lecture20 - ECE 5510 Random Processes Lecture Notes Fall...

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ECE 5510: Random Processes Lecture Notes Fall 2009 Lecture 20 Today: (1) Power Spectral Density; (2) Filtering of R.P.s AA #5 due today. Start HW 9 now (it’s on WebCT); due Thu. Dec. 3. Readings: today: Y&G 11.1, 11.5, 11.8; for next class: Y&G 11.2, 11.6, and 11.8. OH today from 1-2. 1 Power Spectral Density of a WSS Signal Now we make specific our talk of the spectral characteristics of a random signal. What would happen if we looked at our WSS random signal on a spectrum analyzer (set to average)? We first need to remind ourselves what the “frequency domain” is. Def’n: Fourier Transform Functions g ( t ) and G ( f ) are a Fourier transform pair if G ( f ) = integraldisplay t = -∞ g ( t ) e - j 2 πft dt g ( t ) = integraldisplay f = -∞ G ( f ) e j 2 πft df PLEASE SEE TABLE 11.1 ON PAGE 413. Def’n: Power Spectral Density (PSD) The PSD of WSS R.P. X(t) is defined as: S X ( f ) = lim T →∞ 1 2 T E bracketleftBigg vextendsingle vextendsingle vextendsingle vextendsingle integraldisplay T - T X ( t ) e - j 2 πft dt vextendsingle vextendsingle vextendsingle vextendsingle 2 bracketrightBigg Note three things. Firstly, the a spectrum analyzer records X ( t ) for a finite duration of time 2 T (in this notation) before displaying X ( f ), the FT of X ( t ). We define PSD in the limit as T goes large. Secondly, this expected value is “sitting in” for the time average. This is valid if the process has the “ergodic” property, which we do not cover specifically in this course, although it is described on page 378 of Y&G. Thirdly, we don’t directly look at the FT on the
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ECE 5510 Fall 2009 2 spectrum analyzer; we look at the power in the FT. Power for a (possibly complex) voltage signal G is S = | G | 2 = G * · G . Theorem: If X ( t ) is a WSS random process, then R X ( τ ) and S X ( f ) are a Fourier transform pair, where S X ( f ) is the power spectral density (PSD) function of X ( t ): S X ( f ) = integraldisplay τ = -∞ R X ( τ ) e - j 2 πfτ R X ( τ ) = integraldisplay f = -∞ S X ( f ) e j 2 πfτ df Proof: Omitted: see Y&G p. 414-415. This theorem is called the Wiener-Khintchine theorem, please use the formal name whenever talking to non-ECE friends to convince them that this is hard.
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