Chapter (5) Finite- Length Discrete Fourier...
Transcript of Chapter (5) Finite- Length Discrete Fourier...
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Chapter (5)
Finite- Length Discrete Fourier Transform
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Proof of the Orthognality Property
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N – Point DFT and IDFT
Verifying : Examples 5.1 and 5.2
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Verifying : Examples 5.3 and 5.4
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0 5 10 150
2
4
6
8Magnitude of the DFT samples
Frequency index k
Mag
nitu
de
0 5 10 15-2
-1
0
1
2Phase of the DFT samples
Frequency index k
Phas
e
0 1 2 3 4 5 6 70
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Original time-domain sequence
Time index n
Ampl
itude
u = [ones(1,N)];
N=8, the length of the sequence
U = fft(u,M)
M=16, the length of the DFT
stem(t,u)
stem(k,abs(U))
stem(k,angle(U))
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v = ifft(V,N)
Read in the length K=8 of the DFT and the desired % length N =13 of the IDFT
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% Program 5_3 % Numerical Computation of Fourier transform Using DFT k = 0:15; w = 0:511; x = cos(2*pi*k*3/16);% Generate the length-16 sinusoidal sequence X = fft(x); % Compute its 16-point DFT=fft(x,16) XE = fft(x,512); % Compute its 512-point DFT % Plot the frequency response and the 16-point DFT samples plot(k/16,abs(X),'o', w/512,abs(XE)) xlabel('\omega/\pi'); ylabel('Magnitude')