COM5220 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)  0.25x(n  2)  w(n)

where w(n) is a zero-mean white noise sequence with variance 3. For each of the following cases, show that H (e j ) 2  constant.

(1) H(z)  (1 z1z1) , where z1 is a real. (z1 z1)

 w2 .

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

(z1 z )(z1 z*) aa

Also, use these results to infer that a general H(z) expressible in the form of H (z)(1zN11z1) (1zNz1)

ap (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 vnisazeromeaniidsequenceand Evnvn2ij.Findthe

j1

jji varianceof xˆn(1/M)M yj n.

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

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.

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COM 5220 Adaptive Signal Processing Fall 2021 6. Consider the following two estimators for the correlation of a random signal:

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

n0 n0

1. (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 basedoneachofthetwoestimators,i.e.,compute rˆm and rm form=-M+1,…, M1. From the simulation results, show that rˆm has high variability for m > M/4.
2. (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 semi- definite property of R if the true correlation values are replaced with the estimates

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

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