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 Document
Unformatted text preview: 8.7 Power Spectral Density 287 Figure 8.4 Plot of S XX ( w ) for Example 8.10 The crosspower spectral density is generally a complex function even when both X ( t ) and Y ( t ) are real. Thus, since R YX ( τ ) = R XY ( − τ ) , we have that S YX ( w ) = S XY ( − w ) = S ∗ XY ( w ) where S ∗ XY ( w ) is the complex conjugate of S XY ( w ) . Example 8.10 Determine the autocorrelation function of the random process with the power spectral density given by S XX ( w ) = braceleftBig S  w  < w otherwise Solution S XX ( w ) is plotted in Figure 8.4. R XX ( τ ) = 1 2 π integraldisplay ∞ −∞ S XX ( w ) e jw τ dw = 1 2 π integraldisplay w − w S e jw τ dw = S 2 j πτ bracketleftbig e jw τ bracketrightbig w − w = S 2 j πτ bracketleftbig e jw τ − e − jw τ bracketrightbig = S πτ parenleftbigg e jw τ − e − jw τ 2 j parenrightbigg = S πτ sin ( w τ ) trianglesolid Example 8.11 A stationary random process X ( t ) has the power spectral density S XX ( w ) = 24 w 2 + 16 Find the meansquare value of the process. 288 Chapter 8 Introduction to Random Processes Solution Method 1 (BruteForce Method) : The meansquare value is given by E bracketleftbig X 2 ( t ) bracketrightbig = 1 2 π integraldisplay ∞ −∞ S XX ( w ) dw = 1 2 π integraldisplay ∞ −∞ 24 w 2 + 16 dw = 1 2 π integraldisplay ∞ −∞ 24 16 [ 1 + ( w / 4 ) 2 ] dw Let w / 4 = tan θ . Then dw = 4sec 2 ( θ ) d θ 1 + ( w / 4 ) 2 = 1 + tan ( θ ) 2 = sec 2 ( θ ) Also, when w = −∞ , θ = − π / 2; and when w = ∞ , θ = π / 2. Thus, we obtain E bracketleftbig X 2 ( t ) bracketrightbig = 24 32 π integraldisplay π / 2 − π / 2 4sec 2 ( θ ) d θ sec 2 ( θ ) = 3 π integraldisplay π / 2 − π / 2 d θ = 3 π [ θ ] π / 2 − π / 2 = 3 π braceleftbigg π 2 − parenleftbigg − π 2 parenrightbiggbracerightbigg = 3 π braceleftbigg π 2 + π 2 bracerightbigg = 3 Solution Method 2 (SmartMove Method) : From Table 8.1 we observe that e − a  τ  ↔ 2 a a 2 + w 2 That is, e − a  τ  and 2 a /( a 2 + w 2 ) are Fourier transform pairs. Thus, if we can iden tify the parameter a in the given problem, we can readily obtain the autocorrela tion function. Rearranging the power spectral density, we obtain S XX ( w ) = 24 w 2 + 16 = 24 w 2 + 4 2 = 3 braceleftbigg 2 ( 4 ) w 2 + 4 2 bracerightbigg ≡ 3 braceleftbigg 2 a w 2 + a 2 bracerightbigg This means that a = 4 and the autocorrelation function is R XX ( τ ) = 3 e − 4  τ  Therefore, the meansquare value of the process is E bracketleftbig X 2 ( t ) bracketrightbig = R XX ( ) = 3 trianglesolid 8.7 Power Spectral Density 289 8.7.1 White Noise White noise is the term used to define a random function whose power spectral density is constant for all frequencies. Thus, if N ( t ) denotes white noise, S NN ( w ) = N / 2 where N is a real positive constant. The inverse Fourier transform of S NN ( w ) gives the autocorrelation function of N ( t ) , R NN ( τ ) , as follows: R NN ( τ ) =...
View
Full
Document
This note was uploaded on 01/05/2010 for the course STAT 350 taught by Professor Carlton during the Fall '07 term at Cal Poly.
 Fall '07
 Carlton

Click to edit the document details