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Since the DFT is a sampled version of the spectrum of a digital signal, it has certain sampling effects. To explorethese sampling effects more thoroughly, we consider the effect of multiplying the time signal by different window functionsand the effect of using zero-padding to increase the length (and thus the number of sample points) of the DFT. Using thefollowing MATLAB script as an example, plot the squared-magnitude response of the following test cases overthe digital frequencies .
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 betweenthe DFT, , and the DTFT, , of a sequence ?
Download and run the MATLAB file
lab4b.m to observe direct and
autocorrelation-based PSD estimates. A pseudo noise generator isfiltered 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. Theautocorrelation (AC) method is to take the magnitude of the FFT of the autocorrelation of the sequence. In this case rectangular windows wereused in both FFTs. Why do the estimates look exactly the same? Will theestimates 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 directand AC estimates are computed in the same manner described above, except a hamming window has been applied to the sequence in thedirect-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-PSDestimate is computed by summing the hamming-windowed PSD estimates of each 32-sample block. The AC-PSD estimate is computed by taking themagnitude of the FFT of 63 samples of the autocorrelation of the entire 1024-point sequence. Why are 63 samples used in comparing theAC 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 (
) and
direct-PSD window types. Observe the changes. Are there anytradeoffs? Does either the direct method or the autocorrelation method
have an advantage over the other? Is there a direct-method window thatresults in identical block-based estimates?
For simplicity, the autocorrelation window in this lab isrectangular. Can we, however, consider this an unbiased autocorrelation
estimate that is windowed by a rectangular window? (Hint: see theMATLAB documentation for the
xcorr function.) If not,
how would you create an unbiased autocorrelation estimate beforewindowing, and what is the effective window that we have applied to
the unbiased autocorrelation?
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