Section 4 Real and Imaginary Parts It is well known that the DFT components of a

Question

Section 4 Real and Imaginary Parts It is well known that the DFT components of a cosine function are purely real and that of a sine function are purely imaginary. The following 8-point sequences are given for experiment. 0 Practice 1. To simulate the signal xi and find its DFT: >> XI =fft(x1) Perform the same operations on x2, x3, and x4 2. Ignore any slight computer round-off errors, which one of these FFTs is purely real? Which is purely imaginary? 3. According to the above results, deduce a rule by which we can tell whether a real signal has a purely rea l DFT or a purely imaginary DFT.

Explanation / Answer

x = [1 1 1 1 0 0 0 0];
N = length(x);
X = zeros(4,1)
for k = 0:N-1
   for n = 0:N-1
       X(k+1) = X(k+1) + x(n+1)*exp(-j*pi/2*n*k)
   end
end

t = 0:N-1
subplot(311)
stem(t,x);
xlabel('Time (s)');
ylabel('Amplitude');
title('Time domain - Input sequence')

subplot(312)
stem(t,X)
xlabel('Frequency');
ylabel('|X(k)|');
title('Frequency domain - Magnitude response')

subplot(313)
stem(t,angle(X))
xlabel('Frequency');
ylabel('Phase');
title('Frequency domain - Phase response')

X         % to check |X(k)|
angle(X) % to check phase

....................................................

Similarly x2 = [1 1 -1 0 1 0 -1 1 ]

x3 = [1 1 1 1 -1 -1 -1 -1]

x4 = [0 1 1 1 0 -1 -1 -1]

In above code in place of x use these to obtain for others.

...................................................................................................

If x[n] is real and circularly even, then X[k] is also real and circularly even.

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