Random Processes ECE460 Spring, 2012

Random (Stocastic) Processes 2

Random Process Definitions Notation: Mean: 3 Example :

Random Process Definitions Autocorrelation Auto-covariance 4 Example :

Stationary Processes 5 Strict-Sense Stationary (SSS) A process in which for all n, all (t 1, t 2, …,t n,), and all Δ Wide-Sense Stationary (WSS) A process X(t) with the following conditions 1.m X (t) = E[X(t)] is independent of t. 2.R X (t 1,t 2 ) depends only on the difference τ = t 1 - t 2 and not on t 1 and t 2 individually. Cyclostationary A random process X ( t ) is cyclostationary if both the mean, m x ( t ), and the autocorrelation function, R X (t 1 +τ, t 2 ), are periodic in t with some period T 0 : i.e., if and for all t and τ.

Wide-Sense Stationary Mean: Autocorrelation: 6 Example :

Power Spectral Density Generalities : Example: 7

Example Given a process Y t that takes the values ±1 with equal probabilities: Find 8

Ergodic 1.A wide-sense stationary (wss) random process is ergodic in the mean if the time-average of X ( t ) converges to the ensemble average: 2.A wide-sense stationary (wss) random process is ergodic in the autocorrelation if the time-average of R X (τ) converges to the ensemble average’s autocorrelation 3.Difficult to test. For most communication signals, reasonable to assume that random waveforms are ergodic in the mean and in the autocorrelation. 4.For electrical engineering parameters: 9

Multiple Random Processes Defined on the same sample space (e.g., see X ( t ) and Y ( t ) above) For communications, limit to two random processes Independent Random Processes X ( t ) and Y ( t ) – If random variables X ( t 1 ) and Y ( t 2 ) are independent for all t 1 and t 2 Uncorrelated Random Processes X ( t ) and Y ( t ) – If random variables X ( t 1 ) and Y ( t 2 ) are uncorrelated for all t 1 and t 2 Jointly wide-sense stationary – If X ( t ) and Y ( t ) are both individually wss – The cross-correlation function R XY ( t 1, t 2 ) depends only on τ = t 2 - t 1 10 Filter