Homework 4

Textbook exercises: [PDF]

M3.6: Matlab diary, see the figure below:

% Verifying the symmetry relations.  Using the sequence defined for
% example 3.8 in the book:
num = [.008 -.033 .05 -.033 .008];
den = [1 2.37 2.7 1.6 .41];
% but will plot it for -pi<w<pi so we can see the symmetry
w = -pi:pi/k:pi;
k = 256;
w = -pi:pi/k:pi;
h = freqz(num, den, w);
subplot 221
plot(w/pi, real(h))
subplot 222
plot(w/pi, imag(h))
subplot 223
plot(w/pi, abs(h))
subplot 224
plot(w/pi, unwrap(angle(h)))
% even symmetry of real and magnitude; odd symmetry of imag and phase.

M3.8 . I used the following script to generate the plots below, which show that the DFT can be evaluated by samplign the DTFT:

% M3.8
% Adam Berenzweig

% pulse length 
M = 21;
halfM = (M-1)/2;
% overall sequence length (padded)
N=1001;

k = 0:N-1;
w = 2*pi*k/N;

% rectangular pulse
%xn = [zeros(1,(N-M)/2) ones(1,M) zeros(1,(N-M)/2)];

% triangular pulse
n=-(M-1)/2:(M-1)/2;
xn = [zeros(1,(N-M)/2) 1-abs(n)/((M-1)/2) zeros(1,(N-M)/2)];

% by FFT
length(xn)
wn1 = fft(xn);

% by evaluation of DTFT
%wn2 = sin(M/2*w)./sin(w/2);  % for rect
wn2 = (sin(halfM/2*w)./(halfM*sin(w/2))).^2;  % for triangle

subplot 211
plot(w/pi,abs(wn1))
%title(['rectangular pulse, N=' num2str(halfM)]);
title(['triangular pulse, N=' num2str(halfM)]);
xlabel('via fft')
subplot 212
plot(w/pi,abs(wn2))
xlabel('via evaluated DTFT')

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