Question

A discrete-time signal \( x[n] = \sin(\pi^2 n) \), \( n \) being an integer, is ________.

• A. periodic with period \( \pi \)
• B. periodic with period \( \pi^2 \)
• C. periodic with period \( \pi/2 \)
• D. not periodic

Hint:

If argument of sine in discrete domain has irrational multiple → signal is not periodic!

Answer 1

The Correct Option is D

Step 1: Definition of Periodicity in Discrete-Time
A discrete-time signal \( x[n] \) is periodic if: \[ x[n] = x[n + N] \text{for some integer } N \] Step 2: Apply to Given Signal
Given \( x[n] = \sin(\pi^2 n) \), since \( \pi^2 \) is irrational, the value of \( \pi^2 n \) never repeats uniformly.
Step 3: Non-repetition
No integer \( N \) exists such that \( \sin(\pi^2 n) = \sin(\pi^2(n+N)) \) for all \( n \). Hence, not periodic.
Conclusion:
Option (4) is correct — \( \sin(\pi^2 n) \) is not periodic in discrete-time domain.