IEEE Circuits and Systems Magazine - Q3 2022 - 15

ROC defined by :, .zz aa 1 We define
2rj #cm n 1- d ,
Dx n =
b () 12
a
c
hz
z Xz zz
+
-
1
1
a
()
(65)
with the integration path inside the unit disk and the
branchcut line is a segment joining the points
z =!1 .
This implies that
()
b
Z Dx n =
6 cm@
a
+
-
1
1
hz
2 z Xz z
a
(), ;;11.
(66)
Remark III.1. We must note that:
1) In (58) and (59) we have two branchcut points at
.
z =!1 The corresponding branchcut line is any
line connecting these values and being located in
the unit disk. The simplest is a straight line segment
(see figure 1).
2) In (60) and (61) we have the same branchcut points,
but with branchcut line(s) lying outside the unit disk.
For simplifying, we can use two half-straight lines
starting at z
=!1 on the real negative and positive
half lines, respectively (see figure 1).
3) In both of the previous cases, we can extend the
domain of validity to include the unit circumference,
ze
,jn
~
=
;; (, ),0!~r with exception of the
points z =!1 . In these cases, the integration path
in (63) must be deformed around such points.
This deformation is very important when using the
FFT. In such situation, a small numerical trick can
be used: push the branchcut points slightly inside
(outside) the unit circle, that is, to z
z 1 11e ee=- -- +
tive real number.
and
Dx ne
()
b
a
=-1 e+ and
,(, ), with e being a small posi4)
The ROC is independent on the scale graininess, h,
and consequently we can establish a one to one correspondence
between the unit disk, in z, and the left halfplane,
in s given by (/ )(()/(
s hz z )).
=- +--21 111
According to what we just wrote, we can extend the
above definitions to include sinusoids. We define the derivative
of ()
xn ejn
=
De
fb,
jn
;
h
~ , n Z! through
,
~ = 2 ta ,,n` jE e
a
~
2
jn
~
;;1
~r
(67)
independently of considering the forward or backward
derivatives. With this result, we can obtain the derivative
of any function having discrete-time FT, that is expressed
as:
DxnXe
fb,
=
r
r
() 12 ~ a
~ ;
h
2
2r -# () tan` jE e
jj~n
d~.
(68)
In accordance with the existence conditions of the
FT, we can say that, if x(n) is absolutely summable, then
the derivative (68) exists.
THIRD QUARTER 2022
Figure 1. ROC for causal and anti-causal derivatives and
branchcut points and lines.
IEEE CIRCUITS AND SYSTEMS MAGAZINE
15
Branchcut
Line
Anti-Causal
ROC
Causal
ROC
Re
lm
()
=+}
3
h
k=0
j cm / kxn k
ar 2 a
a ().
(70)
As before, we can remove the exponential factor, e ,
jar
in (70). In the following we consider the causal derivative
(69) represented by the simplified notation Da
and
with ZT given by (64).
Some interesting results are [26]
()
1!w =
a /
k
3
=
() () k
k!
" 1 k
-a k
1
ww; 1;,,
we conclude that the TF in (58) and (60) can be expressed
as power series,
cm / } -k
a
1
1
where }k ,a
+
-
z
z
-
-
k ,, ,01f=
quence }k ,a
1
1
a 3
=
k=0
is the inverse ZT of (( -+ a-1111
zz
)/())
and
represents the IR (see Appendix).
In agreement with the meaning attributed to the sek
,, ,01 f=
we define the a-order bilinear
forward and backward derivatives as
()
Dx n
()
f
a
2
h
k
k 0
=
=-}
3
cm / ()axn k
a
(69)
1
kzz; 2;,,
B. Time Formulations
In the previous subsection we introduced the derivatives
using a formulation based on the ZT. However, we
can obtain the corresponding time framework, getting
formulae similar to the GL derivatives [34]. From the binomial
series [35]
Let x(n) be a function with ZT (),Xz analytic in the
!;C 12;
These derivatives enjoy the same properties of the
nabla and delta derivatives [26].
IEEE Circuits and Systems Magazine - Q3 2022

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