Chapter Summary

Chapter 14 Summary

Key Points

• 1.

  The Wiener-Khinchin theorem states that for a WSS process, the PSD $P_x(f)$ is the Fourier transform of the autocorrelation: $P_x(f) = \int r_{xx}(\tau)\,e^{-j2\pi f\tau}\,d\tau$ (CT) or $P_x(f) = \sum_m r_{xx}[m]\,e^{-j2\pi fm}$ (DT).

• 2.

  The PSD is real, non-negative, and even for real-valued processes. It integrates to the total power: $\int P_x(f)\,df = r_{xx}(0)$.

• 3.

  White noise has flat PSD ($N_0/2$ for CT, $\sigma^2$ for DT). CT white noise has infinite power and is an idealization; band-limited white noise with bandwidth $W$ has sinc autocorrelation and finite power $N_0W/2$.

• 4.

  The input-output PSD relation for LTI systems is ${P_x}_{y}(f) = |\check{h}(f)|^2\,{P_x}_{x}(f)$, the most important tool for noise analysis in linear systems.

• 5.

  The cross-spectral density $P_{xy}(f)$ is the FT of $r_{xy}(\tau)$ and is complex in general. The coherence function $\gamma_{xy}(f) \in [0,1]$ measures frequency-by-frequency linear correlation.

• 6.

  For non-WSS processes, the PSD is the FT of the time-averaged autocorrelation. For wide-sense cyclostationary processes, the average is over one period.

• 7.

  The periodogram is the standard finite-data PSD estimator. It is asymptotically unbiased but inconsistent; averaging (Bartlett, Welch) is needed for reliable estimation.