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# handout_week14_c - as T goes to inﬁnity When this happens...

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Stochastic Signals and Systems Analysis and Processing of Random Signals Virginia Tech Fall 2008 Ergodicity Until now we have generally assumed that a statistical description of the random process is available. Of course, this is seldom true in practice; thus we must develop some way of learning the needed statistical quantities from the observed sample functions of the random process of interest. Fortunately, for many stationary random process, we can substitute time averages for the unknown ensemble averages .

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Ergodicity For example, to estimate the mean m X ( t ) of a random process X ( t , ξ ) , we repeat the random experiment and take the following ensemble average : ˆ m X ( t ) = 1 N N X i = 1 X ( t , ξ i ) , where N is the number of repetitions of the experiment, and X ( t , ξ i ) is the realization observed in the i th repetition. In some situations, we are interested in estimating the mean or auto correlation functions from the time average of a single realization, that is, * h X ( t ) i T = 1 2 T Z T - T X ( t , ξ ) dt . In many cases, this time average will tend to the ensemble average
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Unformatted text preview: as T goes to inﬁnity. When this happens for a random process, we say that the random process is ergodic in the mean. * This estimate yields a single number, so obviously it only makes sense to consider processes for which m X ( t ) = m , a constant. Ergodicity Theorem . Let X ( t ) be a WSS process with m X ( t ) = m , then lim T →∞ h X ( t ) i T = m in the mean square sense, if and only if lim T →∞ VAR [ h X ( t ) i T ] = lim T →∞ 1 2 T Z 2 T-2 T ± 1-| u | 2 T ² C X ( u ) du = We can also deﬁne ergodicity in the higher moments, for example, mean square or power. Theorem . A WSS random process X ( t ) is ergodic in mean square if lim T →∞ " 1 2 T Z T-T X 2 ( t ) dt # = R X ( ) Problem Let X ( t ) = A for all t , where A is a zero-mean, unit-variance random variable. Is X ( t ) ergodic?...
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handout_week14_c - as T goes to inﬁnity When this happens...

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