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# ch3 - Chapter 3 Spectral Analysis Dr Lim HS Last Updated 17...

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Chapter 3: Spectral Analysis Dr. Lim HS Last Updated: 17 Jun 2009 c 2009 MMU

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Presentation outline Power Spectral Density Response of Linear Systems to Random Signals Bandlimited Random Processes c 2009 MMU Page 1/59
Power Spectral Density Continuous-Time Random Processes Let X ( t ) be a continuous-time random process. Suppose x T ( t ) is a portion of a sample function x ( t ) in the interval - T < t < T as follows x T ( t ) = x ( t ) , - T < t < T 0 , elsewhere (1) The Fourier transform of x T ( t ) is given by X T ( f ) = Z T - T x T ( t ) e - j 2 πft dt = Z T - T x ( t ) e - j 2 πft dt (2) The energy of x T ( t ) is then E ( T ) = Z T - T x 2 ( t ) dt = Z -∞ | X T ( f ) | 2 df (3) where we have used the Parseval’s theorem. c 2009 MMU Page 2/59

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Power Spectral Density Continuous-Time Random Processes (cont.) By dividing E ( T ) by 2 T , we obtain the average power in x ( t ) over the interval ( - T, T ) : P ( T ) = 1 2 T Z T - T x 2 ( t ) dt = Z -∞ | X T ( f ) | 2 2 T df (4) To find the average power of a sample function x ( t ) we need to extend the T to : P xx = lim T →∞ 1 2 T Z T - T x 2 ( t ) dt = Z -∞ lim T →∞ | X T ( f ) | 2 2 T df (5) Now, recall that x ( t ) is only a sample function selected from an ensemble of allowable time functions. Therefore, P xx is a rv and the average power of the random process X ( t ) is the ensemble average (or expectation) of P xx . c 2009 MMU Page 3/59
Power Spectral Density Continuous-Time Random Processes (cont.) The average power for the random process X ( t ) is then P XX = E [ P xx ] = lim T →∞ 1 2 T Z T - T E [ X 2 ( t )] dt (6) = Z -∞ lim T →∞ E [ | X T ( f ) | 2 ] 2 T df (7) From (6), we see that the average power in X ( t ) is the time average of its second moment E [ X 2 ( t )] . If X ( t ) is WSS, then E [ X 2 ( t )] = R XX (0) and therefore P XX = lim T →∞ 1 2 T Z T - T R XX (0) dt = R XX (0) (8) c 2009 MMU Page 4/59

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Power Spectral Density Continuous-Time Random Processes (cont.) From (7), the average power of X ( t ) can be obtained by a frequency domain integration: P XX = Z -∞ lim T →∞ E [ | X T ( f ) | 2 ] 2 T df (9) the integrand is therefore called the power spectral density (psd) of X ( t ) which is denoted as: S XX ( f ) = lim T →∞ E [ | X T ( f ) | 2 ] 2 T (10) Hereafter, we will assume that X ( t ) is a WSS process. c 2009 MMU Page 5/59
Power Spectral Density Continuous-Time Random Processes (cont.) The psd of a random process X ( t ) is given by the Fourier transform of the autocorrelation function (the proof is given at the end of this section): S XX ( f ) = F{ R XX ( τ ) } = Z -∞ R XX ( τ ) e - j 2 πfτ (11) For real-valued random processes, the autocorrelation is an even function of τ : R XX ( τ ) = R XX ( - τ ) Substitution into (11) implies that S XX ( f ) = Z -∞ R XX ( τ )(cos 2 πfτ - j sin 2 πfτ ) = Z -∞ R XX ( τ ) cos 2 πfτdτ (12) since the integral of the product of an even function ( R XX ( τ ) ) and an odd function ( sin 2 πfτ ) is zero. c 2009 MMU Page 6/59

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Power Spectral Density Continuous-Time Random Processes (cont.) Eq. (12) implies that S XX ( f ) is real-valued and an even function of f .
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