e562a03sf_sol

# e562a03sf_sol - EE562a Final Exam Solutions Spring, 2003...

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EE562a Final Exam Solutions Spring, 2003 Problem 1. (25 points) 1.1) (8 points) Show that R ( τ ) = e -| τ | cos (2 τ ) is a valid correlation function for some wide- sense stationary random processes. Suppose R ( τ ) is the correlation function of X ( u,t ). Find the power spectral density function S X ( ω ) and mean μ X . Solutions: A function is a valid correlation function for some wide-sense stationary random processes if and only if it’s symmetric and non-negative deﬁnite. Because R ( - τ ) = R ( τ ) , it’s symmetric. The Fourier transform of R ( τ ) = e -| τ | · e j 2 τ + e - j 2 τ 2 can be calculated as 1 1 + ( ω - 2) 2 + 1 1 + ( ω + 2) 2 , which is always non-negative, so R ( τ ) is a valid correlation function. The power spectral density function S X ( ω ) = 1 1 + ( ω - 2) 2 + 1 1 + ( ω + 2) 2 . Because there is no delta component in S X ( ω ) , the mean μ X = 0 . 1

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1.2) (9 points) Let X ( u ), Y ( u ) be two independent zero-mean random variables. Suppose that σ 2 X = 9 and σ 2 Y = 4. Let Z ( u,t ) = X ( u ) sin ( t ) + Y ( u ) cos ( t ) + W ( u,t ), where W ( u,t ) is a white noise with unit noise power. Based on the observation of Z ( u,t ) in the interval t (0 , 2 π ], ﬁnd an estimator for the random vector ( X ( u ) ,Y ( u )) t to minimize the mean square estimation error. Solutions: The problem is an application of K-L expansion. ) , ( t u Z π ) sin( t ) cos( t 2 0 2 0 ) ( u Z I ) ( u Z Q 1 9 9 + 1 4 4 + ) ( ˆ u X ) ( ˆ u Y Figure 1: The I-Q estimator for [ X ( u ) ,Y ( u )] t Due to the fact Z ( u,t ) = πX ( u ) · 1 π sin ( t ) + πY ( u ) · 1 π cos ( t ) + W ( u,t ) , let φ 1 ( t ) = 1 π sin ( t ) and φ 2 ( t ) = 1 π sin ( t ) be two orthonormal eigen-functions. We have Z Q ( u ) = πX ( u ) + W Q ( u ) , Z I ( u ) = πY ( u ) + W I ( u ) where W Q ( u ) = Z 2 π 0 W ( u,t
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## This note was uploaded on 09/13/2008 for the course EE 562a taught by Professor Toddbrun during the Spring '07 term at USC.

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e562a03sf_sol - EE562a Final Exam Solutions Spring, 2003...

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