Prerequisites & Notation

Prerequisites for Chapter 19

This chapter brings together the multivariate Gaussian distribution (Ch. 8) with the spectral theory of WSS processes (Ch. 13--15). The key prerequisite is comfort with the fact that a joint Gaussian distribution is fully determined by its mean vector and covariance matrix. We also use LTI system results from Ch. 15.

Multivariate Gaussian distribution: joint PDF, marginals, conditionals(Review ch08)
Self-check: Can you write the PDF of $\mathcal{N}(\boldsymbol{\mu}, \boldsymbol{\Sigma})$ and state that uncorrelated Gaussian RVs are independent?
Wide-sense stationarity and autocorrelation(Review ch13)
Self-check: Can you state the WSS conditions and compute $r_{xx}(\tau)$ for a given process?
Power spectral density and the Wiener-Khinchin theorem(Review ch14)
Self-check: Can you go from $r_{xx}(\tau)$ to $P_x(f)$ via Fourier transform and back?
LTI systems with random inputs: output PSD(Review ch15)
Self-check: Do you know that $P_y(f) = |\check{h}(f)|^2 P_x(f)$ for a WSS input through an LTI filter?
Characteristic functions(Review ch09)
Self-check: Can you state the characteristic function of a Gaussian RV and use it to identify Gaussianity?
Dirac delta function and its Fourier transform
Self-check: Do you know that $\int_{-\infty}^{\infty} \delta(\tau) e^{-j2\pi f\tau} d\tau = 1$ ?

Notation for This Chapter

The following notation is used throughout Chapter 19. Symbols follow the FSP convention established in earlier chapters.

Symbol	Meaning	Introduced
$\{X(t) : t \in \mathcal{T}\}$	A Gaussian process indexed by $\mathcal{T}$
$\mu_X(t)$	Mean function: $\mathbb{E}[X(t)]$
$C_X(t_1, t_2)$	Covariance function: $\text{Cov}(X(t_1), X(t_2))$
$r_{xx}(\tau)$	Autocorrelation of a WSS process
$P_x(f)$	Power spectral density
$N_0$	One-sided noise PSD: $R_N(\tau) = (N_0/2)\delta(\tau)$
$\sigma^2$	Noise variance
$\mathcal{N}(\mu, \sigma^2)$	Gaussian distribution with mean $\mu$ and variance $\sigma^2$
$W(t)$	Standard Wiener process (Brownian motion)
$\check{h}(f)$	Frequency response of an LTI system
$h(t)$	Impulse response of an LTI system
$\text{SNR}$	Signal-to-noise ratio
$W$	Signal bandwidth (Hz)

← Ch 18 Definition and Finite-Dimensional Distributions