Question

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

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

Hint:

For a signal to be periodic, the argument of the sinusoidal function must repeat at regular intervals. In this case, the quadratic term causes irregular behavior.

Answer 1

The Correct Option is D

The given discrete-time signal \( x[n] = \sin(\pi n^2) \) is not periodic. For a discrete-time signal to be periodic, there must be a constant period \( N \) such that \( x[n] = x[n + N] \) for all integer values of \( n \). However, the argument \( \pi n^2 \) grows quadratically, which results in no fixed period for the signal. Therefore, the signal is not periodic, and the correct answer is option (4).