The GATE Grind

GATE 2021 EC – Question 15

Networks, Signals and Systems · Discrete-time Signals · 1 mark · Multiple choice

Consider two 16-point sequences $x[n]$ and $h[n]$. Let the linear convolution of $x[n]$ and $h[n]$ be denoted by $y[n]$, while $z[n]$ denotes the 16-point inverse discrete Fourier transform (IDFT) of the product of the 16-point DFTs of $x[n]$ and $h[n]$. The value(s) of $k$ for which $z[k]=y[k]$ is/are

  1. $k=0,1,2,\dots,15$
  2. $k=0$
  3. $k=15$
  4. $k=0$ and $k=15$

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Correct answer: (C) $k=15$

Explanation

The IDFT of the product is the 16-point circular convolution, $z[k]=y[k]+y[k+16]$. The linear convolution $y[n]$ has length $16+16-1=31$, so in general $y[k+16]\ne0$ for $k=0,\dots,14$. Only for $k=15$ is $y[31]=0$, so $z[k]=y[k]$ only for $k=15$.