Kyung Hee University
Random Processing
Department of Electronics and Radio Engineering
Prof. Hyundong Shin
Communications and Coding Theory Laboratory (CCTLAB)
XII1
C1002900 RP Lecture Handout XII:
Stationary Processes
Reading:
Chapter 9.1
Any pair of samples of the process
X t
with the same temporal separation, i.e.,
X t
and
X s
for any
, might have the same correlation.
This kind of “statistical
timeinvariance
” which is termed
stationary
is in fact a ra
ther natural property of many physical stochastic processes.
12.1. StrictSense Stationarity
The strongest notion of stationarity is referred to as
strictsense
stationarity and is de
fined in terms of the complete statistical characterization of the process. Specifically, a
stochastic process
X t
is said to be
strictsense stationary (SSS)
if all its finite
dimensional distributions are timeinvariant, i.e.,
,
,
,
,
,
,
,
,
,
,
,
,
N
N
N
N
X t
X t
X t
X t
X t
X t
f
x
x
x
f
x
x
x
1
2
1
2
1
2
1
2
(XII.1)
for all choices of
N
, time instants
,
,
N
t
t
1
, and for any
.
The process
X t
does not have absolute time reference, i.e., its statistical prop
erties are invariant to a shift of the origin.
From the definition, it follows that
X t
X t
X
f
x
f
x
f
x
0
.
For the second order,
,
,
,
,
,
,
X t
X t
X t
X t
X t
X t
f
x
x
f
x
x
f
x
x
1
2
1
2
1
2
1
2
1
2
where
t
t
1
2
. Hence, the joint density of
X t
and
X t
is independent
of
t
.
There are many examples of SSS random processes, e.g.,
diiscretetime white Gaussian
noise process and random telegraph process.
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Kyung Hee University
Random Processing
Department of Electronics and Radio Engineering
Prof. Hyundong Shin
Communications and Coding Theory Laboratory (CCTLAB)
XII2
12.2. WideSense Stationarity
A weaker notion of stationarity
–
but one that is typically simpler to verify in practice
–
is
the widesense stationarity. Specifically, we say a process
X t
is
widesense stationary
(WSS)
if its secondorder characterization is timeinvariant, i.e.,:
(i)
Its mean is constant (does not vary with time)
X
X
t
X t
.
(XII.2)
(ii)
Its autocorrelation function depends only on
t
t
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 Spring '10
 HyungdongShin
 Variance, Probability theory, Stochastic process, Stationary process, Kyung Hee University Department of Electronics, Prof. Hyundong Shin

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