[RP]Lecture Note XI

# [RP]Lecture Note XI - Kyung Hee University Department of...

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Kyung Hee University Random Processing Department of Electronics and Radio Engineering Prof. Hyundong Shin Communications and Coding Theory Laboratory (CCTLAB) XI-1 C1002900 RP Lecture Handout XI: Stochastic Processes Reading: Chapter 9.1 In preceding lectures, we have focused on random variables and random vectors, and their manipulation in problems of detection and estimation. We now want to broaden our development to accommodate sequences and waveforms that are random as well. Sequences (discrete-time) and waveforms (continuous-time) of this type are re- ferred to as random or stochastic processes. 11.1. Definitions A random process   X t is an indexed collection , : , X t w t   of random va- riables, all on the same probability space , , P . Example 11.1: AC voltage   cos X t A t    . A random process is a family or an ensemble of time functions depending on the outcome , or equivalently, a function of t and . If   is fixed, it is a single time function, called the sample path correspond- ing to . If t is fixed,   X t is a random variable equal to the state of the given process at time t . If t and are fixed,   X t is a number. 11.2. Statistics of Stochastic Processes In general, probabilistic characterizations of a stochastic process involve specifying the joint probabilistic description of the process at different points in time. In particular,

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Kyung Hee University Random Processing Department of Electronics and Radio Engineering Prof. Hyundong Shin Communications and Coding Theory Laboratory (CCTLAB) XI-2 stochastic processes are completely characterized by the collection of the n th-order densities , , , , n n X t X t f x x 1 1 (XI.1) or distributions , , , , , , n n n n X t X t F x x X t x X t x 1 1 1 1 (XI.2) for every possible choice of n and the time instants , , , n t t t 1 2 . Very difficult. In practice, the development of stochastic processes follows one of two approaches: (i) Only partial statistical descriptions of the processes are pursued; or (ii) The focus is restricted to processes with special structures or properties that substantially simplify their descriptions. One of the simplest stochastic processes is a discrete-time white noise. Example 11.1: A particular simple discrete-time white noise corresponds to the case in which the samples X n are zero-mean and have identical variances: . X n X n X m n m    2 0 (XI.3) If , X n 2 0 , i.e., X n is a discrete-time Gaussian white noise, we have , , , , ; , N N N i X n X n i f x x x 1 1 1 0 (XI.4) which completely characterizes the process.
Kyung Hee University Random Processing Department of Electronics and Radio Engineering Prof. Hyundong Shin Communications and Coding Theory Laboratory (CCTLAB)

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