ECE 734INFT830INFT 978 Detection and Estimation Theory Statistical

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ECE 734INFT830INFT 978 Detection and Estimation Theory Statistical Powered By Docstoc
					                        ECE 734 / INFT 830 / INFT 978
         Detection and Estimation Theory / Statistical Signal Processing
                                   Fall 1999
                                                Homework #2
                                          Due Monday, 9/20/99

Reading: Ch. 3.1-3.4, 3.7-3.8 (skim 3.5-3.6), Ch 2.1-2.2.

Problem 1. The input into a filter is a zero-mean white noise process xt with noise power density No =2.
Let y t denote the filter output. The filter has transfer function
                                                                  1
                                           H1f  =                      :
                                                               + 2j f
(i) Find the impulse response h1  .
(ii) Find Syx f  and Ryx  .
(iii) Find Sy f  and Ry  .
(iv) What is the average power of the output?

Problem 2. The input into a second filter is the output y t of the filter in Problem 1. Let z t denote the
second filter output. The filter has transfer function
                                                                  1
                                           H2f  =                      :
                                                               + 2j f
(i) Find Sz f  and Rz  .
(ii) What is the average power of the output?

Problem 3. The output of a linear time-invariant discrete time system is given by
                                                  1
                                                  X
                                       y n =             hn , kxk
                                                  k=,1

where hn is the unit sample response of the system. Its transfer function is defined as
                                                      1
                                                      X
                                       H f  =                hne,j 2f n :
                                                  n=,1

A first order autoregressive (AR) process is defined by

                                        y n = y n , 1 + xn
where xn is a zero-mean discrete time white noise process with noise power density No=2. The autocor-
relation function of xn is given by
                                                     No
                                     Rx k =          2          k=0
                                                      0           otherwise.

                                                           1
(i) Find the unit sample response hn and transfer function H f  of the system. What condition must
satisfy to ensure a stable system?
(ii) Find Syx f  and Ryx k.
(iii) Find Sy f  and Ry k.

Problem 4. Let z n = y n + y n , 1, where y n is the first order AR process in Problem 3.
(i) Find Szy f  and Rzy k.
(ii) Find Sz f  and Rz k.
(iii) For what value of is z n a white noise process?

Problem 5. Let y t = xt + wt where wt is a zero-mean white noise process with noise spectral
density No=2, and xt and wt are uncorrelated. Let i t be the eigenfunctions of Kxt; u.
(i) Show that i t are also the eigenfunctions of Ky t; u.
(ii) What is the relationship between the eigenvalues of Kx t; u and Ky t; u?

Problem 6. Problem 3.4.9.




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