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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

$ \begin{gathered} X(k)=\{20,-5.828-j 2.414,0,-0.172-j 0.414,0,-0.172+j 0.414,0-5.828+j 2.414\} \\\\ \mathbf{X}(\mathbf{k})=\{\mathbf{2}, \mathbf{0 . 5}-\mathbf{j} \mathbf{1} \mathbf{2 0 7}, \mathbf{0}, \mathbf{0 …

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