[SOLVED] COM5220 - Homework 2

Description

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

_

• H z( )  (1(z^_1z z^a z z^a¹)()(1^_1z z^{a* 1}z^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

y_j n  x n  v_j n, j  0,1,2, ,M 1

where v_j 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)_^M_j1y n_j   .

5. For each of the following signals, show that the sample mean ˆ  (1/ M⁾_i^M₀^1x i  is unbiased and consistent.
   • x n  is an iid sequence with mean value  and variance _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

• 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 m^ˆ  and r m^  for m = –M+1, …, M From the simulation results, show that r mˆ  has high variability for m > M/4.
• 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 m^ˆ  or r m^  for m = 0, 1, 2, …, 24 and m = 0, 1, 2, …, 99? Justify your answers.