Random Processes

1. Random Processes ECE460 Spring, 2012

2. Random (Stocastic) Processes

3. Random Process Definitions Example: Notation: Mean:

4. Random Process Definitions Example: Autocorrelation Auto-covariance

5. Stationary Processes Strict-Sense Stationary (SSS) A process in which for all n, all (t1, t2, …,tn,), and all Δ Wide-Sense Stationary (WSS) A process X(t) with the following conditions • mX(t) = E[X(t)] is independent of t. • RX(t1,t2) depends only on the difference τ = t1 - t2and not on t1 and t2 individually. Cyclostationary A random process X(t) is cyclostationary if both the mean, mx(t), and the autocorrelation function, RX(t1+τ, t2), are periodic in t with some period T0: i.e., ifandfor all t and τ.

6. Wide-Sense Stationary Example: Mean: Autocorrelation:

7. Power Spectral Density Generalities : Example:

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

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

10. Multiple Random Processes 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(t1)and Y(t2) are independent for all t1 and t2 Uncorrelated Random Processes X(t) and Y(t) • If random variables X(t1) and Y(t2)are uncorrelated for all t1 and t2 Jointly wide-sense stationary • If X(t)and Y(t)are both individually wss • The cross-correlation function RXY(t1,t2) depends only on τ = t2 - t1 Filter