ExamRP930110 - 1 ( ) ( ) H z D z = . 5. (15 points) 6. (15...

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Final Exam of Stochastic Processes Jan. 10, 2004 1. (10 points) A process ( ) t x is specified by the following differential equation ( ) 3 ( ) 2 ( ) ( ) , ( ) 4 ii t t t t s i + + = = x x x i S Find ( ) xx s S , its innovations filter ( ) s L and whitening filter ( ) s Γ ; 2. (15 points) Show that, if the process ( ) ϖ X is a white noise with zero mean and autocovariance ( ) ( ) Q u u v δ - , then its inverse Fourier transform ( ) t x is WSS with power spectrum ( ) 2 Q π . 3. (10 points) If we use the following function to estimate the power spectrum ( ) j e S from the autocorrelation [ ], R m N m N - : ( ) [ ] [ ] N j jm T m N e w m R m e - =- = S , explain what are the advantages or disadvantages of using a non-rectangular window, like Bartlett window or Hamming window. 4. A MA filter 1 ( ) 1 ( ) N N k k N k D z a z z - = = - = E can be implemented by the following lattice. (a) (10 points) Show that the forward and backward prediction error filters satisfy 1 1 1 ( ) ( ) ( ) N N N N z z K z z - - - = - E E E 1 1 1 ( ) ( ) ( ) N N N N z z z K z - - - = - E E E . (b) (10 points) Draw the inverse lattice filter, and prove that it implements
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Unformatted text preview: 1 ( ) ( ) H z D z = . 5. (15 points) 6. (15 points) If / ( ) T x R Ie -= and { ( ) ( ), ( )} ( ) ( ) 2 T E t t t T a t b t T--= +-x x x x x , find the constants a and b , and the MS error. 7. (10 points) Let the process [ ] n x be an AR process with the innovations filter 1 1 [ ] 1 N N b z a z a z--= + + + L L Find 1 [ ] { [ ] [ ], 1} n E n n k k =-x x x and the MS prediction error. 8. (20 points) If [ ] [ ] [ ] n n n = + x s and 1 5 ( ) , [ ] 5 [ ] , [ ] (1 0.8 )(1 0.8 ) ss s z R m m R m z z -= = =--S (a) Find the noncausal MS filter of [ ] n s and the resulting MS error; (b) Give a procedure to find the MS estimate of [ 2] n + s in terms of [ ] n x and its past....
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ExamRP930110 - 1 ( ) ( ) H z D z = . 5. (15 points) 6. (15...

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