lect08 - Lecture Notes 8 Random Processes • Definition...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ◦ Gauss-Markov 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 discrete-time 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 . . ....
View Full Document

This note was uploaded on 11/28/2009 for the course EE 278 taught by Professor Balajiprabhakar during the Fall '09 term at Stanford.

Page1 / 12

lect08 - Lecture Notes 8 Random Processes • Definition...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online