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Unformatted text preview: Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.011: Introduction to Communication, Control, and Signal Processing Final Exam — Question Booklet Tuesday, May 20, 2003 Your Full Name: Recitation Instructor & Time: • Write your name in the space above, because we will be collecting your question booklet in addition to you answer booklet but will return it to you when you collect your answer booklet. • This is a closed book exam, but you may use 6 sheets of notes (both sides). • Calculators are allowed, but probably will not be useful. • There are five problems on the exam, with point distributions of 15, 20, 25, 15, 25 , for a total of 100 points . We have also marked for each problem a rough guide to the number of minutes you would spend on the problem if you were dividing your time according to the points on the problem. The time you choose to spend may differ, of course, based on which problems you are more or less comfortable with. However, budget your time carefully ; if you are stuck on a problem, try and move on to another part or another problem, then return if you have time later. Avoid spending inordinately longer than the prorated time on any particular problem. • Be clear and precise in your reasoning and show all relevant work. The grade on each problem is based on our best assessment of your level of understanding, as reﬂected by what you have written. A correct final answer does not guarantee full credit; nor does an incorrect final answer necessarily mean loss of all credit. • If we can’t read it, we can’t/won’t grade it! So please write neatly and legibly. • You are to hand in only the ANSWER booklet that is supplied. No additional pages will be considered in the grading. You may want to first work things through in the blank areas of this question booklet or on scratch paper, and then neatly transfer to the answer booklet the work you would like us to look at. Let us know if you need additional scratch paper. Problem 1 [15 points, 20 minutes] Provide brief answers, with appropriate justifications or computations, for each of the following parts. (a) Can a causal LTI state-space system that is not asymptotically stable become asymptotically stable under appropriate LTI state feedback even if some of its modes are unreachable ? Answer: Yes, as long as unreachable modes are asymptotically stable. All reachable modes can be shifted by state feedback. (b) Consider a continuous-time zero-mean wide-sense-stationary (WSS) random process x ( t ) whose power spectral density S xx ( jω ) is constant at the value N > for frequencies | ω | < ω m and outside this band. By considering the autocorrelation function R xx ( τ ) of this process, one can see that for an appropriately chosen T the samples x ( nT ) of the process constitute a discrete-time zero-mean WSS white process. Determine an appropriate T , and specify what the variance of the corresponding x ( nT ) is....
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- Spring '10
- Variance, Probability theory, Impulse response, power spectral density, HNO, Hyes