Question

| The 8-point DFT of a sequence x[n] is X[k]=102-1047k, Osk57. Use the inverse DFT in MATLAB to find the sequence x[n]. Turn in a copy of your code and the output generated.

Answer 1

Given the 8-point DFT of a sequence x[n] is X[k] = 10*exp(-j0.4*pi*k) where, 0 <= k <= 7.

To find the input sequence x[n] we compute X[k] first and then compute the IDFT of the obtained sequence. MATLAB uses Inverse Fast Fourier Transform algorithm which is much faster than normal DFT and IDFT computations. Hence the function 'ifft' is used here:

MATLAB code:

clear all;
k=0:1:7;
X(k+1)=10*exp(-j*0.4*pi*k);
x = ifft(X);
x_round = round(x,4);

x[n] = { 0.625000000000001 - 1.92355221073453i ,  -0.399552988663392 - 5.07680207690756i , 3.45009188833763 + 6.77118658500953i , 1.73012209374865 + 1.47766386376434i , 1.39754248593737 + 0.454089080003351i , 1.21883234271932 - 0.0959241857050588i , 1.07245059759974 - 0.546440872802577i , 0.905513580320682 - 1.06022018262748i } for n = 0 to 7.

The above is the obtained sequence, rounding it off to four decimal places we get,

x_round[n] = { 0.6250 - 1.9236i   -0.3996 - 5.0768i   3.4501 + 6.7712i   1.7301 + 1.4777i   1.3975 + 0.4541i   1.2188 - 0.0959i   1.0725 - 0.5464i   0.9055 - 1.0602i } for n = 0 to 7.

Note:- X(k+1) is used instead of X(k) because MATLAB doesn't support zero indexing and 'i' is used to represent complex numbers instead of 'j' while displaying results in MATLAB.

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