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Unformatted text preview: Lecture Notes 8 Random Processes Definition and Simple Examples Discrete Time Random Processes IID Random Walk Process Markov Processes Independent Increment Processes GaussMarkov Process Mean and Autocorrelation Function Gaussian Random Processes EE 278: Random Processes 8 1 Random Process A random process (or stochastic process ) is an infinite indexed collection of random variables { X ( t ) : t T } , defined over a common probability space The index parameter t is typically time, but can also be a spatial dimension Random processes are used to model random experiments that evolve in time: Received sequence/waveform at the output of a communication channel Packet arrival times at a node in a communication network Thermal noise in a resistor Scores of an NBA team in consecutive games Daily price of a stock Winnings or losses of a gambler EE 278: Random Processes 8 2 Questions Involving Random Processes Dependencies of the random variables of the process How do future received values depend on past received values? How do future prices of a stock depend on its past values? Long term averages What is the proportion of time a queue is empty? What is the average noise power at the output of a circuit? Extreme or boundary events What is the probability that a link in a communication network is congested? What is the probability that the maximum power in a power distribution line is exceeded? What is the probability that a gambler will lose all his captial? Estimation/detection of a signal from a noisy waveform EE 278: Random Processes 8 3 Two Ways to View a Random Process A random process can be viewed as a function X ( t, ) of two variables, time t T and the outcome of the underlying random experiment For fixed t , X ( t, ) is a random variable over For fixed , X ( t, ) is a deterministic function of t , called a sample function X ( t, w 1 ) X ( t, w 2 ) X ( t, w 3 ) t t t t 1 t 2 X ( t 1 , w ) X ( t 2 , w ) EE 278: Random Processes 8 4 Discrete Time Random Process A random process is said to be discrete time if T is a countably infinite set, e.g., N = { , 1 , 2 , . . . } Z = { . . . , 2 , 1 , , +1 , +2 , . . . } In this case the process is denoted by X n , for n N , a countably infinite set, and is simply an infinite sequence of random variables A sample function for a discrete time process is called a sample sequence or sample path A discretetime process can comprise discrete, continuous, or mixed r.v.s EE 278: Random Processes 8 5 Example Let Z U[0 , 1] , and define the discrete time process X n = Z n for n 1 Sample paths: x n x n x n Z = 1 2 Z = 1 4 Z = 0 n n n 1 2 3 4 5 6 7 . . ....
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 Spring '09
 BalajiPrabhakar

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