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GATE 2026 EE – Question 12

Signals and Systems · Representation of continuous and discrete time signals, shifting and scaling properties · 1 mark · Multiple choice

Let $x_c(t)$ be any continuous-time periodic signal with period $T$. It is sampled uniformly with a sampling period $T_s$, where $T_s\ne T$, resulting in the discrete sequence

$$x[n]=x_c(nT_s),\quad\text{where }n\text{ is an integer.}$$

Which one of the following statements is correct about $x[n]$?

  1. $x[n]$ will always be periodic with period $T/T_s$ for all values of $T/T_s$
  2. $x[n]$ will always be periodic with period 1 for all values of $T/T_s$
  3. $x[n]$ will never be periodic
  4. $x[n]$ will be periodic if and only if $T/T_s$ is a rational number

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Correct answer: (D) $x[n]$ will be periodic if and only if $T/T_s$ is a rational number

Explanation

$x[n+N]=x[n]$ for all $n$ requires $NT_s$ to be an integer multiple of $T$, i.e. $N=kT/T_s$ for some integers $N,k$. This is possible exactly when $T/T_s$ is rational. If $T/T_s$ is irrational, $x[n]$ is not periodic.