final_2009 - ECE 531 Detection and Estimation Final Exam...

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ECE 531 Detection and Estimation - Final Exam May 8, 2009. 8 am - 10 am in ERF 1003. This exam has 6 questions. You will be given 2 hours. You may use the 2 course textbooks but no other aides/notes. No calculators are permitted. No talking, passing notes, copying (and all other forms of cheating) is permitted. Make sure you explain your answers in a way that illustrates your understanding of the problem. Ideas are important, not just the calculation. If something has been proven in class or in the book feel free to cite and use the result without a re-derivation. Use your time wisely, take shortcuts, use what you know! Your name: Your UIN: Your signature: The exam has 6 questions, for a total of 100 points. Question: 1 2 3 4 5 6 Total Points: 20 20 10 15 15 20 100 Score:
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ECE 531 Detection and Estimation - Final Exam Name: 1. Suppose x is an unknown parameter and we have 2 observations y 1 and y 2 with y 1 = x + n 1 y 2 = | x | + n 2 , where n 1 , n 2 are independent, identically distributed zero mean Gaussian random variables of variance σ 2 . HINT: for all these problems you may find it useful to write | x | = x sgn ( x ) , where sgn ( x ) = 1 if x 0 and sgn ( x ) = - 1 if x < 0 . (a) (5 points) Find a bound on the estimation error variance of any unbiased estimate of x . (b) (5 points) Does an efficient estimator exist? If so determine it, if not explain why not. (c) (5 points) Briefly explain or prove why the following is true: if the maximum likelihood estimate ˆ x ML ( y 1 , y 2 ) is nonzero, it has the same sign as y 1 . (d) (5 points) Determine the ML estimate of x based on y 1 and y 2 . HINT: write ˆ x = b sgn ( y 1 ) from part (c) and determine the value of b as a function of y 1 and y 2 . Points earned: out of a possible 20 points
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ECE 531 Detection and Estimation - Final Exam Name: Workspace Points earned: out of a possible 0 points
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ECE 531 Detection and Estimation - Final Exam Name: 2. Suppose we are given y 0 and y 1 as follows: y 0 = a + n 0 , under hypothesis H 1 n 0 , under hypothesis H 0 y 1 = a + b + n 1 , under hypothesis H 1 n 1 , under hypothesis H 0 where n 0 and n 1 are random variables (the noise), and a and b are real constants which we may or may not know (specified in the sub-problems).
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