Chapter Summary

Chapter 19 Summary: Gaussian Processes

Key Points

• 1.

  Definition and characterization: A Gaussian process is one where every finite-dimensional marginal is jointly Gaussian. It is completely characterized by its mean function $\mu_X(t)$ and covariance function $C_X(t_1, t_2)$ — no higher-order statistics are needed.

• 2.

  WSS implies strict stationarity: For Gaussian processes (and only for them), wide-sense stationarity implies strict stationarity, because the Gaussian distribution is determined by its first two moments.

• 3.

  White Gaussian noise: WGN has autocorrelation $R_N(\tau) = (N_0/2)\delta(\tau)$ and flat PSD $P_N(f) = N_0/2$. It has infinite power and cannot be realized as a sample path, but any filtered version is a valid Gaussian process with finite power.

• 4.

  Gaussian + LTI = Gaussian: A GP through an LTI system produces a GP output, fully characterized by $P_Y(f) = |\check{h}(f)|^2P_x(f)$. This makes the matched filter not just the best linear detector but the globally optimal detector for Gaussian noise.

• 5.

  Wiener process: Brownian motion $W(t)$ has independent Gaussian increments, continuous paths, and $\mathbb{E}[W(s)W(t)] = \min(s,t)$. It is not WSS ($\text{Var}(W(t)) = t$ grows without bound) but is the continuous-time limit of random walks (Donsker's theorem).

• 6.

  Applications: Thermal noise modeling ($N_0/2 = kT$), matched filter detection, phase noise in oscillators ($\Phi(t) = \sigma W(t)$), GP regression for channel prediction, and LMMSE estimation exploiting Gaussian channel models.