IEEE Circuits and Systems Magazine - Q2 2022 - 40

where (, )gt x is the IR of the system [55]. The IR characterizes
completely the system. In the time-invariant case
(, )( )
gt gt
xx
=- and the input-output relation assumes
a convolutional form [55]-[57]
() () () .
yt
=-xxx dx
3
#+3
-
Remark II.1. If (, )(/)/, ,, then (7)
gt gt
xx ! R0xx=
+
t
becomes a multiplicative (Mellin) convolution and the
corresponding systems are not time-invariant, but they are
essentially scale invariant. Among these types of systems
we can consider the LS based on the Hadamard [31] and
quantum derivatives [58], [59].
Returning to (8), if () ,, ,
yt Gs et R
st
() =
with () = gt6
!
xt es CRst
() ,,
= !! then
(9)
Gs L ()@ the LT of the IR that is the TF. Therefore,
1)
The exponentials are the eigenfunctions of the
LTIS and the eigenvalues, G(s) are the LT of their
IR [60].
2) If the system is causal, then its IR is a right function
and ()
ht ,0= for t 01 implying that ()
c
,
has a ROC defined by ()Re sa R2 ! .
3) Similarly, if the system is anti-causal, then
() ,,
ht =
a 00tfor 2
Hsa
defined by ()Re sb R1 ! .
4) Consider a bilateral system that is the sum of a
causal and an anti-causal, ()
Re ()
Gs Gs Gs=+ ().
,
ca
()
If its ROC is non void, as b11 the corresponding
IR is two-sided, () () ().
gt gt gt
=+
ca
5) If the region of convergence (ROC) of G(s)
contains the imaginary axis, then we can set
sj ,,R!~~=
so that the response of a LTIS to a
sisoid is also a sisoid with the same frequency. In
this case, the LT becomes the FT and originates
the FR, denoted by ().Gj~ It is frequently represented
by the Bode diagrams that are logarithm
plots of the amplitude, () ()AG j;;
z~ gGj~
~~=
phase, () = ar ().
Therefore, we require that the IR, (),gt [23]
■ is continuous almost everywhere,
■ has bounded variation,
■ is of exponential order.
Concerning the stability of a system, we prefer usually
the BIBO (bounded input-bounded output) stability criterion.
This implies that the IR is absolutely integrable
(AI). Therefore, any stable LS with TF given by G(s) has
frequency response, ().Gj~ For a given stable (unstable)
causal there is always a unstable (stable) anti-causal
with the same G(s) as TF, but different ROC. These considerations
created a framework for the definition of LS
through the IR and corresponding TF.
40
IEEE CIRCUITS AND SYSTEMS MAGAZINE
and
Gs
= ax
+
`
x + a
sa
sa
6) Phase lead/lag compensator [24]
()
implying that () has a ROC
Hsc
gt
(8)
B. Some Examples of TF
The above considerations do not suggest any particular
form that the TF (or FR) can assume. The practical
problem at hand can give insights into the one we must
choose. In Engineering, it is frequent to use the information
depicted by means of the Bode diagrams. Some interesting
models are well-known and were suggested in
different studies of electromagnetic media or electrical
circuits (with suitable ROC):
1) Oscillator
Gs = 2
() ,;
sp
1
+
1
p ! R+
2) Cole-Cole dielectric model
() ,;
Gs = a +
sp
Gs =
sp
s
a +
a
Gs ,;
sp
1
=
()
+ a
p ! R+
Gs =
6 +() @
1 s
1
x
p ! R+
3) Causal highpass filter
() ,;
p R!
+
4) Cole-Davidson dielectric model [61]
()
5) Havriliak-Negami dielectric model [62]
()
a b
;
j ,, ,.
a ! R+ 21
These examples suggest we can introduce some
non common TF. Let N and M be positive integers and
the polynomial coefficients ak
and ,, ,,bk 01= f be
k
real numbers. We introduce also the parameters
ak
and bk k 01 f=
,, ,, that, without loss of generality,
we can assume to form positive real increasing
sequences.
1) The most used model is the rational TF corresponding
to a continuous-time autoregressivemoving
average (CT-ARMA) model:
M
Gs
() = N
/
/
k=0
k=0
where the N and M are the orders of the model that
correspond to the degrees of denominator and numerator
polynomials, respectively.
2) The fractional continuous-time autoregressivemoving
average (CT-FARMA) [63].
SECOND QUARTER 2022
as
bs
k
,
k
k
(10)
k

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