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Compute The DFT of the four point sequences x(n)={0,1,2,3}.
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Solution:

x(n)={0,1,2,3}

$ \begin{aligned}\\ X(\mathrm{k}) & =\sum_{n=0}^{\infty} x(n) e^{-j \pi k n / N} \quad k=0,1,2,3 \\\\ X(0) & =0+1+2+3=6 \\\\ X(1) & =\sum_{n=0}^3 x(n) e^{-j \pi k n / 2} \\\\ & =0+1(-j)+2(-1)+3(j)=-2+j 2 \\\\ X(2) & =\sum_{n=0}^3 x(n) e^{-j \pi k n} \\\\ & =0+1(-1)+2(1)+3(-1)=-2 \\\\ X(3) & =\sum_{n=0}^3 x(n) e^{-j \pi k n} / 2 \\\\ & =0(1)+1(j)+2(-1)+3(-j)=-2-j 2 \\\\ X(k) & =\{6,-2+j 2,-2,-2-j 2\}\\ \end{aligned}\\ $

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