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Unformatted text preview: Chapter 2: Random Processes Dr. Lim HS Last Updated: 12 Jun 2009 c 2009 MMU Presentation outline ◦ The Random Process Concept ◦ Characterization of Random Processes Joint Distributions of Time Samples Mean, Autocorrelation and Autocovariance Functions Multiple Random Processes ◦ Gaussian Random Processes * ◦ Stationary Random Processes ◦ Time Averages and Ergodicity c 2009 MMU Page 1/52 The Random Process Concept ◦ Recall that a rv X is a function for assigning to every outcome s of an experiment a number X ( s ) . ◦ A random process (or stochastic process ) X ( t ) is a function for assigning to every outcome s a function of time X ( t,s ) . ◦ The function X ( t,s ) versus t , for s fixed, is called a realization or sample function of the random process. ◦ Thus, a random process is a family (or ensemble ) of time functions depending on the parameter s . c 2009 MMU Page 2/52 The Random Process Concept (cont.) ◦ We usually suppress the s and use X ( t ) to denote a random process. s 1 s 2 s 3 s 4 1 2 4 8 S x X P ( s 1 ) = 0.2 P ( s 2 ) = 0.4 P ( s 3 ) = 0.1 P ( s 4 ) = 0.3 Figure 1: A random variable mapping of the sample space. s 4 s 3 s 2 s 1 S X ( t ) x ( t ) t t t t Figure 2: A random process mapping of the sample space. c 2009 MMU Page 3/52 The Random Process Concept (cont.) ◦ Figure 3 shows several realizations of a random process. ◦ When t is fixed and s is a variable, the random process represents a rv. ◦ The rv derived from the random process X ( t ) at time t i is denoted X i . t i t t t x n ( t ) x n+ 1 ( t ) x n+ 2 ( t ) Figure 3: Several realizations of a random process. c 2009 MMU Page 4/52 The Random Process Concept (cont.) t x n ( t ) t x n ( t ) t x n ( t ) t x n ( t ) (a) (b) (c) (d) Figure 4: Classification of random processes: (a,c) continuoustime random process, (b,d) discretetime random process. c 2009 MMU Page 5/52 The Random Process Concept (cont.) ◦ The following examples show some instances of random processes. ◦ Example 1. Let ζ be a number selected at random from the interval S = [0 , 1] , and let b 1 b 2 ... be the binary expansion of ζ : ζ = ∞ X i =1 b i 2 i where b i ∈ { , 1 } . Define the discretetime random process X ( n,ζ ) by X ( n,ζ ) = b n , n = 1 , 2 ,... The resulting process is a sequence of binary numbers, with X ( n,ζ ) equal to the n th number in the binary expansion of ζ . c 2009 MMU Page 6/52 The Random Process Concept (cont.) Several realizations of the random process X ( n,ζ ) are shown in Figure 5. Figure 5: Binary expansion of ζ . c 2009 MMU Page 7/52 The Random Process Concept (cont.) ◦ Example 2. Let ζ be a number selected at random from the interval [ 1 , 1] ....
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