Introduction to Stochastic Processes 2.14 2001/08/03 2005/01/05 22:24:24.026 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 distribution random process stationary stochastic process Describes signals that cannot be precisely characterized.

Definitions, distributions, and stationarity Stochastic Process Given a sample space, a stochastic process is an indexed collection of random variables defined for each ω Ω . t t X t ω Received signal at an antenna as in .

For a given t, X t ω is a random variable with a distribution First-order distribution F X t b X t b ω Ω X t ω b First-order stationary process If F X t b is not a function of time then X t is called a first-order stationary process. Second-order distribution F X t 1 , X t 2 b 1 b 2 X t 1 b 1 X t 2 b 2 for all t 1 , t 2 , b 1 , b 2 Nth-order distribution F X t 1 , X t 2 , … , X t N b 1 b 2 … b N X t 1 b 1 … X t N b N Nth-order stationary : A random process is stationary of order N if F X t 1 , X t 2 , … , X t N b 1 b 2 … b N F X t 1 + T , X t 2 + T , … , X t N + T b 1 b 2 … b N Strictly stationary : A process is strictly stationary if it is Nth order stationary for all N. X t 2 f 0 t Θ ω where f 0 is the deterministic carrier frequency and Θ ω : → Ω is a random variable defined over and is assumed to be a uniform random variable; i.e., f Θ θ 1 2 θ 0 F X t b X t b 2 f 0 t Θ b F X t b 2 f 0 t Θ b b 2 f 0 t Θ F X t b θ 2 f 0 t b 2 f 0 t 1 2 θ b 2 f 0 t 2 f 0 t 1 2 2 2 b 1 2 f X t x x 1 1 x 1 1 x 2 x 1 0 This process is stationary of order 1.

The second order stationarity can be determined by first considering conditional densities and the joint density. Recall that X t 2 f 0 t Θ Then the relevant step is to find X t 1 x 1 X t 2 b 2 Note that X t 1 x 1 2 f 0 t Θ Θ x 1 2 f 0 t X t 2 2 f 0 t 2 x 1 2 f 0 t 1 2 f 0 t 2 t 1 x 1

F X t 2 , X t 1 b 2 b 1 x 1 b 1 f X t 1 x 1 X t 1 x 1 X t 2 b 2 Note that this is only a function of t 2 t 1 . Every T seconds, a fair coin is tossed. If heads, then X t 1 for n T t n 1 T . If tails, then X t -1 for n T t n 1 T .

p X t x 1 2 x 1 1 2 x -1 for all t . X t is stationary of order 1. Second order probability mass function p X t 1 X t 2 x 1 x 2 p X t 2 | X t 1 x 2 | x 1 p X t 1 x 1 The conditional pmf p X t 2 | X t 1 x 2 | x 1 0 x 2 x 1 1 x 2 x 1 when n T t 1 n 1 T and n T t 2 n 1 T for some n. p X t 2 | X t 1 x 2 | x 1 p X t 2 x 2 for all x 1 and for all x 2 when n T t 1 n 1 T and m T t 2 m 1 T with n m p X t 2 X t 1 x 2 x 1 0 x 2 x 1 for  n T t 1 , t 2 n 1 T p X t 1 x 1 x 2 x 1 for  n T t 1 , t 2 n 1 T p X t 1 x 1 p X t 2 x 2 n m for  n T t 1 n 1 T m T t 2 m 1 T