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# ps11 - 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 11 Spring 2004 Issued: Thursday, May 6, 2004 Due: Next time the Red Sox win the World Series 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 Problem 11.1 Let y [ n ] = x [ n ] + w [ n ] x [ n ] = h [ k ] v [ n − k ] k be a discrete-time random processes, where v [ n ] and w [ n ] are uncorrelated, zero-mean, white Gaussian noise processes with variances of ν 2 = 2 / 3 and ν 2 = 1, respectively v w and n 2 h [ n ] = u [ n ] . 3 (a) Determine the non-causal Wiener filter for estimating x [ n ] from y [ n ] and the associated mean-square estimation error. (b) Determine the causal Wiener filter for estimating x [ n ] from y [ n ] and the asso- ciated mean-square estimation error. (c) Determine the Kalman filter equations for generating x ˆ[ n | n ], the linear least- squares estimate of x [ n ] at time n based on the data y [0] , y [1] , . . . , y [ n ]. Also, specify a recursion for the mean- square error in the estimate x ˆ[ n | n ]. (d) Determine lim n e [ n | n ], the steady-state estimation error variance for your estimator in part (c). Explain the similarity or differences between your answer and the error variances you calculated in parts (a) and (b). (e) The sequence v [ n ] could represent a message being transmitted through a noisy, dispersive communication channel, in which case this is the signal of interest. Give an eﬃcient algorithm for generating v ˆ[ n | n ], the estimate of v [ n ] based on y [0] , . . . , y [ n ]. 1 x Problem 11.2 Suppose we are transmitting a zero-mean, WSS, white sequence x [ n ] with variance ν 2 through the channel h[ n ] + y [ n ] x [ n ] w [ n ] Figure 2-1 where w [ n ] is zero-mean, WSS white noise with variance ν 2 that is uncorrelated with w x [ n ] and where h [ n ] is the FIR filter 1 h [ n ] = α [ n ] − α [ n − 1] . 2 (a) Write down a state-space model for the problem using x [ n ] s [ n ] = x [ n − 1] as the state vector. (b) Determine a recursive algorithm for computing x ˆ[ n | n ], the linear least-squares estimate of x [ n ] based on y [0] , . . . , y [ n ]. (c) Determine lim n e [ n | n ], the steady-state filtering error variance. (d) Compare your answer to part (c) to the error variances of the causal and non- causal Wiener filters. Problem 11.3 Consider a system with the following state space description x [ n + 1] = a x [ n ] + v [ n ] y [ n ] = x [ n ] + w [ n ] where v [ n ] and w [ n ] are uncorrelated, wide-sense stationary, zero-mean, white random processes with variances 1 and ν 2 respectively....
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ps11 - Massachusetts Institute of Technology Department of...

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