notes2 - ECE 562 Fall 2011 August 25, 2011 Random Processes...

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Unformatted text preview: ECE 562 Fall 2011 August 25, 2011 Random Processes and White Gaussian Noise (WGN) A random process is { X ( t ) ,t T } simply a random signal or a random function of t (which will usually denote time). Just as in the definition of a random variable, there is an underlying sample space in the definition of a random process, which means we should write X ( t ) as X ( t ; ), with . For a fixed value in the sample space , X ( t ; ) is simply a deterministic function of time. The function X ( t ; ) x ( t ; ) is called a sample function . A generic sample function of the process X ( t ) is usually denoted by the lower-case function x ( t ). For for fixed t , X ( t ; ) (or simply X ( t )) is a random variable on . Example 1 : Sinusoid with a random phase (explicit sample space) Suppose = { , 2 ,, 3 2 } with probability distribution P (0) = P ( 2 ) = P ( ) = P ( 3 2 ) = 1 4 . Define a random process on by X ( t, ) =...
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notes2 - ECE 562 Fall 2011 August 25, 2011 Random Processes...

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