IEEE Circuits and Systems Magazine - Q2 2022 - 47

Frequently, we find papers that, assuming the Riemann-Liouville or Caputo derivatives,
use the results we just presented. Nonetheless, this is not consistent with
relations (60) to (62), since they are invalid for such derivatives.
■ If f(t) has LT with a nondegenerate region of convergence,
then the 3 derivatives give the same result,
■ The Liouville-Caputo derivative demands too
much from analytical point of view, since it
needs the unnecessary existence of the Mth
order derivative,
■ If () ,,ft = 1 t R! then the Liouville derivative does
not exist, since the integral is divergent.
There are several properties revealed by these derivatives.
The most important are [92]:
1) Linearity
2) Additivity and Commutativity of the orders
a bb a
DDft DD ft Df t
f ff f== ().
()
3) Neutral element
DD ft Df tf t
aa ==ff
()
f
() ().
(59)
From (59) we conclude that there is always an
inverse operator, that is, for every a there is always
the aorder
derivative that we call antiderivative
given by the same formula and so it is
not needed to join any primitivation constant.
4) Backward compatibility (n N! )
If
a n= , then:
n
-Df
t = lim
h " 0
f
n
()
der derivative.
If
a =- n
Df lim ()·,
k=0
, then:
()
t
-n
f
= "
h
/
n ()k
!
k
n
ft kh hn
that corresponds to a n-th repeated summation [63].
E. The Frequency Response
The property (53) of the derivatives we are using can
be extended to the imaginary axis yielding the following
important result
De je t
a~ = ~~ !
a~
00
jt () jt,,00 R.
then we obtain immediately from (9)
SECOND QUARTER 2022
(60)
In general, and for a system defined by (8), if () =xt ejt0
~
,
and
{~
() gargKmar () / k
k=1
-+
a
Np
k
k=1
ik
=+ +
a
Mz
gk
;cm 1E
j~
/ n arg;cm 1E .
j~
(66)
(67)
As it is easy to verify a simple pseudo-pole/zero
originates a decrease/increase of the amplitude by
less than 20 dB per decade. In figure 2 we show the
Bode plots corresponding to the fractional RC circuit
of example III.1.
IEEE CIRCUITS AND SYSTEMS MAGAZINE
47
/ () ()
k=0
1
k ` jft kh
n
k
n
h
In the following we will assume multiplicities equal to
We obtain this expression repeating the first or1.
For a TF given by (22) the gain and phase are given by
()
Mz
AK m 20
k=1
~
=+ +
-+
20
loglog
log
10()
/
/
Np
n 20
k
k=1
k
10 c
10 c
j~ a
m
gk
j~ a
m
ik
1
1 ,
(65)
where K0
g =-z
kk
()
ab+
f
This result comes immediately from (53).
yt Gj ejt
~0
= ~0
xt =
() () .
When the input is () ()cos ~0 t , we have
() () cos
where
1) () ()AH
~~=
2) () ()z~ {~=
form to the TF:
(58)
Mz
%
%
Gs K k=1
() = 0 Np
k=1
c
`
`
`
s
s
i
g
k
k
j
j
a
a
+
+
1
1
j
m
mk
,
mk
(63)
is called static gain and the integers
m ,, ,12k = f represent the multiplicity of the pseudo1/a
i
=and
pk
poles/zeroes. As it is easy to verify, ()p
1/a
kk and
() . With these changes, (22) assumes a more
classical form. Supposing that all the coefficients ak
bk in (22) are real, then all values of zk
and
are either
real or, being complex, appear in conjugate pairs. In this
case, its is usual to join the corresponding terms:
a
`` jj jj m
a
ss ss11
a
a
i
++ =+ + 1.
|| || || a
``
i ii i
*
c
`
j
(64)
a 2Re()
i
2
and is an even function,
is the phase spectrum, or simply phase,
and is an odd function.
For our objectives, it is preferable to give another
(61)
yt Ht () ,000~~ {~=+^h (62)
is the amplitude spectrum, or gain,

IEEE Circuits and Systems Magazine - Q2 2022

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