This preview shows pages 1–8. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 • ContinuousTime Random Processes ◦ Let X ( t ) be a continuoustime 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 • ContinuousTime 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 • ContinuousTime 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 • ContinuousTime 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 • ContinuousTime 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 realvalued 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...
View
Full
Document
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.
 Spring '09
 ;im

Click to edit the document details