ps10 - Massachusetts Institute of Technology Department of...

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Unformatted text preview: Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.432 Stochastic Processes, Detection and Estimation Problem Set 10 Spring 2004 Issued: Thursday, April 29, 2004 Due: Thursday, May 6, 2004 Final Exam: Our final will take place on May 19, 2004, from 9:00am to 12:00 noon . You are allowed to bring three 8 1 × 11 sheets of notes (both sides). 2 Reading: For this problem set: Chapter 7 Next: Section 4.8, Addenda to Chapters 6 and 7 Problem 10.1 Let y ( t ) = x ( t ) + v ( t ) where x ( t ) and v ( t ) are uncorrelated, zero-mean processes, with 3 S xx ( s ) = 1 − s 2 5 S vv ( s ) = 9 − s 2 (a) Find the noncausal Wiener filter extimating x ( t ) based on y ( t ). Also find the corresponding mean-square estimation error. (b) Find the causal and causally invertible whitening filter for y ( t ). You will find that whitening y ( t ) requires differentiation. (c) Find the causal Wiener filter for estimating x ( t ). You should find that the overall filter doesn’t involve any differentiation. Also, find the associated mean- square estimation error. Problem 10.2 Consider the system depicted in Fig. 2-1 (on the next page), where w ( t ) and v ( t ) are independent, zero-mean noise processes with E [ w ( t ) w ( β )] = λ ( t − β ) 1 . E [ v ( t ) v ( β )] = λ ( t − β ) 5 1 1 −−−−−−− s+2 1 −−−−−−− s+1 x ( t ) w ( t ) z ( t ) ⊕ y ( t ) v ( t ) Figure 2-1 (a) Determine the noncausal Wiener filter for estimation x...
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This note was uploaded on 01/11/2012 for the course EE 6.432 taught by Professor Prof.gregorywornell during the Spring '04 term at MIT.

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ps10 - Massachusetts Institute of Technology Department of...

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