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# Lab 4: Prelab

Module by: Thomas Shen, Douglas L. Jones. E-mail the authors

Summary: You will investigate the effects of windowing and zero-padding on the Discrete Fourier Transform of a signal, as well as the effects of data-set quantities and weighting windows used in Power Spectral Density estimation.

## MATLAB Exercise, Part 1

Since the DFT is a sampled version of the spectrum of a digital signal, it has certain sampling effects. To explore these sampling effects more thoroughly, we consider the effect of multiplying the time signal by different window functions and the effect of using zero-padding to increase the length (and thus the number of sample points) of the DFT. Using the following MATLAB script as an example, plot the squared-magnitude response of the following test cases over the digital frequencies ω c = π8 3π8 ω c 8 3 8 .

1. rectangular window with no zero-padding
2. hamming window with no zero-padding
3. rectangular window with zero-padding by factor of four (i.e., 1024-point FFT)
4. hamming window window with zero-padding by factor of four

Window sequences can be generated in MATLAB by using the boxcar and hamming functions.



1  N = 256;                % length of test signals
2  num_freqs = 100;        % number of frequencies to test
3
4  % Generate vector of frequencies to test
5
6  omega = pi/8 + [0:num_freqs-1]'/num_freqs*pi/4;
7
8  S = zeros(N,num_freqs);                 % matrix to hold FFT results
9
10
11  for i=1:length(omega)                   % loop through freq. vector
12     s = sin(omega(i)*[0:N-1]');          % generate test sine wave
13     win = boxcar(N);                     % use rectangular window
14     s = s.*win;                          % multiply input by window
15     S(:,i) = (abs(fft(s))).^2;           % generate magnitude of FFT
16                                          % and store as a column of S
17  end
18
19  clf;
20  plot(S);                                % plot all spectra on same graph
21



Make sure you understand what every line in the script does. What signals are plotted?

You should be able to describe the tradeoff between mainlobe width and sidelobe behavior for the various window functions. Does zero-padding increase frequency resolution? Are we getting something for free? What is the relationship between the DFT, Xk X k , and the DTFT, Xω X ω , of a sequence xn x n ?

## MATLAB Exercise, Part 2

Download and run the MATLAB file lab4b.m [link] to observe direct and autocorrelation-based PSD estimates. A pseudo noise generator is filtered with a fourth-order IIR filter and various PSD estimates are computed and plotted.

The first plot contains PSD estimates, using a 1024-point FFT, from the first 512 samples of the 1024-sample sequence. The direct method is to take the squared-magnitude of the FFT of the sequence. The autocorrelation (AC) method is to take the magnitude of the FFT of the autocorrelation of the sequence. In this case rectangular windows were used in both FFTs. Why do the estimates look exactly the same? Will the estimates be alike if all 1024 samples are used with a 1024-sample FFT, with all other conditions being equal? Why or why not?

The second plot contains PSD estimates, using a 1024-point FFT, from the first 32 samples of the 1024-sample sequence. The direct and AC estimates are computed in the same manner described above, except a hamming window has been applied to the sequence in the direct-PSD estimate. Why are these estimates different? What will make these estimates identical?

The third plot contains PSD estimates, using a 1024-point FFT, from all 32-sample blocks of the 1024-sample sequence. The direct-PSD estimate is computed by summing the hamming-windowed PSD estimates of each 32-sample block. The AC-PSD estimate is computed by taking the magnitude of the FFT of 63 samples of the autocorrelation of the entire 1024-point sequence. Why are 63 samples used in comparing the AC method to the direct method?

The fourth plot contains the unfiltered spectrum of the PN sequence and the impulse response of the coloring filter.

Try various the block lengths (NpNp) and direct-PSD window types. Observe the changes. Are there any tradeoffs? Does either the direct method or the autocorrelation method have an advantage over the other? Is there a direct-method window that results in identical block-based estimates? For simplicity, the autocorrelation window in this lab is rectangular. Can we, however, consider this an unbiased autocorrelation estimate that is windowed by a rectangular window? (Hint: see the MATLAB documentation for the xcorr function.) If not, how would you create an unbiased autocorrelation estimate before windowing, and what is the effective window that we have applied to the unbiased autocorrelation?

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