Digital Signal Processing Reference
In-Depth Information
d
=
where
means equality for all finite dimensional distributions. The index H is
referred to as the Hurst parameter of the self-similar process Y
(
t
)
.
(
)
Provided that the
q
th order moments of
Y
t
exist, it can be drawn from
Equation 6.7 that
q
q
qH
,
E
|
Y
(
t
)
|
=
E
|
Y
(
1
)
|
|
t
|
(6.8)
which shows that a self-similar process is clearly nonstationary. It also holds
that
Y
(
0
)
=
0
a
.
s
.
,
(6.9)
and
)
}=
σ
2
2
H
2
H
2
H
{
(
)
(
(
|
|
+|
|
−|
−
|
).
E
Y
s
Y
t
s
t
s
t
(6.10)
2
An example of self-similar process is the Brownian motion
B
(
t
)
, which is
defined as follows:
•
B
(
t
)
is a zero-mean Gaussian process.
•
The correlation of
B
(
s
)
and
B
(
t
)
is the minimum of
s
and
t
.
1
2
We can verify that
B
(
t
)
is self-similar with Hurst parameter
H
=
because
a
−
1
/
2
B
a
−
1
/
2
B
a
−
1
min
)
}
.
(6.11)
A more general form of Brownian motion is the fractional Brownian motion
(FBM), defined by
{
(
)
(
)
}=
(
)
=
(
)
=
{
(
)
(
E
as
at
as, at
min
s, t
E
B
s
,B
t
0
−∞
(
|
1
=
H
−
1
/
2
H
−
1
/
2
B
H
(
t
)
t
−
τ
|
−|
τ
|
)
dB
(τ )
(
H
+
1
/
2
)
t
0
|
,
H
−
1
/
2
dB
+
t
−
τ
|
(τ )
t
∈ R
(6.12)
where
B
(
t
)
is a standard Brownian motion,
(.)
is the Gamma function, and
1
2
<
H
<
1. Clearly,
B
H
(
t
)
is a zero-mean Gaussian process and
B
H
(
0
)
=
0.
1
2
,wehave
B
1
/
2
=
(
)
=
Furthermore, if we extend the definition to include
H
t
B
(
t
)
.Itisnot difficult to show that, for each
a
>
0,
d
=
a
H
{
B
H
(
at
)
;
t
∈ R}
{
B
H
(
t
)
;
t
∈ R}
;
that is,
B
H
(
is self-similar.
Of particular interest are self-similar processes with stationary increments.
Aprocess
Y
t
)
(
t
)
is said to have stationary increments if
d
={
{
X
(δ
,t
)
:
=
Y
(
t
+
δ)
−
Y
(
t
)
,t
∈ R}
Y
(δ)
−
Y
(
0
)
}
,
for any
δ.
(6.13)
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