This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 12:051:15pm. 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, weve 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 lowpass filter) Figure 1: Continuous and discretetime 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, were going to be more general, as shown in Figure 1. Let X ( t ) be a WSS random process with autocorrelation R X ( ). Let X ( t ) be the input to a linear timeinvariant (LTI) filter h ( t ). (LTI filters can be represented completely by their impulse response h ( ), and they dont 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 = h ( ) X ( t ) d Y ( t ) = E [ Y ( t )] = E bracketleftbiggintegraldisplay = h ( ) X ( t ) d bracketrightbigg ECE 5510 Fall 2008 2 As long as E Y [ X ( t )] is finite for all t and , you can exchange the order of an integral and an expected value without a problem. Why? An expected value is an integral. Y ( t ) = integraldisplay = h ( ) E [ X ( t )] d = X integraldisplay = h ( ) d For the autocorrelation function, R Y ( ) = E Y [ Y ( t ) Y ( t + )] = E Y bracketleftbiggintegraldisplay = h ( ) X ( t ) integraldisplay = h ( ) X ( t +...
View
Full
Document
 Fall '08
 Chen,R

Click to edit the document details