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GATE 2014 EC (EC4) – Question 55

Networks, Signals and Systems · Discrete-time Signals · 2 marks · Multiple choice

The normalized N-point DFT is $X[k]=N^{-1/2}\sum_{n=0}^{N-1}x[n]e^{-j2\pi nk/N}$ for $0\le k<N$. For N = 4, which sequence satisfies DFT(DFT(x)) = x?

  1. [1,2,3,4]
  2. [1,2,3,2]
  3. [1,3,2,2]
  4. [1,2,2,3]

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Show answer and explanation

Correct answer: (B) [1,2,3,2]

Explanation

With the1/√Nnormalization, applying the DFT twice gives circular time reversal:x[n]→x[(−n)modN]. For N=4, invariance requires x[1]=x[3], while indices0and2map to themselves. Only **[1,2,3,2],B** has that symmetry.