GATE 2024 EC – Question 56

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

The radian frequency value(s) for which the discrete time sinusoidal signal $x[n]=A\cos(\Omega n+\pi/3)$ has a period of 40 is/are ______.

$0.15\pi$
$0.225\pi$
$0.3\pi$
$0.45\pi$

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Correct answer: (A) $0.15\pi$; (D) $0.45\pi$

Explanation

A discrete sinusoid is periodic with fundamental period $N$ when $\Omega=2\pi m/N$ with $m$ and $N$ having no common factor. For $N=40$, $\Omega=2\pi m/40=0.05\pi m$ with $\gcd(m,40)=1$. $0.15\pi$: $m=3$, coprime to 40, fundamental period 40. $0.225\pi$: $m=4.5$, not an integer. $0.3\pi$: $m=6$, but $\gcd(6,40)=2$, so the fundamental period is 20. $0.45\pi$: $m=9$, coprime to 40, fundamental period 40.