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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 ?
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