Second-order Description 2.12 2001/08/06 2005/01/06 14:57:08.549 US/Central Behnaam Aazhang aaz@ece.rice.edu Dinesh Rajan dinesh@ece.rice.edu Mohammad Jaber Borran mohammad@ece.rice.edu Roy Ha rha@rice.edu Shawn Stewart mrshawn@alumni.rice.edu Behnaam Aazhang aaz@ece.rice.edu autocorrelation autocovariance crosscorrelation mean random process stochastic process wide sense stationary Describes signals that cannot be precisely characterized.

Second-order description Practical and incomplete statistics Mean The mean function of a random process X t is defined as the expected value of X t for all t's. μ X t X t x x f X t x continuous k x k p X t x k discrete Autocorrelation The autocorrelation function of the random process X t is defined as R X t 2 t 1 X t 2 X t 1 x 2 x 1 x 2 x 1 f X t 2 X t 1 x 2 x 1 continuous k l x l x k p X t 2 X t 1 x l x k discrete If X t is second-order stationary, then R X t 2 t 1 only depends on t 2 t 1 . R X t 2 t 1 X t 2 X t 1 x 1 x 2 x 2 x 1 f X t 2 X t 1 x 2 x 1 R X t 2 t 1 x 1 x 2 x 2 x 1 f X t 2 - t 1 X 0 x 2 x 1 R X t 2 t 1 0 If R X t 2 t 1 depends on t 2 t 1 only, then we will represent the autocorrelation with only one variable τ t 2 t 1 R X τ R X t 2 t 1 R X t 2 t 1 Properties R X 0 0 R X τ R X τ R X τ R X 0 X t 2 f 0 t Θ ω and Θ is uniformly distributed between 0 and 2 . The mean function μ X t X t 2 f 0 t Θ θ 0 2 2 f 0 t θ 1 2 0 The autocorrelation function R X t τ t X t + τ X t 2 f 0 t τ Θ 2 f 0 t Θ 12 2 f 0 τ 12 2 f 0 2 t τ 2 Θ 12 2 f 0 τ 12 θ 0 2 2 f 0 2 t τ 2 θ 1 2 12 2 f 0 τ Not a function of t since the second term in the right hand side of the equality in is zero. Toss a fair coin every T seconds. Since X t is a discrete valued random process, the statistical characteristics can be captured by the pmf and the mean function is written as μ X t X t 12 -1 12 1 0 R X t 2 t 1 k k l l x k x l p X t 2 X t 1 x k x l 1 1 12 -1 -1 12 1 when n T t 1 n 1 T and n T t 2 n 1 T R X t 2 t 1 1 1 14 -1 -1 14 -1 1 14 1 -1 14 0 when n T t 1 n 1 T and m T t 2 m 1 T with n m R X t 2 t 1 1 n T t 1 n 1 T n T t 2 n 1 T 0 A function of t 1 and t 2 . Wide Sense Stationary A process is said to be wide sense stationary if μ X is constant and R X t 2 t 1 is only a function of t 2 t 1 . If X t is strictly stationary, then it is wide sense stationary. The converse is not necessarily true. Autocovariance Autocovariance of a random process is defined as C X t 2 t 1 X t 2 μ X t 2 X t 1 μ X t 1 R X t 2 t 1 μ X t 2 μ X t 1 The variance of X t is Var X t C X t t Two processes defined on one experiment ().

Crosscorrelation The crosscorrelation function of a pair of random processes is defined as R X Y t 2 t 1 X t 2 Y t 1 y x x y f X t 2 Y t 1 x y C X Y t 2 t 1 R X Y t 2 t 1 μ X t 2 μ Y t 1 Jointly Wide Sense Stationary The random processes X t and Y t are said to be jointly wide sense stationary if R X Y t 2 t 1 is a function of t 2 t 1 only and μ X t and μ Y t are constant.