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Find the DFT of the sequence x[n] = {1, 2, 3, 4, 4, 3, 2, 1} using radix-2 decimation in time FFT algorithm.
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Solution:

Given ={1,2,3,4,4,3,2,1}

We knowthat WkN=ej(2πN)k

Given N=8

Hence, W18=ej(2π8)0=1

=W18=ej(2π8)1=cosπ4jsinπ4=0.707j0.707

=W28=ej(2π8)2=cosπ2jsinπ2=j

=W38=ej(2π8)3=cos3π4jsin3π4=0.707j0.707

enter image description here

X(k)={20,5.828j2.414,0,0.172j0.414,0,0.172+j0.414,05.828+j2.414}X(k)={2,0.5j1207,0,0.5j0.207,0,0.5j0.207,0,0.5j1.207}

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