Chapt 8b

# Chapt 8b - 8.5 Stationary Random Processes 277 = E...

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Unformatted text preview: 8.5 Stationary Random Processes 277 = E bracketleftbig A 2 bracketrightbig cos ( t ) cos ( s ) + E [ AB ]{ cos ( t ) sin ( s ) + sin ( t ) cos ( s ) } + E bracketleftbig B 2 bracketrightbig sin ( t ) sin ( s ) = 2 { cos ( t ) cos ( s ) + sin ( t ) sin ( s ) } = 2cos ( t − s ) Since the mean is constant and the autocorrelation function is a function of the difference between the two times, we conclude that the random process X ( t ) is wide-sense stationary. trianglesolid Example 8.5 Assume that X ( t ) is a random process defined as follows: X ( t ) = A cos ( 2 π t + Phi1) where A is a zero-mean normal random variable with variance σ 2 A = 2 and Phi1 is a uniformly distributed random variable over the interval − π ≤ φ ≤ π . A and Phi1 are statistically independent. Let the random variable Y be defined as follows: Y = integraldisplay 1 X ( t ) dt Determine 1. the mean E [ Y ] of Y . 2. the variance of Y . Solution The mean of X ( t ) is given by E [ X ( t ) ] = E [ A cos ( 2 π t + Phi1) ] = E [ A ] E [ cos ( 2 π t + Phi1) ] = Similarly the variance of X ( t ) is given by σ 2 X ( t ) = E bracketleftbig { X ( t ) − E [ X ( t ) ]} 2 bracketrightbig = E bracketleftbig X 2 ( t ) bracketrightbig = E bracketleftbig ( A cos ( 2 π t + Phi1)) 2 bracketrightbig = E bracketleftbig A 2 bracketrightbig E bracketleftbig { cos ( 2 π t + Phi1) } 2 bracketrightbig = 2 E bracketleftbigg 1 + cos ( 4 π t + 2 Phi1) 2 bracketrightbigg = 2 × 1 2 braceleftbigg 1 + integraldisplay π − π cos ( 4 π t + 2 φ ) f Phi1 ( φ ) d φ bracerightbigg = 1 + 1 2 π integraldisplay π − π cos ( 4 π t + 2 φ ) d φ = 1 278 Chapter 8 Introduction to Random Processes (1) The mean of Y is given by E [ Y ] = E bracketleftbiggintegraldisplay 1 X ( t ) dt bracketrightbigg = integraldisplay 1 E [ X ( t ) ] dt = (2) The variance of Y is given by σ 2 Y = E bracketleftbig { Y − E [ Y ]} 2 bracketrightbig = E bracketleftbig Y 2 bracketrightbig = E bracketleftbiggparenleftbiggintegraldisplay 1 X ( t ) dt parenrightbigg 2 bracketrightbigg = E bracketleftbiggbraceleftbiggintegraldisplay 1 A cos ( 2 π t + Phi1) dt bracerightbigg 2 bracketrightbigg = E bracketleftbiggbraceleftbigg A sin ( 2 π t + Phi1) 2 π vextendsingle vextendsingle vextendsingle 1 bracerightbigg 2 bracketrightbigg = 1 4 π 2 E bracketleftbig { A sin ( 2 π + Phi1) − A sin (Phi1) } 2 bracketrightbig = 1 4 π 2 E bracketleftbig { A sin (Phi1) − A sin (Phi1) } 2 bracketrightbig = trianglesolid Note that Y = integraldisplay 1 X ( t ) dt = integraldisplay 1 A cos ( 2 π t + Phi1) dt = A [ sin ( 2 π + Phi1) − sin (Phi1) ] 2 π = which is why we got the results for the mean and variance of Y . 8.5.2.1 Properties of Autocorrelation Functions for WSS Processes As defined earlier, the autocorrelation function of a wide-sense stationary ran- dom process X ( t ) is defined as R XX ( t , t + τ ) = R XX ( τ ) The properties of autocorrelation functions of wide-sense stationary processes include the following:...
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Chapt 8b - 8.5 Stationary Random Processes 277 = E...

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