Chapter 3 — Stochastic Processes
Companion material for Chapter 3. Covers definitions, wide-sense stationarity, ergodicity, power spectral density, and LTI systems.
§ 3.1 Definition
Stochastic Process
Animated ensemble: multiple realizations X[\eta, k] drawn simultaneously. Highlight the ensemble slice (fixed k) vs. a single trajectory (fixed \eta).
Ensemble vs. Time Averaging
Compare temporal averaging (along one realization) to ensemble averaging (across many realizations). Shows when they agree (ergodic) and when they differ.
§ 3.2–3.3 Expected Values
Mean and Correlation Functions
For a random-phase sinusoid X[k] = A\cos(\omega_0 k + \Phi), \Phi \sim \mathcal{U}[0, 2\pi): compute the theoretical ACF analytically and verify by Monte Carlo.
§ 3.4 Properties of Random Processes
Summary diagram of key relationships: ACF ↔︎ PSD (Wiener–Khinchin), stationarity conditions, ergodicity link.
§ 3.5 Stationarity
Wide-Sense Stationary (WSS)
AR(1) process autocorrelation with adjustable pole coefficient. Displays sample vs. theoretical ACF and compares ensemble sizes. Shows WSS conditions in action.
Demonstrate ACF properties: R_{XX}[0] \geq |R_{XX}[\ell]|, even symmetry, and value at lag 0 equals power.
Sine in Noise: ACF Reveals Periodicity
Add a sinusoid to white noise at low SNR (sine invisible in time domain). Plot the ACF and show the periodic ripple that reveals the sinusoid.
§ 3.6 Ergodicity
When Does Time Average = Ensemble Average?
Side-by-side: ensemble average at fixed time vs. time average over a single realization. Shows convergence for ergodic processes and divergence for non-ergodic ones.
Non-Ergodic Example
Simulate the Pólya urn process. Different realizations converge to different long-run proportions — a clear non-ergodic example.
Measurement of Correlation
§ 3.8 Power Spectral Density
Wiener–Khinchin Theorem
Observe how the ACF shape determines the PSD shape via the Fourier transform relationship (Wiener–Khinchin theorem).
3Blue1Brown — But what is a Fourier series? Rotating vectors, circle drawings, and the frequency decomposition of periodic signals (24 min).
3Blue1Brown — But what is the Fourier transform? Frequency analysis via winding graphs around circles (21 min).
Visualize the Fourier transform: build a signal from sinusoids and observe the frequency content.
See how finite observation windows affect ACF and PSD estimation — spectral leakage and bias vs. variance trade-off.
Colored Noise
Generate white noise, pass it through a shaping filter, and compare input vs. output PSD.
§ 3.9 LTI Systems and WSS Processes
LTI Output Statistics
Step-by-step animation of the convolution integral — the operation that defines an LTI system’s response.
Apply LTI systems to audio signals and hear the effect — connects system theory to the stochastic processing setting.
Full IIR/FIR filter analysis: impulse response, pole-zero plot, magnitude and phase response. Apply to audio or images.
Pass a WSS process through a known FIR filter. Compute the output ACF and PSD analytically, then verify by simulation.
Cross-PSD and System Identification
Identify an unknown LTI system from input/output cross-correlation. Compare estimated impulse response to ground truth.