Question

Which of the followinng can be a valid power spectral density of a real random process $X\left(t\right)$?

• A. $S_X\left(\omega \right)=\delta \left(\omega \right)+\frac{\omega }{{\omega }^2+16}$
• B. $S_X\left(\omega \right)=\frac{{\omega }^2}{{\omega }^2-4}$
• C. $S_X\left(\omega \right)=j\left[\delta \right(\omega +{\omega }_0)-\delta (\omega -{\omega }_0\left)\right]$
• D. $S_X\left(\omega \right)=\frac{{\omega }^2}{{\omega }^2+12}$

Answer 1

The correct option is D $S_X\left(\omega \right)=\frac{{\omega }^2}{{\omega }^2+12}$

For a real random process $X\left(t\right)$, the power spectral density has to satisfy the following condition
$S_X\left(\omega \right)\ge 0$
$S_X\left(\omega \right)=S_X(-\omega )$
Only option (c) satisfies the required condition.