Dr. Behnaam Aazhangl
ELEC 430
Department of Electrical and Computer Engineering
Rice University
Due 31 Jan 2008
HOMEWORK 3 — Random Processes II
Exercise
1. Sampling Random Processes
A discretetime stochastic process
Y
n
≡
X
nT
is obtained by periodic sampling of a continuous
time zeromean stationary process
X
t
where
T
is the sampling interval, i.e.
f
s
=
1
T
is the
sampling rate.
(a) Determine the relationship between the autocorrelation function of
X
t
and the autocor
relation sequence of
Y
n
.
(b) Express the power density spectrum of
Y
n
in terms of the power density spectrum of
X
t
.
(c) Determine the conditions under which the power density spectrum of
Y
n
is equal to that
of
X
t
.
Exercise
2. Processing Gaussian Noise
Suppose that white Gaussian noise
X
t
is the input to a linear system with a transfer function
deﬁned by
H
(
f
) =
±
1

f
 ≤
2
0

f

>
2
Suppose further that the input process is zero mean and has spectral height
N
o
2
= 5. Let
Y
t
denote the resulting output process.
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 Spring '08
 Aazhang
 Signal Processing, Stationary process, stationary process Xt, power density spectrum, LTI Processing Xt

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