ch12_randsys

ch12_randsys - MIT OpenCourseWare http/ocw.mit.edu HST.582J...

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MIT OpenCourseWare http://ocw.mit.edu HST.582J / 6.555J / 16.456J Biomedical Signal and Image Processing Spring 2007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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( 1 2 Harvard-MIT Division of Health Sciences and Technology HST.582J: Biomedical Signal and Image Processing, Spring 2007 Course Director: Dr. Julie Greenberg HST582J/6.555J/16.456J Biomedical Signal and Image Processing Spring 2007 Chapter 12 - RANDOM SIGNALS AND LINEAR SYSTEMS ± c Bertrand Delgutte 1999 Introduction In Chapter 2, we saw that the impulse response completely characterizes a linear, time-invariant system because the response to an arbitrary, but known, input can be computed by convolving the input with the impulse response. The impulse response plays a key role for random signals as it does for deterministic signals, with the important difference that it is used for computing time averages of the output from averages of the input. SpeciFcally, we will show that knowing the impulse response suffices to derive the mean and autocorrelation function of the output from the mean and autocorrelation function of the input. That autocorrelation functions are involved is to be expected, since we showed in Chapter 11 that these functions naturally arise when processing random signals by linear Flters. In Chapter 3, we introduced ±ourier analysis for deterministic signals, and showed that this concept leads to simpliFcations in the analysis and design of linear, time invariant systems. ±requency-domain techniques are as powerful for stationary random signals as they are for deterministic signals. They lead to the concepts of power spectrum and Wiener filters, which have numerous applications to system identiFcation and signal detection in noise. 12.1 Response of LTI systems to random signals Our goal in this section is to derive general formulas for the mean and autocorrelation of the response of a linear system to a stationary random signal, given both the system impulse response and the mean and autocorrelation function of the input. 12.1.1 Mean of y [ n ] Let x [ n ] be a random signal used as input to an LTI system with impulse response h [ n ]. The mean of the output y [ n ]is : <y [ n ] > = <x [ n ] h [ n ] > = < ± h [ m ] x [ n m ] > n = h [ m ] [ n m ] > n (12 . 1) m = −∞ m ± = −∞ In this expression, the notation <.> n is used to specify that averaging is over the time variable n rather than the parameter m . Because we average over n , h [ m ] is constant, and we can write <h [ m ] x [ n m ] > n = h [ m ] [ n m ] > n . By stationarity, we further obtain: [ n ] > = [ n ] > n ± h [ n ] . 2 a ) = −∞ Cite as: Bertrand Delgutte. Course materials for HST.582J / 6.555J / 16.456J, Biomedical Signal and Image Processing, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
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As a special case of (12.2), if x [ n ] has zero mean and the system is stable, the output also has zero mean.
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This note was uploaded on 11/07/2011 for the course AERO 16.422 taught by Professor Juliegreenberg during the Spring '07 term at MIT.

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ch12_randsys - MIT OpenCourseWare http/ocw.mit.edu HST.582J...

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