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Unformatted text preview: ECE 5510: Random Processes Lecture Notes Fall 2008 Lecture 21 Today: (1) Power Spectral Density (2) LTI Filtering • HW 9 due today at 5pm in HW locker. HW 10: assigned today. Delay due date until Tue., Dec. 9 (?) • OH today from 12:05-1:15pm. • Reading: Today: 11.1, 11.5, 11.8. Tue Dec 2: 11.2, 11.3, 11.6. • Discussion items? 1 LTI Filtering of WSS Signals In our examples, we’ve been generally talking about filtered signals. For example: 1. Brownian Motion is an integral (low pass filter) of White Gaussian noise 2. Some of the HW 9 problems. 3. Problems like this: X 1 ,X 2 ,... an i.i.d random sequence. Let Y 2 ,Y 3 ,... be Y k = X k + X k- 1 (also a low-pass filter) Figure 1: Continuous- and discrete-time filtering of a random process X ( t ) or X [ n ] results in output Y ( t ) = X ( t ) ⋆ h ( t ) or Y [ n ] = X [ n ] ⋆ h [ n ]. Here, we’re going to be more general, as shown in Figure 1. Let X ( t ) be a WSS random process with autocorrelation R X ( τ ). Let X ( t ) be the input to a linear time-invariant (LTI) filter h ( t ). (LTI filters can be represented completely by their impulse response h ( τ ), and they don’t change impulse response over time.) Let Y ( t ) be the output of the filter. What is the mean function and autocorrelation of Y ( t )? Y ( t ) = ( h ⋆ X )( t ) = integraldisplay ∞ τ =-∞ h ( τ ) X ( t − τ ) dτ μ Y ( t ) = E [ Y ( t )] = E bracketleftbiggintegraldisplay ∞ τ =-∞ h ( τ ) X ( t − τ ) dτ bracketrightbigg ECE 5510 Fall 2008 2 As long as E Y [ X ( t − τ )] is finite for all t and τ , you can exchange the order of an integral and an expected value without a problem. Why? An expected value is an integral. μ Y ( t ) = integraldisplay ∞ τ =-∞ h ( τ ) E [ X ( t − τ )] dτ = μ X integraldisplay ∞ τ =-∞ h ( τ ) dτ For the autocorrelation function, R Y ( τ ) = E Y [ Y ( t ) Y ( t + τ )] = E Y bracketleftbiggintegraldisplay ∞ α =-∞ h ( α ) X ( t − α ) integraldisplay ∞ β =-∞ h ( β ) X ( t +...
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- Fall '08