Prerequisites & Notation

Prerequisites for Chapter 15

This chapter combines LTI system theory with the spectral analysis of random processes developed in Chapters 13--14. The reader should be comfortable with convolution (time and frequency domain), the Wiener-Khinchin theorem, and the notion of wide-sense stationarity. We will also use Cauchy-Schwarz and basic optimization.

Wide-sense stationarity, autocorrelation, cross-correlation(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, impulse response, frequency response, convolution
Self-check: Can you compute $y(t) = h(t) * x(t)$ and relate it to $\check{y}(f) = \check{h}(f)\check{x}(f)$ ?
Cauchy-Schwarz inequality
Self-check: Can you state and apply Cauchy-Schwarz for integrals: $|\int f g^*|^2 \leq \int |f|^2 \int |g|^2$ ?
White noise and its PSD(Review ch14)
Self-check: Do you know that ideal white noise has $P_x(f) = N_0/2$ for all $f$ ?

Notation for This Chapter

The following notation is used throughout Chapter 15. Symbols follow the FSP convention: lowercase subscripts for PSD arguments, boldface for vectors/matrices.

Symbol	Meaning	Introduced
$h(t)$ , $h[n]$	Impulse response of an LTI system (CT and DT)
$\check{h}(f)$	Frequency response (transfer function) of the LTI system
$P_x(f)$	Power spectral density of input process $X$
$P_y(f)$	Power spectral density of output process $Y$
$P_{xy}(f)$	Cross-power spectral density
$r_{xx}(\tau)$	Autocorrelation of a CT WSS process
$r_{xx}[m]$	Autocorrelation of a DT WSS process
$N_0$	One-sided noise PSD (W/Hz)
$\text{SNR}$	Signal-to-noise ratio
$\sigma^2$	Noise variance / noise power
$E_s$	Signal energy
$B_N$	Noise equivalent bandwidth
$s(t)$	Known deterministic signal (matched filter context)
$\mathcal{N}(\mu, \sigma^2)$	Gaussian distribution with mean $\mu$ and variance $\sigma^2$

← Ch 14 WSS Processes Through LTI Systems