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

(

)

(t) =

∑

i=0

. t

(

)

(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 i=0, 1, 2, …, N

and

= β+i i=0, 1, 2, …, N

we obtain a simpler equation

∑

i=0

α+i

(

α+i)

(t) =

∑

i=0

. t

β+i

(

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) =

⌡

⌠

∞

f(t/u)g(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

2 THE EULER-CAUCHY

EQUATION

2.1 The Integer Order Case

Consider a linear system represented by the

differential equation

∑

i=0

(i)

(t) =

∑

i=0

. t

(i)

(t)

(4)

where x(t) is the input, y(t) the output, and N and M

are positive integers (M≤N). Usually a

is chosen to

be 1. We will assume that this equation is valid for

every t∈R

. Applying the Mellin transform to both

sides of (3) we obtain (Gerardi, 1959;Bertran et al,

2000)

∑

i=0

(-1)

(s)

Y(s) =

∑

i=0

. (-1)

(s)

X(s),

(5)

from where we obtain a transfer function

H(s) =

Y(s)

X(s)

∑

i=0

(-1)

(s)

∑

i=0

(-1)

(s)

(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)

∑

i=0

(-1)

k-i

s(k,i) x

(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

and s(n,1) = (-1)

n-1

(n-1)!

With some manipulation, we obtain:

∑

i=0

(-1)

(x)

∑

i=0

∑

k=i

(-1)

s(k,i) x

∑

i=0

(9)

with the A

coefficients given by

∑

i=k

(-1)

s(k,i)

(10)

or in a matricial format

A = S.a

(11)

where

A = [A

… … A

]

(12)

S=[ s(i,j), i,j=0,1, …,N] (13)

and

a = [a

… … a

]

(14)

With this formulation, the transfer function is given

by:

H(s) =

∑

i=0

∑

i=0

M≤N

(15)

that is the quotient of two polynomials in s. In

general H(s) has the following partial fraction

decomposition

H(s) =

∑

i=1

∑

j=1

(s-p

)

(16)

The constant term only exists when M=N and its

inversion gives a delta at t=1:

-1

[

] =

δ(t-1)

(17)

For inversion of a given partial fraction, we must fix

the region of convergence Re(s) > Re(p

) or Re(s) <

Re(p

) 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

-1

[

(s-p)

] = u(1-t).t

-p

(18)

and

-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

-1

[

(s-p)

] = u(1-t).

(-1)

k-1

[log(t)]

k-1

(k-1)!

-p

(20)

and

-1

[

(s-p)

] =u(t-1).

(-1)

k-1

[log(t)]

k-1

(k-1)!

-p

(21)

We conclude that the response corresponding to an

input δ(t-1) is given by:

h(t)=

δ(t-1)+

∑

i=1

∑

k=1

(-1)

k-1

[log(t)]

k-1

(k-1)!

-p

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)

(t). This is not the situation in the

fractional case. In fact we have two fractional

derivatives given by {see appendix}:

f(t) = lim

q→1

∑

j=0

∞

⎣

⎡

⎦

⎤

(-1)

j(j+1)/2

-jα

f(q

(1 − q)

(23)

and

-1

f(t) = lim

q→1

∑

j=0

∞

⎣

⎡

⎦

⎤

(-1)

j(j-1)/2

f(q

-j

(1 − q

-1

)

(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

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

-1

f(t) ] = (-1)

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

α+i

(

α+i)

(t) =

∑

i=0

. t

β+i

(

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

(-1)

(s+β)

∑

i=0

(-1)

(s+α)

Γ(1-s-α)

Γ(1-s)

Γ(1-s-β)

(28)

Proceeding as in 2.1 we have

H(s) =

∑

i=0

(s+β)

∑

i=0

(s+α)

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)=

δ(t-1)+t

∑

i=1

∑

k=1

(-1)

k-1

[log(t)]

k-1

(k-1)!

-p

u(t-1)

(30)

where the p

, 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

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198

(s) =.

(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:

(t) =

Γ(α-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) =

Γ(α-β)

()

t - 1

α-β-1

u(t-1) +

+ t

∑

i=1

Γ(1-p

)

Γ( α-β-p

+1)

-p

u(t-1)

(33)

Choosing the other region of convergence we have

H(s) =

∑

i=0

(s+β)

∑

i=0

(s+α)

.(-1)

β-α

Γ(s+β)

Γ( s+α)

(34)

The first factor has as inverse the expression:

h(t) =

δ(t-1) +

+ t

∑

i=1

∑

k=1

(-1)

[log(t)]

k-1

(k-1)!

-p

u(1-t)

(35)

For the second we proceed as before. Now the

integration path is in the right half complex plane.

We obtain

(t) = -

Γ(α-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) = -

Γ(α-β)

()

t - 1

α-β-1

u(1-t) -

∑

i=1

Γ(β-α+p

)

Γ (p

)

-p

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

) < 0, (30) increases without bound while (35)

decreases. If Re(p

) > 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

= 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

= 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

(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

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

f(t) = lim

h→0

f(t) – f(t-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):

f(z) = lim

h→0+

∑

k = 0

∞

(-1)

⎝

⎛

⎠

⎞

f(z - kh)

(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

be the following incremental ratio:

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

f(t) = lim

q→1

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

-1

f(t) = lim

q→1

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

order derivative

(Ash et al, 2002;Koornwinder, 1999):

f(t) =

lim

q→1

∑

j=0

⎣

⎡

⎦

⎤

(-1)

j(j+1)/2

-jN

f(q

(1 − q)

(a.6)

where we introduced the q-binomial coefficients

⎣

⎡

⎦

⎤

[α]

[i]

![α-i]

(a.7)

with [α]

given by

[α]

1 − q

(a.8)

Using the q-binomial theorem (Kac and Cheug,

2002), the Mellin transform, and the Pochhamer

symbol we conclude that:

⎣

⎢

⎡

⎦

⎥

⎤

lim

q→1

∑

j=0

⎣

⎡

⎦

⎤

(-1)

j(j+1)/2

-jN

f(q

(1 − q)

= (1-s)

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:

f(t) =

lim

q→1

∑

j=0

∞

⎣

⎡

⎦

⎤

(-1)

j(j+1)/2

-jα

f(q

(1 − q)

(a.10)

that is similar to the one proposed by Salam (1966).

In (a.10) the fractional q-binomial coefficients are

given by

⎣

⎡

⎦

⎤

[]

1 − q

[j]

(a.11)

The Mellin transform of (a.10) reads

M [D

f(t) ] =

−

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

order derivative:

-1

f(t) =

lim

q→1

∑

j=0

⎣

⎡

⎦

⎤

(-1)

j(j-1)/2

f(q

-j

(1 − q

-1

)

(a.13)

The Mellin transform gives:

M [D

-1

f(t) ] = (1-s)

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):

-1

f(t) =

lim

q→1

∑

j=0

∞

⎣

⎡

⎦

⎤

(-1)

j(j-1)/2

f(q

-j

(1 − q

-1

)

(a.15)

and finally

M [D

-1

f(t) ] = (-1)

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

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functions.

The inverse h

(t) of

Γ(s)

Γ(s − α)

is obtained from

(Andrews et al,1999):

Γ(s)Γ(-α)

Γ(s − α)

⌡

⌠

s-1

(1- τ)

-α-1

dτ

(a.17)

Provided that Re(s)>0 and Re(α)<0. This leads

immediately to

(t) =

(-1)

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

(t) of

Γ(1 − s + α)

Γ(1 − s)

gives

Γ(1 − s + α)

(-α)

Γ(1 − s)

⌡

⌠

1-s+α

(1- τ)

-α-1

dτ

(a.19)

With a variable change inside the integral, we

obtain:

(t) =

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:

-1

[F(s-

)] = 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.

f(t) = −

-α

Γ(-α)

⌡

⌠

f(t/τ) (1 − τ

-1

)

-α-1

dτ

(a.22)

and

f(t) =

-α

Γ(-α)

⌡

⌠

∞

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.

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