Prerequisites & Notation

Prerequisites for Chapter 9

The Wiener filter sits at the intersection of three mathematical streams: Hilbert-space geometry (orthogonal projection), wide-sense stationary (WSS) process theory (autocorrelations and spectra), and linear systems (Fourier transforms and z-transforms). You should be comfortable with each before beginning this chapter.

LMMSE estimation in finite dimensions(Review ch02)
Self-check: You can derive $\hat{\mathbf{x}} = \boldsymbol{\Sigma}_{xy}\boldsymbol{\Sigma}_y^{-1}\mathbf{y}$ from the orthogonality principle and know that this is the optimal estimator when $(\mathbf{X},\mathbf{Y})$ is jointly Gaussian.
WSS processes and power spectral densities(Review fsp-ch14)
Self-check: You can state the Wiener-Khinchin theorem: $P_x(f) = \sum_k r_{xx}[k] e^{-j 2\pi f k}$ and know that $P_x(f) \geq 0$ .
LTI systems and the frequency response(Review fsp-ch13)
Self-check: You know that if $Y_n$ is the output of an LTI filter $h[n]$ driven by a WSS input $X_n$ , then $P_y(f) = |\check{h}(f)|^2 P_x(f)$ .
Orthogonality principle and Hilbert-space projection(Review ch02)
Self-check: You can state: the MMSE estimator is characterized by $\mathbb{E}[(X - \hat{X}) Y^*] = 0$ for every observation used in the estimate.
Hermitian symmetry and complex exponentials(Review ch01)
Self-check: You are fluent with $z = e^{j 2\pi f}$ , Hermitian conjugate $(\cdot)^H$ , and reciprocal symmetry of autocorrelation: $r_{xx}[-k] = r_{xx}^*[k]$ .

Notation for Chapter 9

This chapter handles two jointly WSS processes — the desired signal $X_n$ and the observation $Y_n$ . We use lowercase subscripts on autocorrelations and PSDs, following the FSP/FSI convention. The hat symbol denotes an estimate; the check symbol $\\check{\\cdot}$ denotes a frequency-domain (Fourier) representation of a discrete-time filter.

Symbol	Meaning	Introduced
$X_n, Y_n$	Desired signal and observation, both jointly WSS discrete-time processes	s01
$r_{xx}[k], r_{yy}[k], r_{xy}[k]$	Autocorrelation and cross-correlation sequences	s01
$P_x(f), P_y(f), P_{xy}(f)$	Power spectral density and cross-PSD, $f \in [-1/2, 1/2]$	s01
$\hat{X}_n$	Estimate of $X_n$ (via Wiener filter applied to $\{Y_m\}$ )	s01
$E_n = X_n - \hat{X}_n$	Estimation error	s01
$h[k], \check{h}(f)$	Filter impulse response and its frequency response	s01
$\check{h}_{\text{nc}}(f)$	Non-causal Wiener filter transfer function	s02
$\check{h}_c(f)$	Causal Wiener filter transfer function	s04
$P_y^+(f), P_y^-(f)$	Causal (minimum-phase) and anti-causal spectral factors	s03
$J_n$	Innovations process (whitened observation)	s03
$[\cdot]_+$	Causal projection operator: keep non-negative-lag Fourier components	s04
$\sigma_{\text{nc}}^2, \sigma_c^2, \sigma_p^2$	MMSE of non-causal, causal, one-step prediction filters	s02, s04, s05

← Ch 8 The Wiener-Hopf Equation and the Orthogonality Principle