Question

Which of the following properties of Power spectral density is/are correct?

• A. The power spectral density of a stationary process is always non-negative
• B. The mean square value of a stationary process equals the total area under the graph of the power spectral density.
• C. The power spectral density of a real-valued random process is an even function of frequency.
• D. The zero frequency value of the power spectral density of a stationary process equals the total area under the graph of the autocorrelation function.

Answer 1

Answer: (A), (B), (C), (D)

The zero frequency value of the power spectral density of a stationary process equals the total area under the graph of the autocorrelation function.

$\begin{array}{cc}\left(a\right)S_x\left(0\right)& ={\int }_{\infty }^{\infty }{R}_x\left(\tau \right)d\tau \\ \text{where}{R}_x\left(\tau \right)& =\text{Autocorrelation function}\\ S_x\left(f\right)& =\text{Power spectral density}\\ \left(b\right)E\left[X^2\right(t\left)\right]& =R_x\left(0\right)={\int }_{-\infty }^{\infty }{S}_x\left(f\right)df\\ \left(c\right)S_x\left(f\right)& \ge 0\\ \left(d\right)S_x(-f)& =S_x\left(f\right)\end{array}$