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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, ω ) of two variables, time t ∈ T and the outcome of the underlying random experiment ω ∈ Ω ◦ For fixed t , X ( t, ω ) is a random variable over Ω ◦ For fixed ω , X ( t, ω ) 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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This note was uploaded on 11/28/2009 for the course EE 278 taught by Professor Balajiprabhakar during the Fall '09 term at Stanford.
 Fall '09
 BalajiPrabhakar
 Signal Processing

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