I.
Digital Channelized Receiver Based on Time-Frequency Analysis for Signal Interception
´ ˜ GUSTAVO LOPEZ-RISUE NO, Member, IEEE ´ GRAJAL JESUS ´ ALVARO SANZ-OSORIO Universidad Polite´ cnica Madrid Spain
A digital channelized receiver is presented for the interception of a wide variety of signals of complex structure, including those with low probability of interception. The receiver is designed from the perspective of the time-frequency analysis. It uses an extended time-frequency representation based on the noncoherent integration of the short-time Fourier transform (STFT) on which the detection system and the encoder work. The encoder includes robust frequency estimation, automatic modulation recognition, and clustering, to handle broadband and simultaneous signals
INTRODUCTION
Modern electronic interception systems must perform the tasks of detection, classification, and identification in a difficult environment consisting of noise, interference, and multiple nonstationary signals. Moreover, some waveforms are intentionally designed to reduce the probability of interception (low probability of interception (LPI) signals [1]). This environment demands advanced signal processing algorithms running on digital receivers, which have been attracting considerable attention over the past years [2—4]. We propose an advanced digital channelized receiver (ADCRx), which is a particular case of a more general structure known as time-frequency receiver (TFRx). The TFRx main feature is the use of time-frequency analysis before detection and encoding (Fig. 1). Time-frequency analysis [5, 6] allows simultaneous description of a signal in time and frequency, so that the temporal evolution of the signal spectrum can be analyzed. It has become essential for nonstationary signal analysis; therefore, it is an appealing tool for the interception of many radar and communications signals. The TFRx detector works on the time-frequency representation and builds feature vectors containing the information relative to the time and frequency where each detection occurred. The encoder clusters all the feature vectors belonging to the same signal and estimates the pulse descriptor word (PDW).
and to prevent out-of-channel detections (a typical phenomenon in channelized receivers). The receiver has been evaluated for a wide range of signals and shows a good performance in terms of detection, estimation, and processing of simultaneous signals. Signals collected from real-life systems and synthetic signals have been utilized.
Fig. 1. Architecture of time-frequency receiver.
Manuscript received October 20, 2003; revised March 30 and October 4, 2004, and January 11 and January 25, 2005; released for publication March 1, 2005. IEEE Log No. T-AES/41/3/856439. Refereeing of this contribution was handled by J. P. Y. Lee. This work was supported in party by project TIC2002-04569-C02-01 of the Science and Technology Ministry, and the Navy R&D Center (CIDA). This work was presented in part at the International Conference on Radar, Adelaide, Australia, September 2003. Authors’ addresses: G. Lo´ pez-Risuen˜ o, European Space Research and Technology Center, European Space Agency, Noordwijk, The Netherlands; J. Grajal and A. Sanz-Osorio, Grupo de Microondas y Radar, Departamento de Sen˜ ales, Sistemas y Radiocomunicaciones, ETSI de Telecomunicacio´ n, Universidad Polite´ cnica Madrid, Ciudad Universitaria s/n, 28040 Madrid, Spain. E-mail: (
[email protected]). c 2005 IEEE 0018-9251/05/$17.00 °
The ADCRx is based on an extension of the short-time Fourier transform (STFT) or sliding-window Fourier transform [5, 6]. The receiver is mainly intended for radar signal interception; nevertheless, its performance on some communications signals is also analyzed. The encoder’s PDW comprises the following parameters [3]: time of arrival (TOA), pulsewidth (PW), carrier (or mean) frequency (f), pulse amplitude (PA), and intrapulse modulation. The modulation is classified into one out of four categories: no modulation, linear frequency modulation (LFM), phase-shift keying (PSK), and frequency-shift keying (FSK). Digital channelized receivers based on the STFT were first proposed by Fields et al. [7], and
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Zahirniak et al. [8],1 as an attempt to benefit from the advantages of digital technology and the properties of channelized receivers [9], such as: simultaneous-signal interception, high sensitivity for narrowband signals, and high dynamic range. Those receivers were designed as mere “digital replicas” of the previous analog channelized receivers, and were intended for conventional radar pulses of moderate signal-to-noise ratio (SNR around 10 dB) rather than for the wide variety of signals appearing in current interception scenarios. These new signals have motivated this new time-frequency approach leading to the ADCRx. The ADCRx uses a new enriched STFT-based representation including noncoherent integration of different lengths; thus the integration length becomes the third dimension of this extended time-frequency representation. This improves the performance in detection, since the receiver adapts itself to signals with different lengths; it also improves the performance in encoding, since signals are described in a more highly-dimensional space. The ADCRx provides a more complete PDW by means of a new encoder design featuring a clustering algorithm adapted to the channelized architecture, a novel robust frequency estimation technique, and an automatic modulation classifier based on the instantaneous frequency estimation. The receiver works on a block-by-block basis, namely the signal is divided into blocks of samples separately analyzed by the receiver. Afterwards, the PDWs of every block are passed to a data processor that performs deinterleaving and user-interfacing [3], which is beyond the scope of this paper. The block-by-block operation is motivated by the practical implementation as subsequently clarified. The paper is organized as follows. Sections II, III, and IV describe the ADCRx time-frequency processor, detector, and encoder, respectively. Section V shows the performance for a wide range of signals, including high time-bandwidth product signals, such as the LPI ones. Section V also discusses the implementation aspects of the ADCRx. The advantages of ADCRx become apparent by the comparison with more traditional approaches. Finally, the conclusions are drawn in Section VI. Part of the content of this paper was previously presented in [10]. II.
TIME-FREQUENCY PROCESSOR
A. Time-Frequency Representations Time-frequency analysis aims to provide a physically-meaningful representation of the time-varying spectrum of a signal. A time-frequency representation is a mapping of the signal under analysis onto the two-dimensional space of time and frequency. There are many ways of performing such a mapping. Among them, three important categories arise [5, 6]: linear, quadratic and adaptive representations. The most important linear techniques are the STFT and the wavelet transform. The most important quadratic representation is the Wigner-Ville distribution, which gives rise to Cohen’s class by the convolution with different kernels. Adaptive representations are adaptive versions of the previous categories. The most important ones are the signal-dependent time-frequency representation [6, 11] (adaptive version of Cohen’s class) and the adaptive approximation techniques [6, 12, 13], such as the atomic decomposition2 or the basis pursuit. Adaptive approximations perform an adaptive linear expansion of the signal using a redundant set of elementary functions. The most suitable time-frequency techniques for signal interception are the linear and the adaptive-approximation ones, since they deal with simultaneous signals without the appearance of the cross-terms inherent to the quadratic distributions [5]. Moreover, within the linear techniques, the STFT is particularly more adequate than the Wavelet Transform: The STFT can be interpreted as a uniform filter bank with constant noise power at the output of each STFT channel. However, the Wavelet Transform is a filter bank with constant relative bandwidth [5], so that higher frequency channels hold a higher noise power and a worse detection performance. Regarding the computational burden, the STFT is more efficient than the adaptive approximations [12, 13]. Consequently, it becomes the most appropriate time-frequency technique for fast signal interception. B. STFT Implementation The discrete version of the STFT is defined as [5, 6] STFTx(w) (n, k) =
n+(L¡1) X m=n
We first present a brief review of the time-frequency representations and the suitability of the STFT for interception. Then, the discrete STFT and the extension used in the time-frequency processor are described. 1 These
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systems are also described in [3].
x(m)w(m ¡ n)e¡j(2¼k=L)m , k = 0, : : : , L ¡ 1
(1)
where w(m) is the analysis window defined from 0 to L ¡ 1, and x(m) the signal under analysis. Index 2 Also known as matching pursuit [13] or adaptive gabor representation [6].
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Fig. 2. Analysis window for 64-channel receiver. 256 samples, §1 dB passband ripple and 60 dB losses within attenuation band. (a) Window. (b) Window frequency response.
n refers to the discrete time and k to the channel of normalized center frequency k=L. As in [3], [7], and [8], the design of the analysis window is based on a filtering approach: The Parks-McClellan method [14], which computes the minimax optimum linear-phase filter that meets a proposed filter mask. To achieve a high dynamic range, the filter must possess abrupt transition bands and low sidelobes leading to a window much longer than the required number of channels. For instance, for a 64-channel receiver, with a §1 dB ripple passband and a 60 dB attenuation band starting at the center of the adjacent filters, a 256-tap window is required. This window is shown in Fig. 2. Let K be the number of desired channels, and L the window length, several techniques can be applied to compute discrete Fourier transforms (DFTs) of K out of L coefficients, e.g. time data folding [3] or polyphase filters [8]. Actually, only channels k = 1 to k = K=2 ¡ 1 are used since the signal under analysis, x(m), is real-valued. Channels k = 0 and k = K=2 hold different statistical properties and are not used; thus the number of effective channels is K=2 ¡ 1. The filter-bank interpretation allows the STFT decimation in time. The decimation factor M is upper-bounded by M · K=2
(2)
due to both the Nyquist sampling criterion and the ambiguity bandwidth of the instantaneous frequency estimate [8]. This estimation is performed by the digital instantaneous frequency measurement method (DIFM) (treated in Section III). Decimation in time alleviates the STFT computation since the analysis window is shifted M samples instead of one as in (1). For an N-sample block, the size of the decimated STFT matrix becomes (1 + (N ¡ L)=M) £ (K=2 ¡ 1).
C. Extension of STFT The STFT holds a fixed time-frequency resolution that depends on the analysis window. Furthermore, time and frequency resolutions are related by the uncertainty principle [5, 6]: A good time resolution implies poor frequency resolution and vice versa. This lack of flexibility, as a consequence of the STFT nonadaptive characteristic, has a strong impact on the detection performance [15]: If a signal has a wider bandwidth than the frequency resolution, the signal energy is spread out over several coefficients in frequency, so that the probability of detection based on individual coefficients reduces. Similarly, a signal with a greater duration than the time resolution spreads out its energy over several coefficients in time. The ADCRx implements a more flexible time-frequency representation which is an extension of the STFT. This extension relies on different-length noncoherent integration and maintains the low computational burden of the STFT. The motivation of this extension is addressed in the following. The STFT processing gain for a narrowband signal becomes [3]3 : Gp = K=(2Lins Bn ), where Lins is the channel insertion losses at the signal frequency, and Bn the relative noise bandwidth with respect to a K-tap rectangular window. This gain exclusively comes from the channelization, and does not benefit from the fact that a signal with a greater duration than the analysis window stays longer in the same channel, as usually happens to the signals with a high time-bandwidth product, such as the LPI ones. In other words, we are losing processing gain whereas LPI signal interception paradoxically demands a high processing gain. 3 The
processing gain of an algorithm is the ratio between the SNR required to detect a signal without and with it at the same conditions of probability of false alarm and detection.
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To increase the processing gain by noncoherent integration, we define the smoothed spectrogram as Ii (m, k) =
Li ¢m X
r=1+Li ¢(m¡1)
jSTFT(rM, k)j2
(3)
where Li is the integration length, and M the decimation factor.4 A finite set of smoothed spectrograms with different integration lengths must be used to account for signals with different durations. The definition (3) assumes that the integrators work in an integration and dump mode for the sake of computational efficiency. The integrator lengths Li are assumed to be divisible by one another. With regard to the number of integrators, at least two integration types should be considered. First, the one-sample integrator (I1 ), which holds the highest resolution in time; and second, the full-length integrator (IF ), which achieves the highest processing gain for long-duration signals. The full-length integrator is the longest integrator possible within the block of samples, i.e., LF = 1 + (N ¡ L)=M. Intermediate-length integrators are also needed in order to better match the intermediate-duration signals and prevent the collapsing losses of the longest integrators [16]. This is illustrated in Fig. 3, where the processing gain of integrators with different lengths is drawn. For each integrator, the collapsing losses appear when the signal is shorter than the integrator length. For longer signals, the processing gain is constant and equal to the gain for a signal matching the integrator length. For each signal length, an optimum processing gain exists that corresponds to the integrator matching the signal length. This gain has been approximated in Fig. 3 as a straight line.5 According to Fig. 3, the difference in the processing gain of the noncoherent integrators used in the ADCRx defines an upper bound for the mismatching losses. Mismatching losses are defined as the difference in detecting a signal by means of the matched integrator instead of the ADCRx. In practice, they can be set to 3 dB. To summarize, the ADCRx time-frequency processor outputs the following extended representation: fSTFT, I1 , I2 , : : : , IF g.
Fig. 3. Noncoherent integration processing gain versus signal duration.
vector per detection. The detection problem is addressed as a hypothesis testing problem: H0 : H1 :
x=n X x= sr + n
(4) (5)
r
where n is a real-valued zero-mean white Gaussian noise of known power. The noise power is known since it mainly comes from the receiver internal noise [3]. Every vector sr corresponds to a signal present in the analyzed N-sample block. As the signals are unknown, the detection is carried out by a local procedure, i.e., a detection test is performed on each point of the time-frequency representation, namely each coefficient of the smoothed spectrograms. The local tests turn out to be H1
Ii (m, k) ? thi ,
i = 1, 2, : : : , F
(6)
H0
where the thresholds are set to meet a desired value of global false alarm probability (PFAg ): 0 1 [ PFAg = PH0 @ fIi (m, k) > thi gA : (7) i,m,k
III. DETECTION AND FEATURE EXTRACTION A. Detection The ADCRx detection stage works on the previous time-frequency representation and builds a feature 4 The classic spectrogram of a signal is the squared absolute value of the STFT (jSTFT(n, k)j2 ) [5, 6]. 5 The processing gain is approximately proportional to the squared root of the integrator length [17], i.e., it is a straight line in logarithmic units. In the context of the ADCRx, this approximation is valid for Li M=K À 1 [17]; nevertheless, we use the approximation in a general way for a didactic purpose.
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That is, the global PFA is the probability of having at least a local detection under the hypothesis H0 . A local PFA can also be defined for every local detection. For the time-frequency points of the ith smoothed spectrogram, it becomes PFAi (m, k) = PH0 (Ii (m, k) > thi ), i = 1, 2, : : : , F:
(8)
As the noise is white, the local PFA is constant for the local decisions in the same smoothed spectrogram. To find the thresholds fthi g meeting the global PFA
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Fig. 4. Detection stage of ADCRx.
(PFAg ), we assume for simplicity that the local PFA is the same for all the smoothed spectrograms. For a given local PFA , the thresholds thi can be analytically computed as in [18]. The relationship between the local PFA and PFAg has to be obtained by simulation (see Appendix C). Reciprocally, the global probability of detection (Pdg ) can be defined as the probability of having at least a local detection under the hypothesis H1 . As in (8), a local probability of detection (Pdi (m, k)) can be defined for each coefficient of the smoothed spectrograms. B. Feature Vector Construction As mentioned above, there is a corresponding feature vector per local detection. Apart from the time-frequency location of the local detection, the feature vector includes the instantaneous frequency estimation by using the DIFM. For the channel k, the DIFM is defined as DIFM(r, k) argfSTFT((r + 1)M, k)g ¡ argfSTFT(rM, k)g = 2¼M (9) where M is again the STFT decimation factor, and argf¢g is the phase angle. The DIFM was used in former digital channelized receivers [3, 7, 8] to estimate the carrier frequency. It is used here to 1) estimate the carrier frequency in a more robust way, and 2) recognize the signal modulation (both tasks treated in Section IV). The exact structure of the feature vector is detailed as follows. If a local detection occurs at Ii (m, k), the feature vector (µ) contains the smoothed spectrogram index (i), its numerical value (Ii (m, k)), time location (m) and channel (k), and the DIFMs within the
TABLE I Example of ADCRx Time-Frequency Processor. Analysis Window of Fig. 2 Parameter
Symbol
Value
Block length (Total) number of channels Effective number of channels Decimation Integration lengths for I1 , I2 , I3
N K K=2 ¡ 1 M L1 L2 L3
1024 64 31 32 1 5 25
Parks-McClellan Analysis Window Length Pass-band ripple
L Rp
Attenuation
Ra
256 §1 dB 60 dB
integration length (Li ) for channel k, i.e., µ = [i, Ii (m, k), m, k, DIFM(r1 , k), DIFM(r1 + 1, k), : : : , DIFM(r2 , k)]T
(10)
where indices r1 and r2 are r1 = 1 + (m ¡ 1) ¢ Li r2 = m ¢ Li :
(11)
Table I shows an example of a time-frequency processor used in subsequent sections. It includes three integrators of 1, 5, and 25 samples, so that the maximum loss in the noncoherent processing gain using the ADCRx will be around 5 log10 L3 =L2 = 5 log10 L2 =L1 = 3:5 dB, according to the discussion in the previous section. A complete diagram of detection stage of this system is also shown in Fig. 4. The diagram follows the filter-bank perspective for the sake of clarity. At the output of every channel, the integrators work in parallel, in integration and dump mode.
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IV. ENCODER From the feature vectors built in the detection stage, the encoder estimates the number of signals in the block under analysis, and their PDWs. Depending on its SNR, bandwidth, and duration, a signal can be simultaneously detected in different points of different smoothed spectrograms, so that several feature vectors can correspond to the same signal. Therefore, clustering is required to group all the feature vectors of the same signal. The PDW of every signal is estimated from the feature vectors of the corresponding cluster. The encoder takes advantage of the STFT filter-bank interpretation by working in two steps: in-channel and in-block. First, the in-channel processing clusters the feature vectors from the same channel, and computes an in-channel PDW by assuming only one signal per channel. Second, the in-block processing clusters the in-channel PDWs of the same signal, fuses those PDWs into an in-block PDW, and removes the out-of-channel detections. The one-signal-per-channel assumption is valid for short- or medium-duration blocks, i.e., blocks of samples lasting microseconds or hundreds of microseconds. For instance, in a dense environment of 105 signals per second in the band from 9 to 9:5 GHz, the mean number of signals within a 100 ¹s interval is 10; thus, on average, there would be a signal every 50 MHz-bandwidth channel.6 In practice, this kind of blocks is required to achieve low latency times. A. In-Channel PDW In-channel PDW construction is divided into several steps which are repeated for each channel where, at least, a local detection occurred. 1) Signal Duration: Let £k(i) be the set of feature vectors with smoothed spectrogram index i and channel k, the in-channel PA estimate for each smoothed spectrogram becomes p ½ maxf2 I(µ)=Li : µ 2 £k(i) g, if £k(i) 6= Ø c PAi = 0, otherwise: (12) I(µ) is the amplitude of the smoothed spectrogram corresponding to the feature vector µ. The estimates c s are unbiased and consistent for sinusoidal signals PA i at a high SNR [19, sect. 6.1.2].7 Using these estimates, the signal is classified into a number of categories corresponding to the different smoothed spectrograms: signals of duration similar to the integration length of I1 , signals of duration similar 6 A uniform distribution of the signals in time and frequency is assumed. 7 No insertion losses in the channel are assumed. In the context of [19, sect. 6.1.2.], Ii (n, k) can be viewed as the signal smoothed periodogram.
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to the integration length of I2 , etc. The classification is carried out by successive hypothesis tests: c Shorter than IF PA F¡1 ? th , c Duration as IF PAF¡1,F PA F
c Shorter than IF¡1 PA F¡2 ? thPAF¡2,F¡1 , : : : c Duration as IF¡1 PA F¡1
c Duration PA 1 ? c Duration PA 2
as I1
as I2
thPA1,2
(13)
that rely upon the fact that, when the signal is shorter c tends to be smaller than the integrator length, PA i than the true value due to the additional integration of noise-only samples. Each threshold thPAi¡1,i is obtained by simulating radar pulses with a similar length to the integrators Ii¡1 and Ii and by minimizing c (see c =PA the probability of error in the test PA i¡1 i Appendix C). For illustration purposes, Fig. 5 shows the probability of misclassification for several conventional radar pulses. We use the system configuration of Table I, which has three noncoherent integrators and discriminates among three types of duration: short (as the integrator I1 ), medium-length (as the integrator I2 ) and long signals (as the integrator I3 ). As can be noticed, the pulses are properly classified for SNRs above 0 dB. 2) PA Estimate: If the signal is classified into the category of signals of length similar to the integrator c is defined as Ii , the in-channel PA estimate (PA) c the one corresponding to the integrator Ii , i.e., PA i 2 (eqn. (12)). As the noise power is known (¾ ), the SNR estimate becomes c2 d = PA : SNR 2¾2
(14)
3) TOA and PW Estimates: Temporal parameters (TOA and PW) are estimated according to the prior classification. That is, if the signal has a duration similar to the integrator Ii , the feature vectors coming from the smoothed spectrogram Ii are used. In order d and PW, d the initial (n ) and final to compute TOA init (nend ) decimated samples of the signal at channel k are required: ninit = minf(m(µ) ¡ 1) ¢ Li + 1 : µ 2 £k(i) g nend = maxfm(µ) ¢ Li : µ 2
£k(i) g
(15) (16)
where Ii is the smoothed spectrogram corresponding to the signal duration, and m(µ) is the time location of each feature vector µ. Definitions (15) and (16) are appropriate for low SNR. For high SNR, the smoothed spectrogram I1 is used, instead of the Ii , in order to provide a more accurate estimation. Smoothed d is over a certain spectrogram I1 is used when SNR threshold (SNRth ) that assures a high probability of detection (in practice, this threshold can be defined
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Fig. 5. Probability of misclassification in terms of duration for several conventional radar pulses. System configuration of Table I; thPA12 = 4 and thPA23 = 2; block of 1024 samples.
as the SNR to detect a sinusoid with a probability of 99% by means of the I1 ). For the sake of clarity, TOA and PW estimators are defined in Appendix A, where the effect of the analysis window transient is also considered. 4) Frequency Estimate: The frequency estimate is computed by a weighted average of the DIFM samples from the time sample ninit (eqn. (15)) to nend ¡ 1 (eqn. (16)) at the considered channel. The weights are the values of the smoothed spectrogram I1 and the expression of the frequency estimate becomes Pnend ¡1 I1 (r, k)DIFM(r, k) ˆf = r=ninit : (17) Pnend ¡1 r=ninit I1 (r, k) This expression also turns out to be the mean frequency estimate in the framework of the time-frequency analysis [6]. The weighted DIFM average has been compared with other methods working with instantaneous frequency estimates as well, such as Kay’s estimate [20], the single-DIFM estimate, and the nonweighted DIFM average [8]. The weighted DIFM average has the best performance for nonstationary signals as illustrated in Fig. 6, where the root-mean-square error (RMSE) versus the SNR is shown for a BPSK signal intercepted by the system of Table I. The RMSE of those estimates does not decrease as the SNR increases since they are biased. The bias is due to the DIFM ambiguity bandwidth [8], which amounts to §1=(2M) for a given decimation factor (M). Assuming ideal symbol-to-symbol transitions, a §180± phase transition results in a frequency equal to half the ambiguity bandwidth, i.e., §1=(2M). Thus these transitions are always computed
with the same sign, so that a bias is introduced. We have checked that the bias is below the 10% of the channel bandwidth for different PSKs (with smaller bandwidth than the channel) and configurations (channel bandwidth and decimation factors). If a smaller error is required, PSK-oriented carrier frequency estimators [21, ch. 3] can be used after the modulation classification. The modulation classifier is frequency-shift invariant; hence, it is not affected by the frequency estimation bias (see Section IVB). For sinusoids, Kay’s estimate is better and reaches the Cramer-Rao bound for high SNR [20]. This is also shown in Fig. 6. The weighted and nonweighted DIFM average exhibit the same performance, which is slightly worse than the Cramer-Rao bound. The maximum likelihood frequency estimate is not considered since it does not work with DIFM samples but with the signal Fourier transform, which is more computationally demanding. Moreover, both Kay’s and the maximum likelihood estimate have the same performance for high SNR [20]. On the whole, the weighted DIFM average is the most robust estimate for both nonstationary and stationary signals, and becomes very suitable for the encoder implementation. B.
Modulation Recognition
The in-channel automatic modulation classifier (AMC) follows a decision-theoretic approach inspired by [22] and [23]. It distinguishes among four categories: no modulation, LFM, PSK, and FSK, which are the typical modulations encountered in radar and digital communications systems. AMC uses the DIFM outputs of the considered channel within the signal duration, i.e., between
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Fig. 6. RMSE of frequency estimated by several techniques using DIFM samples for BPSK with 62 samples/symbol and a sinusoid. Table I system configuration. BPSK and channel bandwidths are approximately the same (for 250 MHz sampling rate, 4 and 3.9 MHz, respectively).
the decimated samples ninit and nend ¡ 1 (eqns. (15) and (16)). In the preprocessing steps, a number of initial and final samples are removed to avoid peaks from the signal transients, and the signal length is verified to be greater than a minimum value. If lower, the signal is considered to be nonmodulated. Then, AMC proceeds as depicted in Fig. 7. First, a frequency linear model is obtained by least squares. The model error (") helps separate LFM and nonmodulated signals from PSK- and FSK-modulated ones. The discrimination between LFM and nonmodulated signals is made by the magnitude of the estimated chirp-rate (aˆ ). AMC discriminates between PSK and FSK by the maximum of the 1st-order difference of the DIFM sequence (¢f), i.e., ¢f(r, k) = DIFM(r + 1, k) ¡ DIFM(r, k)
(18)
since PSK phase transitions are expected to be greater than FSK frequency transitions. This relies upon the fact that, for PSKs with ideal symbol-to-symbol transitions, the amplitude of the frequency impulse corresponding to a phase step ¢Á is: ¢f = ¢Á=2¼M (M is the decimation factor). We restrict the FSK symbol separation to the channel bandwidth in order that the AMC succeeds in the classification. For broader band FSKs, the correct recognition should be done by the in-block processing. However, the variety of FSK signals makes the design of rules for in-block recognition rather difficult, and the only modulation-related task performed by the in-block clustering is the recognition of fast LFM signals (see Section IVC). 886
Fig. 7. In-channel AMC flow chart.
The thresholds for the estimators " and aˆ hold the following general structure: d PW) d PW) d PW) d = ¹ (SNR, d + c ¢ ¾ (SNR, d thT (SNR, T T T
(19)
where ¹T and ¾T are the mean and standard deviation of the considered statistic T. To compute thLFM , ¹" and ¾" come from the distribution of the model error " for both nonmodulated and LFM signals. Regarding tha , ¹a and ¾a come from the distribution of the chirp-rate estimate aˆ for nonmodulated signals only. Thresholds tha and thLFM can be analytically computed as indicated in Appendix B. Constant cT in (19) minimizes the probability of error among the different modulation classes. This probability is computed by simulation as the sum of the
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probability of wrong classification for several signals representative of all the modulation categories (see Appendix C). For the sake of simplicity, the threshold th¢f is defined as a constant and also computed by simulation to minimize the misclassification error (Appendix C). C.
In-Block PDW
The in-channel PDWs are clustered according to a number of heuristic rules. First of all, the d which in-channel PDW with the greatest SNR, we call main PDW, is selected and its adjacent channels are grouped into the cluster according to d and main PDW modulation. time overlapping, SNR, For each cluster, the final PDW is the corresponding main PDW except for clusters considered to come from LFM signals. In that case, a reestimation of the parameters is performed. The remaining nongrouped in-channel PDWs undergo the same process by selecting a new main PDW. d is required in order to consider The estimate SNR the overlapping in time, since temporal parameter estimation degrades for low SNR. A threshold SNRf is used, so that overlapping in time is not considered d < SNR . The detailed description of the for SNR f in-block rules is beyond the scope of the paper. Nevertheless, there are two aspects that deserve further consideration: The improvement of PDW estimation for LFMs by in-block clustering, and the cancellation of out-of-channel detections. 1) PDW Estimation for LFM Signals: In the in-block processing, a cluster is considered as LFM if the main PDW is found to be LFM modulated. However, for fast LFM signals, there could be not enough samples to carry out the in-channel modulation analysis. In those cases, the LFM character becomes apparent after examining the time-frequency arrangement of the in-channel PDWs forming the cluster. In this paper, the rule for automatic recognition of fast LFMs is as simple as checking that there is a significant shift in time and frequency between the main PDW and the PDW of an adjacent channel. The chirp-rate estimate is then obtained as the frequency separation-time separation ratio between these two PDWs. For LFM signals, the final PDW takes into account the in-channel PDWs of the cluster. The in-block TOA is reestimated as the minimum in-channel TOA (similarly for the PW) and the in-block mean frequency is the weighted average of the in-channel frequency estimates. The weights are the c The in-block PA c is the corresponding in-channel PAs. c Regarding the chirp-rate estimation, main PDW’s PA. if the main PDW is found to be LFM modulated, the chirp-rate estimate is simply the main PDW chirp-rate.
Otherwise, the chirp-rate estimation is as described in the previous paragraph. To illustrate the benefits of the in-block PDW estimation, a chirped pulse sweeping 40 MHz in 4 ¹s have been analyzed by the system of Table I at a 250 MHz sampling rate. The signal features a fast LFM modulation, and sweeps 10:24 channels in 1000 samples. Fig. 8 shows the reduction in the TOA RMSE by means of in-block clustering with respect to the main in-channel PDW (on the left). The in-block probability of right modulation recognition is also depicted (on the right). It becomes 90% for 4 dB SNR. However, the in-channel probability of recognition is zero since the minimum number of samples to run the AMC is not reached. 2) Cancellation of Out-of-Channel Detections: Once a cluster has been formed, other nongrouped channels may also be removed to prevent out-of-channel detections. Two types of out-of-channel detections are considered: rabbit-ear effect and signal sidelobe detections. The cancellation algorithm uses two thresholds determined by simulation and proceeds as follows. a) The lowest threshold is related to the rabbit-ear effect. If the number of channels in the cluster are greater than the threshold, the nongrouped overlapped-in-time channels with short duration8 are removed. This is the case of high-SNR pulses, since their leading and trailing edges usually cause short-duration detections in adjacent channels (high-SNR pulses are used to compute this threshold). b) The greatest threshold eliminates signal sidelobe detections. If the number of channels in the cluster are greater than this, the signal has a very broad band and all nongrouped, overlapped-in-time channels are removed (independently of their duration). This is the case of high-SNR very short pulses and high-SNR very broadband PSKs. For these cases, the signal sidelobes cause detections in channels so far away from the signal frequency that they are not clustered with the corresponding main PDW (high-SNR broadband PSKs are used to compute this threshold). In [7] and [8], another cancellation method is proposed based on the DIFM. It is suitable for nonmodulated narrowband pulses with high SNR. For low SNR or modulated signals, the DIFM errors become so high, as was shown in Fig. 6, that this method cannot apply. V. RESULTS AND DISCUSSION This section describes the performance of the ADCRx for a wide variety of signals related to the 8n
end
¡ ninit + 1 · 2.
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Fig. 8. In-block clustering for fast chirped pulse sweeping 40 MHz in 4 ¹s: TOA RMSE of in-block and main PDW are compared on the left. On the right, in-block probability of modulation recognition is shown. (System configuration in Table I, sampling rate 250 MHz.)
radar field: conventional pulses, (linear) chirped pulses, CW signals and phase-coded signals. To complete the analysis, some digital communications signals have been evaluated as well. Among the phase-coded waveforms, this section studies Barker pulses and long-duration BPSK signals with random modulation. Long-duration BPSKs with random modulation are a generalization of the pseudorandom-code radars [24, ch. 10] and the BPSK communications signals, since neither the code in radar nor the information in communications are a priori known by the interception receiver. Several minimum shift keying (MSK) signals [25] are also utilized as representatives of FSK communications signals. Regarding the system, a 250 MHz sampling rate has been selected. The parameters of the time-frequency representation are the same as in Table I: a 3.9 MHz channelization and a 4.1 ¹s signal block. For the encoder, the main parameters are listed in Table II. In the subsequent sections, the receiver is evaluated in terms of sensitivity and estimation error of the in-block PDW. Some of the signals in this section were collected by an X-band receiver at 250 Msamples/s and quantized with 8 bits. These real-life signals correspond to existing radar and communications systems operating in real environments. Hence, the data are affected by the linear and nonlinear distortions of the sampling receiver. These distortions have turned out to be unimportant since analyses with purely synthetic signals with the same characteristics as the real-life ones yielded the same results. The real-life signals were collected at a high SNR condition, so that 888
TABLE II Example of ADCRx Encoder (for Time-Frequency Processor in Table I) Parameter
Symbol
Value
Threshold for long/medium-length signal Threshold for medium-length/short signal SNR threshold to compute TOA and PW by I1 Guard samples for AMC (left and right) Minimum length (after guard sample removal) Constant for threshold tha Constant for threshold thLFM Threshold for PSK/FSK
thPA2,3
2
thPA1,2
4
SNRth
3 dB
–
L=2M ¡ 1 = 3
–
SNR threshold for in-block clustering Number of channels for rabbit-ear cancellation Number of channels for broadband signals (sidelobe cancellation)
SNRf
d < SNRth 5, if SNR 3, otherwise 9 20 6:94 £ 10¡4 (80± )
ca cLFM th¢f
0 dB
–
4
–
20
additional computer-generated noise has been added to meet the desired SNR values. A. Sensitivity ADCRx is compared with other typical systems in terms of sensitivity for a wide variety of signals, some of them exhibiting the LPI features. The other receivers are: A channelized receiver based solely on the STFT like the approaches of Fields et al. [7]
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TABLE III Sensitivity (dB) Signal
ADCRx
STFT (only)
DFT
ED
MF
Conventional pulse 500 ns Conventional pulse 1 ¹s CWLFM 100 MHz/1 ms CWLFM 500 MHz/1 ms Chirped pulse 1 MHz/1 ¹s Chirped pulse 2 MHz/2 ¹s Chirped pulse ¡40 MHz/4 ¹s(¤) BPSK 2 MHz BPSK 4 MHz Barker-13 pulse 4.8 ¹s(¤) BPSK 10 MHz MSK 1 MHz (freq. sep.)/500 ns (bit duration) MSK 3.3 MHz (freq. sep.)/150 ns (bit duration)(¤)
¡1:6 ¡4 ¡8:5 ¡8:5 ¡4 ¡5:5 ¡4 ¡8 ¡8 ¡6 ¡5:5 ¡6 ¡5:5
¡0:4 ¡1:5 ¡1:5 ¡2 ¡1:1 ¡1:6 ¡3:4 ¡3:2 ¡3:2 ¡3:7 ¡2:2 ¡2 ¡2:9
3.9 ¡1:4 ¡11:5 ¡6 ¡1 ¡2:5 4.5 ¡6:5 ¡6:5 ¡5:5 ¡2 ¡9:5 ¡6
3.6 0.6 ¡5:3 ¡5:3 0.6 ¡2:2 ¡5:3 ¡5:3 ¡5:3 ¡5:3 ¡5:3 ¡5:3 ¡5:3
¡5 ¡8 ¡14 ¡14 ¡8 ¡11 ¡14 ¡14 ¡14 ¡14 ¡14 ¡14 ¡14
Note: SNR at PFAg = 10¡6 , Pdg = 90%, 250 MHz sampling rate, 1024-sample block. Signals with asterisk (¤) are from real-life systems in their typical operating environments. Pulsed signals are time centered in block and, if nonmodulated, its frequency changes from a realization to another according to a uniform distribution within the channel bandwidth. For other synthetic signals, carrier frequency coincides with center of 16th STFT channel.
and Zahirniak et al. [8]; a receiver based on the DFT with a quadratic detector after every bin (it can be considered as a channelized receiver with an analysis window as long as the block length); the energy detector (ED); and, finally, the matched filter (MF), which is the optimum detector when the signal parameters are known, and the signal phase offset is uniformly distributed over [0, 2¼). The comparison is shown in Table III, where the sensitivity is defined as the minimum SNR for PFAg = 10¡6 and Pdg = 90%. CWLFM refers to the continuous-wave linear frequency modulated signals. Signals with an asterisk (¤) come from the real-life collecting system. The overall ADCRx performance is better than the others (apart from the MF). Its processing gain with respect to the receiver based on the single STFT is due to the noncoherent integration. The improvement ranges from 1 to 7 dB. The DFT receiver is only suitable for very narrowband signals due to its narrow channelization. For instance, the 100 MHz/1 ms CWLFM sweeps a very small bandwidth in the analyzed interval (around 400 KHz). It should be noted that the DFT has the best performance for the MSKs herein used as well, since MSKs typically feature a very concentrated spectrum [25]. Obviously, the DFT will perform worse than the ADCRx for higher MSK symbol rates due to the increase in bandwidth. For signals with very broad bandwidths, the ADCRx can become worse than the ED since channelization no longer provides an improvement and the only improvement factor comes from the noncoherent integration (note that ED is a noncoherent integrator without channelization). Finally, note that neither the DFT nor the ED are able to extract temporal information about the signals
(TOA, PW and modulation). Furthermore, the ED is not suited to handle simultaneous signals. B.
Parameter Estimation Performance
Relative estimation errors of the in-block PDW parameters are in general below 20%, and, in most cases, below 10% for a SNR greater than 0 dB. c This is illustrated in Fig. 9, where the RMSE of PA, ˆ d f, and the chirp-rate estimate are depicted for PW, d the signals of Table III. Although not shown, TOA d performance is quite similar to that of PW. The exceptions to the general good performance are the broadband signals, i.e., broadband PSKs/FSKs c RMSE becomes and fast LFMs. For instance, PA relatively higher for the 10 MHz BPSK, since the estimator is not intended for signals occupying several channels. The ¡40 MHz/4 ¹s chirped pulse has such a fast modulation that the in-channel modulation analysis is not possible and the chirp-rate estimation must be estimated in the in-block analysis, which presents a greater error than the in-channel one. This chirped pulse was already used to illustrate the in-block processing (Section IVC, Fig. 8). Despite the overall good performance, the estimation errors do not always tend to zero as the c this is due to the channel SNR increases. For PA, insertion losses, which are not considered in the estimator because of the difficulty to be computed d and PW d for broadband signals. The estimates TOA are biased due to the decimation uncertainty and the window transient (see Appendix A). Concerning the chirp-rate estimation, a bias appears that increases as chirp-rate does. It is caused by the channelization, since the filtering of a linear chirp by the STFT adds a nonlinear frequency modulation to the linear chirp. Obviously, this additional term becomes zero for
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c c in Fig. 9. Parameter estimation for signals in Table III versus SNR. (a) RMSE of P A normalized to true value. (b) RMSE of PW samples (note that block length is 1024). PSKs and FSKs have same RMSE as CWLFMs. (c) RMSE of fˆ relative to channel bandwidth. (d) RMSE of chirp-rate relative to true value (only for LFM signals). nonmodulated signals. For the frequency estimation, this issue has been already discussed in Section IVA; nevertheless, the errors reported herein can be slightly greater than those presented in Section IVA due to the errors in the previous estimation of the beginning (ninit ) and ending (nend ) of the signal (ninit and nend were known a priori in the simulations of Section IVA). C. Modulation Recognition Performance In general, the probability of correct classification approaches 100% as the signal SNR increases. The SNR value reaching the 100% of correct classification depends on the signal and the system architecture. The classifier performance for the signals in Table III is shown in Table IV by means of the confusion matrix at 10 dB SNR. The SNR for 90% of probability of correct classification is provided as well. 890
The classifier thresholds have been set to favor the recognition of nonmodulated pulses. Thus, the required SNR for the 90% of correct classification appears very low for conventional pulses. Obviously, for LFM signals sweeping a narrow bandwidth within the analyzed time interval (e.g., the 100 MHz/1 ms CWLFM) the correct classification occurs at higher SNRs than for broader band LFMs. When misclassified, they are usually classified as nonmodulated. The case of short chirped pulses, like the 1 MHz/1 ¹s one, is discussed later in this section. Concerning the PSKs and FSKs, a degradation in performance occurs for those occupying several channels, so that 100% of correct classification could not be achieved. In those cases, the in-channel AMC cannot properly recognize the modulation since filtering destroys the modulation information carried in the phase. This is the case of the 3.3 MHz/150 ns MSK, whose 3 dB bandwidth is 7 MHz. Although not illustrated, another drawback caused by channelization
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Fig. 10. Modulation recognition for multicomponent signal composed of chirped pulse ¡40 MHz/4 ¹s and CW signal without modulation. Separation: 1 channel. (Probability conditioned to detection). (a) Multicomponent signal with several spectral separations. (b) Number of estimated PDW versus SNR. TABLE IV Confusion Matrix for Modulation Recognition (Conditioned to Detection) at 10 dB SNR Signal Pulse 500 ns Pulse 1 ¹s Chirped pulse 1 MHz/1 ¹s Chirped pulse 2 MHz/2 ¹s Chirped pulse ¡40 MHz/4 ¹s(¤) CWLFM 100 MHz/1 ms CWLFM 500 MHz/1 ms BPSK 2 MHz BPSK 4 MHz Barker-13 pulse 4.8 ¹s(¤) BPSK 10 MHz MSK 1 MHz/500 ns MSK 3.3 MHz/150 ns(¤)
No Modulation
LFM
PSK
FSK
SNR (dB) 90%
98% 97% 89% 0 0 62% 0 0 0 0 16% 0 2%
2% 3% 11% 100% 100% 38% 100% 0 0 0 0 0 15%
0 0 0 0 0 0 0 100% 100% 100% 82% 0 48%
0 0 0 0 0 0 0 0 0 0 2% 100% 35%
¡5 ¡1 – 4 4 14 ¡1 0 0 0 15 8 –
Note: Input SNR for 90% of right modulation classification is also provided. Signals with asterisk (¤) were collected from real-life systems in their typical operating environments. Symbol – means that the 90% probability cannot be achieved.
is the degradation in the modulation recognition when the PSK or FSK carrier frequency exhibits an important offset from the channel center frequency. Both problems can be overcome by reconstructing the signal using the Gabor expansion [6] of the STFT coefficients from the channels where it was detected. However, this operation can be very time consuming for fast signal interception. The performance for short chirped pulses, such as the 1 MHz/1 ¹s one, is very poor as well: there are not enough samples due to their relatively short duration and the use of the maximum decimation factor (M = 32). This problem may be overcome by the use of a lower decimation factor. For example, if M = 16 were used, a 90% of correct classification would be achieved at 20 dB SNR. Nevertheless, this reduction in the decimation approximately doubles the computational burden. Experiments also shows that, below the decimation factor K=4, no further improvement is achieved in the modulation
recognition due to the transient of the analysis window. Typical modulation recognition approaches, like Azzouz and Nandi’s [22] and Liedtke’s [26], are intended for isolated communications signals and require long records of them. Moreover, due to the processing gain of channelization, the use of the ADCRx results in a higher recognition performance in terms of SNR (except for the above-referred case of very broadband PSKs and FSKs). D. Multicomponent Example and Estimation of the Number of PDWs The ADCRx has also been tested for simultaneous signals. In this section two simultaneous signals are considered: a ¡40 MHz/4 ¹s chirped pulse and a nonmodulated CW signal in three relative arrangements with a 1-, 2- and 3-channel separation, respectively. Fig. 10(a) shows the probability of right
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modulation recognition for both signals. Only the results for 1-channel separation are displayed since the others are similar. Both signals have the same SNR. The modulation recognition is similar to the case of the signals separately considered except for high SNRs, at which both signals tend to occupy their adjacent channels, so that in-block clustering groups both signals in the same cluster. Due to the arrangement of the in-channel PDWs, the cluster tends to be classified as LFM. Thus, the probability of recognizing the modulation of the CW signal decreases. Likewise, the probability of estimating two PDWs, i.e., two signals, decreases with the SNR, whereas the probability of finding only one PDW increases (Fig. 10(b)). Regarding the estimation of the number of signals, it is just the number of in-block PDWs. Fig. 10(b) shows the probability of estimating one or two signals and the probability of detecting false signals (more than two PDWs). There are some important peaks caused by in-block clustering errors when processing fast LFMs at particular SNR values. For other signals, the in-block clustering performs properly, and shows a very low probability of estimating false signals. Nevertheless, small peaks in the error (usually below 10%) may appear at certain SNR values. E. Computational Load and Practical Implementation The novel features of the ADCRx slightly increase its computational load in comparison to the previous digital channelized architectures [3, 7, 8]. Simulations using MATLAB on a 1.2 GHz, 256 MB RAM, Pentium IV showed that the smoothed spectrograms increase the computational load only by 5%. It should also be noted that the in-channel processing of the encoder can be computationally as demanding as the previous ADCRx stages. This depends on the number of local detections in the detection stage and is similar to the results obtained in the simulation of other channelized architectures [3, 7, 8].9 Concerning the practical implementation of the ADCRx, a recent work [27] on the implementation of the fast Fourier transform (FFT) in a commercial field programmable gate array (FPGA) has shown pipelined architectures working up to 960 Msamples/s for different FFT sizes, and performing consecutive windowed FFTs of 64 samples every 70 ns with a cost in area lesser than 15%. The time-frequency processor of Table I, at a sampling rate of 250 MHz, 9 Nevertheless, the encoder’s computational load is expected to decrease when implemented in a real-time oriented computing language, since most of the time in the current version is devoted to transferring data among the modules forming the encoder software.
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can be implemented in one FPGA since, due to the decimation factor (M = 32), we require a time between consecutive FFTs of less than 128 ns. The rest of the FPGA area can be utilized to implement the smoothed spectrograms, the detection, the feature building, and even the heaviest part of the in-channel PDW estimation. To assure real-time operation, a few digital signal processors (DSP) can work in parallel afterwards to complete the in-channel PDWs and construct the in-block PDWs of each block of samples. This multiprocessor architecture represents a feasible real-time implementation of an ADCRx with an instantaneous bandwidth of 125 MHz. To facilitate the implementation, there are already FPGAs including a PowerPC processor, such as the Xilinx’s Virtex Pro family, and many manufacturers provide multiprocessor boards (see for example [28]).
VI.
CONCLUSIONS
This paper describes an ADCRx for automatic signal interception, whose design is based on time-frequency analysis. It uses an extension of the STFT consisting of a set of smoothed spectrograms obtained by noncoherent integration of the representation generated by the STFT. Noncoherent integration improves the processing gain and is carried out along the time axis. Different integration lengths are used in order to adapt to different signal durations. The ADCRx includes a novel encoder with the following notable aspects: 1) the use of two levels of processing (in-channel and in-block), 2) the fusion of the information from the various smoothed spectrograms for a particular channel, 3) the use of weighted DIFM averaging to obtain a robust frequency estimate independently of the modulation, 4) the AMC, and 5) the cancellation of out-of-channel detection at the in-block level. The ADCRx is an efficient algorithm, which is technologically feasible for fast signal interception (using a multiprocessor architecture, an instantaneous bandwidth of several hundreds of MHz can be analyzed in real time). It combines both the advantages of digital technology and channelized receivers, and outperforms previous approaches in terms of sensitivity and variety of signals that can be intercepted. Additionally, it is able to handle simultaneous signals with very different properties. Nevertheless, there are a number of limitations inherent to the channelization and the two-level encoder architecture: Modulation recognition of broadband PSKs and FSKs, estimation of extra PDWs for fast LFMs occupying many channels, and estimation errors for broadband signals.
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d can be written estimator PW
Fig. 11. Possible values of TOA for decimation factor M (M · K=2).
APPENDIX A. IN-CHANNEL TEMPORAL PARAMETER ESTIMATION The in-channel TOA and PW are estimated by using the parameters ninit and nend (eqns. (15) and (16)). In doing so, two issues should be taken into account: the uncertainty due to the decimation, and the transient due to the analysis window. Both problems and the solutions adopted in the paper are described as below. A. Decimation Uncertainty Using the approximate relation between time and bandwidth (time ¼ 1=bandwidth), the effective length of the STFT analysis window can be assumed to be equal to the total number of channels (K). Given a decimation factor of M samples, a typical situation for the pulse leading edge is shown in Fig. 11. Considering ninit (the first decimated sample of the STFT where the signal was detected), there is a range of possible TOAs depending on the window length and the decimation factor. This uncertainty is modeled here as a discrete uniform distribution. In particular, TOA becomes a random variable, defined as L=2 + (ninit ¡ 2)M + K=2 + Ud (1, M)
(20)
where L is the (true) window length, and Ud (1, M) stands for a discrete uniform random variable ranging from 1 to M. For ninit = 1, the expression is slightly different (Ud (1, (K + M)=2)). d utilizes the mean of the The TOA estimator (TOA) corresponding discrete uniform distribution, i.e., 8 2+L+K < , ninit = 1 4 d = TOA : L+K M ¡1 2
+ (ninit ¡ 1)M ¡
2
,
otherwise.
d = maxfM, PW0 g PW 8 (nend ¡ ninit + 1)M ¡ K, > > > > ninit 6= 1, nend 6= LF > > > > > > L=2 + (nend ¡ 1)M ¡ K=2, > > > > < ninit = 1, nend = LF : PW0 = > (L ¡ 3K)=4 + (nend ¡ ninit + 1)M ¡ M=2, > > > > > > ninit 6= 1, nend = LF > > > > > (L ¡ 3K)=4 + nend M ¡ M=2, > > : ninit = 1, nend 6= LF
(22)
B.
Analysis Window Transient
d and PW d from (21) and (22) The use of TOA results in an increase in the error for both parameters as SNR increases. This is due to the fact that the analysis window length (L) is greater than the number of channels (K). Figs. 12(a) and (b) show this effect for the configuration in Table I and a nonmodulated pulse of 500 samples. Note the staircase behavior of the TOA and PW RMSE (without correction) caused by the bias in the estimation of the TOA and the PW. The bias is due to the transients induced by the analysis window: The TOA estimate tends to occur earlier than the true value and the PW estimate becomes greater than the true one. The behavior of the TOA and PW estimates suggests the use of a staircase-like correction term for both ninit and nend before TOA and PW computation (by (21) and (22)). This correction depends on the SNR, ranges from 0 to L=2M (half of the maximum transient duration), and has to be adjusted for every system. For the system in Table I, this correction becomes 8 d < 15 0, if SNR > > > > > d < 20 < 1, if 15 · SNR : (23) > d < 40 > 2, if 20 · SNR > > > : d ¸ 40 3, if SNR
As can be noticed in Figs. 12(a) and (b), the correction results in a significant improvement in the estimation of both in-channel TOA and PW. Both figures also show the greater importance of the estimate bias over its variance, also caused by the two issues discussed above, i.e., decimation and window transient.
(21)
d = TEND d ¡ The PW estimate is defined as: PW d + 1, where TEND d is the pulse trailing edge, TOA d The which is computed in a similar way as TOA.
APPENDIX B. AMC THRESHOLD COMPUTATION This section summarizes the deduction of the analytical expressions for thresholds tha and thLFM
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Fig. 12. Mean error and RMSE of TOA and PW estimation with and without transient correction. ADCRx of Table I and nonmodulated pulse of 500 samples. (a) Error in TOA estimation. (b) Error in PW estimation.
of the in-channel AMC (Section IVB). Let x(r) be a narrowband signal corrupted by a real, zero-mean white Gaussian noise n(r), i.e., x(r) = A cos(2¼f0 r + Á(r)) + n(r)
(24)
where f0 is the carrier frequency, and Á(r) stands for the frequency/phase modulation. For the channel where the signal is, the phase of the decimated STFT can be approximated by argfSTFT(w) x (rM, k0 )g
(25) for high SNR [20]. This result is obtained by a first-order approximation. nÁk0 is the phase error, which follows a real, zero-mean Gaussian distribution. Its covariance depends on the aperiodic correlation of the analysis window, i.e., L¡1 Lf0 X w(r) ¢ w(r + mM) SNR
(26)
r=0
where Lf0 accounts for the insertion losses of channel k0 at frequency f0 . The SNR is defined at the input of the STFT, i.e., SNR = A2 =2¾2 . Due to the relationship between the STFT phase and the instantaneous frequency estimate (DIFM, Section III), the phase error gives rise to a real, zero-mean Gaussian frequency error. Assuming an R-dimensional vector of instantaneous frequency samples, fi = [fi (0), fi (M), : : : , fi ((R ¡ 1)M)]T
(27)
its covariance matrix turns out to be Cnf = ©CnÁ ©T
(28)
where CnÁ is the (R + 1)st-order covariance matrix of the corresponding phase error vector, and © is the 894
0
¢¢¢
¢¢¢
¢¢¢
0
1
0C C C .. C : .A
(29)
¡1 1
In the following, the mathematical expressions of both thresholds are treated. A. Threshold tha
¼ 2¼(f0 ¡ k0 =K)rM + Á(rM) + nÁk0 (rM)
CnÁ (mM) =
R £ (R + 1) conversion matrix 0 ¡1 1 0 B 0 ¡1 1 1 B B ©= . 2¼M B @ .. ¢¢¢
A least squares linear model for the instantaneous frequency vector of (27) is given by · ¸ aˆ = A+ fi (30) bˆ where A+ is the Penrose pseudoinverse of 1 0 0 1 B M 1C C B A=B .. .. C C B @ . .A (R ¡ 1) ¢ M
(31)
1
ˆ and aˆ , bˆ are the model parameters: fi (rM) = aˆ rM + b. Provided that fi follows a linear model plus a Gaussian noise of covariance (28), the estimate aˆ also exhibits a Gaussian distribution with variance ¾a2ˆ = A+ (1)Cnf (A+ (1))T
(32)
where A+ (1) means the first row of A+ . If the signal does not have modulation, the mean of aˆ becomes zero. Therefore, the threshold tha has the following expression q tha = ca A+ (1)Cnf (A+ (1))T (33) and ca is set to minimize the error probability (Appendix C).
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B. Threshold thLFM
A. PFA Thresholds
From the least squares equation (30), the linear model error " becomes
For signal detection, there is a threshold thi for every smoothed spectrogram Ii . The set of thresholds fthi gFi=1 is defined to meet a specified global PFA or PFAg (eqn. (7)). On the one hand, this requirement is not fulfilled for a unique set of thresholds; hence, the additional condition that the local PFA (eqn. (8)) is the same for all the smoothed spectrograms is imposed. On the other hand, an analytical expression relating the local PFA and PFAg is rather difficult to find, since the points of the smoothed spectrograms are correlated, and this correlation depends on the time-frequency processor configuration. Therefore, the correspondence between the local PFA and the PFAg is determined by simulation. Let the local PFA value be denoted p.
"=
1 (f ¡ AA+ fi )T (fi ¡ AA+ fi ): R i
(34)
It can be equivalently written as a quadratic form of the frequency error vector (nf ), i.e., "=
1 T n (I ¡ AA+ )T (I ¡ AA+ )nf : R f
(35)
Therefore, the mean of " becomes ¹" =
1 Tr((I ¡ AA+ )T (I ¡ AA+ )Cnf ) R
(36)
where Tr means the trace operator. Diagonalizing Cnf , Cnf = V¤VT , and considering the unitary transformation nf = VT vf , " can be expressed as "=
1 T v V(I ¡ AA+ )T (I ¡ AA+ )VT vf : R f
(37)
Thus, the variance of " becomes 2 f(diag(¤))T Bu − Bu diag(¤)g R2 (38) where Bu is the unitary-transformed matrix ¾"2 = E["2 ] ¡ ¹2" =
Bu = V(I ¡ AA+ )T (I ¡ AA+ )VT
(39)
Bu − Bu means element-by-element multiplication, and diag(¤) is the diagonal of the eigenvalue matrix. Using (36) and (38), the threshold becomes thLFM = ¹" + cLFM ¾" :
(40)
Constant cLFM is set to minimize the error probability (see Appendix C). Both tha and thLFM depend on the d signal SNR and duration, so that the estimates SNR d and PW will be used in practice. APPENDIX C. SIMULATION-BASED THRESHOLD COMPUTATION This appendix briefly describes the computation of some of the most important thresholds utilized by the ADCRx. In particular, it is explained the procedure to set the thresholds for signal detection on the different smoothed spectrograms (Section IIIA) so that a given global PFA can be met; the thresholds for the classification of signals according to their duration (Section IVA); and the constants ca (of threshold tha ) and cLFM (of threshold thLFM ), and the threshold th¢f of the AMC (Section IVB and Appendix B).
1) The local PFA axis is explored for different values of p. 2) For each value p, the thresholds fthi gFi=1 corresponding to every smoothed spectrogram are analytically obtained by Maranda’s method [18]. 3) The ADCRx is simulated with an adequate number of Monte Carlo realizations by using only noise at the input (only the time-frequency processor and the detection system must be simulated). By means of those Monte Carlo simulations, the PFAg corresponding to this value of p can be estimated. Once the curve PFAg versus p is constructed, the required value of p for a specified PFAg is obtained, and so the required thresholds fthi gFi=1 . For low-sidelobe analysis windows and decimation factors (M) close to the maximum (K=2), it has been found by simulation that the smoothed spectrogram coefficients are nearly independent; a low-sidelobe window approximately guarantees independence among frequency channels, and a high decimation factor approximately provides independence in time. Thus the following analytical correspondence between PFAg and the local PFA value (p) holds: PFAg ¼
µ
¶ F X LF K ¡1 ¢p¢ 2 Li
(41)
i=1
and the above-described simulation procedure is avoided in this case. B.
Thresholds for Duration-Based Signal Classification
The thresholds treated in this and the following section have been obtained following a common methodology. Every threshold is used to distinguish between two hypotheses (hypotheses Ha and Hb ) in the framework of a hypothesis test: Hb
©(x) ? th©
(42)
Ha
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where ©(x) is a given statistic of the input signal x to be tested and th© is the threshold to be determined. To obtain its value, we follow the Bayes criterion, i.e., minimize the probability of error. Since the probability of error depends on the signal and the SNR considered, several signals representing each hypothesis (Ha and Hb ) are evaluated for different SNR values. For each signal, the probability of error is defined as the mean of the probability of error computed at different SNR values. The probability of error at a given SNR is the probability that the signal is misclassified when it holds this particular SNR. This probability is computed by Monte Carlo analysis. Once the probability of error for every selected signal is obtained (after averaging for the designated SNR values), the error probability for each hypothesis is the mean of the probability of error of all the signals representing this hypothesis. That is, the signals representing a hypothesis are assumed to be equiprobable. Finally, both hypotheses are assumed to be equiprobable as well; therefore, the total probability of error is computed as the mean of the probability of error of both hypotheses. To obtain the adequate threshold value, the total probability of error is obtained for a range th© , and the one corresponding to a minimum probability of error is selected as the desired threshold. In the case of the thresholds fthPAi¡1,i gFi=2 , they are used to separate signals of similar duration to the length of the smoothed spectrogram Ii from signals of shorter duration according to the hypothesis test c H : Shorter than Ii PA i¡1 b ? th : c Ha : Duration as Ii PAi¡1,i PA
(43)
i
For each threshold thPAi¡1,i , the signal representing the hypothesis Ha is a nonmodulated pulse lasting K + M(Li ¡ 1) samples; K, M, and Li are the total number of channels, the decimation factor and the smoothed spectrogram integration length, respectively. The parameter K accounts for the effective length of the analysis window (the definition and motivation of the effective length can be found in Appendix A). The representative of the hypothesis Hb is also a nonmodulated pulse of duration: K + M(Li¡1 ¡ 1) samples; Li¡1 is the integration length of the smoothed spectrogram Ii¡1 . Note that Li is greater than Li¡1 , according to the definition of the time-frequency processor. The frequency of both pulses is centered in a channel and their respective probabilities of error are computed for only one SNR value: the one assuring a 90% detection probability for the shortest pulse. Below this value of SNR, the errors in the estimation of the PA tend to be very high, and it is not possible to achieve low probability of errors. For greater SNR values, the errors in the PA estimate decrease, so that the probability of error will decrease as well and become very small. 896
For the thresholds fthPAi¡1,i gFi=2 , we have used nonmodulated pulses as representatives 1) for the sake of simplicity, and 2) due to the fact that the errors in the PA estimation for modulated signals do not differ much from the errors for nonmodulated pulses if their bandwidths are limited to a channel (see Fig. 9). Thus, they would give rise to similar probabilities of error and thresholds. C. AMC Constant Determination The computation of the constants cLFM and ca , and the threshold th¢f is similar to the method presented in the previous section. The constant cLFM is part of the threshold to distinguish LFM-modulated and nonmodulated signals from PSK- and FSK-modulated ones (Section IVB). This threshold is defined as: thLFM = ¹" + cLFM ¾" ; the values for ¹" , ¾" were obtained in Appendix B. The corresponding hypothesis test is "
Hb : PSK or FSK
?
Ha : LFM or No Modulation
thLFM
(44)
where " is the error of the least-squares linear fit of the frequency estimate (Section IVB). The representatives of the hypothesis Ha are nonmodulated and chirped pulses with different durations; in the case of the chirped pulses, their bandwidth is equal to the channel bandwidth. For the ADCRx in Table I the nonmodulated and chirped pulses have 1024 (the block length) and 500 samples duration. For lower durations, the signal length estimation would be smaller than the minimum length to perform the modulation analysis (see in Table II the minimum d > SNR = length plus the guard samples when SNR th 3 dB); the signal would then be classified as a nonmodulated pulse. The pulses are centered in time in the block, and their mean frequency is the channel center frequency. We have selected four SNR values: 5, 10, 15, and 20 dB over the SNR at which a single DIFM features a 5% error with respect to the channel bandwidth for a sinusoid (in this case, this SNR value is 0 dB approximately according to Fig. 6). As the hypothesis Hb representatives, BPSKs and MSKs with smaller bandwidth than the channel are chosen. The BPSKs’ symbol periods are 64 samples (3 dB bandwidth equal to the channel) and 1024 samples (the full block length; but at least one bit transitions must be assured in within the block). The MSKs feature symbol durations of 64 samples (approximately 99% of the signal energy within the channel [25]) and 512 samples (separation in frequency equal to 1=1024, which would be the channel bandwidth if the whole processing were coherent, i.e., by a 1024-point DFT). The signals are present during the whole sample block and their
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 41, NO. 3
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carrier frequencies are centered in a channel. The SNR values are the same as in the hypothesis Ha . The constant ca is associated with the threshold that helps to distinguish LFM-modulated from nonmodulated signals (Section IVB). This threshold is defined as: tha = ca ¾aˆ (see Appendix B for a definition of ¾aˆ ), and the corresponding hypothesis test is aˆ
[8]
[9]
Hb : LFM
?
Ha : No Modulation
tha :
(45)
The representatives of the hypothesis Ha are the nonmodulated pulses used in the computation of cLFM , and the representatives of the hypothesis Hb are the chirped pulses also used to compute cLFM plus two chirped pulses both with bandwidth 1=1024 (the channel bandwidth if the whole processing were coherent, i.e., if a 1024-point DFT were used) and duration 1024 and 500, respectively. These new signals represent the smallest LFM modulation that should be detected. The SNR values are the same as for the constant cLFM . Threshold th¢f is defined as a constant for the sake of simplicity, and is utilized to distinguish PSKfrom FSK-modulated signals (Section IVB). The corresponding hypothesis test is max j¢f(n)j
[7]
Hb : PSK
?
Ha : FSK
th¢f
[10]
[11]
[12]
[13]
[14]
(46)
where ¢f(n) is the first-order difference of the DIFM sequence. The representatives of the hypothesis Hb are the same BPSK signals used to compute cLFM , and the representatives of the hypothesis Ha are the MSK signals also used to compute cLFM . The SNR values are the same as well.
[15]
[16]
[17]
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Sa´ nchez-Marcos, M. A., Garrido-Ga´ lvez, M., Lo´ pez-Vallejo, M. L., and Grajal, J. Implementing the FFT algorithm on FPGA platforms: A comparative study of parallel architectures. In XIX International Conference on Design of Circuits and Integrated Systems (DCSI 2004), Bourdeax, France, Nov. 2004. Dy4 Systems Digital Signal Processors Catalog (July 2004). http://www.dy4.com/products/dsp/index.htm.
˜ (SM’99–M’03) was born in Barcelona, Spain, in Gustavo Lo´ pez-Risueno 1974. He received the Ingeniero de Telecomunicacio´ n and the Ph.D. degrees from Universidad Polite´ cnica de Madrid, Madrid, Spain, in 1998 and 2002, respectively. From 2000 to 2003, he was an assistant professor at the Department of Signals, Systems and Radiocommunications, Universidad Polite´ cnica de Madrid, working in the areas of statistical signal processing for active and passive radars, and passive radar design and technology. He spent the fall of 2000 and the summer of 2002 as a visiting researcher at the Adaptive System Lab, McMaster University, Hamilton, Canada. Since 2004, he has been with the European Space Research and Technology Centre (ESTEC) of the European Space Agency (ESA), Noordwijk, Netherlands, working on advanced receivers for global navigation satellite systems. His current research interests include statistical signal processing, spread spectrum communications, and global navigation satellite systems.
´ Grajal was born in Toral de los Guzmanes (Leo´ n), Spain, in 1967. He Jesus received the Ingeniero de Telecomunicacio´ n degree and the Ph.D. degree from the Technical University of Madrid, Madrid, Spain in 1992 and 1998, respectively. Since 2001 he has been associate professor at the Signals, Systems, and Radiocommunications Department of the Technical School of Telecommunication Engineering of the same university. His research activities are in the area of hardware design for radar systems, radar signal processing, and broadband digital receivers for radar and spectrum surveillance applications.
´ Alvaro Sanz-Osorio was born in Madrid, Spain, in 1979. He received his telecommunication engineering degree from the Technical University of Madrid (UPM) in 2003, where he carried out his master thesis on the design of a digital radar receiver at the Microwaves and Radar Group (GMR). His research activities are focused on the area of signal processing for radar receivers. 898
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