Random Processes and Noise

Syllabus: Random Processes: Auto correlation and power spectral density, properties of white noise, filtering of random signals through LTI systems.

4 hand-picked visual resources on this topic, 1 of them little-known gems — under each card, look for See it. Every link is checked against the live site before it ships.

Stationarity, autocorrelation and PSD

A process is wide-sense stationary if its mean is constant and its autocorrelation depends only on the lag $\tau$ . Everything else in this topic assumes WSS, and the PSD is simply the Fourier transform of the autocorrelation (Wiener–Khinchine).

$R_X(\tau)=E[X(t)X(t+\tau)]\ \xleftrightarrow{\ \mathcal{F}\ }\ S_X(f)$
Wiener–Khinchine
$R_X(0)=E[X^2]=\text{total average power}=\int_{-\infty}^{\infty}S_X(f)\,df$
$|R_X(\tau)|\le R_X(0),\qquad R_X(-\tau)=R_X(\tau)$
$S_X(f)\ge0\ \text{and even (real process)}$
a candidate PSD that goes negative anywhere is not a PSD
$\sigma_X^2=R_X(0)-\mu_X^2$

Where marks are lost

$R_X(\infty)\to\mu_X^2$ , not zero, for a process with a DC component. Questions that ask for the mean from a given $R_X(\tau)$

See it

A random process through an LTI system

The single most-used result in the topic: the filter shapes the PSD by $|H(f)|^2$ . Power at the output is then the area under that product, which for white noise into a filter is $N_0/2$ times the filter's energy.

White noise and noise figure

White noise has a flat two-sided PSD $N_0/2$ , so its autocorrelation is an impulse: samples at any two distinct instants are uncorrelated. Its total power is infinite, which is why it only ever appears after a filter.

$S_N(f)=\dfrac{N_0}{2},\qquad R_N(\tau)=\dfrac{N_0}{2}\delta(\tau)$

Analog Modulation →

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