We cannot extend the same two concepts of simulating and whitening to the case of continuous time random signals or processes. For simulating, we create a filter into which we feed a white noise signal. We choose the filter so that the output signal simulates the 1st and 2nd moments of any arbitrary random process. For whitening, we feed any arbitrary random signal into a specially chosen filter so that the output of the filter is a white noise signal.
Simulating a continuous-time random signal
edit
White noise fed into a linear, time-invariant filter to simulate the 1st and 2nd moments of an arbitrary random process. White noise can simulate any wide-sense stationary , continuous -time random process
x
(
t
)
:
t
∈
R
{\displaystyle x(t):t\in \mathbb {R} \,\!}
mean
μ
{\displaystyle \mu }
covariance function
K
x
(
τ
)
=
E
{
(
x
(
t
1
)
−
μ
)
(
x
(
t
2
)
−
μ
)
∗
}
where
τ
=
t
1
−
t
2
{\displaystyle K_{x}(\tau )=\mathbb {E} \left\{(x(t_{1})-\mu )(x(t_{2})-\mu )^{*}\right\}{\mbox{ where }}\tau =t_{1}-t_{2}}
and power spectral density
S
x
(
ω
)
=
∫
−
∞
∞
K
x
(
τ
)
e
−
j
ω
τ
d
τ
.
{\displaystyle S_{x}(\omega )=\int _{-\infty }^{\infty }K_{x}(\tau )\,e^{-j\omega \tau }\,d\tau .}
We can simulate this signal using frequency domain techniques.[clarification needed
Because
K
x
(
τ
)
{\displaystyle K_{x}(\tau )}
Hermitian symmetric and positive semi-definite , it follows that
S
x
(
ω
)
{\displaystyle S_{x}(\omega )}
real and can be factored as
S
x
(
ω
)
=
|
H
(
ω
)
|
2
=
H
(
ω
)
H
∗
(
ω
)
{\displaystyle S_{x}(\omega )=|H(\omega )|^{2}=H(\omega )\,H^{*}(\omega )}
if and only if
S
x
(
ω
)
{\displaystyle S_{x}(\omega )}
Paley-Wiener criterion .
∫
−
∞
∞
log
(
S
x
(
ω
)
)
1
+
ω
2
d
ω
<
∞
{\displaystyle \int _{-\infty }^{\infty }{\frac {\log(S_{x}(\omega ))}{1+\omega ^{2}}}\,d\omega <\infty }
If
S
x
(
ω
)
{\displaystyle S_{x}(\omega )}
rational function , we can then factor it into pole -zero form as
S
x
(
ω
)
=
Π
k
=
1
N
(
c
k
−
j
ω
)
(
c
k
∗
+
j
ω
)
Π
k
=
1
D
(
d
k
−
j
ω
)
(
d
k
∗
+
j
ω
)
.
{\displaystyle S_{x}(\omega )={\frac {\Pi _{k=1}^{N}(c_{k}-j\omega )(c_{k}^{*}+j\omega )}{\Pi _{k=1}^{D}(d_{k}-j\omega )(d_{k}^{*}+j\omega )}}.}
Choosing a minimum phase
H
(
ω
)
{\displaystyle H(\omega )}
s-plane , we can then simulate
x
(
t
)
{\displaystyle x(t)}
H
(
ω
)
{\displaystyle H(\omega )}
We can simulate
x
(
t
)
{\displaystyle x(t)}
linear , time-invariant filter
x
^
(
t
)
=
F
−
1
{
H
(
ω
)
}
∗
w
(
t
)
+
μ
{\displaystyle {\hat {x}}(t)={\mathcal {F}}^{-1}\left\{H(\omega )\right\}*w(t)+\mu }
where
w
(
t
)
{\displaystyle w(t)}
continuous -time, white-noise signal with the following 1st and 2nd moment properties:
E
{
w
(
t
)
}
=
0
{\displaystyle \mathbb {E} \{w(t)\}=0}
E
{
w
(
t
1
)
w
∗
(
t
2
)
}
=
K
w
(
t
1
,
t
2
)
=
δ
(
t
1
−
t
2
)
.
{\displaystyle \mathbb {E} \{w(t_{1})w^{*}(t_{2})\}=K_{w}(t_{1},t_{2})=\delta (t_{1}-t_{2}).}
Thus, the resultant signal
x
^
(
t
)
{\displaystyle {\hat {x}}(t)}
moment properties as the desired signal
x
(
t
)
{\displaystyle x(t)}
Whitening a continuous-time random signal
edit
An arbitrary random process x(t) fed into a linear, time-invariant filter that whitens x(t) to create white noise at the output. Suppose we have a wide-sense stationary , continuous -time random process
x
(
t
)
:
t
∈
R
{\displaystyle x(t):t\in \mathbb {R} \,\!}
mean
μ
{\displaystyle \mu }
covariance function
K
x
(
τ
)
{\displaystyle K_{x}(\tau )}
power spectral density
S
x
(
ω
)
{\displaystyle S_{x}(\omega )}
We can whiten this signal using frequency domain techniques. We factor the power spectral density
S
x
(
ω
)
{\displaystyle S_{x}(\omega )}
Choosing the minimum phase
H
(
ω
)
{\displaystyle H(\omega )}
s-plane , we can then whiten
x
(
t
)
{\displaystyle x(t)}
H
i
n
v
(
ω
)
=
1
H
(
ω
)
.
{\displaystyle H_{inv}(\omega )={\frac {1}{H(\omega )}}.}
We choose the minimum phase filter so that the resulting inverse filter is stable . Additionally, we must be sure that
H
(
ω
)
{\displaystyle H(\omega )}
ω
∈
R
{\displaystyle \omega \in \mathbb {R} }
H
i
n
v
(
ω
)
{\displaystyle H_{inv}(\omega )}
singularities .
The final form of the whitening procedure is as follows:
w
(
t
)
=
F
−
1
{
H
i
n
v
(
ω
)
}
∗
(
x
(
t
)
−
μ
)
{\displaystyle w(t)={\mathcal {F}}^{-1}\left\{H_{inv}(\omega )\right\}*(x(t)-\mu )}
so that
w
(
t
)
{\displaystyle w(t)}
random process with zero mean and constant, unit power spectral density
S
w
(
ω
)
=
F
{
E
{
w
(
t
1
)
w
(
t
2
)
}
}
=
H
i
n
v
(
ω
)
S
x
(
ω
)
H
i
n
v
∗
(
ω
)
=
S
x
(
ω
)
S
x
(
ω
)
=
1.
{\displaystyle S_{w}(\omega )={\mathcal {F}}\left\{\mathbb {E} \{w(t_{1})w(t_{2})\}\right\}=H_{inv}(\omega )S_{x}(\omega )H_{inv}^{*}(\omega )={\frac {S_{x}(\omega )}{S_{x}(\omega )}}=1.}
Note that this power spectral density corresponds to a delta function for the covariance function of
w
(
t
)
{\displaystyle w(t)}
K
w
(
τ
)
=
δ
(
τ
)
{\displaystyle K_{w}(\tau )=\,\!\delta (\tau )}
