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Random Processes. ECE460 Spring, 2012. Random ( Stocastic ) Processes. Random Process Definitions. Example :. Notation: Mean:. Random Process Definitions. Example :. Autocorrelation Auto-covariance. Stationary Processes. Strict-Sense Stationary (SSS)
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Random Processes ECE460 Spring, 2012
Random Process Definitions Example: Notation: Mean:
Random Process Definitions Example: Autocorrelation Auto-covariance
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.
Wide-Sense Stationary Example: Mean: Autocorrelation:
Power Spectral Density Generalities : Example:
Example Given a process Yt that takes the values ±1 with equal probabilities: Find
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:
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