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Compute the DFT of the following two sequences, $h[n]=\{1,3,-1,-2\} \quad$ and $\quad x[n]=\{1,2,0,-1\}$
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Solution:

where $N=4 \Rightarrow e^{j \frac{2 \pi}{N}}=e^{j \frac{2 \pi}{4}}=e^{j \frac{\pi}{2}}=j$

$H(k)=\sum_{n=0}^3 h[n] e^{-j \frac{\pi}{2} n k} \quad$ for $\quad k=0,1,2,3$

$ \begin{aligned} & H(0)=h[0]+h[1]+h[2]+h[3]=1 \\\\ & H(1)=h[0]+h[1] e^{-j \pi / 2}+h[2] \cdot e^{-j \pi}+h[3] \cdot e^{-j 3 \pi / 2}=2-j 5 \\\\ & H(2)=h[0]+h[1] e^{-j \pi}+h[2] \cdot e^{-j 2 \pi}+h[3] \cdot e^{-j 3 \pi}=-1 \\\\ & H(3)=h[0]+h[1] e^{-j 3 \pi / 2}+h[2] \cdot e^{-j 3 \pi}+h[3] \cdot e^{-j 9 \pi / 2}=2+j 5\\ \end{aligned}\\ $

$ \begin{aligned} & X(0)=x[0]+x[1]+x[2]+x[3]=2 \\\\ & X(1)=x[0]+x[1] e^{-j \pi / 2}+x[2] \cdot e^{-j \pi}+x[3] \cdot e^{-j 3 \pi / 2}=1-j 3 \\\\ & X(2)=x[0]+x[1] e^{-j \pi}+x[2] \cdot e^{-j 2 \pi}+x[3] \cdot e^{-j 3 \pi}=0 \\\\ & X(3)=x[0]+x[1] e^{-j 3 \pi / 2}+x[2] \cdot e^{-j 3 \pi}+x[3] \cdot e^{-j 9 \pi / 2}=1+j 3\\ \end{aligned} $

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