•1
±
Properties of autocorrelation function
•
If a random process is WSS, then its
autocorrelation function satisfies the
following
a)
(even function)
b)
c)
is periodic if
d) When a random process is real valued,
continuous time and WSS, its
autocorrelation
is positive definite .
)
(
)
(
τ
−
=
X
X
R
R
)
0
(
)
(
X
X
R
R
≤
)
(
X
R
)
0
(
)
(
/
X
X
R
T
R
T
=
∃
²
Cyclostationary Process;
Definition:
A random process
is
said to be cyclostationary if
and a constant T.
Remark:
Similarly to stationarity,
may be wide sense cyclostationary. It is
WCS if
and T>0 we have
a)
b)
)}
(
{
t
X
)
,...
,
(
)
,...
,
(
2
1
)....
(
)
(
2
1
)
(
)....
(
)
(
1
1
2
1
n
lT
t
X
lT
t
X
n
t
X
t
X
t
X
x
x
x
F
x
x
x
F
n
+
+
=
N
l
∈
∀
)}
(
{
t
X
N
l
∈
∀
I
t
t
m
lT
t
m
X
X
∈
∀
=
+
1
1
1
),
(
)
(
I
t
t
t
t
C
lT
t
lT
t
C
X
X
∈
∀
=
+
+
2
1
2
1
2
1
,
),
,
(
)
,
(
Example:
1. In binary transmission, X(t) takes on
with equal probability in each interval
we
assume the values between two
intervals are independent.
1
±
nT
t
T
n
T
n
≤
≤
−
)
1
(
:
n
T
t
t
if
t
X
t
X
E
∈
=
2
1
2
1
,
1
))
(
)
(
(
otherwise
0
⇒
Or
2. X(t)=Acos(t)+Bsin(t)
A and B are R.V.
a) Is
stationary? WSS?
Cyclostationary?
WCS?
b) What are the conditions on A and B for all
modes of stationarity?
)}
(
{
t
X
=
+
)
,
(
t
t
R
X
nT
t
t
T
n
<
+
<
−
)
,
(
)
1
(
1
w
o
.
0
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document•2
The distribution may be written as:
•
Hence not stationary since the
distribution is varying with t.
•
Not W.S.S , since
•
Clearly cyclostationary and WCS.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 Krim
 Stationary process, random process

Click to edit the document details