IEEE Circuits and Systems Magazine - Q3 2022 - 10

4) Behaviour on C
The nabla exponential increases/decreases, as follows
(for the delta exponential the results are similar)
[15], [16].
■ It is real for real s.
■ It is positive for
sx h x /,1
■ It oscillates for sx h x /,1
= 1
= 2
! R.
! R.
■ For values of s inside the right Hilger circle it is
bounded and goes to zero as
s /.h1 "
■ On the Hilger circle it has absolute value equal to
1 and it degenerates into a sisoid.
■ Outside the Hilger circle its absolute value increases
as s; ; increases and goes to infinite as
;s " 3;
5) Delayed exponentials
Let
n Z0
!
.
following relations
(, )(,) (, )
dd T " $ !=and
et
nh se ts en hs00 (18)
TT
(, )(,) (, ). " $ !=-d
6) Product of exponentials
■ Different types at the same time
(, )(,) ,, ,``jj (19)
et se tv et sv
dd T=et sv
11 sh
+
$
D
with v C!
■ Different types at distincts times
(, )( ,) (, )
(, )( ,) (, ),
et se se ts
et se se ts
$
d
$
T -= -
TT
-= -
dd
with x !T .
■ Same type
et se tv et sv svh)
dd$ =+ -
(, )(,)
and
et se tv et sv svh).
TT$ =+ +
(, )(,)
7) Cross derivatives
De (, )( ,)
De (, )( ,)
T dd
d =-$TT
ts sh et hs=+$
ts sh et hs
C. Nabla Laplace Transform
The nabla and delta exponentials allow us to define two
transforms [22]. Let f(nh) be a signal and assume that it
has a Nabla Laplace transform (NLT), ().Fsd
sis equation for the NLT is given by
() ()
N fnhF sh fnhe nh s
n
6
@== / () ,.T^h
-
d
=+
3
Its
inverse transform (synthesis equation) is given by
10
IEEE CIRCUITS AND SYSTEMS MAGAZINE
3
(21)
The ROC is defined by all points distancing from /h1 less
than
and /.h
;;p 112ph1
/. A simple criterion imposes that ;;1 1The
pole must stay outside the Hilger circle.
;;sh
/
1
THIRD QUARTER 2022
T(,
d(,
Remark II.3. As it can be verified by computation,
the NLT of the correlation requires the introduction of the
Delta LT that is defined by [22]
Fs hf nh enhs
n
T ()
=3
=+
d
3
/
() (, ).
The corresponding inverse is given by
()
1 #Fs en hs s
() (( 1d
fnh=-), ).
2rj TT
A nabla causal exponential is defined by
(, )( ,) ().
et pe th pt=+ $ f
cdd
The analyfor
any p C! Its NLT reads [22]:
.
N (, )().et hp t
sp
1
6 d
+=$ f
@
-
(28)
(26)
(27)
Example II.4 (Causal and anti-causal exponentials).
xx
xx
=
+vh
-
From (8) we obtain immediately the impulse response
(IR) of the " differintegrator "
ticular, when a 1 ,=- we get N sn
s .a
-- = f which
11
(),
(20)
leads to a classic result: the step response is the
accumulation ( " integral " ) of the IR
n
rnhh fkh
k
f()= / ().
=0
In par+
.
+
The delayed exponentials verify the
et nh se ts en hs00 (17)
fn NhFsF se nh sds
2 j
()=
-1 6
dd +d
c
r
1
s /.h1=
()@=- 1 # ^h (22)
() () ,,
where the integration path, c , is any simple closed contour
in a region of analyticity of the integrand that includes
the point
with centre at s /.h1=
The simplest path is a circle
Attending to the properties of the
exponential that we listed before, the limit as h 0 " in
(21) leads to the usual two-sided LT.
We assume that s is inside the ROC of the transform. The
NLT enjoys the following properties
■ Linearity
■ Transform of the derivative
N fnhs Fs
6 d ()@ = a
a
()
d(),
(23)
reproducing a well known property of the CT Laplace
transform. The ROC is the disk inside the
Hilger circle.
■ Time shift
The NLT of (),
N () (, )( ).
6fnhn he nh sF s00
fnhn h0
@
-
N hf kh gnhkhF sG s
k
= /
3
=+
-=
$dd
3
G
-= -
T
d
■ Convolution in time
() () () ().
with n Z0
! is given by:
,
(24)
(25)
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