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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 $W_N^k=e^{-j\left(\frac{2 \pi}{N}\right) k}$

Given $N=8$

Hence, $W_8^1=e^{-j\left(\frac{2 \pi}{8}\right)^0}=1$

$=W_8^1=e^{-j\left(\frac{2 \pi}{8}\right)^1}=\cos \frac{\pi}{4}-j \sin \frac{\pi}{4}=0.707-j 0.707$

$=W_8^2=e^{-j\left(\frac{2 \pi}{8}\right)^2}=\cos \frac{\pi}{2}-j \sin \frac{\pi}{2}=-j$

$=W_8^3=e^{-j\left(\frac{2 \pi}{8}\right)^3}=\cos \frac{3 \pi}{4}-j \sin \frac{3 \pi}{4}=-0.707-j 0.707$

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$ \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 . 5}-\mathbf{j} \mathbf{0 . 2 0 7, 0, 0 . 5 - \mathbf { j } 0 . 2 0 7, 0, 0 . 5}-\mathbf{j} \mathbf{1 . 2 0 7}\} \end{gathered} $

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