Question

A discrete- time signal $x\left[n\right]=sin\left({\pi }^{2}n\right)$, n being an integer, is

• A.
  $\text{not periodic}$
• B.
  $\text{periodic with period}{\pi }^2$
• C.
  $\text{periodic with period}\pi$
• D.
  $\text{periodic with period}\pi /2$

Answer 1

The correct option is A

$\text{not periodic}$
$x\left[n\right]=sin\left({\pi }^{2}n\right)$

${\omega }_o={\pi }^2$

$\therefore N=\frac{2\pi }{{\omega }_o}.m$

where m is the smallest integer that converts $\frac{2\pi }{{\omega }_o}$ into an integer value.

$\therefore N=\frac{2\pi }{{\pi }^2}.m=\frac{2}{\pi }.m$

So, there exists no such integer value of m which could make the N integer, so the system is not periodic.