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70 Chapter 3 THE DISCRETE-TIME FOURIER ANALYSIS
>> X = x * (exp(-j*pi/500)).^(n’*k); % DTFT of x
>> % verification
>> X_check = alpha*X1 + beta*X2; % Linear Combination of X1 & X2
>> error = max(abs(X-X_check)) % Difference
error =
7.1054e-015
Since the maximum absolute error between the two Fourier transform arrays
is less than 10 −14 , the two arrays are identical within the limited numerical
precision of MATLAB.
□
□ EXAMPLE 3.8 Let x(n) bearandom sequence uniformly distributed between [0, 1] over 0 ≤
n ≤ 10 and let y(n) =x(n − 2). Then we can verify the sample shift property
(3.6) as follows.
>> x = rand(1,11); n = 0:10;
>> k = 0:500; w = (pi/500)*k;
>> X = x * (exp(-j*pi/500)).^(n’*k); % DTFT of x
>> % signal shifted by two samples
>> y = x; m = n+2;
>> Y = y * (exp(-j*pi/500)).^(m’*k); % DTFT of y
>> % verification
>> Y_check = (exp(-j*2).^w).*X; % multiplication by exp(-j2w)
>> error = max(abs(Y-Y_check)) % Difference
error =
5.7737e-015
□
□ EXAMPLE 3.9 To verify the frequency shift property (3.7), we will use the graphical approach.
Let
x(n) =cos(πn/2), 0 ≤ n ≤ 100 and y(n) =e jπn/4 x(n)
Then using MATLAB,
>> n = 0:100; x = cos(pi*n/2);
>> k = -100:100; w = (pi/100)*k; % frequency between -pi and +pi
>> X = x * (exp(-j*pi/100)).^(n’*k); % DTFT of x
%
>> y = exp(j*pi*n/4).*x; % signal multiplied by exp(j*pi*n/4)
>> Y = y * (exp(-j*pi/100)).^(n’*k); % DTFT of y
% Graphical verification
>> subplot(2,2,1); plot(w/pi,abs(X)); grid; axis([-1,1,0,60])
>> xlabel(’frequency in pi units’); ylabel(’|X|’)
>> title(’Magnitude of X’)
>> subplot(2,2,2); plot(w/pi,angle(X)/pi); grid; axis([-1,1,-1,1])
>> xlabel(’frequency in pi units’); ylabel(’radiands/pi’)
>> title(’Angle of X’)
>> subplot(2,2,3); plot(w/pi,abs(Y)); grid; axis([-1,1,0,60])
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