ON THE LINEAR SCALE FRACTIONAL SYSTEMS
An Application of the Fractional Quantum Derivative
Manuel Duarte Ortigueira
UNINOVA and DEE of Faculdade de Ciências e Tecnologia da UNL
Campus da FCT da UNL, Quinta da Torre, 2825 – 114 Monte da Caparica, Portugal
Keywords: Linear Scale Invariant Systems, Fractional Linear Systems, Fractional Quantum Derivative.
Abstract: The Linear Scale Invariant Systems are introduced for both integer and fractional orders. They are defined
by the generalized Euler-Cauchy differential equation. It is shown how to compute the impulse responses
corresponding to the two regions of convergence of the transfer function. This is obtained by using the
Mellin transform. The quantum fractional derivatives are used because they are suitable for dealing with this
kind of systems.
1 INTRODUCTION
Braccini and Gambardella (1986) introduced the
concept of “form-invariant” filters. These are
systems such that a scaling of the input gives rise to
a scaling of the output. This is important in detection
and estimation of signals with unknown size
requiring some type of pre-processing: for example
edge sharpening in image processing or in radar
signals. However in their attempt to define such
systems, they did not give any formulation in terms
of a differential equation. The Linear Scale Invariant
Systems (LSIS) were really introduced by Yazici
and Kashyap (1997) for analysis and modelling 1/f
phenomena and in general the self-similar processes,
namely the scale stationary processes. Their
approach was based on an integer order Euler-
Cauchy differential equation. However, they solved
only a particular case corresponding to the all pole
case. To insert a fractional behaviour, they proposed
the concept of pseudo-impulse response. Here we
avoid this procedure by presenting a fractional
derivative based general formulation of the LSIS.
We assume that the fractional LSIS is described by
the general Euler-Cauchy differential equation
i=0
N
a
i
t
α
i
.y
(
α
i
)
(t) =
i=0
M
b
i
. t
β
i
.y
(
β
i
)
(t)
(1)
This equation is difficult to solve for any values for
N or M and any derivative orders. However, when
the derivative orders have the format
α
i
= α+i i=0, 1, 2, …, N
and
β
i
= β+i i=0, 1, 2, …, N
we obtain a simpler equation
i=0
N
a
i
t
α+i
.y
(
α+i)
(t) =
i=0
M
b
i
. t
β+i
x
(
β+i)
(t)
(2)
that we can solve with the help of the Mellin
transform and using the fractional quantum
derivative (Ortigueira, 2007, 2008). As we will
show, the above equation allows us to obtain two
transfer functions. Each of them has two terms that
lead to two inverse functions. The impulse response
is obtained by using the multiplicative convolution
defined by (Bertran et al, 2000):
f(t)٧g(t) =
0
f(t/u)g(u)
du
u
(3)
Before going into the solution of equation (2),
we are going to obtain the solution of the integer
order equation corresponding to put α=β=0 in (2).
Then we will solve equation (2) for any α and β.
This will be done in section 1. Other interesting
results will be introduced in section 3. Finally we
will present some conclusions.
196
Duarte Ortigueira M.
ON THE LINEAR SCALE FRACTIONAL SYSTEMS - An Application of the Fractional Quantum Derivative.
DOI: 10.5220/0002246901960202
In Proceedings of the 6th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2009), page
ISBN: 978-989-674-001-6
Copyright
c
2009 by SCITEPRESS Science and Technology Publications, Lda. All rights reserved
2 THE EULER-CAUCHY
EQUATION
2.1 The Integer Order Case
Consider a linear system represented by the
differential equation
i=0
N
a
i
t
i
.y
(i)
(t) =
i=0
M
b
i
. t
i
x
(i)
(t)
(4)
where x(t) is the input, y(t) the output, and N and M
are positive integers (MN). Usually a
N
is chosen to
be 1. We will assume that this equation is valid for
every tR
+
. Applying the Mellin transform to both
sides of (3) we obtain (Gerardi, 1959;Bertran et al,
2000)
i=0
N
a
i
(-1)
i
(s)
i
Y(s) =
i=0
M
b
i
. (-1)
i
(s)
i
X(s),
(5)
from where we obtain a transfer function
H(s) =
Y(s)
X(s)
=
i=0
M
b
i
(-1)
i
(s)
i
i=0
N
a
i
(-1)
i
(s)
i
(6)
In this expression we need to transform both
numerator and denominator into polynomials in the
variable s. To do it we use the well known relation
(Abramowitz and Stegun, 1972 )
(x)
k
=
i=0
k
(-1)
k-i
s(k,i) x
i
(7)
where s( , ) represent the Stirling numbers of first
kind that verify the recursion
s(n+1,m) = s(n,m-1) – ns(n,m) (8)
for 1 m n and with
s(n,0) = δ
n
and s(n,1) = (-1)
n-1
(n-1)!
With some manipulation, we obtain:
i=0
N
a
i
(-1)
i
(x)
i
=
i=0
N
k=i
N
a
k
(-1)
k
s(k,i) x
i
=
i=0
N
A
i
x
i
(9)
with the A
i
coefficients given by
A
i
=
i=k
N
a
k
(-1)
k
s(k,i)
(10)
or in a matricial format
A = S.a
(11)
where
A = [A
0
A
1
… … A
N
]
T
(12)
S=[ s(i,j), i,j=0,1, …,N] (13)
and
a = [a
0
a
1
… … a
N
]
T
(14)
With this formulation, the transfer function is given
by:
H(s) =
i=0
M
B
i
s
i
i=0
N
A
i
s
i
MN
(15)
that is the quotient of two polynomials in s. In
general H(s) has the following partial fraction
decomposition
H(s) =
B
M
A
N
+
i=1
N
j=1
m
i
a
ij
(s-p
i
)
j
(16)
The constant term only exists when M=N and its
inversion gives a delta at t=1:
M
-1
[
B
M
A
N
] =
B
M
A
N
δ(t-1)
(17)
For inversion of a given partial fraction, we must fix
the region of convergence Re(s) > Re(p
i
) or Re(s) <
Re(p
i
) similar to identical situation found in the
usual shift invariant systems with the Laplace
transform. Let us assume that the poles are simple.
Accordingly to each region of convergence we have
(Bertran et al, 2000) respectively
M
-1
[
1
(s-p)
] = u(1-t).t
-p
(18)
and
M
-1
[
1
(s-p)
] =u(t-1).t
-p
(19)
ON THE LINEAR SCALE FRACTIONAL SYSTEMS - An Application of the Fractional Quantum Derivative
197
By successive derivation in order to p we obtain the
solution for higher order poles
M
-1
[
1
(s-p)
k
] = u(1-t).
(-1)
k-1
[log(t)]
k-1
(k-1)!
t
-p
(20)
and
M
-1
[
1
(s-p)
k
] =u(t-1).
(-1)
k-1
[log(t)]
k-1
(k-1)!
t
-p
(21)
We conclude that the response corresponding to an
input δ(t-1) is given by:
h(t)=
B
M
A
N
δ(t-1)+
i=1
N
k=1
m
i
a
ik
.
(-1)
k-1
[log(t)]
k-1
(k-1)!
t
-p
i
w(t)
(22)
where w(t) is equal to u(1-t) or to u(t-1), in
agreement with the region of convergence adopted
to invert (15). To compute the output to any function
x(t) we only have to use the multiplicative
convolution.
We must call the attention to the fact the
point of application of the impulse is t=1 and not
t=0, as it is the case of the shift-invariant systems.
2.2 The Fractional Quantum
Derivative
To consider a more general case we must introduce
the notion of fractional quantum derivative. This
was not needed in the previous section because in
the integer order case we only have one Mellin
transform for t
K
f
(K)
(t). This is not the situation in the
fractional case. In fact we have two fractional
derivatives given by {see appendix}:
D
α
q
f(t) = lim
q1
j=0
α
j
q
(-1)
j
q
j(j+1)/2
q
-jα
f(q
j
t)
(1 q)
α
t
α
(23)
and
D
α
q
-1
f(t) = lim
q1
j=0
α
j
q
(-1)
j
q
j(j-1)/2
f(q
-j
t)
(1 q
-1
)
α
t
α
(24)
These derivatives have the same Mellin transform in
the integer order case, but in the general their Mellin
transforms are given by:
M [D
α
q
f(t) ] =
Γ(1 s + α)
Γ(1 s)
F(s-α)
(25)
valid for Re(s) < min(0,α)+1, in the first case and by
M [D
α
q
-1
f(t) ] = (-1)
α
.
Γ(s)
Γ(s α)
F(s-α)
(26)
valid for Re(s) > max(0,α), in the second case. It is
interesting that the first corresponds to the anti-
causal case when working in the Laplace transform
context, while the second corresponds to the causal
one.
2.3 The Fractional Order Equation
Consider now a linear system represented by the
fractional differential equation
i=0
N
a
i
t
α+i
.y
(
α+i)
(t) =
i=0
M
b
i
. t
β+i
x
(
β+i)
(t)
(27)
where α and β are real numbers. With the Mellin
transform we obtain two different transfer functions
depending on the derivative we use, (23) or (24).
From (23) we have:
H(s) =
i=0
M
b
i
(-1)
i
(s+β)
i
i=0
N
a
i
(-1)
i
(s+α)
i
.
Γ(1-s-α)
Γ(1-s)
Γ(1-s)
Γ(1-s-β)
(28)
Proceeding as in 2.1 we have
H(s) =
i=0
M
B
i
(s+β)
i
i=0
N
A
i
(s+α)
i
.
Γ(1-s-α)
Γ(1-s-β)
(29)
So, the transfer function in (29) has two parts; the
first is similar to (25) aside a translation on the pole
and zero positions. Its inverse has the format:
h(t)=
B
M
A
N
δ(t-1)+t
α
i=1
N
k=1
m
i
C
ik
.
(-1)
k-1
[log(t)]
k-1
(k-1)!
t
-p
i
u(t-1)
(30)
where the p
i
, i=1,2, …, N are the poles. We must
remark that it does not depend explicitly on β. The
second factor in (29) leads to a new convolutional
factor needed to compute the complete solution of
(27). So, we have to compute the inverse Mellin
transform of
ICINCO 2009 - 6th International Conference on Informatics in Control, Automation and Robotics
198
H
a
(s) =.
Γ
(1-s-α)
Γ(1-s-β)
(31)
To do it we can always choose an integration path
on the left of all the poles. Computing this integral,
we obtain:
h
a
(t) =
1
Γ(α-β)
t
β
()
t - 1
α-β-1
u(t-1)
(32)
So, the impulse response corresponding to (29) is the
convolution of (30) and (32). By simplicity, assume
that all the poles are simple. In this case, the impulse
response is given by:
h(t) =
B
M
A
N
1
Γ(α-β)
t
β
()
t - 1
α-β-1
u(t-1) +
+ t
α
i=1
N
C
i
.
Γ(1-p
i
)
Γ( α-β-p
i
+1)
t
-p
i
u(t-1)
(33)
Choosing the other region of convergence we have
H(s) =
i=0
M
B
i
(s+β)
i
i=0
N
A
i
(s+α)
i
.(-1)
β-α
Γ(s+β)
Γ( s+α)
(34)
The first factor has as inverse the expression:
h(t) =
B
M
A
N
δ(t-1) +
+ t
α
i=1
N
k=1
m
i
C
ik
.
(-1)
k
[log(t)]
k-1
(k-1)!
t
-p
i
u(1-t)
(35)
For the second we proceed as before. Now the
integration path is in the right half complex plane.
We obtain
h
a
(t) = -
1
Γ(α-β)
t
β
()
t - 1
α-β-1
u(1-t)
(36)
To compute the final impulse response we only have
to convolve the two expressions as we did in the
other case. We obtain, for the simple pole case
h(t) = -
B
M
A
N
1
Γ(α-β)
t
β
()
t - 1
α-β-1
u(1-t) -
t
α
i=1
N
C
i
.
Γ(β-α+p
i
)
Γ (p
i
)
t
-p
i
u(1-t)
(37)
It is interesting to verify that (33) and (37) behavior
like the usual anti-causal and causal systems. When
Re(p
i
) < 0, (30) increases without bound while (35)
decreases. If Re(p
i
) > 0, we verify the reverse
situation. This means that we can use the well
known Routh-Hurwitz test to study the stability of
LSIS.
2.4 Particular Cases
2.4.1 α = β
If α=β, the second terms in (29) and (34) is equal to
1, implying that the complete impulse response is
given by (30) and (35).
2.4.2 α = 0 and β 0
This case is very interesting since it is similar to the
situation treated by Yazici and Kashyap. With α=0,
(30) and (35) do not depend explicitly on β and they
are similar to the integer order case. The dependence
on β appears only in the second therm.
2.4.3 α 0 and β = 0
This situation is more involved, since both terms of
the impulse response depend on α. We can obtain
the general impulse response by putting β=0 into
(30), (32), (35), and (36).
3 THE EIGENFUNCTIONS
AND FREQUENCY RESPONSE
Consider relation (3) and assume that one of the
functions is the impulse response of the system (1)
and the other is a power function t
-σ
, σ∈C. It is not
hard to show that
h(t)٧t
-
σ
= H(σ).t
-σ
(38)
Leading us to conclude that the power function is the
eigenfunction of the LSIS. In particular we can
write:
h(t)٧t
-j
ν
= H(ν).t
-jν
(39)
and H(ν) will be the frequency response of the
system, considering that our “cisoids” have the
format
c(t) = e
-jνlog(t)
(40)
that verify:
c(t) = c(at)
(41)
provided that
a = e
2π
/
ν
(42)
defining the scale periodicity. These results show
that the output of a LSIS to a cisoid is a cisoid. For a
ON THE LINEAR SCALE FRACTIONAL SYSTEMS - An Application of the Fractional Quantum Derivative
199
cosine signal, as input, the output y(t) is given by
y(t) = |H(ν)|.cos[2
π
ν
log(t)+
(ν)]
(43)
where ϕ(ν) is the phase spectrum of the system.
4 CONCLUSIONS
In this paper, we introduced the general formulation
of the linear scale invariant systems through the
fractional Euler-Cauchy equation. To solve this
equation we used the fractional quantum derivative
concept and the help of the Mellin transform. As in
the linear time invariant systems we obtained two
solutions corresponding to the use of two different
regions of convergence. We presented other
interesting features of the LSIS, namely the
frequency response. We made also a brief study of
the stability.
ACKNOWLEDGEMENTS
This work was supported by the Portuguese
Foundation for Science and Technology through the
program FEDER/POSC.
REFERENCES
Abramowitz, M. and Stegun, I. (1972) Stirling Numbers of
the First Kind., §24.1.3 in Handbook of Mathematical
Functions with Formulas, Graphs, and Mathematical
Tables, 9th printing. New York: Dover, p. 824.
Andrews, G.E., Askey, R., and Roy, R. (1999). Special
functions. Cambridge University Press.V.
Ash, J.M., Catoiu, S., and Rios-Collantes-de-Terán, R.,
(2002), “On the nth Quantum Derivative,” J. London
Math. Soc. (2) 66, 114-130.
Bertran, J. Bertran, P., and Ovarlez, J.P., (2000), “The
Mellin Transform”, in “The Transforms and
Applications Handbook”, 2nd ed,, Editor-in-Chief
A.D. Poularikas, CRC Press,.
Braccini, C. and Gambardella, G., (1986), “Form-invariant
linear filtering: Theory and applications,” IEEE Trans.
Acoust., Speech, Signal Processing, vol.ASSP-34, no.
6, pp. 1612–1628,.
Gerardi, F.R., (1959) “Application of Mellin and Hankel
transforms to networks with time varying parameters,”
IRE Trans. Circuit Theory, vol. CT-6, pp. 197–208.
Kac, V. and Cheung, P. (2002) “Quantum Calculus”,
Springer,.
Koornwinder, T.H., (1999) “Some simple applications and
variants of the q-binomial formula,” Informal note,
Universiteit van Amsterdam,.
Ortigueira, M. D., (2006), A coherent approach to non
integer order derivatives, Signal Processing Special
Section: Fractional Calculus Applications in Signals
and Systems, vol. 10, pp. 2505-2515.
Ortigueira, M.D., (2007) “A Non Integer Order Quantum
Derivative”, Symposium on Applied Fractional
Calculus (SAFC07), Industrial Engineering School
(University of Extremadura), Badajoz (Spain),
October 15-17.
Ortigueira, M.D., (2008) “A fractional Quantum
Derivative”, proceedings of the IFAC Fractional
Differentiation and its Applications conference,
Ankara, Turkey, 05 - 07 November.
Yazici, B. and Kashyap, R.L., (1997) “A Class of Second-
Order Stationary Self-Similar Processes for 1/f
Phenomena,” IEEE Transactions on Signal Processing,
vol. 45, no. 2.
APPENDIX - QUANTUM
DERIVATIVE FORMULATIONS
Incremental Ratio Formulation
The normal way of introducing the notion of
derivative is by means of the limit of an incremental
ratio that in the forward case reads
D
h
f(t) = lim
h0
f(t) – f(t-h)
h
(a.1)
By repeated application, this definition leads to the
derivative of any integer order that can be
generalized to any real or complex order by the well
known forward Grünwald-Letnikov fractional
derivative (Ortigueira, 2006):
D
α
h
f(z) = lim
h0+
k = 0
(-1)
k
α
k
f(z - kh)
h
α
(a.2)
An alternative derivative valid only for t>0 or t<0 is
the so-called quantum derivative (Kac and Cheug,
2002). Let Δ
q
be the following incremental ratio:
Δ
q
f(t) =
f(t) – f(qt)
(1 q)t
(a.3)
where q is a positive real number less than 1 and f(t)
is assumed to be a causal type signal. The
corresponding derivative is obtained by computing
the limit as q goes to 1
D
q
f(t) = lim
q1
f(t) – f(qt)
(1 q)t
(a.4)
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This derivative uses values of the variable below t.
We can introduce another one that uses values above
t. It is defined by
D
q
-1
f(t) = lim
q1
f(q
-1
t) – f(t)
( q
-1
1)t
(a.5)
The repeated application of (a.3) followed by the
limit computation leads to the N
th
order derivative
(Ash et al, 2002;Koornwinder, 1999):
D
N
q
f(t) =
lim
q1
j=0
N
N
j
q
(-1)
j
q
j(j+1)/2
q
-jN
f(q
j
t)
(1 q)
N
t
N
(a.6)
where we introduced the q-binomial coefficients
α
i
q
=
[α]
q
!
[i]
q
![α-i]
q
!
(a.7)
with [α]
q
given by
[α]
q
=
1 q
α
1 q
(a.8)
Using the q-binomial theorem (Kac and Cheug,
2002), the Mellin transform, and the Pochhamer
symbol we conclude that:
M
lim
q1
j=0
N
N
j
q
(-1)
j
q
j(j+1)/2
q
-jN
f(q
j
t)
(1 q)
N
t
N
= (1-s)
N
F(s N)
=
Γ(1 s + N)
Γ(1 s)
F(s-N)
(a.9)
The previous results are readily generalised for the
case of a real order, α, (Ortigueira,2007;
Ortigueira,2008) leading to a Grunwald-Letnikov
like fractional quantum derivative:
D
α
q
f(t) =
lim
q1
j=0
α
j
q
(-1)
j
q
j(j+1)/2
q
-jα
f(q
j
t)
(1 q)
α
t
α
(a.10)
that is similar to the one proposed by Salam (1966).
In (a.10) the fractional q-binomial coefficients are
given by
α
j
q
=
[]
1 q
α
j
q
[j]
q
(a.11)
The Mellin transform of (a.10) reads
M [D
α
q
f(t) ] =
Γ
(1
s + α)
Γ(1 s)
F(s-α)
(a.12)
valid for Re(s) < min(0,α)+1. This relation allows us
to obtain an integral representation of the fractional
quantum derivative, as we will see later. As referred
before, in (a.10) we are using values of the variable
less than t. In the following we will consider the
other case. The repeated application of (a.5) leads to
the N
th
order derivative:
D
N
q
-1
f(t) =
lim
q1
j=0
N
N
j
q
(-1)
j
q
j(j-1)/2
f(q
-j
t)
(1 q
-1
)
N
t
N
(a.13)
The Mellin transform gives:
M [D
N
q
-1
f(t) ] = (1-s)
N
F(s-N)
(a.14)
that coincides with (a.9) as expected. To generalize
the above results for any order, we substitute α for N
in the above expressions. We have from (a.10):
D
α
q
-1
f(t) =
lim
q1
j=0
α
j
q
(-1)
j
q
j(j-1)/2
f(q
-j
t)
(1 q
-1
)
α
t
α
(a.15)
and finally
M [D
α
q
-1
f(t) ] = (-1)
α
.
Γ(s)
Γ(s α)
F(s-α)
(a.16)
valid for Re(s) > max(0,α). Remark the difference
relatively to (a.12) mainly in the region of
convergence.
Integral Formulations
The two Mellin transforms in (a.12) and (a.16) lead
to different integral representation of fractional
derivatives by computing the corresponding inverse
ON THE LINEAR SCALE FRACTIONAL SYSTEMS - An Application of the Fractional Quantum Derivative
201
functions.
The inverse h
b
(t) of
Γ(s)
Γ(s α)
is obtained from
(Andrews et al,1999):
Γ(s)Γ(-α)
Γ(s α)
=
0
1
τ
s-1
(1- τ)
-α-1
dτ
(a.17)
Provided that Re(s)>0 and Re(α)<0. This leads
immediately to
h
b
(t) =
(-1)
α
Γ(-α)
()
1 - t
-α-1
u(1-t)
(a.18)
u(t) is the Heaviside unit step. A similar procedure
to obtain the inverse h
a
(t) of
Γ(1 s + α)
Γ(1 s)
gives
Γ(1 s + α)
Γ
(-α)
Γ(1 s)
=
0
1
τ
1-s+α
(1- τ)
-α-1
dτ
(a.19)
With a variable change inside the integral, we
obtain:
h
a
(t) =
1
Γ(-α)
()
t - 1
-α-1
u(t-1)
(a.20)
To compute in integral formulations of the
derivatives corresponding to (a.12) and (a.16) we
remark that the inverse Mellin transform of F(s-α) is
given by:
M
-1
[F(s-
α
)] = t
-α
f(t)
(a.21)
and use the convolution (3). With (a.12) and (a.16)
we obtain the following integral formulations, valid
for Re(α) < 0.
D
α
b
f(t) =
t
-α
Γ(-α)
.
0
1
f(t/τ) (1 τ
-1
)
-α-1
dτ
(a.22)
and
D
α
a
f(t) =
t
-α
Γ(-α)
1
f(t/τ) (τ
-1
1)
-α-1
dτ
(a.23)
signals. Although we obtained these results for α<0,
they remain valid for other values of α, since
Γ(s)
Γ(s α)
and
Γ(1 s + α)
Γ(1 s)
are analytic in the regions
of convergence and we can fix an integration path
independent of α. This can be confirmed by
expanding (a.22) and (a.23) and transforming each
term of the series.
ICINCO 2009 - 6th International Conference on Informatics in Control, Automation and Robotics
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