# ECE 734INFT830INFT 978 Detection and Estimation Theory Statistical

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```					                        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 xt with noise power density No =2.
Let y t denote the ﬁlter output. The ﬁlter has transfer function
1
H1f  =                      :
+ 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
H2f  =                      :
+ 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 =             hn , kxk
k=,1

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

A ﬁrst order autoregressive (AR) process is deﬁned by

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

1
(i) Find the unit sample response hn 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 = xt + wt where wt is a zero-mean white noise process with noise spectral
density No=2, and xt and wt are uncorrelated. Let i t be the eigenfunctions of Kxt; 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.

2

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