Second Order Approximation of the Fractional
Laplacian Operator for Equal-Ripple Response
Todd J. Freeborn, Brent Maundy, and Ahmed Elwakil*
Department of Electrical and Computer Engineering, University of Calgary, Calgary, Alberta, Canada.
*Department of Electrical and Computer Engineering, University of Sharjah, P.O. Box 27272, Emirates (elwakil@ieee.org).
Abstract—In this paper we propose a modification to a second
order approximation of the fractional-order Laplacian operator,
s
α
, where 0 <α< 1. We show how this proposed modification
can be used to change the ripple error of both the magnitude
and phase responses of the approximation when compared to the
ideal case. Equal-ripple magnitude and phase responses that have
both less cumulative error and less maximum ripple deviation
are presented using this modification. A 1
st
order lowpass filter
with fractional step of 0.8, that is of order 1.8, is implemented
using the proposed approximation. Experimental results verify
the operation of this approximation in the realization of the
fractional step filter.
I. I NTRODUCTION
Traditionally the Laplacian operator, s, is raised to an
integer order, i.e s, s
2
, ...s
n
, when used in the design of analog
circuits. However, it is also mathematically valid to raise to
a non-integer order, s
α
, where 0 <α< 1; representing a
fractional order system. A fractional derivative may be defined,
according to the Riemann-Liouville definition [1], as
d
α
dt
α
f (t) ≡ D
α
f (t)=
d
dt
1
Γ(1 − α)
t
ˆ
0
f (τ )
(t − τ )
α
dτ
(1)
where Γ(·) is the gamma function. Applying the Laplace
transform, with zero initial conditions, to (1) yields
L {
0
d
α
t
f (t)} = s
α
F (s) (2)
This fractional Laplacian operator has practical applications in
numerous areas of engineering [2]. However, at this time there
are no commercial fractance devices available. Therefore, to
physically realize circuits that make use of the advantages of
s
α
, integer order approximations have been used. There are
many methods to create an approximation of s
α
that include
Continued Fraction Expansions (CFEs) as well as numerous
rational methods [3]. These methods present a large array
of approximations with varying order and accuracy, with the
accuracy and approximated frequency band increasing as the
order of the approximation increases. The importing of these
concepts into circuit theory is relatively new, and has shown
applications in power electronics [4], integrator [5], [6] and
differentiator circuits [7], multivibrator circuits [8], and filter
theory [9], [10] with potentially many other applications.
In this paper, we propose a modification to a second-
order approximation for the fractional Laplacian, s
α
, which
can be used to manipulate the error of both the magnitude
and phase response. We show how this approximation can
10
-2
10
-1
10
0
10
1
10
2
-40
-20
0
20
40
Frequency (Hz)
Magnitude (dB)
10
-2
10
-1
10
0
10
1
10
2
0
20
40
60
80
Frequency (Hz)
Phase (degrees)
Figure 1. Magnitude and phase of ideal (solid) and 2nd order approximation
(dashed) of s
α
when α =0.8
be manipulated to create an equi-ripple error response when
compared to the ideal case. A fractional lowpass filter of order
1.8 is verified experimentally to show the application of this
approximation.
II. LAPLACIAN OPERATOR APPROXIMATION
Previous work on the approximation of the fractional Lapla-
cian operator has yielded a second order approximation using
the CFE method [11] as,
s
α
≈
(α
2
+3α + 2)s
2
+ (8 − 2α
2
)s +(α
2
− 3α + 2)
(α
2
− 3α + 2)s
2
+ (8 − 2α
2
)s +(α
2
+3α + 2)
(3)
The magnitude and phase response of (3) compared with
s
α
when α = 0.8 is shown in Fig. 1. From this
figure, the magnitude error of the approximation com-
pared to the ideal case does not exceed 1.338 dB for
ωǫ [0.023, 42.85] rad/s; while the phase error does not exceed
2.173 for ωǫ [0.127, 7.87] rad/s.
We note also both the magnitude and phase responses have
inner and outer ripples, as highlighted in Fig. 1. To compare
the peaks of the inner and outer ripples for both the magnitude
and phase, the error of (3) compared to the ideal s
α
was
calculated as
Error = s
α
−
(α
2
+3α + 2)s
2
+ (8 − 2α
2
)s +(α
2
− 3α + 2)
(α
2
− 3α + 2)s
2
+ (8 − 2α
2
)s +(α
2
+3α + 2)
(4)
Using (4) the peaks of the magnitude and phase ripple errors
were calculated numerically as shown in Fig. 2, for 0 <α< 1,
illustrating that both the inner and outer ripple errors vary
significantly with α. Note the size of the outer ripple is
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