48089_q1spr05soln - Massachusetts Institute of Technology...

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Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.011: Introduction to Communication, Control and Signal Processing QUIZ 1, March 15, 2005 Answer Booklet Your Full Name: Recitation Instructor & Time : at o’clock This quiz is closed book , but two sheets of notes are allowed. Calculators will not be necessary and are not allowed. Put your name in the space indicated above, and your recitation time next to the name of your instructor. Check that this answer booklet has pages numbered up to 16. The booklet contains spaces for all relevant reasoning and answers. Neat work and clear explanations count; show all relevant work and reasoning! You may want to Frst work things through on scratch paper and then neatly transfer to this booklet the work you would like us to look at. Let us know if you need additional scratch paper. Only this booklet will be considered in the grading; no additional an- swer or solution written elsewhere will be considered. Absolutely no exceptions! There are two problems , weighted as indicated on the question booklet . DO NOT DISCUSS THIS QUIZ WITH 6.011 STUDENTS WHO HAVE NOT YET TAKEN IT TODAY! Problem Your Score 1 (20 points) 2 (30 points) Total (50 points) 1
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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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