Problem 1
Suppose that a white Gaussian noise
X
t
X
t
is input to a linear system with transfer function given by
Hf=1if|f|≤20if|f|>2
H
f
1
f
2
0
f
2
(1)
Suppose further that the input process is zero mean and has spectral
height
N
0
2=5
N
0
2
5
.
Let
Y
t
Y
t
denote the resulting output process.
- Find the power spectral density of
Y
t
Y
t
.
Find the autocorrelation of
Y
t
Y
t
(i.e.,
R
Y
τ
R
Y
τ
).
-
Form a discrete-time process (that is a sequence of
random variables) by sampling
Y
t
Y
t
at time instants T T seconds
apart. Find a value for T T
such that these samples are uncorrelated. Are these samples
also independent?
-
What is the variance of each sample of the output process?
Problem 2
Suppose that
X
t
X
t
is a zero mean white Gaussian process with spectral height
N
0
2=5
N
0
2
5
. Denote
Y
t
Y
t
as the output of an integrator when the input is
Y
t
Y
t
.
- Find the mean function of
Y
t
Y
t
. Find the autocorrelation function of
Y
t
Y
t
,
R
Y
t+τt
R
Y
t
τ
t
- Let
Z
k
Z
k
be a sequence of random variables that have been obtained by
sampling
Y
t
Y
t
at every T T seconds and
dumping the samples, that is
Z
k
=∫k-1TkT
X
τ
dτ
Z
k
τ
k
1
T
k
T
X
τ
(2)
Find the autocorrelation of the discrete-time processes
Z
k
Z
k
's, that is,
R
Z
k+mk=E
Z
k+m
Z
k
R
Z
k
m
k
E
Z
k+m
Z
k
- Is
Z
k
Z
k
a wide sense stationary process?
Problem 3
Proakis and Salehi, problem 3.63, parts 1, 3, and 4
Problem 4
Proakis and Salehi, problem 3.54
Problem 5
Proakis and Salehi, problem 3.62
Comments, questions, feedback, criticisms?
Send feedback