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