Ch3 - Chapter 3 Spectral Analysis Dr Lim HS Last Updated 17 Jun 2009 c 2009 MMU Presentation outline ◦ Power Spectral Density ◦ Response of

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Unformatted text preview: Chapter 3: Spectral Analysis Dr. Lim HS Last Updated: 17 Jun 2009 c 2009 MMU 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 , 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 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 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 dτ (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 )(cos2 πfτ- j sin2 πfτ ) dτ = Z ∞-∞ R XX ( τ )cos2 πfτdτ...
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This note was uploaded on 08/21/2009 for the course FET 44 taught by Professor ;im during the Spring '09 term at Multimedia University, Cyberjaya.

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Ch3 - Chapter 3 Spectral Analysis Dr Lim HS Last Updated 17 Jun 2009 c 2009 MMU Presentation outline ◦ Power Spectral Density ◦ Response of

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