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Chapt 8c

Chapt 8c - 8.7 Power Spectral Density 287 Figure 8.4 Plot...

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8.7 Power Spectral Density 287 Figure 8.4 Plot of S XX ( w ) for Example 8.10 The cross-power 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 0 | w | < w 0 0 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 0 w 0 S 0 e jw τ dw = S 0 2 j πτ bracketleftbig e jw τ bracketrightbig w 0 w 0 = S 0 2 j πτ bracketleftbig e jw 0 τ e jw 0 τ bracketrightbig = S 0 πτ parenleftbigg e jw 0 τ e jw 0 τ 2 j parenrightbigg = S 0 πτ sin ( w 0 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 mean-square value of the process.

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288 Chapter 8 Introduction to Random Processes Solution Method 1 (Brute-Force Method) : The mean-square 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 (Smart-Move 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 mean-square value of the process is E bracketleftbig X 2 ( t ) bracketrightbig = R XX ( 0 ) = 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 0 / 2 where N 0 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 ) = ( N 0 / 2 ) δ ( τ ) where δ ( τ ) is the impulse function. The two functions are shown in Figure 8.5.

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