IEEE Circuits and Systems Magazine - Q2 2022 - 50

above presented. We define the variable order (VO) forward
GL derivative as
Df limth ()- a t /
k= 0
a t
f
()
() =
h " 0+
3 ((t))k
k!
- a
ft kh
-
().
(76)
of the constant order GL FD. Let () =ft es C,
3 (( ))k ()
,
De lim+h- a t /
st
a t
f
()
=
h "
In particular, if () = ejt~
= se ,().Re s 2 0
ft
a()
tst
, then
()
k= 0
- a t
k !
a t
f
-
e
!
st kh
(77)
Dftj e
()
= ~ tj t
() ()a~
()
,
and the derivative of a co-sine (or sine) is an amplitudephase
modulated co-sine (or sine), then
Dt tt B
a t
f =+ ~ 2 0
2
()
coscos~~ ~a() ,.
()
a t
()
8
t
Dt =
a t
f
()
f
=
C
C
((t)) #() ()a t
-
1
-a
t
-a t
a
()
((t))
-+1
f t
(),
(78)
which is similar to the constant order case [63].
For functions with LT, we can define a regularized Liouville
VOFD by [23], [105]
Df t =
a
f ()
#3 --a
x
()
t
-a
1
C((t))
=ft xfax x,
()
() ((t))/
-where
() () .
N
Nt () ()
m!
() -1 mm
f
t
mGd
(79)
Nt = a t6@ With this derivative, we can define
VO LS. For example, the VO FARMA reads
()
M
// (80)
t
aD yt
k
ab=
k00
==
kk ()
k
()
bD xt
k
()
t
with t R! . We can introduce the VO IR and VO TF [23],
[105].
2) Fractional Stochastic Processes
Consider a continuous-time linear system with TF given
by G(s), having no poles on the imaginary axis (regular
system). Assume that the input x(t) to the system is a stationary
stochastic process with autocorrelation function
Rt Ex tx=+6
=
xx xx @
() () () .
yt gt xt)
responding autocorrelation is
() () () ().
Rt gt gt Rt
yy =- xx
))
50
IEEE CIRCUITS AND SYSTEMS MAGAZINE
(81)
The output is given by () () (), and the corDf
() F () () ()@,
ab
ab
-
+
(82)
tj jFj
1 ~~ ~
=
- 6 ab
-
we obtain a differential equation
SECOND QUARTER 2022
(90)
r
Example III.2. The VOFD of the unit step is
()
t xx
-- d
1
Ss Gs Gs Ss
yy =- xx
In the integer order case, ()
yy
leads us to define
()
yy
() () () (),
(84)
Ss has a non empty ROC
that includes the imaginary axis, but, in general, the
ROC is empty, since H(s) exists only for ()
Sj limGs Gs Sj
Gj Sj
==
sj
"
~
xx
Sj
xx
() (),
~~$
~~
() () ()
2
xx ~ and ().Sj
yy ~
Re s 2 0. This
This definition preserves most important properties
st
Let Ss L ()@ represent the LT of the autoxx
() = 6Rt
xx
correlation, and let us define the power spectral density
(or simply the spectrum) of x(t) as
Sj F ()@,Rt
xx ()
~ = 6
xx
(83)
obtained restricting s to the imaginary axis, i.e. s j~=
.
The relation (83) states the Wiener-Khintchin-Einstein
theorem [106]. We get
(85)
that relates the input with the corresponding output
power spectral densities, ()
In applications, mainly in modelling real data, we assume
that the input is white noise, w(t). In this case, the
autocorrelation is an impulse, usually written as
Rt
ww = vd2
yy
=
().
Sj~v22~
Gj
(86)
Therefore, the output spectrum of a linear system is
() () ,
(87)
stating an important relation suitable for stochastic
modeling and identification. Let us go back to equation
(84) and substitute there the expression of the TF (10).
We have
M
Ss
yy () = N
/
/
k=0
k=0
as
bs
k
$
ka
k
ka
M
/
/
k=0
N
k=0
However, we must take into account that, if a ! 1 ,
these relations are only valid in the limit when sj . " ~
Consequently, we have
() () ()
() () ().
N
N
/ / k m
k=0 m 0
=
M
aa jj Sj
bb jj Sj
M
=
/ / k m
k=0 m 0
~~ ~
~~ ~
km
aa
-=
-
yy
mk
aa
xx
(89)
With the inverse FT and introducing a two-sided derivative
Di
c
as
bs
k
k
-
-
()
()
ka
Ss
ka
xx().
(88)
IEEE Circuits and Systems Magazine - Q2 2022

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