New discrete-time fractional derivatives based on the bilinear transformation: Definitions and properties

Research Article

New discrete-time fractional derivatives based on the bilinear transformation: Definitions and properties

Manuel D. Ortigueira

^a,⇑

, J.A. Tenreiro Machado

^b

aCTS–UNINOVA and NOVA Faculty of Sciences and Technology of Nova University of Lisbon, Campus da FCT da UNL, Quinta da Torre, 2829 – 516 Caparica, Portugal

bInstitute of Engineering, Polytechnic of Porto, Dept. of Electrical Engineering, Porto, Portugal

h i g h l i g h t s

The paper introduces new discrete- time derivative concepts based on the bilinear transformation.

Forward and backward derivatives having a high degree of similarity with the usual continuous-time Grunwald-Letnikov derivatives are introduced.

Corresponding linear discrete-time systems are defined.

g r a p h i c a l a b s t r a c t

a r t i c l e i n f o

Article history:

Received 18 December 2019 Revised 12 February 2020 Accepted 16 February 2020 Available online 25 February 2020

Keywords:

Discrete-time Fractional derivative Time scale

bilinear transformation

a b s t r a c t

In this paper we introduce new discrete-time derivative concepts based on the bilinear (Tustin) transfor- mation. From the new formulation, we obtain derivatives that exhibit a high degree of similarity with the continuous-time Grünwald-Letnikov derivatives. Their properties are described highlighting one impor- tant feature, namely that such derivatives have always long memory.

Ó2020 The Authors. Published by Elsevier B.V. on behalf of Cairo University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Introduction

The continuous/discrete unification introduced by Hilger[3]led to the definition of two discrete-time fractional derivatives, nabla and delta, that are essentially the usual incremental ratia. In[11]

the fractional versions of such derivatives were proposed together with the corresponding differential equations for discrete-time lin-

https://doi.org/10.1016/j.jare.2020.02.011

2090-1232/Ó2020 The Authors. Published by Elsevier B.V. on behalf of Cairo University.

This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Peer review under responsibility of Cairo University.

⇑ Corresponding author.

E-mail addresses: [email protected] (M.D. Ortigueira), [email protected] (J.A.

Tenreiro Machado).

Contents lists available atScienceDirect

Journal of Advanced Research

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / j a r e

ear systems. These versions have stability domains that are defined by the Hilger circles[11]. Such domains do not coincide with the stability domain of traditional causal discrete-time systems that is defined relatively to the unit circle. As it is well known, the sta- bility domain of causal continuous-time system is the right half complex plane (HCP). Therefore, there is a relation between the left (right) HCP and the interior (exterior) of the unit disk that can be expressed by a particular case the bilinear (or, Möbius) transforma- tion. Such map was proposed by Tustin[17]and used since then for the discrete-time approximation of continuous-time linear sys- tems without considering the definition of any discrete-time derivative[18,13,14]. Hereafter we formulate a discrete-time frac- tional calculus that mimics the corresponding continuous-time version, but that is fully autonomous. The motivation for this study is to have in the discrete-time domain the tools and results avail- able for the continuous-time fractional signals and systems[10].

Another important characteristic of the proposed derivatives is that they are suitable to be implemented through the FFT with the corresponding advantages, from the numerical and calculation time perspectives.

The new derivatives

On the Z and Discrete-Time Fourier Transforms

In the following we consider that our domain is thetime scale Th¼ ðhZÞ ¼f. . .;nh. . .;2h;h;0;h;2h;. . .;nh;. . .g

withh2R^þ, that is called graininess[1,11]. In the following the symbolnwill represent any generic point inT. In engineering appli- cations where discrete signals are result of sampling continuous- time signals,his the sampling interval.

LetxðnÞdenote any function defined onT, leaving implicit the graininess, unless it is convenient to display it. The Z transform (ZT) is defined by

XðzÞ ¼Z½xðnÞ ¼ X¹

n¼1

xðnÞzⁿ; z2C: ð1Þ

In some scientific domains, as Geophysics,zinstead ofz¹is used. In some domains, the ZT is often called ‘‘generating function” or ‘‘char- acteristic function”.Definition (1)is the bilateral ZT that leads to the particular case of the unilateral ZT, defined by

XuðzÞ ¼X¹

n¼0

xðnÞzⁿ;

often adopted in the study of systems. The existence conditions of the ZT are similar to those of the bilateral Laplace transform (LT) [7,15,16]

YðsÞ ¼L½yðtÞ ¼ Z₁

1yðtÞe^stdt; s2C: ð2Þ

Therefore, the existence conditions can be stated as follows.

If functionxðnÞis such that there are finite positive real num- bers,randrþ, for which

X¹

n¼0

jxðnÞjrⁿ<1 andX¹

n¼1

jxðnÞjrⁿ_þ<1;

ð3Þ

then the ZT exists and the range of values for which those series converge defines a region of convergence (ROC) that is an annulus.

We must have in mind that this condition is sufficient, but not necessary. The signals that verify(3)are theexponential ordersig- nals[16].

Definition 1. A discrete-time signalxðnÞis called anexponential order signal if there exist integers n1 and n2, and positive real numbers a;b;A, and B, such that A aⁿ¹<jxðnÞj<B bⁿ² for n1<n<n2.

For these signals the ZT exists and the ROC is an annulus cen- tred at the origin, generally delimited by two circles of radiusr

and rþ, such that r<jzj<rþ. However, there are some cases where the annulus can become infinite:

If the signal is right (i.e.,xðnÞ ¼0;n<n02Z), then the ROC is the exterior of a circle centered at the origin (rþ¼ 1):jzj>r. If the signal is left (i.e.,xðnÞ ¼0; n>n02Z), then the ROC is the

interior of a circle centered at the origin (r¼0):jzj<rþ. If the signal is a pulse (i.e., non null only on a finite set), then the

ROC is the whole complex plane, possibly with the exception of the origin. In the ROC, the ZT defines an analytical function.

It should be noted that the ROC is included in the definition of a given ZT. This means that we may have different signals with the same function as ZT, but different ROC.

If the ROC contains the unit circle, then by making z¼e^ix;^{j j}

x

<

p

; i¼ ffiffiffiffiffiffiffi

p1

, we obtain the discrete-time Fourier transform, which we will shortly call Fourier transform (FT). This means that not all signals with ZT have FT. The signals with ZT and FT are those for which the ROC is non-degenerate and contains the unit circle (r<1;rþ>1). For some signals, such as sinusoids, the ROC degenerates in the unit circumference (r¼rþ¼1), and there is no ZT.

Definition 2.The inverse ZT can be obtained by the integral defined by

xðnÞ ¼ 1 2

p

ⁱ

I

c

XðzÞzⁿ¹dz; ð4Þ

where

c

is a circle centred at the origin, located in the ROC of the transform, and taken in a counterclockwise direction.

In such situation the integral in(4)converges uniformly. The calculation uses the Cauchy’s theorem of complex variable func- tions[16].

Definition 3. For functions that have a ROC including the unit circle or for functions having a degenerate ROC, as it is the case of the periodic signals, it is preferable to work with the discrete-time Fourier transform that can be obtained from the Z transform through the transformationz¼e^ix; ^{j j}

x

<

p

Xðe^ixÞ ¼ X¹

n¼1

xðnhÞe^ixn ð5Þ

with the inversion integral xðnÞ ¼ 1

2

p

Z _p

p

Xðe^ixÞe^ixnd

x

^ð6Þ

meaning that a discrete-time signal can be considered as a synthe- sis of elementary sinusoidsXðe^ixÞe^ixnd

x

^.

Remark 1. In fractional applications, we have branchcut points at z¼ 1. Therefore, we have to avoid them by using an integration circle in(4)having a radius,r, greater (smaller) than 1 for the cau- sal (anti-causal) cases. We have then

fðnÞ ¼ rⁿ 2

p

Z _p

p

Xðre^ixÞe^ixnd

x

^ð7Þ

Forward and backward derivatives based on the bilinear transformation

The Tustin transformation is usually expressed by s¼2

h 1z¹

1þz¹; ð8Þ

wherehis the sampling interval,sis the derivative operator associ- ated with the (continuous-time) Laplace transform and z¹ the delay operator tied with the Z transform.

Definition 4. LetxðnhÞbe a discrete-time function, we define the order 1forward bilinear derivative DxðnhÞofxðnhÞas the solution of DxðnhÞ þDxðnhhÞÞ ¼2

h½xðnhÞ xðnhhÞ ð9Þ

Definition 5. Similarly, we define the order 1 backward bilinear derivative DxðnhÞofxðnhÞas the solution of

DxðnhþhÞ þDxðnhÞ ¼2

h½xðnhþhÞ xðnhÞ ð10Þ

Definition 6.The bilinear exponentiale_sðnhÞis the eigenfunction of Eq.(9) or(10). If we set xðnhÞ ¼e_sðnhÞ;yðnhÞ ¼se_sðnhÞ; s2C, withesð0Þ ¼1, then

esðnhÞ ¼ 2þhs 2hs n

; n2Z; s2C: ð11Þ

Properties of the bilinear exponentialesðnhÞ.

Whenn! 1, this exponential is – Increasing, ifReðsÞ>0, – Decreasing, ifReðsÞ<0,

– Sinusoidal, ifReðsÞ ¼0, withs–0, – Constant equal to 1, ifs¼0, It is real for reals,

It is positive fors¼ jxj<²h; x2R, It oscillates fors¼ jxj>²h; x2R.

Following the procedure in[11]we could use this exponential to construct a bilinear discrete-time Laplace transform. However, formula(8)suggests havingz¼^2þhs_2hsthat leads to the Z transform, since such transformation sets the unit circlejzj ¼1 as the image of the imaginary axis ins, independently of which value of his used. Therefore, the exponential has the usual properties.

Whenn! 1, this exponential is – Increasing, ifjzj>1, – Decreasing, ifjzj<1,

– Sinusoidal, ifjzj ¼1, withz–1, – Constant equal to 1, ifz¼1, It is real for realz,

It is positive forz¼x>0; x2R, It oscillates forz–R^þ0.

In what concerns to the derivative definitions, instead of con- sidering(9)or(11), as in[1,11]where the nabla (causal) and delta (anti-causal) derivatives were introduced, we start from the ZT formulations.

Definition 7. Let z2C and h2R^þ. Consider the discrete-time exponential function, zⁿ;n2Z. We define the forward bilinear derivative (Df) as a discrete-time linear operator such that Dfzⁿ¼2

h 1z¹

1þz¹zⁿ: ð12Þ

The operatorH_fðzÞdefined by HfðzÞ ¼2

h 1z¹

1þz¹; j jz >1; ð13Þ

will be called foward transfer function (TF) of the derivative, bor- rowing the nomenclature used in signal processing[7,16].

Definition 8. The backward bilinear derivative (D_b) is defined as a discrete-time linear operator verifying

Dbzⁿ¼2 h

z1

zþ1zⁿ: ð14Þ

whereHbðzÞis the operator HbðzÞ ¼2

h z1

zþ1; j jz <1; ð15Þ

called backward transfer function of the derivative.

By the repeated application of the above operators we obtain the forward and backward derivatives for any positive integer order. However, we introduce the corresponding fractional deriva- tives, valid for any real order.

Definition 9. Let

a

²R. The

a

-order forward bilinear fractional derivative is a discrete-time operator with TF

HfðzÞ ¼ 2 h

1z¹ 1þz¹ a

; jzj>1 ð16Þ

such that D^a_fzⁿ¼ 2

h 1z¹ 1þz¹ a

zⁿ; jzj>1: ð17Þ

Definition 10. The backward bilinear fractional derivative has TF

HbðzÞ ¼ 2 h

z1 zþ1 _a

; jzj<1 ð18Þ

such that D^a_bzⁿ¼ 2

h z1 zþ1 _a

zⁿ; jzj<1: ð19Þ

Having defined the derivative of an exponential we are in con- ditions of defining the derivative of any signal having ZT.

Definition 11. From (4) and (17) we conclude that, ifxðnÞis a function with Z transform XðzÞ, analytic in the ROC defined by z2C:jzj>a; a<1;then

D^a_fxðnÞ ¼ 1 2

p

ⁱ

I

c

2 h

1z¹ 1þz¹ a

XðzÞzⁿ¹dz; ð20Þ

with the integration path outside the unit disk. This implies that

ZhD^a_fxðnÞi

¼ 2 h

1z¹ 1þz¹ a

XðzÞ; jzj>1: ð21Þ

Definition 12. Let xðnÞ be a function with Z transform XðzÞ, analytic in the ROC defined byz2C:jzj<a; a>1:We define

D^a_bxðnÞ ¼ 1 2

p

ⁱ

I

c

2 h

z1 zþ1 a

XðzÞzⁿ¹dz; ð22Þ

with the integration path inside the unit disk and the branchcut line is a segment joinning the pointsz¼ 1. This implies that

ZD^a_bxðnÞ

¼ 2 h

z1 zþ1 _a

XðzÞ; jzj<1: ð23Þ

Remark 2. We must note that:

1. In(16) and (17)we have two branchcut points atz¼ 1. The corresponding branchcut line is any line connecting these val- ues and being located in the unit disk. The simplest is a straight line segment (seeFig. 1).

2. In(18) and (19)we have the same branchcut points, but with branchcut line(s) lying outside the unit disk. For simplifying, we can use two half-straight lines starting atz¼ 1 on the real negative and positive half lines, respectively (seeFig. 1).

3. In both previous cases, we can extend the domain of validity to include the unit circumference, z¼e^ixn; ^{j j 2 ð0}

x

;

p

^{Þ, with}

exception of the pointsz¼ 1. In these cases the integration path in(4)must be deformed around such points, as it can be seen atFig. 2for the causal case.

This deformation is very important in applications where we use the fast Fourier transform (FFT). In such cases a small numerical trick can be used: push the branchcut points slightly inside (outside) the unit circle, that is, to z¼ 1þ

e

^and

z¼1

e

^ð1

e

;1þ

e

^{Þ, with}

e

being a small positive real number.

4. The ROC is independent on the scale graininess,h, and conse- quently we can establish a one to one correspondence between the unit disk, inz, and the left half-plane, ins¼²_h^1z_1þz¹1.

Example 1. In Figs. 3 and 4 we represent the bilinear causal derivatives of orders

a

^¼0:5 and

a

^¼0:8 of a triangle function.

According to what we just wrote we can extend the above def- initions to include sinusoids. We define the derivative of xðnÞ ¼e^ixn; n2Z, through

Df;be^ixn¼ 2 htan

x

2 _a

e^ixn; j

x

^j<

p

^{; ð24Þ}

Fig. 1.ROC for causal and anti-causal derivatives and branchcut points and lines.

Fig. 2.Integration path modification for causal derivative.

-400 -200 0 200 400

0 50 100 150 200 250 300

Triangle function

Magnitude

-1000 -500 0 500 1000

-10 -5 0 5 10

Derivative of order 0.5

Magnitude

t

Fig. 3.Derivative of ordera^¼0:5 of a triangle function withh¼1.

-400 -200 0 200 400

0 50 100 150 200 250 300

Magnitude

Triangle function

-1000 -500 0 500 1000

-6 -4 -2 0 2 4 6

t

Magnitude

Derivative of order 0.8

Fig. 4.Derivative of ordera^¼0:8 of a triangle function withh¼1.

independently of considering the forward or backward derivatives.

Definition 13. For a function having discrete-time Fourier trans- form(6), the bilinear derivative is expressed as:

Df;bxðnÞ ¼ 1 2

p

Z _p

p Xðe^ixÞ 2

htan

x

2 _a

e^ixnd

x

^ð25Þ

that is suitable for implementations with the FFT. According to the existence conditions of the FT, we can say that, ifxðnÞis absolutely sommable, then the derivative(25)exists.

Properties of the derivatives

We present the main properties of the above derivatives. The proofs are easily obtained from the corresponding FT.

1. Linearity

The linearity property of the fractional derivative is straightfor- ward from the above formulae.

2. Time shift

The derivative operators are shift invariant:

Df;bxðnn0Þ ¼Df;bxðmÞ

m¼nn0

This property is immedately obtained from(20)or(22)as a con- sequence of the shift property of the Z transform Z½xðnn0Þ ¼XðzÞzⁿ⁰; n02Z

3. Additivity and Commutativity of the orders Let

a

^andbbe two real values. Then

D^ahD^bxðnÞi

¼D^bD^axðnÞ

¼D^aþbxðnÞ

To prove this relation it is enough to observe that, in the forward case, we have ²_h^1z_1þz¹1

a 2 h1z¹

1þz¹

b

¼ ²_h^1z_1þz¹1

a^þb

and that the product is commutative. For the backward derivative, the situa- tion is identical.

4. Neutral element

This comes from the additivity property by puttingb¼

a

^, D^a_f;bhD_f;^a_bfðnÞi

¼D⁰fðnÞ ¼fðnÞ:

This result is very important because it states the existence of inverse derivative.

5. Inverse element

The existence of neutral necessarily implies that there is always an inverse element: for every

a

order derivative, there is always a

a

order derivative given by the same formula and so it does not need joining any primitivation constant. We adopt the des- ignation ‘‘derivative” for positive orders and ‘‘anti-derivative”

for negative ones.

6. Associativity of the orders

Let

a

;b, and

j

be three real values. Therefore, we can write:

D^jhD^aD^bi

xðnÞ ¼D^jþaþbxðnÞ ¼D^aþbþjfðnÞ ¼D^ahD^bþji xðnÞ as a consequence of the additivity.

7. Derivative of the convolution LetxðnÞ yðnÞ ¼P1

k¼1xðkÞyðnkÞbe the discrete-time convo- lution. Its ZT isXðzÞYðzÞ. Since we can write

2 h1z¹

1þz¹

_a

XðzÞYðzÞ

½ ¼ ²_h^1z_1þz¹_1a n XðzÞo

YðzÞ

¼XðzÞ ²_h^1z_1þz¹1

_a n YðzÞo

; we conclude that

Df½xðnÞ yðnÞ ¼DfxðnÞ

yðnÞ ¼xðnÞ DfyðnÞ :

For the backward derivative of the convolution, we obtain an identical result.

Time formulations

In the previous sub-section, we introduced the derivatives using a formulation based on the ZT. Here we obtain the corresponding time framework, getting formulae similar to the Grünwald- Letnikov derivatives. From the binomial series[2]

ð1wÞ^a¼X¹

k¼0

ð1Þ^kðaÞ_k

k! w^k; jwj<1;

we conclude that the TF in(16) and (18)can be expressed as power series,

1z¹ 1þz¹ a

¼X¹

k¼0

w^akz^k; jzj>1;

wherew^ak;k¼0;1; , is the inverse ZT of ^1z_1þz¹1

a

and represents the impulse response (IR) corresponding to the TF.

Let the discrete convolution be defined by xðnÞ yðnÞ ¼ X¹

k¼1

xðkÞyðnkÞ; n2Z:

The IR,w^ak;k¼0;1; , is obtained as the discrete convolution of the binomial coefficients sequence:

w^ak¼ð

a

^Þk

k! ð1Þ^kð

a

^Þk

k! ¼X^k

m¼0

ð

a

^Þm

m!

ð1Þ^kmð

a

^Þkm

ðkmÞ! ; k2Z^þ0: ð26Þ Performing this discrete convolution we obtain the following results

1. The sequencew^ak;k¼0;1; , that is obtained as the discrete convolution of two causal sequences, is causal and, there- fore, is null fork<0. We will assume it below.

2. For any

a

²R, we have

wk^a¼ ð1Þ^kw^ak k2Z^þ0 ð27Þ The proof is immediate from(26).

3. Initial value

From the initial value theorem of the ZT, it is immediate that w^a0¼1 independently of the order.

4. Final value

Let

a

60. From the final value of the ZT, w^a₁¼lim

z!1ðz1Þ 1z¹ 1þz¹ a

that is 0, if16

a

60, and 2, if

a

^{¼ 1. For}

a

<1 the sequence grows up to1. For

a

>0 we apply(27).

5. If

a

²Rbut

a

R Z, then w^ak ¼ ð1Þ^kð

a

^Þk

k!

X^k

m¼0

ð

a

^ÞmðkÞ_m ð

a

^kþ1Þm

ð1Þ^m

m! ; k2Z^þ0 ð28Þ 6. Letting

a

^¼Nin(28), we get

w^Nk ¼ ð1Þ^kðNÞ_k k!

X

minðk;NÞ m¼0

ðNÞ_mðkÞ_m ðNkþ1Þ_m

ð1Þ^m

m! ; k2Z^þ0 ð29Þ 7. If

a

²Z, set

a

^{¼ N};N2Z^þ. We use

w^Nk ¼X^k

m¼0

ð1Þ^mðNÞ_m m!

ðNÞ_km ðkmÞ!

to obtain

w^N_k ¼ðNÞ_k k!

X

minðk;NÞ m¼0

ðNÞ_mðkÞ_m ðNkþ1Þ_m

ð1Þ^m

m! ; k2Z^þ0: ð30Þ Comparing(30)with(29), we conclude that they differ only in the factorð1Þ^k; k2Z^þ0

8. A recursion

LetW^{ðzÞ ¼}Zw^ak ¼ ^1z_1þz¹1

a

. AsDW^{ðzÞ ¼ P}nðn1Þw^an1zⁿ and DW^{ðzÞ ¼}

a

^1z1þz¹¹

_a

1þz¹ 1z¹

D ^1z_1þz¹1

, after some algebraic manipulation we obtain:

w^ak¼ 2

a

k w^a_k1þ 12 k

w^a_k2; kP2; ð31Þ withw^a0¼1 and w^a1¼ 2

a

^.

This recursion shows that, if

a

<0, thenw^akis a positive sequence.

As consequence, attending to(27), the sequence corresponding to positive orders is always oscillating: successive values have different sign.

9. Relation with the Hypergeometric function

The first factor in(28), namelyð1Þ^kða_k!^Þk, represents the bino- mial coefficient, while the second is a sequence from the Gauss Hypergeometric function

fn¼X^k

m¼0

ða^ÞmðkÞm

ða^kþ1Þm

ð1Þ^m

m! ¼2F1ða;k;1ka;1Þ; n2Z^þ0: ð32Þ This sequence verifies a second order recurrence[18]:

ð

a

^nþ2Þð

a

^nþ1Þfn

¼ 2

a

^ð

a

^nþ2Þfn1þ ðn1Þðn2Þf_n2; nP2;

ð33Þ with initial valuesf₀¼1 andf₁¼2.

We can rewrite(33)as fn¼ 2

a

a

^þn1fn1

ðn1Þðn2Þ

ð

a

^þn1Þ

a

^{þn2fn2: ð34Þ}

If we considerg_n¼^ða_ðn1Þ!^þ1Þn1f_n; nP1; g₀¼0 andg₁¼2, then we obtain[18]

g_n¼ 2

a

n1g_n1þg_n2; nP2; ð35Þ that is a polynomial of degreen1 in

a

. Inserting(35)into(34) and the resulting expression in(28), we obtain

w^ak¼ ð1Þ^k

a

kg_k; k>0; ð36Þ

wherew^a0¼1 andg_nis given by(35)withg₁¼2.

Substituting(35)in(36)we obtain(31), as expected.

10. For a fixedk2Z;w^ak is a polynomial in

a

^{of degreek.}

As pointed above,w^a0and w^a1are polynomials of degrees 0 and 1, respectively. Assume thatw^ak1has degreek1. Then, the first term in right hand side in(31)ensures thatw^akhas degreek. Asw^a0¼1 and w^a1¼ 2

a

, recursion(31)shows that the independent coefficient of such polynomial is null for k>0.

11. The coefficient of

a

^kdecreases with increasingk.

For simplifying the proof, let w^ak¼Pk

m¼1p_m

a

^{m and}

w^ak1¼Pk1

m¼1q_m

a

^{m. From (31) we conclude that}

p_k¼ ²_k^aq_k1, because the second term in the right hand side of(31)only affects the lower order coefficients of the poly- nomial. As this happens fork¼2;3; , we can write

pk¼ ð1Þ^k2^k k!

that decreases withk. In fact, after simplifying the common fac- tors between 2^kandk!, the denominator is the largest odd divi- sor ofn!. The numerator is always a power of 2 corresponding to the factors that were not used when removing the common fac- tors (see below37)[6].

Example 2.We are going to presentw^Nk for some values ofN2Z^þ and for any real order obtained by recursive computation.

1. N¼1

w¹k¼ 0 k<0

1 k¼0

2ð1Þ^k k>0 8<

:

w¹k ¼ 0 k<0 1 k¼0 2 k>0 8<

: 2. N¼2

w²k¼

0 k<0

1 k¼0

ð1Þ^k4k k>0 8<

:

w²k ¼ 0 k<0 1 k¼0 4k k>0 8<

:

3. For any negative order

a

^{, with}

a

>0

Using the recursion(35)withw0^a¼1 andw1^a¼2

a

, we obtain successively:

w2^a ¼2

a

²

w3^a ¼⁴₃

a

^3þ23

a

w4^a ¼²₃

a

^4þ43

a

²

w5^a ¼₁₅⁴

a

^5þ2015

a

^3þ15⁶

a

w6^a ¼₄₅⁴

a

^6þ4045

a

^4þ4645

a

²

w7^a ¼₃₁₅⁸

a

^7þ140315

a

^5þ392315

a

^3þ315⁹⁰

a

w8^a ¼₃₁₅²

a

^8þ315⁵⁶

a

^6þ308315

a

^4þ264315

a

²

ð37Þ

In Fig. 5 we depict the values of w^ak;k¼0;1; ;500 and

a

^{¼ 0}:5k; k¼1;2; ;6:

Similarly, for positive orders

a

^¼0:2k; k¼1;2; ;6, the bilin- ear sequences are plotted inFig. 6.

Remark 3. In previous works, ARMA approximations to these sequences were proposed[8,9,5]. Nonetheless, we will not con- sider them here.

Definition 14. In agreement with the meaning attributed to the sequencew^ak; k¼0;1; , we define the

a

-order forward and back- ward derivatives as

D^ð_f^aÞxðnÞ ¼ 2 h

aX¹

k¼0

w^akxðnkÞ ð38Þ

and

D^ð_b^aÞxðnÞ ¼e^iap 2 h

aX¹

k¼0

w^akxðnþkÞ: ð39Þ

The use of the termsforwardandbackwardis due to the ‘‘time flow”, from past to future or the reverse[10]. This terminology is the reverse of the one used in some mathematical literature.

We can remove the exponential factor,e^iap, in(39)to obtain a right derivative. In the following we will consider the causal

0 50 100 150 200 250 0.20

0.40.6 0.81

Alpha= -0.5

Magnitude

0 50 100 150 200 250

1.21 1.41.6 1.82

Alpha= -1

Magnitude

0 50 100 150 200 250

100 2030 4050

Alpha= -1.5

Magnitude

0 50 100 150 200 250

2000 400600 1000800

Alpha= -2

Magnitude

0 50 100 150 200 250

0 5000 10000 15000 20000

Alpha= -2.5

Magnitude

k

Fig. 5.Bilinear sequences corresponding to ordersa¼ 0:5k;k¼1;2; ;5:.

0 50 100 150 200 250

-1 -0.5 0 0.5 1

Alpha= 0.25

Magnitude

0 50 100 150 200 250

-1 -0.5 0 0.5 1

Alpha= 0.5

Magnitude

0 50 100 150 200 250

-1.5-1 -0.50.51.501

Alpha= 0.75

Magnitude

0 50 100 150 200 250

-3 -2 -10123

Alpha= 1

Magnitude

k

Fig. 6.Bilinear sequences corresponding to ordersa¼0:25k;k¼1;2; ;4;.

derivative(38)represented by the simplified notationD^aand with ZT given by(21).

Other properties.

The first is causal while the second is anti-causal.

In fact, ifxðnÞ ¼0;n<n02Z, thenD^ð_f^aÞxðnÞ ¼0;n<n0and we obtain

D^ð_f^aÞxðnÞ ¼ 2 h

aⁿⁿX⁰

k¼0

w^akxðnkÞ ð40Þ

that is null forn<n₀. For the backward the proof is similar using xðnÞ ¼0;n>n02Z, leading to

D^ð_b^aÞxðnÞ ¼e^iap 2 h

a^nþnX⁰

k¼0

w^akxðnþkÞ ð41Þ

that is null forn>n₀.

Fractional derivative of the impulse

Let introduce the Kroneckker impulse,dðnÞ; n2Z, by dðnÞ ¼ 1 n¼0

0 n–0

:

The Heaviside discrete unit step is usually defined by

e

^{ðnÞ ¼ 1 nP0}

0 n<0

:

and its ZT is given by Z^½

e

^{ðnÞ ¼}₁¹_z1; jzj>1:

As we can see, the derivative of any order of the Kroneckker impulse is essentially given by thew^ancoefficients. In fact, from (38)we get

D^adðnÞ ¼ 2 h

a

w^an

e

ðnÞ; ð42Þ

where

e

^ðnÞis used to express the right behaviour of the deriva- tive of the delta, stating the causality of the operator.

Fractional derivative of the unit step

The functionw¹n , introduced inExample 2, is a modified version of the unit step. It is straightforward to confirm that

w¹n ¼2

e

^ðnÞ dðnÞ:

with ZTZ w¹n

¼^h₂^1þz_1z¹1; jzj>1, as expected. According to the above properties, we can obtain the fractional derivative of the unit step function. We have

e

^{ðnÞ ¼1}₂w¹n þ1 2dðnÞ:

Consequently D^a

e

^{ðnÞ ¼1}₂ ²_h

a1

w^an¹þ1 2

2 h

a

w^an:

Fractional derivative of thewfunction

We are interested in computing the derivative ofw^an, for any

a

^withn2Z. From(42)and the additivity property, we can write

D^bD^adðnÞ ¼D^b 2 h

a

w^an

¼ 2 h

aþb

w^an^þb

e

^ðnÞ

that leads to

D^b w^a_n ¼ 2 h

b

w^a_n^þb

e

ðnÞ: ð43Þ

Backward compatibility

Often, discrete-time systems are viewed as mere approxima- tions to the continuous-time counterpart. However, and as seen above, the discrete-time systems exist by themselves and have properties that are independent from, although similar to, the continuous-time analogues. Nonetheless, this observation does not prevent us from establishing a continuous path from each other. In fact, we can go from the discrete into the continuous domain by reducing the graininess. To see it, let us return to(20) and rewrite it as

D^ð_f^aÞxðnhÞ ¼ 2 h

aX¹

k¼0

w^akxðnhkhÞ:

Assume thatxðnhÞresulted from a continuous-time functionxðtÞ and define a new function,yðtÞ, by

yðtÞ ¼ 2 h

aX¹

k¼0

w^akxðtkhÞ: ð44Þ

The LT of(44)is YðsÞ ¼ 2

h

aX¹

k¼0

w^ake^khsXðsÞ ¼ 2 h

1e^hs 1þe^hs a

XðsÞ; ð45Þ

where YðsÞ ¼L½yðtÞ and XðsÞ ¼L½xðtÞ. Knowing that limh!01e^hs

h ¼s, we can write YðsÞ ¼s^aXðsÞ; ReðsÞ>0;

meaning thatYðsÞis the LT of the (continuous-time) derivative of xðtÞ. This relation states a compatibility between the new formula- tion described above and the well known results from the continuous-time derivative formulation[12]. If we used the back- ward formulation, we would obtain the same result, but with a ROC valid for ReðsÞ<0. Taking in account the above equations and(38), we conclude that, fort2R, we can write:

D^ð_f^aÞxðtÞ ¼lim

h!0

2 h

aX¹

k¼0

w^akxðtkhÞ: ð46Þ

Similarly, we can obtain from(39) D^ð_b^aÞxðtÞ ¼e^iaplim

h!0

2 h

aX¹

k¼0

w^akxðtþkhÞ ð47Þ

Relations (46) and (47) state two new ways of computing the continuous-time fractional derivative that are similar to the Grünwald-Letnikov derivatives. However, it may be interesting to remark that we can compute derivatives with(44)instead of(22).

The differential discrete-time linear systems

The above derivatives lead us to consider systems defined by constant coefficient differential equations with the general form X^N

k¼0

akD^akyðnÞ ¼X^M

k¼0

bkD^bkuðnÞ ð48Þ

withaN¼1. The operatorDis the forward (or backward) derivative above defined, assuming orders

a

kandbk;k¼0;1;2; . The coeffi- cientsakandbk;k¼0;1;2; are real numbers andNandMrepre- sent any given positive integers. LetgðnÞbe the IR of the system defined by(48)that is,

v

^{ðnÞ ¼}dðnÞ. The output is the convolution of the input and the IR,

yðnÞ ¼gðnÞ

v

^{ðnÞ: ð49Þ}

If

v

^{ðnÞ ¼zn}, then the output is given by:

yðnÞ ¼zⁿ X¹

n¼1

gðnÞzⁿ

" # :

The summation expression will be calledtransfer functionas usu- ally and it is the ZT,GðzÞ, of the IR.

With the definition of forward derivative and mainly formula (21)we write

GðzÞ ¼ X^M

k¼0

bk 2 h1z¹

1þz¹

_bk

X^N

k¼0

ak 2 h

1z¹ 1þz¹

_ak; jzj>1; ð50Þ

for the causal case, and

GðzÞ ¼ X^M

k¼0

bk 2 hz1

zþ1

_bk

X^N

k¼0

ak 2 hz1

zþ1

_ak; jzj<1; ð51Þ

for the anti-causal case. We can give to expressions(50) and (51)a form that states their similarity with the classic fractional linear systems[13,4]. For example, for the first, let

v

^{¼ 2}h1z¹

1þz¹

. We have

Gð

v

^{Þ ¼}

X^M

k¼0

bk

v

^bk

X^N

k¼0

ak

v

^ak

ð52Þ

Remark 4. It is important to note that the factors

2 h

ak

;k¼1;2; , do not have any important role in the compu- tations. Therefore, they can be merged withakandbkcoefficients.

Example 3. Consider the simple system with transfer function

Gð

v

^{Þ ¼}

v

^{a1þ1: ð53Þ}

InFig. 7we represent the impulse responses for several values of the order,

a

^¼0:25k;k¼1;2; ;6.

It is interesting to verify that all the IR assume a finite value at the origin, contrarily to the continuous-time system analog to(53) described byGð

v

^{Þ ¼}_va¹þ1; Reð

v

^Þ>0.

Conclusions

In this paper, we introduced new discrete-time fractional derivatives based on the bilinear transformation. We obtained both time and frequency representations. The corresponding impulse responses are always finite, contrarily to their continuous-time analogs. We illustrate the behaviour of the forward derivative through the computation of the impulse response of a simple system.

Compliance with Ethics Requirements

This article does not contain any studies with human or animal subjects.

0 50 100 150 200 250 300

-0.1 0 0.1 0.2 0.3 0.4 0.5

IR

Alpha= 0.5

0 50 100 150 200 250 300

-0.1 0 0.1 0.2 0.3 0.4 0.5

IR

Alpha= 1

0 50 100 150 200 250 300

-0.2 0 0.2 0.4 0.6 0.8

IR

Alpha= 1.5

0 50 100 150 200 250 300

--0.8-0.4-0.20.60.20.40.60.80

t

IR

Alpha= 2

Fig. 7.Impulse responses corresponding to(53)for ordersa^¼0:5k;k¼1;2; ;4.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influ- ence the work reported in this paper.

Aknowledgements

This work was partially funded by National Funds through the Foundation for Science and Technology of Portugal, under the pro- jects UIDB/00066/2020.

References

[1]Bohner M, Peterson A. Dynamic Equations on Time Scales: an introduction with applications. Boston: Birkäuser; 2001.

[2] Henrici P. Applied Computational Complex Analysis. vol. 2, Wiley-Interscince Publication; 1991.

[3]Hilger S. Analysis on measure chains - a unified approach to continuous and discrete calculus. Res. Math. 1990;18(1–2):18–56.

[4]Magin R, Ortigueira MD, Podlubny I, Trujillo J. On the fractional signals and systems. Signal Process 2011;91(3):350–71.

[5] Maione G, Lazarevic´ MP. On the symmetric distribution of interlaced zero-pole pairs approximating the discrete fractional tustin operator. In: 2019 IEEE

international conference on Systems, Man and Cybernetics (SMC), Bari, Italy;

2019. p. 2578–83.

[6] The on-line Encyclopedia of Integer Sequences, https://oeis.org/book.html [accessed October 2019].

[7]Oppeneim AV, Schafer RW. Discrete-time signal processing. 3rd ed. Upper Saddle River, USA: Prentice Hall Signal Processing; 2009.

[8]Ortigueira MD, Serralheiro AJ. A new least-squares approach to differintegration modeling. Signal Process 2006;86(10):2582–91.

[9]Ortigueira MD, Serralheiro AJ. Pseudo-fractional ARMA modelling using a double Levinson recursion. IET Control Theory Appl 2007;1(1):173.

[10] Ortigueira MD. Fractional calculus for scientists and engineers. Springer:

Lecture Notes in Electrical Engineering, volume 84; 2011. doi:10.1007/978-94- 007-0747-4).

[11]Ortigueira MD, Coito FJV, Trujillo JJ. Discrete-time differential systems. Signal Process 2015;107(2015):198–217.

[12]Ortigueira MD, Machado JT. Which derivative? Fractal Fractional 2017;1(1):3.

[13]Petráš I. Fractional-order nonlinear systems: modeling, analysis and simulation. Springer; 2011.

[14]Petráš I. A new discrete approximation of the fractional-order operator. In:

Proceedings of the 13th International Carpathian Control Conference (ICCC), May. p. 547–51.

[15] Proakis JG, Manolakis DG. Digital signal processing: principles, algorithms, and applications, Prentice-Hall; 2006.

[16]Roberts MJ. Signals and systems: analysis using transform methods and matlab. International ed. McGraw-Hill; 2003.

[17]Tustin A. A method of analysing the behaviour of linear systems in terms of time series. J Inst Electrical Eng – Part IIA: Automat Regul Servo Mech 1947;94 (1):130–42.

[18] Strehl V. Personal mail communication; 2003.