Fractional Calculus for Scientists and Engineers pp 123-144 | Cite as

# The Fractional Quantum Derivative and the Fractional Linear Scale Invariant Systems

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## Abstract

The normal way of introducing the notion of derivative is by means of the limit of an incremental ratio that can assume three forms, depending the used translations as we saw in Chaps. 1 and 4. On the other hand, in those derivatives the limit operation is done over a set of points uniformly spaced: a linear scale was used. Here we present an alternative derivative, that is valid only for *t* > 0 or *t* < 0 and uses an exponential scale

## Keywords

Impulse Response Fractional Derivative Fractional Order System Integer Order Incremental Ratio## 6.1 Introduction

*t*> 0 or

*t*< 0 and uses an exponential scale. We are going to introduce the so-called Quantum Derivative [1, 2]. We proceed as in Chap. 2. Let. \( \Updelta_{q} \) be the following incremental ratio:

*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 (to be more precise, we should state

*q*?1

^{?})

*t*. We can introduce another one that uses values above

*t*. It is defined by

We will generalize these derivatives, first for integer orders, and afterwards for real ones as we did before. We will present the two formulations that come naturally from (6.2) and (6.3) and using values below and above the independent variable. We can define also “two-sided” derivatives as we did in Chap. 5, but we will not do it here.

From the Mellin transform of both derivatives we will obtain two integral formulae similar to the Liouville derivatives presented in Chap. 2. Although we will not study here the properties of these derivatives, it may be advanced that they can be used in scale variation problems and to deal with systems defined by Euler–Cauchy type differential equations as we will see later. For now, we will present the steps leading to the fractional quantum derivative and its relation with the Mellin transform (MT).

## 6.2 The Summation Formulations

### 6.2.1 The “Below *t*” Case

*R*

^{+}. We introduce the multiplicative convolution defined by

*multiplicative convolution*is equal to the product of the transforms of both functions. So we obtain:

*q*-binomial formula [1]

*q*-binomial coefficients

*u*(

*t*) is the Heaviside unit step. We have:

The right hand side in (6.18) or (6.19) is the well known Mellin transform of the *N*th order derivative. The left side is a new way of expressing such derivative. This expression suggests that we may work with the “derivative”\( t^{n} D_{q}^{N} f(t) \), also called *scale derivative*.

*N*in the left hand expression. In the numerator we obtain the fractional

*q*-binomial \( [1 - q^{s + 1} ]_{q}^{\alpha } \). The generalised Gauss binomial formula [1]

*q*-binomial coefficients are given by

*q*-gamma function by

*t*) > 0. With this function,

*q*-binomial in (6.20) is given by

To maintain the coherence we will consider (6.18) as the correct solution in the integer order case, for the “below *t*” situation.

### 6.2.2 The “Above *t*” Case

*t*. We proceed as in the last section. Let \( \Updelta_{{q^{ - 1} }} \) be the following incremental ratio:

*q*-binomial formula leading to

*N*in the above expressions. We have from (6.35):

*q*-binomial theorem, we have:

## 6.3 Integral Formulations

The two Mellin transforms in (6.29) and (6.42) lead to different integral representation of fractional derivatives by computing the corresponding inverse functions. To do it, we will use well known results of the Beta function. To start, we are going to obtain the inverse \( h_{b} \left( t \right) \) of \( {\frac{\Upgamma ( - s)}{\Upgamma ( - s - \alpha )}} \).

## 6.4 On the Fractional Linear Scale Invariant Systems

### 6.4.1 Introduction

Braccini and Gambardella [8] introduced the concept of “form-invariant” filters. These are systems such that a scaling of the input gives rise to the same 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 (LScIS) were really introduced by Yazici and Kashyap [9, 10] 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 LScIS. These are described by fractional Euler–Cauchy equations. The fractional quantum derivatives are suitable for dealing with these systems. The use of the Mellin transform allowed us to define the multiplicative convolution and, from it, it is shown that the power function is the eigenfunction of the LScIS and the eigenvalue is the transfer function.

The computation of the impulse response from the transfer function is done following a procedure very similar to the used in the shift-invariant systems in Chap. 4. We will follow a two step procedure. In the first we solve a particular case with integer differentiation orders. Later we solve for the fractional case.

### 6.4.2 The General Formulation

*t*” (analogue to anti-causal) and “above

*t*” (analogue to causal) derivatives. If

*t*were a time, we would talk on anti-causal and causal. We saw that working in the context of the Mellin transform we obtain two different regions of convergence: left and right relatively to a vertical straight line. This is not needed when dealing with integer order systems because we only have one Mellin transform for \( t^{n} f^{(n)} \left( t \right) \) if

*n*is integer. We rewrite here the two fractional quantum derivatives we are going to use

### 6.4.3 The Eigenfunctions and Frequency Response

*t*= 1 and not

*t*= 0, as it is the case of the shift-invariant systems. \( \delta \left( {t - 1} \right) \) is the inverse of \( \Updelta \left( s \right) \, = \, 1 \). On the other hand, using the derivative definitions presented above, it is easy to show that:

*h*(

*t*) be the impulse response of the system,

*x*(

*t*). We conclude immediately that

*x*(

*t*) =

*g*(

*at*). It is a simple task to show that the output is

*y*(

*at*) showing that the system is really scale invariant.

## 6.5 Impulse Response Computations

### 6.5.1 The Uniform Orders Case

### 6.5.2 The Integer Order System with \( \alpha = \beta = \, 0 \)

*x*(

*t*) is the input,

*y*(

*t*) the output, and

*N*and

*M*are positive integers \( (M \le N) \). Usually

*a*

_{ N }is chosen to be 1. We will assume that this equation is valid for every \( t \in R^{ + } \). The system defined by (6.62) with

*M*= 0 was already studied [see 9, 10]. However, it is interesting to repeat the computations here to acquire some background into the general case.

*s*. To do it we use the well known relation [11]

*v*(,) represent the Stirling numbers of first kind that verify the recursion

for \( 1 \le m \le n \) and with

\( v(n,0) = \delta_{n} \) and \( v(n,1) = ( - 1)^{n - 1} (n - 1)! \)

*A*

_{ i }coefficients given by

*s*. In the integer order case, it is indifferent which derivative we use, because they lead to the same result (6.17). This is a consequence of two facts:

*H*(

*s*) has the following partial fraction decomposition

*M*=

*N*and its inversion gives a delta at

*t*= 1:

*w*(

*t*) is equal to \( u\left( {1 - t} \right) \) or to \( u\left( {t - 1} \right) \) in agreement with the adopted the region of convergence. By successive derivation in order to

*p*we obtain the solution for higher order poles

To compute the output to any function *x*(*t*) we only have to use the multiplicative convolution. As in the shift-invariant systems, we have several ways of choosing the region of convergence. We can have all right signals, all left signals or, mixed right and left signals. In [9, 10] the first term does not appear, since only the all-pole case was discussed.

It is interesting to make here an important remark. Verify that (6.79) behaviours like the usual responses of the anti-causal and causal systems. When \( \text{Re} \left( {p_{i} } \right) \, > \, 0 \) and \( t \, > \, 1, \) it increases without bound as \( t \to \infty \), while it decreases as \( t \to 0. \, \)if \( \text{Re} \left( {p_{i} } \right) \, < \, 0, \) (6.79) increases without bound as \( t \to 0 \), while it decreases as \( t \to \infty \). This means that we can use the well known Routh–Hurwitz test to study the stability of LScIS.

### 6.5.3 The Fractional Order System

*n*a non negative integer. To invert it we can always choose an integration path on the right of all the poles similarly to the path shown in Fig. 6.1, but with the most left segment infinitely far. The residues are given by

*N*partial fractions and invert

*N*transforms with the format \( {\frac{{\Upgamma \left( {1 + s - \alpha } \right)}}{{\left( {s - \alpha - p} \right)\Upgamma \left( {1 + s - \beta } \right)}}} \). By simplicity, we assumed that all the poles are simple. We proceed as above to compute the residues. Collecting them the impulse response is given by

We must remark that the above results are valid even if \( \alpha \) and \( \beta \) are positive integers. Of course, we could obtain other solutions by choosing other integration paths such that there were poles on the left and on the right of it. In these cases we would obtain “two-sided” responses. It is interesting to remark that:

If\( \alpha = \beta \), the second terms in (6.82) and (6.87) are equal to 1, implying that the complete impulse response is given by (6.83).

When \( \alpha = 0 \) and \( \beta \ne 0 \) in (6.86) we obtain a situation very similar to the one treated by Yazici and Kashyap [9, 10].

If \( \alpha \, = \, \beta + 1 \), (6.85) and (6.88) become merely power functions and so self-similar.

### 6.5.4 A Simple Example

*w*(

*t*) is equal to ?

*u*(1?

*t*) or to

*u*(

*t*?1), in agreement with the adopted region of convergence. The analogue shift invariant corresponding system

As seen, we made a substitution \( t \to e^{t} .\)

### 6.5.5 Additional Comments

The impulse responses stated (6.86) and (6.89) depend directly on the differential equation (6.62) not on the way we followed to obtain them. This means that we are not obliged to use the quantum derivative. In fact we could also use another derivative like Grunwald–Letnikov, Riemann–Liouville or Caputo, but it would be very difficult to arrive at the results we obtained. The quantum derivative allows us to obtain such impulse responses more easily. On the other hand, those derivatives are suitable for dealing with shift-invariant systems defined over *R*, not \( R^{ + } .\)

In the integer order case, we can switch from the LScIS to the corresponding linear shift-invariant systems: we only have to perform a logarithmic transformation. However, this is not evident neither correct in the fractional case, due to the first term (6.86) and (6.89), as the example presented above shows. This fact may come from the difficulty in defining fractional derivative of a composite function. The lack of emphasis on this fact is due to the desire of presenting a linear system that exists by itself and not because can be the transformation of another one. It is more or less the same situation that we find when introducing difference equations. They exist and do not need to be presented as transformations of the ordinary differential equations (with bilinear or other mapping). It is curious to refer that we can obtain the corresponding shift invariant system, by considering that the transfer function in (6.62) is now a transfer function of a shift invariant system and use the Laplace transform to go back into a new differential equation.

The LScIS, being scale invariant, but not shift invariant, can be useful in detection problems and in image processing. Their conjunction with the Wavelet transform can be interesting [9, 10].

## 6.6 Conclusions

We presented the quantum fractional derivative as an alternative to the common Grünwald–Letnikov and Liouville derivatives. It was described in two formulations: summation and integral. Its Mellin transform was also presented and use to establish the relation between the two formulations. The summation formulations are similar to the Grünwald–Letnikov fractional derivatives. The main difference lies in the use of an exponential scale for the independent variable. The Grünwald–Letnikov derivatives use a linear scale. The integral formulations are similar to the Liouville derivatives. This derivative is useful to solve fractional Euler–Cauchy differential equations and can be useful in dealing with scale problems.

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 LScIS, namely the frequency response. We made also a brief study of the stability.

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