•1
Stationarity of Normal Processes
•I
f
X
(
t
) is a Gaussian process, then by
definition
are jointly Gaussian random variables
• Their
joint characteristic function is
11
2 2
( ),
,
( )
nn
XX
tXX
t
t
==
=
"
1,
()
(,)
/
2
12
(, ,, )
XX
kk
i
ki
k
kl
k
X
jt
C
t
t
n
e
μω
ωω
φωω
ω
=
−
∑∑
∑
=
"
Gaussian Processes Properties
f
X
(
t
) is widesense stationary, then we have
• If we now shift the time indices to get
• Their joint characteristic is as above for all n and c
establishing strict stationarity from WSS for Normal
processes
1
2
1
1
XX
n
k
i
k
X
jC
t
t
n
e
μ
=
−−
∑
=
"
112
2
,
,
,
X
Xt c X
Xt c
X
′′
′
=
+=
+
=
+
"
Systems with Stochastic Inputs
•
A deterministic system transforms each
input waveform
into an output
waveform
through time (t)
operation
•
Thus a set of realizations at the input
corresponding
to a process
X
(
t
)
generates a new set of realizations at the
output associated with a new process
Y
(
t
).
(, )
i
X t
ξ
[ (, )]
ii
Yt
TXt
=
)}
,
(
{
t
Y
Our goal is to study the output process statistics in terms
of the input process statistics and the system function.
]
[
⋅
T
⎯
⎯ →
⎯
)
(
t
X
⎯
⎯→
⎯
)
(
t
Y
t
t
)
,
(
i
t
X
)
,
(
i
t
Y
Stochastic Input/Output
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View Full Document•2
Deterministic Systems
Systems with Memory
TimeInvariant
systems
Linear systems
LinearTime Invariant
(LTI) systems
Memoryless Systems
)]
(
[
)
(
t
X
g
t
Y
=
)]
(
[
)
(
t
X
L
t
Y
=
Timevarying
systems
.
)
(
)
(
)
(
)
(
)
(
∫
∫
∞
+
∞
−
∞
+
∞
−
−
=
−
=
τ
d
t
X
h
d
X
t
h
t
Y
()
ht
Xt
LTI system
Memoryless Systems:The output
Y
(
t
) in this case depends
on the present value of the input
X
(
t
).
i.e.,
)}
(
{
)
(
t
X
g
t
Y
=
Memoryless
system
Memoryless
system
Memoryless
system
Strictsense
stationary input
Widesense
stationary input
X
(
t
) stationary
Gaussian with
)
(
XX
R
Strictsense
stationary output.
Need
not
be
stationary in
any sense.
Y
(
t
) stationary,but
not
Gaussian with
).
(
)
(
η
XX
XY
R
R
=
Theorem:
If
X
(
t
) is a zero mean stationary Gaussian process, and
Y
(
t
) =
g
[
X
(
t
)], where
represents a nonlinear memoryless device
then
Proof:
where
are jointly Gaussian random
variables, and hence
)
(
⋅
g
)}.
(
{
),
(
)
(
X
g
E
R
R
XX
XY
′
=
=
2
1
2
1
2
1
)
,
(
)
(
)}]
(
{
)
(
[
)}
(
)
(
{
)
(
2
1
dx
dx
x
x
f
x
g
x
t
X
g
t
X
E
t
Y
t
X
E
R
X
X
XY
∫∫
=
−
=
−
=
)
(
),
(
2
1
−
=
=
t
X
X
t
X
X
*
1
12
/2
1
2
*
*
1
2

(0)
( )
(
0
)
(, )
(
,
) ,
( ,
)
{
}
XX
XX
XX
XX
XX
xA x
TT
A
RR
fx
x
e
X
xx
x
A EXX
L
L
π
−
−
=
==
⎛⎞
=
⎜⎟
⎝⎠
Δ
where
L
is an upper triangular factor matrix with positive diagonal
entries. i.e.,
• Consider the transformation
so that
•
Z
1
,
Z
2
are hence zero mean independent Gaussian random
variables. Also
and hence
.
0
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 Fall '08
 Krim
 Autocorrelation, Stationary process, Memoryless Systems

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