ExamRP940111 - Final Exam of Stochastic Processes Jan. 11,...

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Final Exam of Stochastic Processes Jan. 11, 2005 1. (6 points) Let ( ) t X be a SSS process with mean η . Form an unbiased estimator of the variance V and list the conditions that make ( ) t X be a variance-ergodic process. 2. (5 points) (a) Point out the draw back of using periodogram 2 2 ( ) ( ) T j T T S e d ϖτ ϖ τ - - = T R as an estimate of the power spectrum of a process ( ) t X , where ( ) T R is an estimate of the autocorrelation ( ) R of ( ) t X . (10 points) (b) Briefly describe the principle of smoothed spectrum using lag window ( ) w and discuss the criteria of selecting a good window. 2 2 ( ) ( ) ( ) T j w T T S w R e d - - = 3. (15 points) An MA filter 1 ( ) 1 N N k k k D z a z - = = - can be implemented by a lattice filter with forward prediction error filter ˆ ( ) ( ) N E z D z = and backward prediction error filter ( ) N E z ( . We know that ˆ ( ) N E z and ( ) N E z ( satisfy ˆ ( ) N E z 1 1 1 ˆ ( ) ( ) N N N E z K z E z - - - = - ( 1 1 1 ˆ
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ExamRP940111 - Final Exam of Stochastic Processes Jan. 11,...

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