IEEE Circuits and Systems Magazine - Q3 2022 - 8

The paper outlines as follows. In Section II the nabla
and delta linear time invariant systems (LTIS) are introduced
and studied. We focus our attention in the nabla
versions, because they correspond to the causal case.
We introduce the nabla and delta discrete-time derivatives
(subsection II-A) and study their eigenfunctions
that will be called nabla and delta exponentials (subsection
II-B). From these concepts, we can define two discrete
Laplace transforms. We discuss the Nabla Laplace
transform only (subsection II-C), as well as its properties
and the backward compatibility with the continuous-time
Laplace transform. The nabla linear systems
are introduced and studied in subsection II-E where the
transient and steady-state responses are obtained. The
stability and initial-conditions are also considered.
The linear systems based on the bilinear transformation
are studied in Section III. The formulation is essentially
performed in the Z transform domain which makes
easier the introduction and study of the forward/backward
derivatives (subsection III-A). The time formulation of the
derivatives is presented in subsection III-B. With them,
we define the bilinear differential discrete-time linear systems
(subsection III-C). Some illustrative examples are
presented. In Section IV we introduce the state-variable
formulation in a general setup valid also for the continuous-time
systems. Finally some conclusions are drawn in
Section V.
A. Abbreviations
The following abbreviations are used in this manuscript:
ARMA
CT
DT
Autoregressive-Moving Average
Continuous-Time
Discrete-Time
FARMA
FD
FIR
FT
FFT
FR
IC
IR
IIR
GL
LTIS
LS
LT
MLF
NLT
ROC
TF
ZT
Fractional Autoregressive-Moving Average
Fractional derivative
Finite Impulse Response
Fourier transform
Fast Fourier transform
Frequency response
Initial-conditions
Impulse Response
Infinite Impulse Response
Grünwald-Letnikov
Linear time-invariant system
Linear system
Laplace transform
Mittag-Leffler function
Nabla Laplace transform
Region of convergence
Transfer function
Z transform
mds
Month 00, 2021
8
IEEE CIRCUITS AND SYSTEMS MAGAZINE
ft e-=0
a
()
D () =
jar n
II. On the Nabla and Delta Linear Time
Invariant Systems
A. Fractional Nabla and Delta Derivatives
Consider that our working domain is the time scale
TZ 32 02 3ff== -- -
with h R! +
() ,,, ,, ,, ,,hh hh hh h
. We can consider a slided time scale by a
"
,
given value, ah ,1 but it does not bring any new relevant
notion for this paper.
Set t = nh. We define the nabla derivative by:
ftld () =
ft ft h
-h
and
the delta derivative by
()
ftl
D
=
() ()
(1)
ft hf t() ()
+h
.
(2)
As
it can be seen, the first derivative is causal, while
the second is anti-causal. The repeated application of
these derivatives allows us to obtain the
NN )
th ,( !N
order derivatives and from them the general non integer
order (i. e.,
a ! R )0
+
formulations [22]:
3 ()
/
ft = =n 0
a
()
d ()
and
3 ()
/
-a n
n!
h
ft nh
+
a
()
,
^h-a n
(4)
obtained from the generalised Grünwald-Letnikov (GL)
derivative (see [2]). The symbol
chhamer representation of the raising factorial: ^h
^
-=P -+=
-
aa1
hn
k
n
^
stands for the Po-=a
0 1 ,
k .h We will call these derivatives forward
and backward respectively (this terminology is the
reverse of the one used in some mathematical literature).
The exponential factor in (4) is frequently removed, mainly
when the variable is not time. These formulations for
the fractional derivatives state different forms from those
we find in some current texts on discrete fractional calculus
(see [16], [17]), but are suitable for generalizing classic
tools like the impulse and frequency responses.
Example II.1. Consider the Heaviside unit step:
()
f nh = '
1
,
,
n
n
10
$
.
- a n
n!
h
ft nh
-
a
()
(3)
(5)
with n Z! . It is straightforward to show that the nabla derivative
of the unit step is the discrete-time impulse
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IEEE Circuits and Systems Magazine - Q3 2022

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