Homework Eleven

Let . This is not a bandlimited signal. Let . In this problem you will explore the DFT for various combinations of sampling rates and zero padding.

1. Make a plot of for < f < 5$" align="middle" border="0" height="30" width="72" /> Hz. (where .) For example,

   freqrange = 0:.1:5;   % make the range of frequency variables
   Xf = 1 ./ sqrt((2*pi*freqrange).^2 + 1);  % magnitude Fourier tranform
   plot(freqrange,Xf);             % plot F.T.
   

   You should overlay this plot over every spectrum computed below for comparison.
   
2. Let be the sampling interval (i.e., ). Let , and sample 128 points of the function. Then plot the FFT of the sampled data:

   N0=128;  % number of sample points
   T = 0.1; % sample interval
   Fs = 1/T; % sample rate
   k = (0:(N0-1));  % index range
   k_2 = (0:(N0/2-1));  % half the points
   x = T*exp(-(k*T));   % the sampled function.  (Scale by T)
   xout = fft(x,N0);    % compute the FFT
   plot(k_2*Fs/N0, abs(xout(1:(N0/2))),'-',freqrange, Xf,':');
                        % plot sampled spectrum and true spectrum
   % Note: the factor fs/N0 converts from bin number to frequency in Hz.
   

   Print the plot. Comment on discrepancies between the true spectrum and the spectrum computed by the DFT.
3. Now try some variations.
   1. Take 128 point of data, with zero-pad to 512 points.
   2. Take 128 points of data, with .
   3. Take 128 points of data, with .
   4. Take 512 points of data, with
   5. Take 512 points of data, with
   Turn in appropriate plots for each variation. Interpret all the results in light of aliasing and spectral leakage.
   

1. Using MATLAB, find the linear convolution of and :

   x = ones(1,16);   % set up x data
   y = [1 2 3 4 5 6 7 8 7 6 5 4 3 2 1];  % set up y data
   z = conv(x,y);    % convolve
   stem(0:length(z)-1,z);   % plot the convolved results
   

2. Now test the convolution theorem by taking transforms of the signals:

   X = fft(x,32);  % take the DFT of x data
   Y = fft(y,32);  % take the DFT of y data
   Z = X .* Y;     % Multiply.  .* is point-for-point multiply
   znew = real(ifft(Z,32));  % compute the inverse transform
                             % use real() to get rid of the imaginary part
                             % (which should be very small)
   stem(0:length(znew)-1,znew);  % plot the convolution.
   

3. Compare the plot from part (a) with the plot from part (b).
4. Now we will examine cyclical convolution by dealing with 16 point FFTs.

   X1 = fft(x,16);
   Y1 = fft(y,16);
   Z1 = X1 .* Y1;
   z1 = ifft(Z1,16);
   stem(0:15, real(z1));
   

   Using cyclical convolution, account for this strange result.