Understanding the Concept:
The Power Spectral Density (PSD), denoted as $S_x(f)$, describes how the power of a random signal or random process is distributed across different frequencies.
According to Wiener-Khinchin theorem and standard stochastic relations, the total average power $P_{\text{avg}}$ of a Wide-Sense Stationary (WSS) random signal can be recovered by evaluating its autocorrelation function $R_x(\tau)$ at a time lag of zero ($\tau = 0$):
$$P_{\text{avg}} = R_x(0)$$
Since the autocorrelation function $R_x(\tau)$ and the Power Spectral Density $S_x(f)$ form a Fourier transform pair:
$$R_x(\tau) = \int_{-\infty}^{\infty} S_x(f) e^{j2\pi f \tau} \, df$$
Substituting $\tau = 0$ into this inverse Fourier relation yields:
$$P_{\text{avg}} = R_x(0) = \int_{-\infty}^{\infty} S_x(f) e^{0} \, df = \int_{-\infty}^{\infty} S_x(f) \, df$$
Step-by-step Evaluation:
• The mathematical integral expression $\int_{-\infty}^{\infty} S_x(f) \, df$ is precisely defined geometrically as the complete area bounded under the Power Spectral Density curve evaluated over the entire frequency spectrum.
• Hence, integrating or finding the total area under the PSD directly yields the average total power contained inside the system.
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