Question

Which of the following statement(s) is/are correct about the power spectral density (PSD) of a weakly stationary process?

• A. The zero-frequency value of the power spectral density of a weakly stationary random process is equal to the area under the graph of autocorrelation function.
• B. The power spectral density of a stationary process $X\left(t\right)$ is always non-negative.
• C. The power spectral density of a real-valued weakly stationary process is an odd function of frequency.
• D. The individual frequency components of the power spectral density $S_{XX}\left(f\right)$ of a weakly stationary process $X\left(t\right)$ are uncorrelated with each other.

Answer 1

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

The individual frequency components of the power spectral density $S_{XX}\left(f\right)$ of a weakly stationary process $X\left(t\right)$ are uncorrelated with each other.

The power psectral density of a real valued WSS process can be calculated as
$\begin{array}{cc}{S}_{XX}\left(f\right)& ={\int }_{-\infty }^{\infty }{R}_{XX}\left(\tau \right)exp(-j2\pi f\tau )d\tau \\ S_{XX}(-f)& ={\int }_{-\infty }^{\infty }{R}_{XX}\left(\tau \right)exp\left(j2\pi f\tau \right)d\tau \\ \text{Now, substituting}\tau \to -\tau & \\ S_{XX}(-f)& ={\int }_{-\infty }^{\infty }{R}_{XX}\left(\tau \right)exp(-j2\pi f\tau )d\tau \\ & =S_{XX}\left(f\right)\end{array}$