IEEE Circuits and Systems Magazine - Q3 2022 - 17

Z () ,.nf =
6
@
1
1- z
-1
;;21
z
As we can see, the derivative of any order of the
Kroneckker impulse is essentially given by the coefficients
n}a
(69) we get
a () cm a
Dn 2 a
=
h
where ()nf
d}nf(),
n
(71)
is used to express the right behaviour
of the derivative of the delta, stating the causality
of the operator.
■ Fractional derivative of the unit step
The function
}n ,1n
2 () (),nn
}f1
with ZT given by Z hzz21 (/ )(()/(
,
- =-d
1
6
} =+ -
-
n @
1
f} 1
()
Consequently
Dn
a =+ }
a
f} 1
a
() cc-mm a
a
2 hh
12
-1
nn
12
2
.
Dx te lim 2
a
()
b
■ Fractional derivative of the } function
We are interested in computing the derivative of
}n ,a
.
tivity property, we can write
()
which leads to
D }}n f n
b 6 cm@n = 2 b
a
h
ab+
().
(72)
1) Backward Compatibility
Usually, DT systems are considered as mere approximations
of their CT counterparts. Nevertheless, and as
shown above, the DT systems exist by themselves and
have properties that, although similar to, are independent
from the CT analogues. However, this observation
does not prevent us from establishing a continuous path
from each other. In fact, we can go from the discrete into
the continuous domain by reducing the graininess. To
see it, let us return to (69) and rewrite it as
Dx nh
()
f
a
THIRD QUARTER 2022
() cm / kxnhkh
a
=-}
3
2
h
k 0
=
a ().
DD nD 22n
a
bad} n}f==b
a
; mmE
cc
hh
ab
+
ab
+
()
n
()=+} xt kh
k=0
iar
h " 0 h
for any a with n Z! From (71) and the addicm
/ k
a 3
a ().
(76)
Relations (75) and (76) state two new ways of computing
the continuous-time FD that are similar to the
Grünwald-Letnikov derivatives. However, it may be interesting
to remark that we can compute derivatives
with (73) instead of (65).
C. The Bilinear Discrete-Time Linear Systems
The above derivatives lead us to consider systems defined
by constant coefficient differential equations with the general
form (33), where the operator Dd
the forward (or backward) derivative previously defined.
Let g(n) be its IR. The output is () () ().
yn gn vn)
=
With the definition of forward derivative and mainly formula
(64) we write the TF
2
M
Gz
/
/
bk
()= N
k=0
ak
k=0
M
c
c
/
/
bk
h
2
h
for the causal case, and
()
Gz = N
k=0
ak
k=0
c
c
2
1
1
1
1
+
-
+
-
hz
z
2
hz
z
z
z
z
z
+
-
+
-
-
-
1
1
-
-
1
1
m
m
1
1
1
1
m
m
bk
k
z
1
a ,,; 2;
(77)
bk
z
1
a ,,; 1;
k
(78)
IEEE CIRCUITS AND SYSTEMS MAGAZINE
17
nn
2
=+ d
-
1
n
2
1
--)),
11
;;z 12 as expected. According to the above properties,
we can obtain the FD of the unit step function.
We have
().
introduced in the example A.1, is
a modified version of the unit step. It is straightforward
to confirm that
"
Assume that x(nh) resulted from a CT function x(t) and
define a new function, (),yt by
yt
defined in the Appendix. In fact, from
() cm / kxt kh
2
=-}
a 3
h
k 0
=
The LT of expression (73) is
()
Ys==}
k=0
c
h 0
e
22 e Xs
h
m / k
a 3
where ()= yt6
lim eh s ()/
,
aeX s
-khs
()
1-= we can write
a
Ys L ()@ and ()
-hs
=
c
h
1
1
+
-
-
-
hs
hs
Xs L () .xt= 6
() (),( ),Re
Ys sX ss 02
meaning that Y(s) is the LT of the (continuous-time)
derivative of ().xt This relation expresses the compatibility
between the new formulation described above
and the well known results from the continuous-time
derivative formulation [34] (see the companion paper
[2]). With the backward formulation, we would obtain
the same result, but with a ROC valid for () .
Re s 01 The
above equations together with (69), lead to the conclusion
that, for
Dx limt=-} xt kh
k=0
()
f
a
t R! we can write:
()
,
h " 0 h
cm / k
a 3
2
a ().
(75)
Similarly, from the backward formulation, mainly (70),
we obtain
m
a
(),
(74)
@ Knowing that
a ().
(73)
is substituted by
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