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# ps3 - t sin(2 πt dt Explain why the knowledge of l 1 and l...

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EE 806, Detection and Estimation Theory Apr. 21, 2008 OSU, Spring 2008 Due: May 5, 2008 Problem Set 3 Problem 1 - (Poor, Ch. 3, Pr. 3) Hint: Recall the M -ary Bayes problem, and the fact that minimum-error-probability implies a particular cost assignment. Problem 2 - (Poor, Ch. 3, Pr. 9) For the LMP stated in part (b), you can ignore the randomization γ since it is not needed. Problem 3 In this problem we consider a simple phase shift keying (PSK) system. Speci±cally, suppose that we observe a signal Y ( t ) that consists of one of four equally likely signals in additive white Gaussian noise. That is, we have 4 hypotheses H 0 , H 1 , H 2 and H 3 : H m : Y ( t ) = A cos ± 2 πt + 2 ² + W ( t ) , 0 6 t 6 1 where A is a given constant and W ( t ) is zero-mean white Gaussian noise with E [ W ( t ) W ( τ )] = δ ( t - τ ) . (a) Given the waveform Y ( t ) = y ( t ), consider the two statistics l 1 = Z 1 0 y ( t ) cos(2 πt ) dt l 2 = Z 1 0 y

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Unformatted text preview: ( t ) sin(2 πt ) dt. Explain why the knowledge of l 1 and l 2 is su²cient to form the optimum decision rule (i.e., explain why l 1 and l 2 are su²cient statistics for the hypothesis testing problem). Also determine the pdf of [ l 1 l 2 ] T under each hypothesis. (b) Specify the decision regions ( l 1 , l 2 ) space for the minimum probability of error decision rule. (c) Show that α 6 P (error) 6 2 α, where α = 1-Φ( A/ 2). ( Hint: Examine the decision regions determined in part (b). What is the region corresponding to an error if, say, H is true? Can you represent this region as a union of two half spaces?) 1 Problem 4 (Poor, Ch. 3, Pr. 13) Problem 5 (Poor, Ch. 3, Pr. 14) Problem 6 (Poor, Ch. 3, Pr. 17) 2...
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ps3 - t sin(2 πt dt Explain why the knowledge of l 1 and l...

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