An Operational Approach to Fractional Scale-Invariant Linear Systems

: The fractional scale-invariant systems are introduced and studied, using an operational formalism. It is shown that the impulse and step responses of such systems belong to the vector space generated by some special functions here introduced. For these functions, the fractional scale derivative is a decremental index operator, allowing the construction of an algebraic framework that enables to compute the impulse and step responses of such systems. The effectiveness and accuracy of the method are demonstrated through various numerical simulations


Introduction
The mathematical framework often used to define systems is based on shift-invariant derivatives [1,2], resulting from the works of Leibniz, Euler, Lagrange, and Liouville.However, a different concept was introduced by C. Braccini and G. Gambardella: the form-invariant linear filtering.It was a new kind of processing that was applied to several different fields such as optical pattern recognition, image restoration and reconstruction from projections [3].This was the first step into the introduction of the scale-invariant linear systems, really done by B. Yazici and R. L. Kashyap for analysis and modelling 1/f phenomena and in general the self-similar processes, namely the scale stationary processes [4,5].In parallel, physicists started studying the importance of scale in several physical systems [6][7][8][9][10][11][12][13].Although the concept of scale is not very well defined, since it is a parameter expressing relative relations [6], the concept of scale-invariant system is well defined.While the shift-invariant systems are related and use in their definition the usual convolution [14] (D'Alembert's) and corresponding derivatives, the scale-invariant systems are based on the Mellin's convolution [15].To fully define these systems the fractional scale-derivatives were introduced and studied [16], generalizing the classic Hadamard definitions [17].These derivatives allow the formalization of linear scale-invariant systems of the autoregressive-moving average (ARMA) type [16,18].With the use of the Mellin transform, the transfer functions of these systems assume a form identical to the one got from the shift-invariant systems through the use of the Laplace transform [19].
The objective of this paper is the study and search for the impulse and step responses of these systems.For this purpose, we first find a sequence of functions for which the fractional scale derivative behaves as a decremental index operator and then develop an algebraic framework similar to the one we used in the shift-invariant systems [20].This approach allows us to closed forms for both the impulse and step responses of the systems.
The paper is organized as follows.Section 2 contains some preliminary results [16].In Section 3 the sequence of functions {u k (τ)} k for which Hadamard right (left) derivative and fractional scale derivatives of type Grünwald-Letnikov are decremental index operators is founded.Section 3.2 contains the algebraic framework needed to solve fractional scaleinvariant systems.In Section 4 we present an operational method based on algebraic framework introduced in Section 3. Numerical examples are solved in Section 5. Finally, Section 6 contains the main conclusions.

The Mellin Convolution
Definition 1.We call a linear system scale-invariant or dilation-invariant (DI) if its input-output relation is given by the Mellin convolution [16] where τ ∈ R + , and g(τ) is the impulse response: the response to x(τ) = δ(τ − 1).
We demand that the impulse response, g(t), be at least with bounded variation.
Similarly to the shift-invariant case, where the exponentials are the eigenfunctions, the powers where G(v) is the transfer function given by which is a modifed version of the Mellin transform (MT) of the impulse response.The Mellin transform in (1), denoted by M[g(τ)](v), has a parameter sign change −v → v relatively to the usual Mellin transform [15,21,22].
where F(v) and G(v) are the Mellin transforms of f and g, respectively.The inverse Mellin transform related to (1) is where γ is vertical straight line in the ROC of the transform.

Scale-Derivatives
Definition 2. Let α ∈ R. We define the α-order scale derivative (SD) as the operator D s obeying the rule [16, If a function x(τ) has Mellin transform X(v), then it has fractional scale-derivative that is given by for a suitable ROC.The way how we express v α imposes a form for the derivative.To start, we notice that we can consider two situations corresponding to the sign of Re(v).If it is positive, we obtain stretching derivatives, while if it is negative, we obtain the shrinking one.For both, we can express the inverse Mellin transform of v α X(v) in two different forms: summation or integral.
2.2.1.Stretching Derivatives: Re(v) > 0 [16] Let ε(τ) be Heaviside step function.We have two ways of expressing v α : These expressions lead to the following scale derivative [16]: (the last expression is valid for any real order, provided that we assume the summation to be null for N ≤ 0).These relations can alternatively be expressed by the Hadamard derivatives: 1.
Hadamard right derivative [17,24] Hadamard-Liouville right derivative [16] These derivatives are equivalent from the Mellin transform point of view, although not from numerical aspects.2.2.2.Shrinking Derivatives: Re(v) < 0 [16] We have again two ways of expressing v α : These expressions lead to the following scale derivative [16]: that can alternatively be expressed by the Hadamard derivatives 1.

ARMA Type Systems
Definition 3. We define the dilation (scale)-invariant fractional autoregressive-moving average (DI-FARMA) system through where The results in [25] can be easily adapted.As the power τ v is the eigenfunction of ( 7), we obtain easily the transfer function which is a bit difficult to manipulate [26].In the following, we shall be considering the so-called "commensurate" systems described by differential equations with the format The corresponding transfer function is , where we assume that M 0 < N 0 , for simplicity.The objective of our work is to find the impulse and step responses of systems.As expected, we have two cases in agreement with the assumed ROC.

The Algebraic Framework for
We say that S is decremental index operator of the sequence of functions {u k (τ)} k∈Z .
Remark 1.A well known example involves the generalized shift-invariant derivatives, D, and the power functions, stating It has been extended and studied by several researchers [27][28][29].
The main goal in this section is to find sequence of functions {u k (τ)} k∈Z verifying (8) for the scale derivatives, mainly the Hadamard's.

Hadamard Right (Left) Derivative
In the following we shall be addressing the stretching derivative case (Re(v) > 0); the case of the shrinking derivative (Re(v) < 0) is analogous.We remember the result shown in Section 2.2.1 which will be used in this subsection.
Remark 2. For the next result, we need to observe that [30] u Proof.From Remark 2 and ( 5) we obtain Therefore The previous Lemma tells us that D α s+ u 0 (τ) = u −1 (τ).So, we wonder what about D α s+ u −1 (τ).For this, from the additivity of operator D α s+ we have that The penultimate equality follows from Lemma 1 with the order of the derivative equal to 2α.Following this reasoning we obtain that Therefore Hadamard right (left) derivative is a decremental index operator on the sequence of functions (10).
Remark 3. The condition Re(v) < 0 leads to the 0 < τ ≤ 1 case.The construction of the u k (τ)'s is similar to case 1 ≤ τ.Here we get that It is not difficult to verify that Remark 4. As we previously mentioned the Hadamard right (left) derivative and scale derivative of type Grünwald-Letnikov are equivalent.Despite this, in Appendix A we verify that the scale derivative of type Grünwald-Letnikov is a decremental index operators on the same sequence of functions (10).
In some situations, it is more convenient to use the step response instead of the impulse response because this one has a singularity at τ = 1.Therefore, another definition of u k (τ) for which both Hadamard right (left) derivative and fractional scale derivative of type Grünwald-Letnikov are decremental index operators is possible and desired.For 1 ≤ τ The justification can be seen in Appendix B.

The Framework
and Therefore, we extend this product to any k, m ∈ Z as follows Remark 5. Observe that Consider {u k (τ)} k∈Z as a sequence of basic functions.Let F be the set of all the formal Laurent series Several properties of F and the product (15) can be founded in [20,31].Observe that This product is associative and commutative.Under this multiplication F is a field.

Definition 5. Let us define the function
that we will call α-log-exponential function.
We could define an analogue to the Mittag-Leffler function [16], but this one is more useful.
Let us introduce also the sequential convolution and in general, The α-log-exponential function has some useful properties [31]: 1.
Derivative on a parameter where D γ means usual derivative with respect to γ.

Convolution of two different γ parameters α-log-exponential functions
For Next, we will introduce and prove some other properties which will allow us to deduce that functions E γ,m (τ) are the generating elements of solution space of fractional scale-invariant systems.
Using the basic recurrence Remark 6. Observe that letting n = 0 in Theorem 2 we get Theorem 3. Let N and n be positive integers such that N ≤ n.Then we have Proof.The proof is done by induction with N ≤ n.For the base step of the induction (N = 1), we appeal to Lemma 2. For N = 2, due to Suppose that theorem is valid for N − 1.This is Now, we will prove that theorem is valid for N. Observe that Proof.The proof is by induction on l ∈ N.For the base step of the induction (l = 1) we have that In the penultimate equality we apply the Remark 6 to the first term of the sum.Suppose that the theorem is valid for l − 1, this is Now, we will prove that theorem is valid for l.Observe that Again, Remark 6 is applied in the last equality.
Theorem 4 suggests that impulse (step) response to a fractional scale-invariant system is a linear combination of functions E γ,n (t).

The AR Case
Consider an AR system.Its impulse response is given by the solution of the equation where and its roots γ 1 , γ 2 , . . ., γ m have multiplicity 1.Consider that y(τ) ∈ F .In terms of the convolution , the Equation ( 22) can be rewritten as Assume that impulse response is given by a linear combination of α-log-exponential functions From (19), it is not difficult to deduce that for any m ∈ N.
A simple computation leads to Now, in order to find the impulse response we obtain a linear system with m equations which can be reduced recursively to The coefficient matrix is invertible, since it has Vandermonde format and the roots γ j are distinct.Therefore, the system has a unique solution.Finally, using the c i 's, we obtain r δ (τ), the solution to fractional system (22).
For the case of roots with multiplicity greater than one, we must propose an alternative solution.Consider that a given root, γ k , has multiplicity m k > 1.To obtain the solution, we need to add another linear combination of α-log-exponentials to the previous solution With the obtained guess of the solution we are led to a linear system with m equations.The coefficient matrix is of generalized Vandermonde-type [32] having non null determinant.To fix ideas, consider the system (22) but with γ 4 , γ 5 , . . ., γ m simple roots of p(x) and γ 1 a root with multiplicity 3 (γ 1 = γ 2 = γ 3 ).The proposed solution assumes the form As in the previous case, from we obtain a system of m linear equations that can be recursively reduced to the following system This system has a coefficient matrix of generalized Vandermonde-type whose determinant is equal to ∏ 3<i<j≤m (γ i − γ j ) m i m j , where m i , m j are the multiplicities of the roots γ i , γ j , respectively.It follows that the system has a unique solution.
We have proven that the solution to system ( 22) is an element of vector space generated by the set {E γ,n (τ Furthermore, in (p.338, [31]) it is proved that the set ( 24) is linearly independent.

The ARMA Case
By means of the method above described, we can solve the more general problem are polynomials of degree m and r, respectively, with constant coefficients in C. We procedure as follows.Firstly, we compute the impulsive response r δ (τ).In different words, we solve the fractional system Later, the solution is given by the convolution  For 0 < τ ≤ 1, the impulse response is and step response and The Figures 3 and 4 show the graphical representation of the solutions with 0 < τ ≤ 1 and several values of α.

• Mellin transform:
We only consider the case τ ≥ 1, the case 0 < τ ≤ 1 is analog.By means of Mellin transform, the system (26) can be transformed to equation Using the geometric series, with √ 2 < v α , we obtain that Computing the inverse Mellin transform we obtain the impulsive response For the step response we apply (A1) and obtain  Example 2. Consider the fractional scale-invariant linear system p(D α s+ )y(τ) = q(D α s+ )δ(τ − 1), with p(x) = x 2 + 2 and q(x) = x − √ 2. Observe that system is related with the Example 1.Following (25) we only need to compute the convolution Simplifying we obtain that For τ ≥ 1, the impulse response is and step response The Figures 5 and 6 show the graphical representation of the solutions with τ ≥ 1 and several values of α.
For 0 < τ ≤ 1, the impulse response is and step response  The Figures 7 and 8 show the graphical representation of the solutions with 0 < τ ≤ 1 and several values of α.

Conclusions
We have proved that both Hadamard right (left) derivative and fractional scale derivative of type Grünwald-Letnikov are index operators on the same sequence of functions {u k } k∈Z .The Mellin convolution suggested to us how to define an algebraic product, which allowed us to construct a simple mathematical method to resolve fractional scale-invariant systems.The method relies on the roots of characteristic polynomial and the resolution of a linear system of equations.As expected, in our simulations, there is no convergence issue.
Using the Mellin transform and its inverse the previous relation can be rewritten as n! q −nv ln α (q) M ln kα−1 (τ) Γ(kα) τ v dv.
By binomial theorem and properties of limits it is not difficult to verify that It follows that The last equal follows from (3).By properties of delta function, δ(τq −n − 1) = q n δ(τ − q n ).It follows that By Mellin transform we get that δ(τ − q n ) = 1 2πi γ q n(v−1) τ v dv.So n! ln −α (q)q nv τ v dv.

Solving DI-FARMA Systems 3 . 1 .Definition 4 .
Sequence of Basic Functions for Fractional Scale Derivatives Let S be an operator and {u k (τ)} k∈Z a sequence of functions, verifying By means of Cauchy series product, we can extend the Mellin product, , to any elements of F as follows.Let a j , b j , two sequences of real numbers and define a = j (τ), as elements of F .Then

Figure 1 .
Figure 1.Impulse response of Example 1 with 1 ≤ τ and several values of α.

Figure 2 .
Figure 2. Step response of Example 1 with 1 ≤ τ and several values of α.

Figure 5 .
Figure 5. Impulse response of Example 2 with 1 ≤ τ and several values of α.

Figure 6 .
Figure 6.Step response of Example 2 with 1 ≤ τ and several values of α.