IEEE Circuits and Systems Magazine - Q3 2022 - 11

There are two exponentials, eigenfunctions of nabla and delta derivatives, that
allow the introduction of the nabla and delta discrete Laplace transforms.
The anti-causal exponential is defined by
(, )( ,) ()
(, )( ).
et pe th pt h
NLTe th pt h
a
dd
d
=- +- -
@
$ f
6-+ -- =
$ f
sp
1
-
The ROC of (29) is the inverse of the one for the causal: the
pole must be inside the Hilger circle.
We can generalise the above results for multiple poles
by considering /(1 sp)(29)
E.
The Nabla Linear Systems
1) Steady-State Response
We define a nabla linear system through the following
differential equation [22]
N
M
// (33)
a
k==
k
k 01 f=
00
k
aD yt =
k
d
()
as a function of p and computing
sucessive integer order derivatives.
Letting p 0= in the expressions (28) and (29) we obtain
immediately the NLT of the unit step and powers. For
the causal case, we obtain
N ha a +1 n
;
()
n !
f nh = 1
s
()E
a+1
.
The ROC is the disk inside the Hilger circle.
D. The Discrete-Time Fourier Transform
Similarly to the CT case, the exponential degenerates into
a sisoid when its absolute value is equal to 1. From (12),
sh
11
;;
11
enhe n
Fe hf nh e
d
(, ), ,
()
~
jhn
~~
fnhFee
2
()
=
=
=
1 #r -
r
r
/
/
h
h
()
-j
jhn
3
~
! Z
/ () ,
-3
jj hn~~ d~.
showing that the TF is the NLT of the IR [22]. With these
results we can write the TF as
With the substitution h~ X= we obtain expressions that
are independent of the sampling interval (graininess) h
j n
en e
Fe hf nh e
d
(, ),
()
X =
=
3
X
j
XX
-
and
fnhFee
2
r -
()= 1 # () .
r
r XX
jj dn
X
(32)
Expressions (31) and (32), define the discrete-time Fourier
transform pair (DTFT), namely the direct and inverse
transforms, respectively [24], [28]. The presence
of the factor h in (31) ensures a back compatibility with
the CT Fourier transform.
THIRD QUARTER 2022
/ () -j n
3
(31)
M
Gs
d = N
()
/
/
k=0
k=0
As in the CT case, we conclude that:
■ The exponentials are the eigenfunctions of the linear
systems (33)
■ The eigenvalues are the transfer function values.
Let us analyse the following example.
Example II.5. Let h 1= and consider the differential equation
yt yt yt yt xt42+- +=
nm l
()
()
()
() ().
If ()=xn 2 n- , that corresponds to set s =-1 in (12), then
the solution is given by:
IEEE CIRCUITS AND SYSTEMS MAGAZINE
11
as
bs
k
k
bk
.
ak
(36)
;;
variable ()/ehs 1
jwh
=- -
(30)
bD xt
k
bk
d
()
where ak and bk (, ,, ) with a 1N
= are real numbers.
The operator Dd is the nabla derivative defined
previously. The orders N and M are any positive integers.
The positive real numbers ka and kb with
k ,, ,01 f=
form strictly increasing sequences. It is interesting to remark
that when
ak k ,= relation (33) can be transformed
into a difference equation, by using (1) and the binomial
decomposition.
Let g(t) be the IR of the system defined by (33) with
xt = d nh The output is the convolution (25) of the
input and the IR
() ().
yt gt xt)
=
-= defines a circle centred at /h1 and has radius
equal to /h1 (right hand Hilger circle). Similarly, the left Hilger
circle comes from sh
+= (13). With the change of
in (12), (21), and (22), we obtain
xt enhs
= d
() () ().
If () (, ), then the output is given by:
3
yt enhs hg nh enhs
n
() (, )( )( ,) .
==-d
3
;
/
3
Gs hg nh enhs
n
d =3
()
T
=/
() (, ),
(35)
T
E
The summation expression is the transfer function as
usually. We have
(34)
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