Chapter 9: The Discrete-Time Wiener Filter

Learning Objectives

• Formulate the Wiener filtering problem for jointly wide-sense stationary processes and derive the Wiener-Hopf normal equations from the orthogonality principle
• Derive the non-causal Wiener filter transfer function in the frequency domain and interpret it as a frequency-selective attenuator
• State and apply the Paley-Wiener condition and derive the spectral factorization of a WSS process
• Derive the causal Wiener filter via the innovations representation and the causal projection operator
• Compute the Kolmogorov-Szego prediction error variance and connect it to the geometric mean of the PSD
• Analyze the AR(1) signal in white noise in closed form and quantify the gap between causal and non-causal MMSE
• Position the Wiener filter within the broader family of estimators: as the infinite-horizon limit of LMMSE, as the precursor to the Kalman filter, and as the target of adaptive algorithms (LMS/RLS)