COM 5220 - Adaptive Signal Processing  - Homework 2 -Solved

1.      Consider a zero-mean stationary sequence x n  with a correlation sequence r i . Show that, if r i  is a real and even function of i, then its Fourier transform R e j is a real, even, and non-negative function of .

 

2.      Find the autocorrelation function associated with the AR(2) process given by

                                                                x n( ) x n(  1) 0.25 (x n 2) w n( )

where w n( ) is a zero-mean white noise sequence with variance w2.

 

3.      For each of the following cases, show that  H e( j) 2  constant.

(1)    H z( )   (1(z1zz1 z11)) , where z1 is a real.



(2)     H z( )  (1(z1z za z za1)()(11z za* 1za*)) , where * denotes complex conjugate.

Also, use these results to infer that a general H z( ) expressible in the form of  

                                                                                       (1 zN11z1)    (1 z zN 1)

                                                                      Hap( )z  (z1  zN11)     (z1  zN)

is all-pass.

 

4.      A deterministic signal is estimated by averaging M noise corrupted measurements

                                                      yj n  x n  vj n, j  0,1,2, ,M 1

where vj n  is a zero mean iid sequence and E v n v n j   i  2 i  j . Find the variance of x nˆ  (1/ M)Mj1y nj   .

 

5.      For each of the following signals, show that the sample mean ˆ  (1/ M)iM01x i  is unbiased and consistent.

(1)    x n  is an iid sequence with mean value  and variance 2.

(2)    x n   w n  aw n 1 , where w n  is an iid sequence with zero mean and unit

variance.




6.      Consider the following two estimators for the correlation of a random signal:

    1 M m 1              1 M m 1 r mˆ  M m  x n x n m   ; rm M n0 xnxn  m.

                                                                n 0

(1)    Generate a stationary Gaussian random signal with samples x(n) for n = 0, 1, 2, …, M1 using MATLAB, where M=100. Then estimate the correlation of the random signal based on each of the two estimators, i.e., compute r mˆ  and r m  for m = -M+1, …, M1. From the simulation results, show that r mˆ  has high variability for m > M/4.

(2)    Let R be a correlation matrix of the random signal x(n). It is known that R is a positive semi-definite matrix if the true correlation values are used. How is the positive semidefinite property of R if the true correlation values are replaced with the estimates

r mˆ  or r m  for m = 0, 1, 2, …, 24 and m = 0, 1, 2, …, 99? Justify your answers.