14.21 202.1 1000 1.2 0.12%

1220.8 7.96 63.39 1000 220.8 22.1%

.0106 .00043 0 .0125 0.0019 15.2%

251.38 38.74 1501.39 N/A N/A 10.06% of Y-Axis

% detection = 100%

305

Test 126

Figure 161: Test 126 Plot – Scalogram, 512, -3

Table 122: Test 126 Metrics – Scalogram, 512, -3 TEST 126 Scalogram FSK 8-component fs=5KHz fc=1.5KHz, 1KHz, 1.25KHz, 1.5KHz, 1.75KHz, 1.25KHz, .75KHz, 1KHz modBW=1KHz modperiod=12.5ms #samples=512 SNR=-3dB 25 runs

Plot Time (sec)

fc1 (Hz)

fc2 (Hz)

fc3 (Hz)

fc4 (Hz)

fc5 (Hz)

fc6 (Hz)

fc7 (Hz)

fc8 (Hz)

Mod BW (Hz)

Mod Period (sec)

T-F localization

mean std dev var. actual error % error

5.99 0.329 0.108 N/A N/A N/A

1521.2 12.19 148.5 1500 21.2 1.41%

1002.8 10.82 117.06 1000 2.8 0.28%

1229.9 11.71 137.06 1250 20.1 1.63%

1510.6 14.02 196.56 1500 10.6 0.71%

1755.7 27.93 779.94 1750 5.7 0.33%

1237.8 17.49 305.73 1250 12.2 0.98%

754.4 4.75 22.65 750 4.4 0.59%

983.3 25.88 669.56 1000 16.7 1.67%

1267.2 4.99 24.9 1000 267.2 26.7%

.0102 .0007 0 .0125 0.0023 18.4%

321.93 17.95 322.19 N/A N/A 12.88% of Y-Axis

% detection = 16%

306

Test 127

Figure 162: Test 127 Plot – Reassigned Spectrogram, 512, 10

Table 123: Test 127 Metrics – Reassigned Spectrogram, 512, 10 TEST 127 Reassigned Spectrogram FSK 8-component fs=5KHz fc=1.5KHz, 1KHz, 1.25KHz, 1.5KHz, 1.75KHz, 1.25KHz, .75KHz, 1KHz modBW=1KHz modperiod=12.5ms #samples=512 SNR=10dB 25 runs

Plot Time (sec)

fc1 (Hz)

fc2 (Hz)

fc3 (Hz)

fc4 (Hz)

fc5 (Hz)

fc6 (Hz)

fc7 (Hz)

fc8 (Hz)

Mod BW (Hz)

Mod Period (sec)

T-F localization

mean std dev var. actual error % error

34.85 0.888 0.789 N/A N/A N/A

1501.5 4.98 24.8 1500 1.5 0.10%

1000.8 5.63 31.72 1000 0.8 0.08%

1247.9 3.97 15.76 1250 2.1 0.16%

1499.1 7.57 57.32 1500 0.9 0.06%

1753.1 5.78 33.41 1750 3.1 0.18%

1248.6 1.78 3.19 1250 1.4 0.11%

749.5 4.95 24.51 750 0.5 0.06%

998.6 6.06 36.67 1000 1.40 0.14%

1057.4 4.76 22.7 1000 57.4 5.74%

.01061 .00021 0 .0125 0.00189 15.12%

51.3 12.11 146.63 N/A N/A 2.05% of Y-Axis

% detection = 100%

307

Test 128

Figure 163: Test 128 Plot – Reassigned Spectrogram, 512, 0

Table 124: Test 128 Metrics – Reassigned Spectrogram, 512, 0 TEST 128 Reassigned Spectrogram FSK 8-component fs=5KHz fc=1.5KHz, 1KHz, 1.25KHz, 1.5KHz, 1.75KHz, 1.25KHz, .75KHz, 1KHz modBW=1KHz modperiod=12.5ms #samples=512 SNR=0dB 25 runs

Plot Time (sec)

fc1 (Hz)

fc2 (Hz)

fc3 (Hz)

fc4 (Hz)

fc5 (Hz)

fc6 (Hz)

fc7 (Hz)

fc8 (Hz)

Mod BW (Hz)

Mod Period (sec)

T-F localization

mean std dev var. actual error % error

46.64 1.77 3.13 N/A N/A N/A

1494.1 21.14 446.89 1500 5.9 0.39%

999.1 11.54 133.1 1000 0.9 0.09%

1253.0 9.36 87.61 1250 3.0 0.24%

1517.3 13.73 188.42 1500 17.3 1.15%

1743.4 9.75 95.06 1750 6.6 0.38%

1252.7 10.19 103.89 1250 2.7 0.22%

761.6 13.91 193.39 750 11.6 1.55%

995.7 9.69 94.08 1000 4.3 0.43%

1120.1 18.52 342.97 1000 120.1 12.0%

.0112 .00062 0 .0125 0.0015 12.0%

119.63 20.19 407.73 N/A N/A 4.78% of Y-Axis

% detection = 100%

308

Test 129

Figure 164: Test 129 Plot – Reassigned Spectrogram, 512, -2

Table 125: Test 129 Metrics – Reassigned Spectrogram, 512, -2 TEST 129 Reassigned Spectrogram FSK 8-component fs=5KHz fc=1.5KHz, 1KHz, 1.25KHz, 1.5KHz, 1.75KHz, 1.25KHz, .75KHz, 1KHz modBW=1KHz modperiod=12.5ms #samples=512 SNR=low detect (-2dB) 25 runs

Plot Time (min: sec)

fc1 (Hz)

fc2 (Hz)

fc3 (Hz)

fc4 (Hz)

fc5 (Hz)

fc6 (Hz)

fc7 (Hz)

fc8 (Hz)

Mod BW (Hz)

Mod Period (sec)

T-F localization

mean std dev var. actual error % error

1:09 1.25 1.56 N/A N/A N/A

1520.7 10.72 115.0 1500 20.7 1.38%

1018.8 9.87 97.42 1000 18.8 1.88%

1261.9 15.21 231.34 1250 11.9 0.95%

1500.0 16.43 269.95 1500 0.0 0.0%

1754.7 12.29 151.04 1750 4.7 0.27%

1261.6 15.54 241.49 1250 11.6 0.93%

755.2 22.80 520.08 750 5.2 0.69%

1012.3 18.41 538.93 1000 12.3 1.23%

1152.5 15.51 240.7 1000 152.5 15.3%

.0113 .0014 0 .0125 0.0012 9.60%

153.88 29.01 841.56 N/A N/A 6.16% of Y-Axis

% detection = 20%

309

Test 130

Figure 165: Test 130 Plot – Reassigned Scalogram, 512, 10

Table 126: Test 130 Metrics – Reassigned Scalogram, 512, 10 TEST 130 Reassigned Scalogram FSK 8-component fs=5KHz fc=1.5KHz, 1KHz, 1.25KHz, 1.5KHz, 1.75KHz, 1.25KHz, .75KHz, 1KHz modBW=1KHz modperiod=12.5ms #samples=512 SNR=10dB 25 runs

Plot Time (sec)

fc1 (Hz)

fc2 (Hz)

fc3 (Hz)

fc4 (Hz)

fc5 (Hz)

fc6 (Hz)

fc7 (Hz)

fc8 (Hz)

Mod BW (Hz)

Mod Period (sec)

T-F localization

mean std dev var. actual error % error

7.35 0.08 0.006 N/A N/A N/A

1502.0 7.08 50.13 1500 2.0 0.13%

1002.3 5.04 25.4 1000 2.3 0.23%

1250.7 6.73 45.29 1250 0.7 0.06%

1501.3 7.29 53.14 1500 1.3 0.09%

1748.7 9.94 98.8 1750 1.3 0.07%

1251.3 6.92 47.89 1250 1.3 0.10%

751.2 3.22 10.37 750 1.2 0.16%

1001.6 4.93 24.31 1000 1.6 0.16%

1054.9 12.23 149.57 1000 54.9 5.49%

.0109 .0017 0 .0125 0.0016 12.80%

49.0 11.92 142.09 N/A N/A 1.96% of Y-Axis

% detection = 100%

310

Test 131

Figure 166: Test 131 Plot – Reassigned Scalogram, 512, 0

Table 127: Test 131 Metrics – Reassigned Scalogram, 512, 0 TEST 131 Reassigned Scalogram FSK 8-component fs=5KHz fc=1.5KHz, 1KHz, 1.25KHz, 1.5KHz, 1.75KHz, 1.25KHz, .75KHz, 1KHz modBW=1KHz modperiod=12.5ms #samples=512 SNR=0dB 25 runs

Plot Time (sec)

fc1 (Hz)

fc2 (Hz)

fc3 (Hz)

fc4 (Hz)

fc5 (Hz)

fc6 (Hz)

fc7 (Hz)

fc8 (Hz)

Mod BW (Hz)

Mod Period (sec)

T-F localization

mean std dev var. actual error % error

10.58 0.723 0.523 N/A N/A N/A

1496.7 8.27 68.39 1500 3.3 0.22%

1002.7 7.21 51.58 1000 2.7 0.27%

1240.2 9.79 95.84 1250 9.8 0.78%

1500.5 9.03 81.54 1500 0.5 0.03%

1754.6 14.51 210.54 1750 4.6 0.26%

1254.0 7.77 60.37 1250 4.0 0.32%

755.1 5.09 25.91 750 5.1 0.68%

994.3 7.23 52.27 1000 5.7 0.57%

1074.4 15.22 231.65 1000 74.4 7.44%

.0117 .002 0 .0125 0.0008 6.40%

56.3 13.37 178.76 N/A N/A 2.25% of Y-Axis

% detection = 100%

311

Test 132

Figure 167: Test 132 Plot – Reassigned Scalogram, 512, -2

Table 128: Test 132 Metrics – Reassigned Scalogram, 512, -2 TEST 132 Reassigned Scalogram FSK 8-component fs=5KHz fc=1.5KHz, 1KHz, 1.25KHz, 1.5KHz, 1.75KHz, 1.25KHz, .75KHz, 1KHz modBW=1KHz modperiod=12.5ms #samples=512 SNR=low detect (-2dB) 25 runs

Plot Time (sec)

fc1 (Hz)

fc2 (Hz)

fc3 (Hz)

fc4 (Hz)

fc5 (Hz)

fc6 (Hz)

fc7 (Hz)

fc8 (Hz)

Mod BW (Hz)

Mod Period (sec)

T-F localization

mean std dev var. actual error % error

14.03 0.83 0.689 N/A N/A N/A

1487.6 11.07 122.54 1500 12.4 0.49%

1002.7 7.19 51.69 1000 2.7 0.27%

1236.8 14.79 218.74 1250 13.2 1.06%

1512.3 20.22 408.85 1500 12.3 0.82%

1748.2 13.39 179.29 1750 1.8 0.10%

1251.8 17.93 321.48 1250 1.8 0.14%

755.0 5.12 26.21 750 5.0 0.67%

1001.8 6.79 46.10 1000 1.8 0.18%

1098.5 17.71 313.64 1000 98.5 9.85%

.01188 .0022 0 .0125 0.00062 4.96%

66.0 15.29 233.78 N/A N/A 2.64% of Y-Axis

% detection = 20%

312

Test 133

Figure 168: Test 133 Plot – RSPWVD, 512, 10

Table 129: Test 133 Metrics – RSPWVD, 512, 10 TEST 133 RSPWVD FSK 8-component fs=5KHz fc=1.5KHz, 1KHz, 1.25KHz, 1.5KHz, 1.75KHz, 1.25KHz, .75KHz, 1KHz modBW=1KHz modperiod=12.5ms #samples=512 SNR=10dB 25 runs

Plot Time (sec)

fc1 (Hz)

fc2 (Hz)

fc3 (Hz)

fc4 (Hz)

fc5 (Hz)

fc6 (Hz)

fc7 (Hz)

fc8 (Hz)

Mod BW (Hz)

Mod Period (sec)

T-F localization

mean std dev var. actual error % error

22.36 0.838 0.702 N/A N/A N/A

1499.3 5.54 30.69 1500 0.7 0.05%

1001.2 2.19 4.79 1000 1.2 0.12%

1247.8 10.77 115.49 1250 2.2 0.18%

1499.4 6.69 44.76 1500 0.6 0.04%

1749.5 5.25 27.56 1750 0.5 0.03%

1249.7 9.93 98.60 1250 0.3 0.02%

750.3 3.43 11.76 750 0.3 0.04%

1001.3 2.77 7.67 1000 1.3 0.13%

1035.1 13.98 195.44 1000 35.1 3.51%

.0106 .00077 0 .0125 0.0019 15.2%

18.38 9.94 98.8 N/A N/A 0.74% of Y-Axis

% detection = 100%

313

Test 134

Figure 169: Test 134 Plot – RSPWVD, 512, 0

Table 130: Test 134 Metrics – RSPWVD, 512, 0 TEST 134 RSPWVD FSK 8-component fs=5KHz fc=1.5KHz, 1KHz, 1.25KHz, 1.5KHz, 1.75KHz, 1.25KHz, .75KHz, 1KHz modBW=1KHz modperiod=12.5ms #samples=512 SNR=0dB 25 runs

Plot Time (sec)

fc1 (Hz)

fc2 (Hz)

fc3 (Hz)

fc4 (Hz)

fc5 (Hz)

fc6 (Hz)

fc7 (Hz)

fc8 (Hz)

Mod BW (Hz)

Mod Period (sec)

T-F localization

mean std dev var. actual error % error

24.84 0.919 0.845 N/A N/A N/A

1501.8 7.17 51.41 1500 1.8 0.12%

1003.7 3.33 11.09 1000 3.7 0.37%

1246.6 12.23 149.57 1250 3.4 0.27%

1496.5 18.4 338.56 1500 3.5 0.23%

1755.7 8.91 79.39 1750 4.3 0.25%

1248.0 11.04 121.88 1250 2.0 0.16%

755.5 4.49 20.16 750 5.5 0.73%

1002.3 6.73 45.29 1000 2.3 0.23%

1074.1 34.19 1168.9 1000 74.1 7.41%

.0108 .00089 0 .0125 0.0017 13.6%

31.63 22.84 521.67 N/A N/A 1.27% of Y-Axis

% detection = 100%

314

Test 135

Figure 170: Test 135 Plot – RSPWVD, 512, -3

Table 131: Test 135 Metrics – RSPWVD, 512, -3 TEST 135 RSPWVD FSK 8-component fs=5KHz fc=1.5KHz, 1KHz, 1.25KHz, 1.5KHz, 1.75KHz, 1.25KHz, .75KHz, 1KHz modBW=1KHz modperiod=12.5ms #samples=512 SNR=low detect (-3dB) 25 runs

Plot Time (sec)

fc1 (Hz)

fc2 (Hz)

fc3 (Hz)

fc4 (Hz)

fc5 (Hz)

fc6 (Hz)

fc7 (Hz)

fc8 (Hz)

Mod BW (Hz)

Mod Period (sec)

T-F localization

mean std dev var. actual error % error

43.57 1.323 1.75 N/A N/A N/A

1505.3 20.07 402.80 1500 5.3 0.35%

1004.7 41.72 1740.6 1000 4.7 0.47%

1262.1 34.91 1218.7 1250 12.1 0.96%

1495.1 28.43 808.26 1500 4.9 0.33%

1751.5 30.21 912.64 1750 1.5 0.09%

1251.5 14.51 210.54 1250 1.5 0.12%

751.3 53.22 2832.4 750 1.3 0.17%

993.1 11.19 125.22 1000 6.9 0.69%

1104.1 46.97 2206.2 1000 104.1 10.4%

.0102 .00099 0 .0125 0.0023 18.4%

46.5 47.16 2224.07 N/A N/A 1.86% of Y-Axis

% detection = 8%

315

Test 136

Figure 171: Test 136 Plot – HT of WVD, 512, 10

Figure 172: Test 136 Plot – X-Y View – HT of WVD, 512, 10

316

Table 132: Test 136 Metrics – HT of WVD, 512, 10 TEST 136 – Part I HT of WVD FSK 8-component fs=5KHz fc=1.5KHz, 1KHz, 1.25KHz, 1.5KHz, 1.75KHz, 1.25KHz, .75KHz, 1KHz modBW=1KHz modper=.0125sec #samples=512 SNR=10dB (25 runs)

Plot Time (m:s)

θ1 (rad.)

ρ1 (samp.)

mean

16:33

0.009

std dev

0.884

var.

z1 (int.)

θ2 (rad.)

ρ2 (samp.)

50.48

6843.2

3.1445

51.83

0.011

0.634

92.8

0.017

0.781

0.0001

0.402

8611.8

actual

N/A

0.0

51.2

error

N/A

0.009

% error

N/A

N/A

z2 (int.)

θ3 (rad.)

ρ3 (samp.)

z3 (int.)

θ4 (rad.)

ρ4 (samp.)

z4 (int.)

7422

0.014

0.554

8694

0.009

50.48

6843.2

0.60

107.1

0.008

0.68

120.1

0.011

0.634

92.8

0.0003

0.36

1.14e4

0.0

0.462

1.44e4

0.0001

0.402

8611.8

N/A

3.142

51.2

N/A

0.0

0.0

N/A

0.0

51.2

N/A

0.72

N/A

0.0025

0.63

N/A

0.014

0.554

N/A

0.009

0.72

N/A

1.41%

N/A

0.08%

1.23%

N/A

N/A

N/A

N/A

N/A

1.41%

N/A

% detection = 100% # x-term false positives=2

% false positive=100% (for each of the 2 false positives)

TEST 136 – Part II HT of WVD FSK 8-component fs=5KHz fc=1.5KHz, 1KHz, 1.25KHz, 1.5KHz, 1.75KHz, 1.25KHz, .75KHz, 1KHz modBW=1KHz modper=.0125sec #samples=512 SNR=10dB θ5 (rad.)

ρ5 (samp.)

z5 (int.)

θ6 (rad.)

ρ6 (samp.)

z6 (int.)

θ7 (rad.)

ρ7 (samp.)

z7 (int.)

θ8 (rad.)

ρ8 (samp.)

z8 (int.)

0.024

103.7

2703.5

0.014

0.554

8694

3.1463

100.96

2503.6

3.1445

51.83

7422

0.027

1.52

88.8

0.008

0.68

120.1

0.033

1.19

68.43

0.017

0.60

107.1

0.0

2.31

7885.4

0.0

0.462

1.44e4

0.0011

1.42

4682.6

0.0003

0.36

1.14e4

0.0

102.4

N/A

0.0

0.0

N/A

3.142

102.4

N/A

3.142

51.2

N/A

0.024

1.3

N/A

0.014

0.554

N/A

0.0043

1.44

N/A

0.0025

0.63

N/A

N/A

1.26%

N/A

N/A

N/A

N/A

0.13%

1.41%

N/A

0.08%

1.23%

N/A

% detection = 100% # x-term false positives=2

% false positive=100% (for each of the 2 false positives)

317

Test 137

Figure 173: Test 137 Plot – HT of CWD, 512, -3

Figure 174: Test 137 Plot – X-Y View – HT of CWD, 512, -3 318

Table 133: Test 137 Metrics – HT of CWD, 512, -3 TEST 137 – Part I HT of CWD FSK 8-component fs=5KHz fc=1.5KHz, 1KHz, 1.25KHz, 1.5KHz, 1.75KHz, 1.25KHz, .75KHz, 1KHz modBW=1KHz modper=.0125sec #samples=512 SNR=-3dB (25 runs)

Plot Time (sec)

θ1 (rad.)

ρ1 (samp.)

mean

28.4

0.012

50.96

3526.3

3.146

51.75

std dev

1.12

0.016

0.614

80.1

0.02

var.

1.25

0.0003

0.376

6416

actual

N/A

0.0

51.2

error

N/A

0.012

% error

N/A

N/A

z1 (int.)

θ2 (rad.)

ρ2 (samp.)

z2 (int.)

θ3 (rad.)

ρ3 (samp.)

z3 (int.)

θ4 (rad.)

ρ4 (samp.)

z4 (int.)

3425

0.017

0.782

3761

0.012

50.96

3526.3

0.63

77.7

0.009

0.86

84.5

0.016

0.614

80.1

0.0004

0.39

6037.3

0.0

0.612

7140.3

0.0003

0.376

6416

N/A

3.142

51.2

N/A

0.0

0.0

N/A

0.0

51.2

N/A

0.24

N/A

0.004

0.55

N/A

0.017

0.782

N/A

0.012

0.24

N/A

0.47%

N/A

0.13%

1.07%

N/A

N/A

N/A

N/A

N/A

0.47%

N/A

% detection = 20% # x-term false positives=2

% false positive=100% (for each of the 2 false positives)

TEST 137 – Part II HT of CWD FSK 8-component fs=5KHz fc=1.5KHz, 1KHz, 1.25KHz, 1.5KHz, 1.75KHz, 1.25KHz, .75KHz, 1KHz modBW=1KHz modper=.0125sec #samples=512 SNR=-3dB θ5 (rad.)

ρ5 (samp.)

z5 (int.)

θ6 (rad.)

ρ6 (samp.)

z6 (int.)

θ7 (rad.)

ρ7 (samp.)

z7 (int.)

θ8 (rad.)

ρ8 (samp.)

z8 (int.)

0.0275

99.7

2688.3

0.017

0.782

3761

3.15

100.7

2208

3.146

51.75

3425

0.031

1.55

66.7

0.009

0.86

84.5

0.023

1.34

54.31

0.02

0.63

77.7

0.0009

2.40

4448.8

0.0

0.612

7140.3

0.0005

1.79

2949.5

0.0004

0.39

6037.3

0.0

102.4

N/A

0.0

0.0

N/A

3.142

102.4

N/A

3.142

51.2

N/A

0.0275

2.7

N/A

0.017

0.782

N/A

0.008

1.7

N/A

0.004

0.55

N/A

N/A

2.63%

N/A

N/A

N/A

N/A

0.25%

1.66%

N/A

0.13%

1.07%

N/A

% detection = 20% # x-term false positives=2

% false positive=100% (for each of the 2 false positives)

319

Test 138

Figure 175: Test 138 Plot – HT of RSPWVD, 512, -4

Figure 176: Test 138 Plot – X-Y View – HT of RSPWVD, 512, -4

320

Table 134: Test 138 Metrics – HT of RSPWVD, 512, -4 TEST 138 – Part I HT of RSPWVD FSK 8-component fs=5KHz fc=1.5KHz, 1KHz, 1.25KHz, 1.5KHz, 1.75KHz, 1.25KHz, .75KHz, 1KHz modBW=1KHz modper=.0125sec #samples=512 SNR=-4dB (25 runs)

Plot Time (sec)

θ1 (rad.)

ρ1 (samp.)

mean

54.3

0.006

std dev

1.52

var.

z1 (int.)

θ2 (rad.)

ρ2 (samp.)

51.0

6022.3

3.1423

50.2

0.010

0.59

157.0

0.007

2.31

0.0001

0.348

2.46e4

actual

N/A

0.0

51.2

error

N/A

0.006

% error

N/A

N/A

z2 (int.)

θ3 (rad.)

ρ3 (samp.)

z3 (int.)

θ4 (rad.)

ρ4 (samp.)

z4 (int.)

5947

0.01

2.132

5118

0.006

51.0

6022.3

0.63

122.2

0.009

1.09

93.4

0.010

0.59

157.0

0.0

0.39

1.49e4

0.0

1.19

8723.6

0.0001

0.348

2.46e4

N/A

3.142

51.2

N/A

0.0

0.0

N/A

0.0

51.2

N/A

0.20

N/A

0.0003

1.0

N/A

0.01

2.132

N/A

0.006

0.20

N/A

0.39%

N/A

0.01%

1.95%

N/A

N/A

N/A

N/A

N/A

0.39%

N/A

% detection = 24% # x-term false positives=2

% false positive=100% (for each of the 2 false positives)

TEST 138 – Part II HT of RSPWVD FSK 8-component fs=5KHz fc=1.5KHz, 1KHz, 1.25KHz, 1.5KHz, 1.75KHz, 1.25KHz, .75KHz, 1KHz modBW=1KHz modper=.0125sec #samples=512 SNR=-4dB θ5 (rad.)

ρ5 (samp.)

z5 (int.)

θ6 (rad.)

ρ6 (samp.)

z6 (int.)

θ7 (rad.)

ρ7 (samp.)

z7 (int.)

θ8 (rad.)

ρ8 (samp.)

z8 (int.)

0.015

100.4

3221.7

0.01

2.132

5118

3.144

101.7

3208

3.1423

50.2

5947

0.033

1.65

88.8

0.009

1.09

93.4

0.039

1.43

74.31

0.007

0.63

122.2

0.001

2.72

7885.4

0.0

1.19

8723.6

0.0015

2.04

5521.9

0.0

0.39

1.49e4

0.0

102.4

N/A

0.0

0.0

N/A

3.142

102.4

N/A

3.142

51.2

N/A

0.015

2.0

N/A

0.01

2.132

N/A

0.002

0.7

N/A

0.0003

1.0

N/A

N/A

1.95%

N/A

N/A

N/A

N/A

0.06%

0.68%

N/A

0.01%

1.95%

N/A

% detection = 24% # x-term false positives=2

% false positive=100% (for each of the 2 false positives)

321

Test 139 – Real World LPI Radar Signal For details of Test 139 (real-world LPI radar signal) see section 5.3 through section 5.5 of this dissertation.

322