IEEE Circuits and Systems Magazine - Q3 2022 - 14

The traditional bilinear transformations lead to the introduction of two discrete-time
derivatives, that allow the definition of two discrete-time systems.
The bilinear exponential () is the eigenfunction
of the equations (51) or (52). If we set () (),
() (), s C! with () ,= then
enhs
xnhe nhs
=
ynhsenhs
=
,
enh =
s () ,, .
-
cm+
hs
hs
2
2 ZC
n
ns
!!
enhs
(53)
The properties of the bilinear exponential () are
■ When n " 3 this exponential
* Increases, if () ,
* Decreases, if () ,
* Is sinusoidal, if () ,
* Is constant equal to 1, if s 0= ,
,
Re s 02
Re s 01
Re s 0= with s 0! ,
■ It is real for real s,
■ It is positive for
sx h x /,2
■ It oscillates for sx h x /,2
=;!1;
=;!2;
R,
R.
Following the procedure in the previous section,
we could use this exponential to construct a bilinear
discrete-time LT. However, formula (53) suggests that
()/(
z 22 hs) leads to the ZT, since such trans;
; = as the image of
=+ -
hs
formation sets the unit circle z 1
the imaginary axis in s, independently of the value of
h. Accordingly, the bilinear exponential has the usual
properties:
■ When
* Increases, if ;z 12;
* Decreases, if ;z 11;
■ It is real for real z,
■ It is positive for
n " 3 this exponential
,
,
such that
,
* Is sinusoidal, if ;z 1; = with z ,1!
* Is constant equal to 1, if z ,1=
,
■ It oscillates for z R0! +
zx ,02=
.
x R!
Therefore, we do not need to introduce a new transform,
since the ZT is suitable.
In what concerns derivative definitions, instead of
considering (51) or (52), we start from the ZT formulations.
Let z C! and
h R! +
exponential function, z ,n
.
ward bilinear derivative (Df) as an elemental DT system
such that
Dz = 2
h
f
nn
1
1
1
+
-
z
z
-
-
1
z .
The forward TF of such derivative, (),Hzf
Hz = 2
h
14
f () ,.
1
1
+
-
z
z
-
-
1
1
z 21
IEEE CIRCUITS AND SYSTEMS MAGAZINE
(54)
is defined by
(55)
. Consider the discrete-time
n Z! We define the forDz
cm ; 1;zz 1 .
a
a nn
b
= 2
hz
z
+
-
1
1
,
(61)
Once we defined the derivative of an exponential, we
are in conditions of obtaining the derivative of any signal
having ZT. We only have to use the inversion integral
of the ZT
xn =
()
2 j
1
r # () n 1- d .
Xz zz
c
From (62) and (59) we conclude that, if x(n) is a function
with ZT (),Xz analytic in the ROC defined by
;za ,2;
z C! :
Dx n 12
c h
a ,11 then
()
a
f
=
2rj #cm n-1d ,
()
1
1
+
-
z
z Xz zz
-
-
1
1
a
(63)
with the integration path outside the unit disk. This implies
that
6 cm@
Z Dx n = 2
a
f ()
h
1
1
+
-
z
z Xz z
-
-
1
1
a
(),.
;;21
(64)
THIRD QUARTER 2022
(62)
such that
Dz cm ; 2; ,.
= 2
a
f
nn
1
h
1
1
b
= 2
+
-
z
z
hz
z
-
-
1
a
zz 1
Identically, the backward bilinear FD has TF
()
Hz cm z
a
+
-
1
1
;;11
,,
(59)
Hz = 2
h
1
1
+
-
z
z
The repeated application of the above operators lead
to the forward and backward derivatives for any positive
integer order that can be generalized for any real
order. Let
a ! R . The a-order forward bilinear FD is a
DT LS with TF
f () cm 1
,,; 2;
-
-
1
1
a
z
(58)
with backward TF, (),Hzb
Hz = 2
given by
b () ,.
hz
z
+
-
1
1
z 11
(57)
e 01s
Dz = 2
b
nn
hz
z
+
-
1
1
z ,
(56)
The backward bilinear derivative ()Db
the system verifying
is defined as
(60)
IEEE Circuits and Systems Magazine - Q3 2022

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