48089_q1spr05soln

# 48089_q1spr05soln - Massachusetts Institute of Technology...

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6.011 Quiz 1, March 15, 2005 Problem 1 (20 points) Suppose x ( t )= y ( t ) cos( ω o t + Θ), where: y ( t ) is a wide-sense stationary (WSS) process with mean µ y and autocovariance function C yy ( τ ); ω o is a known constant; and Θ is a random variable that is independent of y ( · ) and is uniformly distributed in the interval [0 , 2 π ]. Do part (a) below especially carefully, because (b) and (c) depend on it to some extent! You might Fnd it helpful in one or more parts of the problem to recall that 1 cos( A ) cos( B )= 2 [cos( A + B )+cos( A B )] . (a) (8 points) ±ind the mean µ x ( t ) and autocorrelation function E [ x ( t + τ ) x ( t )] of the process x ( · ). Also Fnd the cross-correlation function E [ y ( t + τ ) x ( t )]. Explain precisely what features of your answers tell you that: (i) x ( · ) is a WSS process; and (ii) x ( · )and y ( · ) are jointly WSS. Begin
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48089_q1spr05soln - Massachusetts Institute of Technology...

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