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Letx(n) be four point sequence withX(k)={1,2,3,4}. Find the DFT of the following sequence using X(k).
1. $p(n)=x(n)cos⁡(n π/2) \\ p(n)=\bigg(\frac{e^{j nπ/2}+e^{-j nπ/2}}{2}\bigg) x(n) \\ p(n)= \frac{1}{2}\bigg[e^{j nπ/2} x(n)+e^{-j nπ/2} x(n)\bigg] \\ By \ Frequency \ Shift \ Property, \\ Q(k)=\frac{1}{2}[X(k-1)+X(k+1)] \\ Q(k)=\frac{1}{2} \bigg(\begin {bmatrix} \ 4 \\ 1 \\ 2 \\ 3\end{bmatrix}+ \begin {bmatrix} \ 2 \\ 3 \\ 4 \\ 1\end{bmatrix}\bigg) \\ Q(k)= \begin{bmatrix} \ 3 \\ 2 \\ 3 \\ 2\end{bmatrix}$
2. $q(n)=2δ(n)+3{Four \ point \ u(n)}+4x(n). \\ By \ applying \ DFT \ we \ get, \\ Q(k)=2DFT{δ(n)}+3DFT{four \ point \ u(n)}+4DFT{x(n)} \\ Q(k)=2\begin {bmatrix} \ 1 \\ 1 \\ 1 \\ 1 \end{bmatrix} + 3\begin {bmatrix} \ 4 \\ 0 \\ 0 \\ 0\end{bmatrix}+4\begin {bmatrix} \ 1 \\ 2 \\ 3 \\ 4 \end{bmatrix} \\ Q(k)=\begin {bmatrix} \ 15 \\ 10 \\ 14 \\ 18 \end{bmatrix}$