Given a sequence $x(n)$ for $0 \leq n \leq 3$, where $x(0)=1, x(1)=3, x(2)=3$, and $x(3)=4$. Evaluate its DFT $X(k)$.

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written 2.3 years ago by

prajapatijaimin • 3.3k

Solution:

Since $N=4, W_4=\mathrm{e}^{-\mathrm{j} \pi / 2}$ , then using:

$X(k)=\sum_{n=0}^3 x(n) W_4^{k n}=\sum_{n=0}^3 x(n) e^{-j \frac{\pi k n}{2}}\\$

Thus, for $k=0$

$ \begin{aligned} & X(0)=\sum_{n=0}^3 x(n) e^{-j 0}=x(0) e^{-j 0}+x(1) e^{-j 0}+x(2) e^{-j 0}+x(3) e^{-j 0} \\\\ &=x(0)+x(1)+x(2)+x(3) \\\\ &=1+2+3+4=10 \\\\ & \text { for } k=1 \\\\ …

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