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handout_week14_a - Stochastic Signals and Systems Analysis...

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Stochastic Signals and Systems Analysis and Processing of Random Signals Virginia Tech Fall 2008 Power Spectral Density For deterministic processes (signals), Fourier series and Fourier transform represent the frequency components as a “frequency spectrum” Frequency components refer to an average rate of change (averaged over possible realizations) The power spectral density S ( f ) of a signal x ( t ) is a function that gives the variation of density of power with frequency, which can be defined as S ( f ) = lim T →∞ 1 2 T Z T - T x ( t ) e - j 2 π ft dt 2
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Power Spectral Density Frequencies are related to autocorrelation function High autocorrelation more “memory” in process, slower changes Low autocorrelation less “memory” in process, faster changes The autocorrelation function R ( τ ) of a power signal x ( t ) is defined as the time average R ( τ ) = lim T →∞ 1 2 T Z T - T x ( t ) x ( t + τ ) dt The autocorrelation function and power spectral density of a power signal are closely related this relationship is stated formally by the Wiener-Khinchine theorem: S ( f ) = F [ R ( τ )] = Z -∞ R ( τ ) e - j 2 π f τ d τ Power Spectral Density The power spectral density (or
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