GATE 2023 IN Q.29: solution and answer

GATE 2023 · INQ.29 · 1 mark · NAT

Let $y (t) = x (4 t)$ , where $x (t)$ is a continuous-time periodic signal with fundamental period of 100 s. The fundamental period of $y (t)$ is ______ s (rounded off to the nearest integer).

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Answer and solution

ANSWER

The fundamental period of $y (t) = x (4 t)$ is $25 s$ .

Official answer: 25Source: Official answer key, GATE 2023 IN (IIT Kanpur)

Periodicity of $x$ : $x (t + 100) = x (t)$ for all $t$ , and $100 s$ is the smallest such positive shift. Every period of $x$ is therefore a positive integer multiple of $100 s$ .
Test a shift $T$ on $y$ : $y (t + T) = x (4 (t + T)) = x (4 t + 4 T)$ .
So $y (t + T) = y (t)$ for all $t$ exactly when $4 T$ is a period of $x$ , i.e. $4 T = 100 k$ for a positive integer $k$ .
Smallest positive choice ( $k = 1$ ): $T_{y} = \frac{T _{x}}{4} = \frac{100}{4} = 25 s$
Time compression by a factor of $4$ makes $y (t)$ complete four cycles in every $100 s$ , so the fundamental period of $y (t)$ is $25 s$ .

KEY IDEA

Time scaling $y (t) = x (a t)$ with $a > 0$ divides the fundamental period by $a$ : $T_{y} = T_{x} / a$ . Compression ( $a > 1$ ) shortens the period and expansion ( $0 < a < 1$ ) lengthens it.

WATCH OUT FOR

Multiplying instead of dividing, which gives $4 \times 100 = 400 s$ . Replacing $t$ by $4 t$ compresses the signal in time, so each cycle takes a quarter as long and the period is $100/4 = 25 s$ .

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GATE 2023 IN Q.29: solution and answer

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