IEEE Circuits and Systems Magazine - Q2 2022 - 52

As it is well known, a stable CT-ARMA (integer order) system has an impulse
response that decreases to zero exponentially when the argument
goes to infinite: it has a short memory.
The above-defined process is a somehow strange process
with infinite power. However, the power inside any
finite frequency band is always finite. Its spectrum is
S ()a ~ =
||
~
v
2
2a ,
(103)
and thus a " /f1 noise " . From the fractional noise, (),rta
we can generate a fractional Brownian motion, using expression
(99).
The process introduced in (99) enjoys the properties
usually attributed to fBm [112], [113], namely
1) ()v 00=
a
and Ev t = 0, for every t 0$ .
6
2) The covariance is
() ()
Ev tv s @aa =+ -6
where
V
=H 21C ()sin
v
+
2
HHr
.
3) The process has stationary increments.
4) The incremental process has a /f1 b
with t R! and T R! +
hH H
2
1
(105)
spectrum.
Consider the process corresponding vt/H 12+
The incremental process defined, for t and tT ,,
by
().
dt vt vt T
()=- -++
2
1
Rt =+ +- - 2 t
2
d
V 6 tT tT
If
b 2 0 , then
() (/ )
;
C bbr
cos
F 111E =
||b22
||b
t
~
the incremental process:
()
Sd ~v
=
mated by
S ~ . v T
d ()
22
~
1
4 || -
H
This result shows that:
■ If 01H /,211 then the spectrum is parabolic
and corresponds to an antipersistent fBm, be52
IEEE
CIRCUITS AND SYSTEMS MAGAZINE
21
.
(110)
() (),
has the following autocorrelation function:
() || || || .
H HH H22 2
@
(106)
(107)
H HH
a ()@
V ts ts
2
6|| || || ,
22 21
a+ @ (104)
cause the increments tend to have opposite
signs; this case corresponds to the integration
of a stationary fractional noise.
■ If /,H12111 then the spectrum has a hyperbolic
nature and corresponds to a persistent
fBm, because the increments tend to have the
same sign; this case corresponds to the integration
of a nonstationary fractional noise.
IV. Tempered Fractional Linear Systems
A. Tempered Fractional Derivatives
As it is well known, a stable CT-ARMA (integer order)
system has an impulse response that decreases to zero
exponentially when the argument goes to infinite: it has a
short memory. A CT-FARMA has an impulse response that
is a sum of one integer and one fractional components
(31). The fractional part decreases like a power function
of the argument and that is the reason why we claim
that fractional systems are of long range. However, we
find phenomena that are neither of short, nor long range.
They are medium range systems and can be obtained
by embedding the two types of responses. Therefore, a
multiplication of the derivative kernels by an exponential
gives the required behavior. This reasoning leads to the
so-called tempered derivatives. Due to the similarity to
the results introduced above, we will consider here only
the forward derivatives. For the backward see [24]. For
a ! R we can write
Df limth / n!
,f () =
h " 0+
,
(108)
so that the FT of Rtd () leads to the spectrum of
2 sin ~
||
~
2(/ )T 2
21
H+
.
(109)
For || /,T%~r the spectrum can be approxithat
has LT
6
n=0
L Dfts Fs Re s
m =+mm2a
,f
() () (),(). (112)
@
a
The inverse LT of this expression can be obtained
from the properties of the LT and of the Gamma function.
It is given by:
L se t a
amt
-- ().
C()
1 6 +=
() @
m !!
a
f
-
Equation (113) and the convolution property of the LT
allow us to introduce the integral version of the TFD as
() ()
Dftfte#3
- t #t
m
a
,f =-x
=ef e
-3
()
x
-mx x a
--1
mmx ()--1
a
C()
t - x
-a
C()
-a
dx,
(114)
SECOND QUARTER 2022
dx
t
--1
(113)
m
a am3
() -nh (),
-a n
ef tnh
-
(111)
IEEE Circuits and Systems Magazine - Q2 2022

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