48092_finalfall04soln - Massachusetts Institute of...

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Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.011: Introduction to Communication, Control and Signal Processing Fall 2004 Final Exam SOLUTIONS YOUR NAME: Recitation Hour: This is a closed book exam, but you may use FOUR 8 1 2 × 11” sheets of notes (both sides). Calculators are not allowed. The questions are in two parts. Part I, worth 60% of the exam, comprises several relatively short questions, which should require only somewhat brief calculations and explanations. We estimate that this will take you around 100 minutes. Part II comprises two longer problems, which are worth 20% each, and which we estimate will take you around 40 minutes each. We would rather see you do 80% of the exam quite well than 100% of the exam quite poorly! Be clear and precise in your reasoning and show all relevant work. 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 this ANSWER booklet . No additional pages will be considered in the grading. You may want to ±rst work things through in the blank areas of the question booklet or on scratch paper, and then neatly transfer to thisr booklet the work you would like us to look at. Let us know if you need additional scratch paper. Problem Your Score 1(8po ints) 2(8po 3(8po 4(8po 5(10po 6(8po 7(10po 8(20po 9(20po Total (100 points) 1
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Problem 1 (8 points) The voltage waveform v ( t ) between the red terminal and ground of a power supply in your lab is equally likely to be either +5V for all time t ,or 5V for all time t , because the power supply is equally likely to have been manufactured (in the distant past) by the Duraplus or Everminus companies. For this random process v ( t ), determine: (i) the mean value of the process; (ii) its autocorrelation function; (iii) its autocovariance function; (iv) whether it is wide-sense stationary; (v) whether it is strict-sense stationary; and (vi) whether it is ergodic in mean value. (i) E [ v ( t )] = µ v = 1 2 (5) + 1 2 ( 5) = 0 (ii) E [ v ( t + τ ) v ( t )] = 1 2 (5)(5) + 1 2 ( 5)( 5) = 25 (iii) E [( v ( t + τ ) µ v )( v ( t ) µ v )] = 1 2 (5)(5) + 1 2 ( 5)( 5) = 25 (iv) YES, it is WSS: Since the mean and autocorrelation are constant, this process is WSS. (v) YES, it is SSS: Since the time origin is irrelevant to the joint densities of samples, this process is SSS. In other words, the probabilistic descriptions are time invariant. (vi) NO, it is not ergodic. The time average is either 5 for all time, with probability 1 2 5 for all time, with probability 1 2 , while the ensemble average is 0. (Continue this problem on next side = = ) 2
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Problem 1 (continued) (i) Mean = 0 (ii) Autocorrelation function = 25 (iii) Autocovariance function = 25 (iv) Is the process wide-sense stationary? YES (v) Is the process strict-sense stationary? YES (vi) Is the process ergodic in mean value? NO 3
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Problem 2 (8 points) Which of the following are valid autocorrelation functions for a continuous-time wide-sense stationary random process x ( t )? Briefly justify your answers. For each case that represents
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48092_finalfall04soln - Massachusetts Institute of...

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