Digital Signal Processing Reference
In-Depth Information
4.4
Stochastic Processes
We now consider the case of random processes. The Wigner distribution of a
random process
x
(
t
)
is defined by
E
x
∗
t
x
t
e
−
i
τω
d
1
2
1
2
τ
1
2
τ
W
x,x
(
t,
ω)
=
−
+
τ
,
(4.17)
π
and is called the
Wigner spectrum
.
9
,
10
,
13
In Equation 4.17,
E
[]
,
represents the
ensemble averaging operator. We proceed as we did in the previous section
by studying the case of the harmonic oscillator
d
2
x
(
t
)
dx
(
t
)
2
+
2
µ
+
ω
0
x
(
t
)
=
F
(
t
)
,
(4.18)
dt
2
dt
where now
F
(
t
)
is a Gaussian random white noise with autocorrelation
function
R
F,F
(
t
1
,t
2
)
=
N
0
δ(
t
1
−
t
2
).
(4.19)
This problem has been widely studied and a fundamental quantity is the
power spectrum of
x
.
3
For this case we expect the spectrum to be constant
in time if we consider the system to be in “stationary” phase, that is, far away
from the initial conditions. Standard methods of stochastic analysis allow
one to compute analytically the power spectrum of
x
(
t
)
. However, to show
the effectiveness of our method we consider the problem where we make
one of the coefficients time varying, a case that the standard methods cannot
handle. But first we consider the constant coefficient case, and we show that
our method is in perfect agreement with the classical result: it returns an
instantaneous spectrum that does not change with time.
The standard result for the power spectrum,
G
x
(ω)
(
t
)
of
x
(
t
)
,defined as the
Fourier transform of the autocorrelation function
R
x
(τ )
,is
2
G
F
G
x
(ω)
=|
H
(ω)
|
(ω)
,
(4.20)
where
H
is the transfer function of the system. It is obtained by evaluating
the polynomial defined in Equation 4.84 in
i
(ω)
ω
, that is,
H
(ω)
=
P
(
i
ω)
,
(4.21)
Notice that with constant coefficients the polynomial is no longer a function
of time. Because the power spectrum of the Gaussian noise is constant and
equal to
G
F
(ω)
=
N
0
, we obtain from Equations 4.20 and 4.21
N
0
G
x
(ω)
=
2
.
(4.22)
ω
2
2
0
−
ω
+
4
µ
2
ω
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