The GATE Grind

GATE 2020 EC – Question 39

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

A finite duration discrete-time signal $x[n]$ is obtained by sampling the continuous-time signal $x(t)=\cos(200\pi t)$ at sampling instants $t=n/400$, $n=0,1,\dots,7$. The 8-point discrete Fourier transform (DFT) of $x[n]$ is defined as

$$X[k]=\sum_{n=0}^{7}x[n]e^{-j\frac{\pi kn}{4}},\quad k=0,1,\dots,7.$$

Which one of the following statements is TRUE?

  1. All $X[k]$ are non-zero.
  2. Only $X[4]$ is non-zero.
  3. Only $X[2]$ and $X[6]$ are non-zero.
  4. Only $X[3]$ and $X[5]$ are non-zero.

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Show answer and explanation

Correct answer: (C) Only $X[2]$ and $X[6]$ are non-zero.

Explanation

The sampled signal is $x[n]=\cos\left(\frac{200\pi n}{400}\right)=\cos\left(\frac{\pi n}{2}\right)$, which is exactly 2 cycles in 8 samples. Its DFT has non-zero values only at $k=2$ and $k=8-2=6$.