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handout_week13_a

# handout_week13_a - a 1 a 2 a N N X k = 1 N X l = 1 a k a l...

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Stochastic Signals and Systems Random Processes Virginia Tech Fall 2008 Properties Autocorrelation Func. Stationary Processes In this lecture we treat random processes that are jointly stationary and of second order, that is, E h X ( t ) 2 i < . Some important properties of the auto- and cross-correlation functions of stationary second-order processes are summarized as follows. They also hold for the respective covariance functions. The autocorrelation function at τ = 0 gives the average power (second moment) of the process R X ( 0 ) = E [ X ( t ) 2 ] for all t .

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Properties Autocorrelation Func. Stationary Processes The autocorrelation function is an even function of τ R X ( τ ) = E [ X ( t + τ ) X ( t )] = E [ X ( t ) X ( t + τ )] = R X ( - τ ) The autocorrelation function is maximum at τ = 0 | R X ( τ ) | ≤ R X ( 0 ) And the maximum of the cross-correlation function is | R XY ( τ ) | ≤ p R X ( 0 ) R Y ( 0 ) Properties Autocorrelation Func. Stationary Processes The covariance function is positive semidefinite for all N > 0, t < 1 < t 2 < . . . < t N and all complex
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Unformatted text preview: a 1 , a 2 , . . . , a N , N X k = 1 N X l = 1 a k a * l R X ( t k-t l ) ≥ • The autocorrelation function is a measure of the rate of change of a random process P [ | X ( t + τ )-X ( t ) | > ± ] = 2 { R X ( )-R X ( τ ) } ± 2 Properties Autocorrelation Func. Stationary Processes • A WSS random process X ( t ) is mean-square periodic if for some T we have R X ( τ ) = R X ( τ + T ) for all τ . We call the smallest such T > 0 the period . Note: If a WSS random process X ( t ) is m.s. periodic, then its power spectral density is a line spectra with impulses at multiples of the fundamental frequency ω = 2 π/ T . • The autocorrelation function can have three types of components: (1) A component that approaches zero as τ → ∞ ; (2) A periodic component; and (3) A component due to a nonzero mean....
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• Fall '08
• DASILVER
• Stochastic process, Autocorrelation, Stationary process, covariance function, Properties Autocorrelation Func

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