GATE 2026 EE Q.12: solution and answer

GATE 2026 · EEQ.12 · 1 mark · MCQ

Let $x_{c} (t)$ be any continuous-time periodic signal with period $T$ . It is sampled uniformly with a sampling period $T_{s}$ where $T_{s} \neq = T$ , resulting in the discrete sequence

x [n] = x_{c} (n T_{s}), where n is an integer.

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

(A)
$x [n]$ will always be periodic with period $T / T_{s}$ for all values of $T / T_{s}$
(B)
$x [n]$ will always be periodic with period $1$ for all values of $T / T_{s}$
(C)
$x [n]$ will never be periodic
(D)
$x [n]$ will be periodic if and only if $T / T_{s}$ is a rational number

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Answer and solution

ANSWER

Option D: $x [n]$ will be periodic if and only if $T / T_{s}$ is a rational number.

Official answer: DSource: Official answer key, GATE 2026 EE (IIT Guwahati)

What discrete-time periodicity needs. $x [n]$ is periodic if some positive integer $N$ satisfies $x [n + N] = x [n]$ for every integer $n$ , that is, $x_{c} (n T_{s} + N T_{s}) = x_{c} (n T_{s})$ . This holds for every $x_{c}$ of period $T$ whenever $N T_{s}$ is a whole number of periods: $N T_{s} = m T$ for some positive integer $m$ .
The "if" part. Suppose $\frac{T}{T _{s}} = \frac{p}{q}$ with $p, q$ positive integers in lowest terms. Then $p T_{s} = q T$ , so $x [n + p] = x_{c} (n T_{s} + p T_{s}) = x_{c} (n T_{s} + q T) = x_{c} (n T_{s}) = x [n] .$ So $x [n]$ is periodic, and $N = p$ samples is a period. It is always an integer. For a particular signal the fundamental period can be a divisor of $p$ .
The "only if" part (DT sinusoid rule). Take $x_{c} (t) = cos (\frac{2 π t}{T})$ , which has period $T$ . Then $x [n] = cos (Ω_{0} n)$ with $Ω_{0} = 2 π \frac{T _{s}}{T}$ . If this has a period $N$ , put $n = 0$ : $cos (Ω_{0} N) = 1$ , so $Ω_{0} N = 2 π m$ for some integer $m$ . That gives $\frac{T _{s}}{T} = \frac{m}{N}$ , which is rational, so $\frac{T}{T _{s}}$ is rational too. If $T_{s} = T / 2$ , for example, the samples never repeat. Some special signals, such as a constant, give a periodic $x [n]$ for any $T_{s}$ . So "only if" means that periodicity is guaranteed for every signal of period $T$ only when $T / T_{s}$ is rational.
Option A is false. If $T / T_{s} = 3/2$ , a period of $1.5$ samples is impossible, because a discrete-time period must be an integer. By step 2 ( $p = 3$ , $q = 2$ ) the true period is $N = 3$ . If $T / T_{s}$ is irrational, the cosine of step 3 has no period at all.
Option B is false. A period of $1$ would make $x [n]$ constant. But with $T_{s} = T /4$ the cosine gives $cos (\frac{π n}{2}) = 1, 0, - 1, 0, \dots$ , which is not constant.
Option C is false. The same example, $T_{s} = T /4$ (allowed, since $T_{s} \neq = T$ ), gives a periodic sequence with $N = 4$ .
Only option D holds in general: $x [n]$ is periodic for every signal of period $T$ exactly when $T / T_{s}$ is rational.

KEY IDEA

A discrete-time period must be a whole number of samples. So $x [n]$ repeats only when some integer number of samples $N$ spans a whole number of continuous-time periods: $N T_{s} = m T$ , which means $T / T_{s} = N / m$ is rational. If $T / T_{s} = p / q$ in lowest terms, then $N = p$ samples is a period.

WATCH OUT FOR

Picking A because the samples per period $T / T_{s}$ looks like the discrete period. That ratio need not be an integer. For $T / T_{s} = 3/2$ the period is $3$ samples, not $1.5$ , and for an irrational ratio (for example $T_{s} = T / 2$ ) a sampled sinusoid never repeats. Another trap is to think sampling a periodic signal always gives a periodic sequence.

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GATE 2026 EE Q.12: solution and answer

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