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The $N$-point DFT $X$ of a sequence $x[n]$, $0 \leq n \leq N-1$ is given by $$X[k] = \frac{1}{\sqrt{N}} \Sigma_{n=0}^{N-1} x[n] e^{-j \frac{2\pi}{N}nk}, \: \: \: 0 \leq k \leq N-1.$$ Denote this relation as $X=DFT(x)$. For $N=4$, which one of the following sequences satisfies $DFT(DFT(x))=x$?

  1. $x  = \begin{bmatrix} 1 & 2 & 3 & 4 \end{bmatrix}$
  2. $x  = \begin{bmatrix} 1 & 2 & 3 & 2 \end{bmatrix}$
  3. $x  = \begin{bmatrix} 1 & 3 & 2 & 2 \end{bmatrix}$
  4. $x  = \begin{bmatrix} 1 & 2 & 2 & 3 \end{bmatrix}$
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