ch10 - Chapter 10 Analysis and Processing of Random Signals...

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Unformatted text preview: Chapter 10 Analysis and Processing of Random Signals ENCS6161 - Probability and Stochastic Processes Concordia University Power Spectral Density For WSS r.p. X ( t ) , the power spectral density (PSD) S X ( f ) defines F [ R X ( τ )] = integraldisplay + ∞-∞ R X ( τ ) e- j 2 πfτ dτ So, R X ( τ ) = integraltext + ∞-∞ S X ( f ) e j 2 πfτ df The average power of X ( t ) E [ X ( t ) 2 ] = R X (0) = integraldisplay + ∞-∞ S X ( f ) df Note: S X ( f ) ≥ (see pg. 412 of textbook for proof) Cross-Power Spectral Density: Fro two WSS r.p. X ( t ) ,Y ( t ) S X,Y ( f ) = F [ R X,Y ( τ )] where R X,Y ( τ ) = E [ X ( t + τ ) Y ( t )] . ENCS6161 – p.1/11 Power Spectral Density Example: a WSS r.p. with S X ( f ) = N 2 for | f | < w N 2 S X ( f ) S X ( f )- w w f R X ( τ ) R X ( τ ) 1 2 w 1 w τ E [ X 2 ( t )] = integraldisplay w- w N 2 df = N w R X ( τ ) = integraldisplay w- w N 2 e j 2 πfτ df = N sin(2 πwτ ) 2 πτ If τ = ± k 2 w , then X ( t ) and X ( t + τ ) are uncorrelated....
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ch10 - Chapter 10 Analysis and Processing of Random Signals...

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