Zω=∫ab
X
t
ωd
t
Z
ω
t
a
b
X
t
ω
(1)
Y
t
=∫−∞∞htτ
X
τ
d
τ
Y
t
τ
h
t
τ
X
τ
(2)
X
t
′=dd
t
X
t
X
t
t
X
t
(3)-
Z-=∫ab
X
t
ωd
t
-=∫ab
μ
X
td
t
Z
t
a
b
X
t
ω
t
a
b
μ
X
t
-
Z2-=∫ab
X
t
2
d
t
2
∫ab
X
t
1
¯d
t
1
-=∫ab∫ab
R
X
t
2
t
1
d
t
1
d
t
2
Z
2
t
2
a
b
X
t
2
t
1
a
b
X
t
1
t
2
a
b
t
1
a
b
R
X
t
2
t
1
μ
Y
t=∫−∞∞htτ
X
τ
d
τ
-=∫−∞∞htτ
μ
X
τd
τ
μ
Y
t
τ
h
t
τ
X
τ
τ
h
t
τ
μ
X
τ
(4)
If
X
t
X
t
is wide sense stationary and the linear system is time invariant
μ
Y
t=∫−∞∞ht−τ
μ
X
d
τ
=
μ
X
∫−∞∞ht′d
t′
=
μ
Y
μ
Y
t
τ
h
t
τ
μ
X
μ
X
t
h
t
μ
Y
(5)
R
Y
X
t
2
t
1
=
Y
t
2
X
t
1
¯-=∫−∞∞h
t
2
−τ
X
τ
d
τ
X
t
1
¯-=∫−∞∞h
t
2
−τ
R
X
τ−
t
1
d
τ
R
Y
X
t
2
t
1
Y
t
2
X
t
1
τ
h
t
2
τ
X
τ
X
t
1
τ
h
t
2
τ
R
X
τ
t
1
(6)
R
Y
X
t
2
t
1
=∫−∞∞h
t
2
−
t
1
−τ′
R
X
τ′d
τ′
=h*
R
X
t
2
−
t
1
R
Y
X
t
2
t
1
τ
h
t
2
t
1
τ
R
X
τ
h
R
X
t
2
t
1
(7)
where
τ′=τ−
t
1
τ
τ
t
1
.
R
Y
t
2
t
1
=
Y
t
2
Y
t
1
¯-=
Y
t
2
∫−∞∞h
t
1
τ
X
τ
¯d
τ
-=∫−∞∞h
t
1
τ
R
Y
X
t
2
τd
τ
=∫−∞∞h
t
1
−τ
R
Y
X
t
2
−τd
τ
R
Y
t
2
t
1
Y
t
2
Y
t
1
Y
t
2
τ
h
t
1
τ
X
τ
τ
h
t
1
τ
R
Y
X
t
2
τ
τ
h
t
1
τ
R
Y
X
t
2
τ
(8)
R
Y
t
2
t
1
=∫−∞∞hτ′−(
t
2
−
t
1
)
R
Y
X
τ′d
τ′
=
R
Y
t
2
−
t
1
=
h
~
*
R
Y
X
t
2
t
1
R
Y
t
2
t
1
τ
h
τ
t
2
t
1
R
Y
X
τ
R
Y
t
2
t
1
h
~
R
Y
X
t
2
t
1
(9)
where
τ′=
t
2
−τ
τ
t
2
τ
and
h
~
τ=h−τ
h
~
τ
h
τ
for all
τ∈R
τ
.
Y
t
Y
t
is WSS if
X
t
X
t
is WSS and the linear system is time-invariant.
X
t
X
t
is a wide sense stationary process with
μ
X
=0
μ
X
0
, and
R
X
τ=
N
0
2δτ
R
X
τ
N
0
2
δ
τ
.
Consider the random process going through a filter with impulse
response
ht=e−(at)ut
h
t
a
t
u
t
.
The output process is denoted by
Y
t
Y
t
.
μ
Y
t=0
μ
Y
t
0
for all tt.
R
Y
τ=
N
0
2∫−∞∞hαhα−τd
α
=
N
0
2e−(a|τ|)2a
R
Y
τ
N
0
2
α
h
α
h
α
τ
N
0
2
a
τ
2
a
(10)
is called a white process.
Y
t
Y
t
is a Markov process.
- Definition 1: Power Spectral Density
The power spectral density function of a wide sense stationary (WSS)
process
X
t
X
t
is defined to be the Fourier transform of the autocorrelation function
of
X
t
X
t
.
S
X
f=∫−∞∞
R
X
τe−(i2πfτ)d
τ
S
X
f
τ
R
X
τ
2
f
τ
(11)
if
X
t
X
t
is WSS with autocorrelation function
R
X
τ
R
X
τ
.
-
S
X
f=
S
X
−f
S
X
f
S
X
f
since
R
X
R
X
is even and real.
-
Var
X
t
=
R
X
0=∫−∞∞
S
X
fd
f
Var
X
t
R
X
0
f
S
X
f
-
S
X
f
S
X
f
is real and nonnegative
S
X
f≥0
S
X
f
0
for all ff.
If
Y
t
=∫−∞∞ht−τ
X
τ
d
τ
Y
t
τ
h
t
τ
X
τ
then
S
Y
f=ℱ
R
Y
τ=ℱh*
h
~
*
R
X
τ=Hf
H
~
f
S
X
f=|Hf|2
S
X
f
S
Y
f
ℱ
R
Y
τ
ℱ
h
h
~
R
X
τ
H
f
H
~
f
S
X
f
H
f
2
S
X
f
(12)
since
H
~
f=∫−∞∞
h
~
te−(i2πft)d
t
=Hf¯
H
~
f
t
h
~
t
2
f
t
H
f
X
t
X
t
is a white process and
ht=e−(at)ut
h
t
a
t
u
t
.
Hf=1a+i2πf
H
f
1
a
2
f
(13)
S
Y
f=
N
0
2a2+4π2f2
S
Y
f
N
0
2
a
2
4
2
f
2
(14)
