# ps9 - Massachusetts Institute of Technology Department of...

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± Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.432 Stochastic Processes, Detection and Estimation Problem Set 9 Spring 2004 Issued: Thursday, April 15, 2004 Due: Thursday, April 29, 2004 Reading: Course notes: For this problem set: Chapter 6, Sections 7.1 and 7.2 Next: Chapter 7 Exam #2 Reminder: Our second exam will take place Thursday, April 22, 2004, 9am - 11am. The exam will cover material through Lecture 16 (April 8) as well as the associated homework through Problem Set 8. You are allowed to bring two 8 1 × 11 sheets of notes (both sides). 2 Problem 9.1 A discrete-time random process y [ n ] is observed for n = , N . There are two 1 , 2 , ··· equally likely hypotheses for the statistical description of y [ n ]: H 0 : y [ n ] = 1 + w [ n ] , N n = 1 , 2 , ··· H 1 : y [ n ] = 1 + w [ n ] where w [ n ] is a non-stationary zero-mean white Gaussian sequence, i.e. ± E w [ n ] w [ k ] = 0 for n = k (a) Assume that E w 2 [ n ] ± = 2 , n = 1 , 2 , ··· , N. Find the minimum probability of error decision rule based on observation of [ N ], and determine the associated error probability in terms of Q ( · ) y [1] , ··· , y and N , where Q ( x ) = 1 ² ± e 2 / 2 d . 2 ± x Find the asymptotic detection performance, i.e. , compute lim Pr(error). N ²± (b) For this part assume that 2 E w 2 [ n ] = n π 2 , , N. n = 1 , 2 , ··· Find the minimum probability of error decision rule based on observation of [ N ], and determine the associated error probability in terms of Q ( · ) y [1] , ··· , y and N . What is the asymptotic performance i.e. , what is lim Pr(error) in this N ²± case? 1

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± ± ² Problem 9.2 Consider the following binary hypothesis testing problem: H 0 : y ( t ) = (6 + v 1 ) s 1 ( t ) + v 2 s 2 ( t ) + w ( t ) 0 t T H 1 : y ( t ) = v 1 s 1 ( t ) + (6 + v 2 ) s 2 ( t ) + w ( t ) where s 1 ( t ) and s 2 ( t ) are given, orthonormal waveforms on [0 , T ], i.e. , ³ T ³ T ³ T s 2 1 ( t ) dt = s 2 2 ( t ) dt = 1 , s 1 ( t ) s 2 ( t
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ps9 - Massachusetts Institute of Technology Department of...

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