EE 341 Lab 4

Lab 4: Properties of DTFT EE 341 – Winter 2013 Emily Allstot (Section AA)

Assignment 1 In Assignment 1, animal sounds in .wav format are read and their time-domain waveforms are plotted directly from the data, along with their frequency spectrum. The discrete-time Fourier transform (DTFT) is used to find the frequency spectrum of each animal sound, which is possible because the sounds are discrete in the time-domain while being continuous in frequency using the built-in MATLAB function freqz(), as detailed in the provided Lab 4 specifications (Lab4.pdf). Table 1. Animal sounds table

Name

Fs

cat.wav bird.wav toad.wav tiger.wav whale.wav

22050 22050 22255 22255 16000

Length (sec) 1.00 1.14 0.67 0.753 3.45

Length (Samples) 22052 25197 14843 16763 55190

Comments on what you see from the DTFT magnitude

Using the max() built-in function, the fundamental frequency (first peak of the magnitude response) of the cat sound occurs at occurs at 733 Hz (0.209 rad). The precise amplitude value of the fundamental frequency is 379.79. These values are confirmed by the bottom two graphs of Figure 1.

Figure 1. Cat sound: time sequence, DTFT (rad), and DTFT (Hz)

Figure 2. Bird sound: time sequence, DTFT (rad), and DTFT (Hz)

Figure 3. Toad sound: time sequence, DTFT (rad), and DTFT (Hz)

Figure 4. Tiger sound: time sequence, DTFT (rad), and DTFT (Hz)

Figure 5. Whale sound: time sequence, DTFT (rad), and DTFT (Hz)

Assignment 2 Using a given file with Microsoft Stock prices over time, convolution is performed using the MATLAB built-in filter() function. The following given moving-average filter is used to smooth the stock:

The original stock and the resultant smooth filtered stock are both shown in the top plot in Figure 6. The frequency spectrum of h[n] is shown by the DTFT magnitude in the bottom plot in Figure 6, which is.

Figure 6. Original and smoothed stock prices, frequency response

The moving average filter makes the stocks look smoother because it is a LPF – see how in the DTFT, the majority of the signal power is preserved near baseband frequency, but higher frequencies are filtered out. So that is why using this LPF has smoothed out the high-frequency “micro-fluctuations” in stock prices.

Assignment 3 Assignment 3 implements a digital graphic equalizer on a given music.wav file, and the role of dB magnitude to characterize gain and using freqz() with sampling rate as input is observed. If stereo (two channel) sound lasts for 10 seconds, the number of samples one would expect to see is 882,000, as calculated using the following formula:

Additionally, in order to transmit at this level of CD audio quality without compression, it would take 1,411,200 bits, as calculated using the following formula:

Table 2. Filter coefficients and type of filter

Filter coefficients Filter 1 Filter 2 Filter 3

B1 = 0.0495 0.1486 A1 = 1.0000 -1.1619 B2 = 0.1311 0 A2 = 1.0000 -0.4824 B3 = 0.0985 -0.2956 A3 = 1.0000 0.5722

0.1486 0.0495 0.6959 -0.1378 -0.2622 0 0.8101 -0.2269 0.2956 -0.0985 0.4218 0.0563

LP, HP, or BP? LP 0.1311 0.2722

BP HP

Figure 7. Simple 3-bands equalizer filter frequency responses

While performing subband filtering, the three filters’ gains are changed to G1 = 0.1, G2 = 0.1, and G3 = 10. This means that the 1st and 2nd filters are attenuated as much as possible (-20 dB) for a typical equalizer while the 3rd filter is boosted to the max (+20 dB) for a typical equalizer. As expected, these new weights for each filter make the song sound much more “tinny” and lacking bass tones – a lot like a cheap and/or small speaker sounds. The gain of each filter is defined by the following formula:

Thus, the gain of each filter is as follows: Filter 1: -20.1 dB Filter 2: -36.6 dB Filter 3: -7.73 dB The reason that the output of each filter isn’t heard as a distinct source is because they are implemented in parallel, attenuating or boosting their own range frequencies. The sound that is heard at the output of the equalizer is the superposition of the effect of all three filters on the same one sound.