Chapter 19: Gaussian Processes

Learning Objectives

• Define a Gaussian process and explain why mean and covariance functions provide a complete characterization
• Prove that a WSS Gaussian process is strictly stationary
• Define white Gaussian noise, compute its autocorrelation and PSD, and explain why it cannot be realized as a sample path
• Prove that a Gaussian process through an LTI system yields a Gaussian output and derive the output statistics
• Explain why the matched filter is the globally optimal detector (not just the best linear one) for Gaussian noise
• Define the Wiener process, derive its covariance function, and connect it to Brownian motion and the random walk limit
• Apply Gaussian process models to phase noise, channel modeling, and Bayesian regression