Chapter 5 — Linear Optimal Filtering
Companion material for Chapter 5. Covers the Wiener filter, matched filter, and the Wiener–Hopf equation, derived via the orthogonality principle.
The full slide deck for this chapter — estimating one process from another: the orthogonality principle, the Wiener–Hopf equation, noise suppression, linear prediction, and the matched filter.
§ 5.1 Problem Setup
Watch a sliding window of the observation X estimate the target Y[k], stack windows into a least-squares problem, and see how stationarity and ergodicity let you estimate the required correlations from a single record.
§ 5.2 Orthogonality Principle
§ 5.3 Wiener Filter
Build the noncausal Wiener gain H(e^{j\Omega})=S_{UU}/(S_{UU}+S_{NN}) from separate PSD estimates of clean speech and noise, and hear it suppress additive noise.
Fit a ballistic trajectory from a window of noisy measurements and read the same optimal filter off at different lags \tau — smoothing (\tau<0), filtering (\tau=0), and prediction (\tau>0).
§ 5.4 Matched Filter
Build the time-reversed matched filter h[k]\propto u[-k], watch its output spike at the pulse arrival, and run a detection experiment trading false alarms against detections on the ROC.