IEEE Circuits and Systems Magazine - Q3 2022 - 16

■ The first is causal while the second is anti-causal.
■ Fractional derivative of the impulse
Let us introduce the Kroneckker impulse, (),nd
d = '
n
()
1
n
n
! 0
=
.
APPENDIX: The }a
The sequence }a
k , k ,, ,01 f=
is obtained as the discrete
convolution of two binomial sequences:
!
a --1 k
} =k kk!
()k () ()
aak
)
+
.
k ! Z0
In previous works [44]-[46], ARMA approximations to
these sequences were also proposed.
Performing this discrete convolution we obtain the following
results [26]. If R!a
k
} =-1
a
and
} =
-N
k
()k
k
N
mi (, )
n kN
! /
=
m 0
when a N ,=- N Z! +
1. The sequence}a
is null for k 01 .
2. For any
a ! R , we have
() k
-aa +
}} ! Z0
k =-1 k
k
*
(115)
3. Initial value
From the initial value theorem of the ZT, it is immediate
that
} =a
0 1 , independently of the order.
4. Final value
Let
*
a # 0 . From the final value of the ZT,
} =-1hc
3
a
then
} 0 ,=3
a
lim z
z 1
^
"
if -10 ,##a
and
1
1
+
-
z
z
-
-
1
1
} 2 ,=3
a
m ,
a
if a 1 .=- For
a1 1- the sequence increases to 3 , otherwise, for
a2 0 we apply (115).
5. A recursion
The IR sequence verifies the recursion
} =- 2a}}a
kk k12 (116)
aa +kk
1
2
with } =a
a =-2 .
1
is a
0 1 and }a
This recursion shows that, if a10 , then k}a
positive sequence. As consequence, attending to (115),
16
IEEE CIRCUITS AND SYSTEMS MAGAZINE
-cm
k
$ 2
,,
()
() ()
-- +
-- -
k, k ,, ,01 f=
. The sequence k verifies the properties:
is causal and, therefore,
}a
Nk ()m
Nk 1 m m!
mm 1
,. (114)
k Z+
! 0
k ()
k a k
()
kk 1 m m!
! /
=
m 0
()
() ()
-- +
-- -
a mm
k
a
but a " Z ,- then
m
()
1
, k Z+
! 0
(113)
and its ZT is given by
k sequence
the sequence corresponding to positive orders is always
oscillating: successive values have alternating sign.
6. Relation with the Hypergeometric function
The second factor in (113) is a sequence drawn from
the Gauss Hypergeometric function
k
/
m 0
=
!
()
() ()
-- +
-- -
a mm
k
a
=- -- -21
aa
Fk k
1
for n Z .0
+
7. For a fixed k Z! , }a
with the coefficient of k
k is a polynomial in a of degree k
a decreasing with increasing k .
Example A.1. We present k}!N
N ,1=
*
k
1
k
} = *
1
1
21
-
} ==*
1
2
2.
N 2=
*
k
k
} = *
2
2
1
() k
-14
k
} ==*
1
4
k
k
k
k
1
2
k
k
k
()
k
1
2
k
k
k
k
k
k
3.
For any negative order -a , with a2 0
Using the recursion (116) with } =a0
we
obtain successively:
2
}a
}a a
}a a
}a aa
}a aa
}a aaa
}a aaa
-
-
-
-
-
-
-
4
5
6
7
8
2 =
3
a
a
a
a
a
a
a
2
=+
=+
=+ +
=+ +
3
4
3
2
15
4
45
4
=+++
=+++
315
8
315
2
gg
75 3
315
140
8642
315
56
315
308
315
264
(118)
THIRD QUARTER 2022
315
392
3
3
2
42
3
4
53
15
20
64 2
45
40
45
46
315
90
15
6
1 and
}a1
- a =
2 ,
1
2
=
1
2
=
for some values of N Z! +
and for any real order, obtained by recursive computation
1.
(, ;; ),
k
11
m
()
m!
1
m
(117)
n ,Z! by
The Heaviside discrete unit step is usually defined by
()n
f = '
1
n
n
10
$

IEEE Circuits and Systems Magazine - Q3 2022

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