Prerequisites & Notation

Prerequisites

This chapter develops the mean-square calculus of random processes — continuity, differentiation, and integration in the $L^2$ sense — and applies it to bandlimited processes and the Karhunen-Loève expansion. The following background is essential.

WSS processes and autocorrelation(Review fsp/ch13)
Self-check: Can you state the definition of WSS and compute $r_{xx}(\tau)$ for a given process?
Power spectral density and the Wiener-Khinchin theorem(Review fsp/ch14)
Self-check: Can you write the Fourier transform pair relating $r_{xx}(\tau)$ and $P_x(f)$ ?
LTI systems with random inputs(Review fsp/ch15)
Self-check: Can you derive the output PSD $P_y(f) = |\check{h}(f)|^2 P_x(f)$ for a WSS input?
Fourier transforms and Parseval's theorem
Self-check: Can you state Parseval's theorem and use it to relate time-domain and frequency-domain integrals?
Eigenvalues and eigenfunctions of integral operators
Self-check: Do you know what it means for $\phi(t)$ to satisfy $\int K(t,s)\phi(s)\,ds = \lambda \phi(t)$ ?

Notation for This Chapter

We introduce several new symbols for mean-square calculus alongside the familiar PSD and autocorrelation notation from previous chapters.

Symbol	Meaning	Introduced
$r_{xx}(\tau)$	Autocorrelation of a WSS process	Ch. 13
$P_x(f)$	Power spectral density	Ch. 14
$\check{h}(f)$	Frequency response of an LTI system	Ch. 15
$X'(t)$	Mean-square derivative of the process $X(t)$	This chapter
$\text{l.i.m.}$	Limit in mean square (convergence in $L^2$ )	This chapter
$W$	Bandwidth of a bandlimited process	This chapter
$\phi_n(t)$	Eigenfunctions of the autocorrelation kernel (KL basis)	This chapter
$\lambda_n$	Eigenvalues of the autocorrelation kernel	This chapter
$Z_n$	Uncorrelated random coefficients in the KL expansion	This chapter

← Ch 15 Continuity, Differentiation, and Integration in Mean Square