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GATE 2024 EC – Question 14

Communications · Digital Communications · 1 mark · Multiple choice

A digital communication system transmits through a noiseless bandlimited channel $[-W,W]$. The received signal $z(t)$ at the output of the receiving filter is given by $z(t)=\sum_nb[n]x(t-nT)$ where $b[n]$ are the symbols and $x(t)$ is the overall system response to a single symbol. The received signal is sampled at $t=mT$. The Fourier transform of $x(t)$ is $X(f)$. The Nyquist condition that $X(f)$ must satisfy for zero intersymbol interference at the receiver is ______.

  1. $\sum_{m=-\infty}^{\infty}X\!\left(f+\dfrac mT\right)=T$
  2. $\sum_{m=-\infty}^{\infty}X\!\left(f+\dfrac mT\right)=\dfrac1T$
  3. $\sum_{m=-\infty}^{\infty}X(f+mT)=T$
  4. $\sum_{m=-\infty}^{\infty}X(f+mT)=\dfrac1T$

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Correct answer: (A) $\sum_{m=-\infty}^{\infty}X\!\left(f+\dfrac mT\right)=T$

Explanation

Zero ISI requires $x(mT)=\delta[m]$. Sampling $x(t)$ every $T$ makes its spectrum periodic: $\frac1T\sum_mX(f+m/T)$ is the transform of $x(mT)=\delta[m]$, which is 1. Hence $\sum_mX(f+m/T)=T$.