MATLAB Exercise

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 =π83π8 ω c 8 3 8 .

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 ?