Prerequisites & Notation

Before You Begin

Power spectral density connects the time-domain autocorrelation to the frequency-domain distribution of power. Before we begin, make sure you are comfortable with the following tools.

Wide-sense stationarity, autocorrelation $r_{xx}[m]$ and $r_{xx}(\tau)$ (Review FSP Ch. 13)
Self-check: Can you state the three conditions for WSS and write the autocorrelation as a function of the lag only?
Fourier transform pairs and the Parseval/Plancherel theorem
Self-check: Can you compute $\\int |x(t)|^2\\,dt$ from $\\check{x}(f)$ ?
Discrete-time Fourier transform (DTFT) and its inverse
Self-check: Can you write the DTFT of a sequence $x[n]$ and state the periodicity in frequency?
LTI systems: convolution in time, multiplication in frequency(Review FSP Ch. 13)
Self-check: If $Y(t) = h(t) * X(t)$ , what is $\\check{Y}(f)$ in terms of $\\check{h}(f)$ and $\\check{X}(f)$ ?
Positive semi-definite sequences and matrices
Self-check: Can you explain why the autocorrelation of a WSS process is a positive semi-definite function?

Notation for This Chapter

Symbols introduced or heavily used in this chapter. Notation tokens $\ntn{key}$ allow the reader to customize display.

Symbol	Meaning	Introduced
$P_x(f)$	Power spectral density of process $X$	s01
$r_{xx}(\tau)$	Continuous-time autocorrelation
$r_{xx}[m]$	Discrete-time autocorrelation
$P_{xy}(f)$	Cross-power spectral density	s03
$\check{h}(f)$	Frequency response (transfer function) of LTI system
$N_0$	One-sided noise PSD	s02
$W$	Signal bandwidth (Hz)
$\sigma^2$	Noise variance / noise power
$\bar{r}_{xx}[m]$	Time-averaged autocorrelation (non-WSS)	s01
$\hat{P_x}(f)$	Periodogram estimator of the PSD	s04

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