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Signal Processing

Signal Processing 107 (2015) 238–245

http://d0165-16

E-mjtrujill@martynyfjvc@fct

journal homepage: www.elsevier.com/locate/sigpro

A generalized power series and its application in the inversionof transfer functions

Manuel D. Ortigueira a, Juan J. Trujillo b, Valery I. Martynyuk c,Fernando J.V. Coito a

a UNINOVA and DEE of Faculdade de Ciências e Tecnologia da UNL, Campus da FCT da UNL, Quinta da Torre, 2829 – 516 Caparica, Portugalb Departamento de Análisis Matemático, Universidad de La Laguna, 38271 La Laguna, Tenerife, Spainc Khmelnytskiy National University, Faculty of Programming, Computer and Telecommunication System,Department of Radio Engineering and Communication, Khmelnytskiy, Ukraine

a r t i c l e i n f o

Article history:Received 2 December 2013Received in revised form18 March 2014Accepted 19 April 2014Available online 28 April 2014

Keywords:Laplace transform inversionGeneralized Taylor series

x.doi.org/10.1016/j.sigpro.2014.04.01884/& 2014 Published by Elsevier B.V.

ail addresses: mdo@fct.unl.pt (M.D. Ortigueirullmat.es (J.J. Trujillo),uk.valeriy@gmail.com (V.I. Martynyuk),.unl.pt (F.J.V. Coito).

a b s t r a c t

The paper proposes a new approach to compute the impulse response of fractional orderlinear time invariant systems. By applying a general approach to decompose a Laplacetransform into a Laurent like series, we obtain a power series that generalizes the Taylorand MacLaurin series. The algorithm is applied to the computation of the impulseresponse of causal linear fractional order systems. In particular for the commensuratecase we propose a table to obtain the coefficients of the MacLaurin series. Through whatwe call the correspondence principle we show that it is possible to obtain a fractionalorder Laplace transform solution from the integer one. Some examples are presented.

& 2014 Published by Elsevier B.V.

1. Introduction

1.1. On the Taylor series

The importance of the Taylor and MacLaurin series isunquestionable. This motivated the search for generalizationsto the fractional case. The first attempt was made by Riemannleading to what is known as the Riemann–Taylor series [15,6].This approach was retaken by Osler [13] which worked in thecomplex domain. A similar approach was followed by Camposthat proposed several generalizations to the Taylor, Lagrange,Laurent and Teixeira series [2]. However, the series resultingfrom these attempts were not satisfactory enough whichmotivated the appearance of new proposals. The first wasdone by Dzherbashyan and Nersesian [4,5] and later by

a),

Trujillo et al. [17] which presented a fractional Taylor seriesdifferent from the Riemann–Taylor series, but which appearedas a more natural generalization. In the same line of reasoningother formulations appeared, but while Trujillo et al. used theRiemann–Liouville derivative definition, Odibat and Shawag-feh [10] used the Caputo derivative. Another differentapproach was proposed by Jumarie [8] which is also similarto the classic integer order series and is based on the modifiedRiemann–Liouville derivative.

We may wonder about the reason that leads us to proposea new formulation alternative to the above-referred seriesexpansions. As is well known the classic Taylor (MacLaurin)series converges in a circle on the complex plane or on asymmetric interval around the centre point. When we go intothe fractional setup this is not the situation, because we aredealing with derivatives that exhibit causality.1 This meansthat the convergence domain will be a half-circle or a left or

1 We will be anware of the two-sided derivatives.

M.D. Ortigueira et al. / Signal Processing 107 (2015) 238–245 239

right part of the symmetric interval. On the other hand wework frequently with functions that have infinite duration. So,our convergence domain must be a half plane or the left orright half axis. We need a very general formulation.

1.2. The inversion of the transfer function of linear systems

A linear system is characterized by the impulseresponse and its Laplace transform called transfer functionof the system. This can be obtained directly from thedifferential equation describing the system. So, we mustperform a Laplace transform inverse computation. Ingeneral we must use the inversion integral, but the taskmay be very difficult, because the poles may be unknownand we may have branch points. For example the solutionof the general non-commensurate fractional differentialequations has been considered as a non-simple task andonly particular cases have been considered. Podlubny [14]proposed a general rather complicated formula that isvalid for equations without derivatives over the input. Thesame subject was also considered by Diethelm [3] in thecontext of the Caputo derivative, but he did not presentexplicitly a general formula for the solution, except for thetwo-term case. Among the most interesting cases we mustconsider the commensurate case. The transfer functioncorresponding to this case is a rational function of sα that isnormally decomposed into partial fractions which needsthe knowledge of the poles.

1.3. Proposed solution

In this paper we will present a simple algorithm fordecomposing a given transfer function, not necessarily arational function of sα, but can assume very generalaspects, as is the case of the function arctan(1/s).

Here we will propose a simple approach based on anegative power series expansion of the transfer function.The term by term Laplace transform inversion of suchseries leads to a power series that generalizes well-knownpower series: MacLaurin, Taylor, and Puiseaux [7]. Thecorresponding Taylor-like series is easily obtained. Thisseries is really a Dirichlet series because the power may beuncommensurate.

1.4. Outline of the paper

The paper is outlined as follows. In the next section wewill revise some important aspects of the Laplace trans-form and its use in studying linear systems. In Section 3we present an outline of previous more important general-izations of the Taylor series. In Section 4 we show how byusing the initial value theorem we can obtain a Laurentlike series for the transfer function of a linear system. Fromit we obtain general MacLaurin and Taylor series. Someexamples are presented to illustrate the proposed theory.This is particularized in Section 5 to the commensurateorders case where a simple implementation algorithm ispresented. In Section 6 we show how to obtain thegeneralized Taylor series from the normal integer orderseries through what we call the correspondence principle.At last we present some conclusions.

2. On the Laplace transform

2.1. Relations between this transform and the linear timeinvariant systems

The importance of the Laplace transform in the study ofthe linear time invariant (LTI) systems is unquestionable.Many books enhance such facts [11,16].

For these kinds of systems, the input–output relation isgiven by the convolution [16]

yðtÞ ¼Z þ1

�1hðτÞxðt�τÞ dτ ð1Þ

where xðtÞ is the input signal and hðtÞ is the impulseresponse. Consider a particular input xðtÞ ¼ est for tAR andsAC. As is easy to understand, the output is given byyðtÞ ¼HðsÞest provided HðsÞ exists. This is given by

HðsÞ ¼Z þ1

�1hðtÞe� st dt ð2Þ

and called a transfer function. As we see HðsÞ is the bilateral(two-sided) Laplace transform of the impulse response ofthe system. We showed that the exponentials, defined on R,are the eigenfunctions of the continuous-time LTI systems. Thisshows why we must use the two-sided Laplace transform,although the one-sided transform has greater popularity.However, the exponentials used in this transform are noteigenfunctions of the LTI systems. Even if the signal to betransformed is causal seeming we recover the one-sided LTwe shall be using the two-sided and the correspondingproperties, in particular LT½f ðαÞ� ¼ sαFðsÞ. In the following wewill be in this situation, since we will be dealing with theimpulse response of causal linear systems. We will not putany restriction on the format of such transfer functions inorder to include a broader class of functions. In particular weshall be interested in functions with the format

G sð Þ ¼∑Mk ¼ 0bks

βk

∑Nk ¼ 0aks

αkð3Þ

where the αk and βk are increasing sequences of positive reals.This implies the use of a fractional derivative definition suchthat Dαest ¼ sαest , for tAR. Without losing generality we shallbe using the Grünwald–Letnikov derivative.

Concerning the behaviour of the time functions weshall be working with exponential order functions toguarantee the existence of the Laplace transform. Thisdoes not mean that the obtained results are not useful forother functions.

2.2. The general initial value theorem

The Abelian initial value theorem [19] is a very impor-tant result in dealing with the Laplace transform. Thistheorem relates the asymptotic behaviour of a causalsignal, ϕðtÞ, as t-0þ to the asymptotic behaviour of itsLaplace transform, ΦðsÞ ¼ LT ½ϕðtÞ�, as s¼ ReðsÞ-1

Theorem 1 (The initial-value theorem). Assume that ϕðtÞ isa causal signal such that in some neighbourhood of the originis a regular distribution corresponding to an integrablefunction and its Laplace transform is ΦðsÞ with region of

M.D. Ortigueira et al. / Signal Processing 107 (2015) 238–245240

convergence defined by ReðsÞ40. Also, assume that there is areal number β4�1 such that limt-0þ ϕðtÞtβ exists and is afinite complex value. Then

limt-0þ

ϕðtÞtβ

¼ lims-1

sβþ1ΦðsÞΓðβþ1Þ ð4Þ

For proof see [19, Section 8.6, pp. 243–248].As referred above, LT ½f ðαÞ� ¼ sαFðsÞ for ReðsÞ40 (see for

instance [11]). On the other hand attending to the currentinitial value theorem

f ð0þÞ¼ lims-1

sFðsÞ

we can write

limt-0þ

ϕðtÞtβ

¼ limt-0þ

ϕðβÞ tð Þ ¼ lims-1

sβþ1Φ sð Þ ð5Þ

Corollary 1.1. Let �1oαoβ. Then

limt-0þ

ϕðtÞtα

¼ 0

If α4β

limt-0þ

ϕðtÞtα

¼1

To prove the first part we only have to write

limt-0þ

ϕðtÞtα

¼ limt-0þ

ϕðtÞtβ

tβ

tα¼ 0

and realize that the first factor has a finite limit given in (5)and the second has zero as the limit. The second part isimmediately evident.

This means that the function, ϕðtÞ, has the behaviourϕðβÞð0þÞtβ near the origin.

3. Past generalizations of the Taylor series

The first attempt to generalize the Taylor series was doneby Riemann leading to what is known as the Riemann–Taylor series [15,6]. This approach was retaken by Osler [13]which worked in the complex domain. A similar approachwas followed by Campos that proposed several general-izations to the Taylor, Lagrange, Laurent and Teixeira series[2]. However, the series resulting from these attempts werenot satisfactory enough which motivated the rise of newproposals. Here we revise the most interesting one.

3.1. Riemann–Liouville and Caputo derivatives

Most formulations of the fractional Taylor series wereobtained using the Riemann–Liouville or Caputo [9,14]derivatives.2 We make here a brief presentation of thesederivatives.

2 The algorithm we present in this paper is incompatible with thesederivatives.

The forward Riemann–Liouville derivative of the orderα40 is defined by

RLDαaþ f

� �tð Þ ¼ 1

Γðn�αÞddt

� �n Z x

a

f ðτÞ dτðt�τÞα�nþ1 n¼ α½ �þ1ð Þ:

ð6ÞWe can rewrite this relation in the form

RLDαaþ f

� �tð Þ ¼ d

dt

� �n

In�αaþ f

� �tð Þ; ð7Þ

where Iαaþ is a forward Riemann–Liouville integral of theorder α40

Iαaþ f� �

tð Þ ¼ 1ΓðαÞ

Z x

a

f ðτÞ dτðt�τÞ1�α

x4að Þ ð8Þ

The Caputo fractional derivative of the order α isdefined by

CDαaþ f

� �tð Þ ¼ In�α

aþddt

� �n

f� �

tð Þ; ð9Þ

where Iαaþ is a forward Riemann–Liouville integral (8) ofthe order α40.

As the Taylor series involves the computation of deri-vatives that are multiples of a given α and in general3

ðCDαaþ

CDαaþ f ÞðtÞaðCD2α

aþ f ÞðtÞthe coefficients of the fractional Taylor series can be foundby repeated differentiation.

3.2. Riemann formal version of the generalized Taylor series

The Riemann formal version of the generalized Taylorseries [15,6] is

f tð Þ ¼ ∑þ1

m ¼ �1

ðRLDαþma f Þðt0Þ

Γðαþmþ1Þ ðt�t0Þαþm; ð10Þ

where RLDαa for α40 is the Riemann–Liouville fractional

derivative, and RLDαa ¼ I�α

a for αo0 is the Riemann–Liou-ville fractional integral of the order jαj.

3.3. Fractional Taylor series in the Riemann–Liouville form

Let f(t) be a real-value function such that the derivativeðRLDαþm

aþ f ÞðtÞ is integrable. Then the following analogue ofthe Taylor formula holds (see [18, Chapter 1, Section 2.6]):

f tð Þ ¼ ∑m�1

j ¼ 0

ðRLDαþ jaþ f ÞðaþÞ

Γðαþ jþ1Þ ðt�aÞαþ jþRm tð Þ α40ð Þ; ð11Þ

where Dαþ jaþ are forward Riemann–Liouville derivatives,

and

RmðtÞ ¼ ðIαþmaþ

RLDαþmaþ f ÞðtÞ: ð12Þ

Trujillo et al. also obtained the following generalizedTaylor formula [17]:

f tð Þ ¼ ∑m

j ¼ 0

cjΓððjþ1ÞαÞðt�aÞðjþ1Þα�1þRm t; að Þ; ð13Þ

3 As we use the Grünwald–Letnikov derivative, this problem doesnot set.

4 Meaning that the error in time is less than tαn =Γðαnþ1Þ.

M.D. Ortigueira et al. / Signal Processing 107 (2015) 238–245 241

where αA ½0;1�, andcj ¼ ΓðαÞ½ðt�aÞ1�αðRLDα

a Þjf ðtÞ�ðaþÞ; ð14Þ

Rm t; að Þ ¼ ððRLDαa Þmþ1f ÞðξÞ

Γððmþ1Þαþ1Þ ðt�aÞðmþ1Þα; ξA a; t½ �: ð15Þ

3.4. Fractional Taylor series in the Dzherbashyan–Nersesianform

Let αk ðk¼ 0;1;…;mÞ be an increasing sequence of realnumbers such that

0oαk�αk�1r1; α0 ¼ 0; k¼ 1;2;…;m: ð16ÞWe introduce the notation [4,5] (see also [18, Section

2.8])

DðαkÞ ¼ I1�ðαk �αk� 1Þ0þ D1þαk� 1

0þ : ð17ÞIn general, DðαkÞaRLDαk

0þ . Fractional derivative DðαkÞ differsfrom the Riemann–Liouville derivative RLDαk

0þ by a finitesum of power functions (see [9, Eq. 2.68]), but coincideswith the Grünwald–Letnikov derivative if the Liouvillederivative is used [11, Chap. 2].

The generalized MacLaurin formula proposed by Dzher-bashyan and Nersesian reads as [4,5]

f ðtÞ ¼ ∑m�1

k ¼ 0Akt

αk þRmðtÞ ðt40Þ: ð18Þ

where

Ak ¼ðDðαkÞf Þð0þ ÞΓðαkþ1Þ ;

Rm tð Þ ¼ 1Γðαmþ1Þ

Z x

0ðt�τÞαm �1 DðαkÞf

� �τð Þ dτ: ð19Þ

3.5. Fractional Taylor series in the Odibat–Shawagfeh form

The fractional Taylor series is a generalization of theTaylor series for fractional derivatives, where α is thefractional order of differentiation, 0oαo1. The fractionalTaylor series with Caputo derivatives [10] has the form

f tð Þ ¼ f að ÞþðCDαaþ f ÞðaÞ

Γðαþ1Þ ðt�aÞαþðCDαaþ

CDαaþ f ÞðaÞ

Γð2αþ1Þ ðt�aÞ2αþ⋯;

ð20Þwhere CDα

aþ is the Caputo fractional derivative of the order α.

3.6. Fractional Taylor series with the modified Riemann–Liouville derivative

Another formulation was proposed by Jumarie [8]. Itcan be stated as follows: Assume that f(t) is defined in Rand has modified Riemann–Liouville derivative of anyorder αk; kAZþ and 0oαo1; then the following relationholds:

f tþhð Þ ¼ ∑1

¼ 0

hkα

Γðkαþ1ÞfðkαÞ tð Þ ð21Þ

4. The transfer function series representation

Let H(s) be a transfer function and its associated regionof convergence be ReðsÞ40. We are going to use the initialvalue theorem to obtain a decomposition of H(s) into asum of negative power functions plus an error term. Let usdo it:

1.

Define R0ðsÞ ¼HðsÞ with inverse r0ðtÞ. 2. Let γ0 be the real value such that

lims-1

sγ0HðsÞ ¼ A0

where A0 is finite and non-null. Then, if

R1ðsÞ ¼HðsÞ�A0s� γ0 ;

as is clear lims-1sγ0R1ðsÞ ¼ 0 and A0 ¼ limt-0þ r0ðtÞ ¼hðγ0 �1Þð0þ Þ.

3.

Now, repeat the process. Let γ1 be the real value suchthat

lims-1

sγ1R1ðsÞ ¼ A1;

where A1 is finite and non-null. Again introduce

R2ðsÞ ¼HðsÞ�A0s� γ0 �A1s� γ1

with lims-1sγ1R2ðsÞ ¼ 0 and A1 ¼ limt-0þ rðγ1 �1Þ1 ðtÞ,

with r1ðtÞ as the inverse of R1ðsÞ.

4. In general, let γn be the real value such that

lims-1

sγnRnðsÞ ¼ An

where An is finite and non-null. We arrive at thefunction

RnðsÞ ¼HðsÞ� ∑n�1

0Aks

� γk ð22Þ

and lims-1sγnRnðsÞ ¼ 0. As above An ¼ limt-0þ rðγn �1Þn�1

ðtÞ ¼ rðαnÞn�1ð0þ Þ is coherent with the initial value theo-rem γn ¼ αnþ1, for nAZþ

0 .

We can write

HðsÞ ¼ ∑n�1

0Aks

� γk þRnðsÞ ð23Þ

leading us to conclude that HðsÞ can be expanded into aLaurent like power series.

Theorem 2 (The generalized Laurent series). As is easy toconclude jRnðsÞj ¼ oðj1=sγ jÞ4 with γ4γn, allowing us to writewhat we consider a generalized Laurent series

HðsÞ ¼ ∑1

0rðαkÞk ð0þ Þs� γk ð24Þ

In the general case it is not easy to state the conver-gence of this series. If the αk ¼ kα, we can apply the theoryof Z transform [16]. In this case, if the sequenceAk ¼ rðkαÞk ð0þ Þ is also of exponential order, the series in(24) converges in the region that is the exterior of a circlewith the centre at s¼0.

M.D. Ortigueira et al. / Signal Processing 107 (2015) 238–245242

According to the development we did above, the succes-sive functions rðαkÞk ðtÞ come from hðtÞ by removing the inversesof rðαmÞm ð0þ Þs� γm for mok. As is easy to verify, the order αkderivative of such terms is singular at the origin. So, the termsrðαkÞk ðtÞ are equal to the analytic part of hðαkÞðtÞ. When thederivative orders are positive integers, this does not happenbecause the derivatives of the removed terms are zero or thederivatives of the impulse δðtÞ null for t ¼ 0þ . In the followingwe will represent them by hðαkÞa ðtÞ.

Theorem 3 (The generalized MacLaurin series). Let hðtÞ bethe impulse response corresponding to H(s). Then

h tð Þ ¼ ∑1

0hðγk �1Þa 0þ� �tγk �1

ΓðγkÞε tð Þ ð25Þ

where εðtÞ is the Heaviside unit step function that can bewritten as

h tð Þ ¼ ∑1

0hðαkÞa 0þ� � tαk

Γðαkþ1Þε tð Þ ð26Þ

This formulation generalizes the Dzherbashyan andNersesian result in the sense that our approach does notneed the restriction (16).

In the commensurate case, αk ¼ αk, for kAZþ0 , we

obtain easily

h tð Þ ¼ ∑1

0hðαkÞa 0þ� � tαk

Γðαkþ1Þε tð Þ ð27Þ

With α¼1 we obtain the traditional MacLaurin series.

Theorem 4 (The generalized Taylor theorem). Let a be anyreal and gðtÞ ¼ 0, for toa. We have

g tð Þ ¼ ∑1

0gðαkÞa aþ� � ðt�aÞαk

Γðαkþ1Þε t�að Þ ð28Þ

To prove this result it is enough to make gðtÞ ¼ f ðt�aÞ.

4.1. Examples

1.

HðsÞ ¼ sα�β=ðsαþ1Þ.It is the Laplace transform of the generalized Mittag–Leffler function. We are going to proceed as pointedout above. We have

A0 ¼ 1 and γ0 ¼ β

A1 ¼ �1 and γ1 ¼ αþβ

Repeating the process we obtain

Rn sð Þ ¼ ð�1Þn 1sβþn�αðsαþ1Þ

with

An ¼ ð�1Þn and γn ¼ n � αþβ

This leads to

HðsÞ ¼ s�β ∑1

0ð�1Þns�nα

Its inverse is easily obtained

h tð Þ ¼ ∑1

0ð�1Þn t

nαþβ�1

ΓðnαþβÞ � ε tð Þ

in agreement with well-known results [9].

2. HðsÞ ¼ arctanð1=sÞ.

We are going to proceed as above, but as the functionis more involved, we are going to do all the steps withmore details. We have

lims-1

s � arctan 1=s� �¼ lim

s-1arctanð1=sÞ

1=s

¼ limv-0

arctanðvÞv

¼ limv-0

11þv2

1¼ 1

Then A0 ¼ 1 and γ0 ¼ 1 and R0ðsÞ ¼ arctanð1=sÞ�1=s. Inthe second step, we obtain

lims-1

s2½s � arctanð1=sÞ�1=s� ¼ 0

So,

lims-1

s3 arctan 1=s� ��1=s

� ¼ limv-0

arctanðvÞ�vv3

¼ limv-0

D3½arctanðvÞ�3!

¼ limv-0

D2½ 11þv2

�3!¼ �2!

3!¼ �13

We obtain A1 ¼ �1=3 and γ1 ¼ 3 and R1ðsÞ ¼arctanð1=sÞ�1=sþ1=ð3s3Þ. In the third step, we obtainas above

lims-1

s4½arctanð1=sÞ�1=sþ1=ð3s3Þ� ¼ 0

leading us to try the next power

lims-1

s5 arctan 1=s� ��1=sþ1= 3s3

� �� ¼ lim

v-0

arctanðvÞ�vþðv3Þ=3v5

¼ limv-0

D5½arctanðvÞ�5!

¼ limv-0

D4 11þv2

�5!

¼ �4!5!

¼ �15

leading to A2 ¼ �1=5 and γ2 ¼ 5 and R2ðsÞ ¼arctanð1=sÞ�1=sþ1=ð3s3Þ�1=ð5s5Þ. Generalizing andassuming that we proceed indefinitely we obtain

H sð Þ ¼ ∑1

0ð�1Þn 1

2nþ1s�2n�1

The inverse Laplace transform gives

h tð Þ ¼ ∑1

0ð�1Þn 1

2nþ1t2n

ð2nÞ! � ε tð Þ ¼∑1

0 ð�1Þn t2nþ1

ð2nþ1Þ!t

�ε tð Þ ¼ sin ðtÞt

� ε tð Þ

that is the result shown in [1].

3. The following example illustrates a very difficult

situation although without practical applications. LetHðsÞ ¼ 1=ðs

ffiffi3

pþs

ffiffi2

pþ1Þ.

The coefficients of the generalized Laurent series are

An ¼ ð�1Þn

5

M.D. Ortigueira et al. / Signal Processing 107 (2015) 238–245 243

and the powers

γn ¼ ðnþ1Þffiffiffi3

p�n

ffiffiffi2

p

for n¼0,1,….Although this is an academic example it is not difficultto devise similar situations with rational powers. Forthat purpose it is enough to use rational approxima-tions for the square roots.

5. The rational format commensurate case

In this section we will consider a special particular casethat is very useful in applications. It is the case of transferfunctions corresponding to commensurate fractional linearequations with constant coefficients that have the format

G sð Þ ¼∑Mk ¼ 0bks

αk

∑Nk ¼ 0aks

αkð29Þ

where α is a positive real. We will assume that MrN andnormalize aN to 1.

In the following we particularize the above algorithmfor this case. We start by introducing a modified sequenceof the numerator coefficients

b0N� i ¼bM� i; irM0; i4M

(i¼ 0;…;N ð30Þ

and represent the denominator in (29) by D(s).It is not difficult to conclude that

A0 ¼ bM and γ0 ¼ ðN�MÞα ð31ÞOn the other hand the repetition of the above-describedalgorithm leads to

Rn sð Þ ¼ ∑N�1k ¼ 0b

nks

αk

sðNþn�MÞαDðsÞ ð32Þ

where

bnN�k ¼ bn�1N�1�k�An�1 � aN�1�k; k¼ 0;…;N�15 ð33Þ

and

An ¼ bnN and γn ¼ ðNþn�Mþ1Þα ð34Þ

The computation can be done using a table similar tothe Routh table that is used to test the stability of linearsystems. The table starts by creating the first two lines.In the first we put the symmetric of the denominatorcoefficients, in the second we put coefficients of thenumerator inserting zeros, if necessary, to have polyno-mials of equal order.

The upper

letters me an su perscr ipts, n ot powe rs.

�aN ¼ �1

�aN�1 … … … �a1 �a0 bN0

b0N�1

… … … b01 b00 A0 ¼ b0N

To continue, assume that we want to compute the nth line:

1.

multiply the first line by An�1 ¼ bn�1N ;

2.

add to the nth line; 3. discard the first coefficient.

If one (or several) zero appear in the beginning of theline, we slide the values to the left. This means that apower of sα that is equal to the number of slides plus 1.

Let us see an example.

Example 1.

H sð Þ ¼ 1s2αþ3sαþ2

¼ 1sαþ1

� 1sαþ2

This is a transfer function studied in [12]. By directinversion the corresponding impulse response is easilycomputed to be given by the expression

h tð Þ ¼ ∑1

2ð�1Þn 2n�1�1

h itnα�1

ΓðnαÞ ε tð Þ

Using the new procedure that we just described, we areable to iteratively obtain the same result. The followingtable presents the first five iterations of the procedure,applied to H(s):

�1

�3 �2 0 0 1 A0¼0 We must slide the line 2 columns 1 0 0 A0¼1 γ0 ¼ 2α

�3

�2 0 A1 ¼ �3 γ1 ¼ 3α 7 6 0 A2 ¼ 7 γ2 ¼ 4α

�15

�14 0 A3 ¼ �15 γ3 ¼ 5α 31 30 0 A4 ¼ 31 γ4 ¼ 6α

As expected, the computed terms of the series ∑10 An

ðtγn �1=ΓðγnÞÞεðtÞ match exactly the same terms of theknown impulse response h(t). Further iterations of theprocedure yield the coefficients for the remainder terms ofthe impulse response series.

Example 2.

HðsÞ ¼ 1=ðs3þs1=3þ1Þ:This example is more difficult than the previous one as

the fraction decomposition involves a larger number ofterms of complex nature:

H sð Þ ¼ 1s3þs1=3þ1

¼ ½ðs2α�2:02sαþ1:17Þ

�ðs2α�0:94sαþ1:13Þ � ðs2αþ0:56sαþ1:04Þ�ðs2αþ1:58sαþ0:87Þ � ðsαþ0:82Þ��1jα ¼ 1=3

The impulse response may still be obtained from thepartial fraction decomposition. However, the complexity ofthe task is largely increased due to the number of terms.The algorithm implementation as a computer programmeis straightforward, allowing us to compute expansion (23)with as many terms as desired. The first 10 coefficientscorresponding to this example are presented in the

M.D. Ortigueira et al. / Signal Processing 107 (2015) 238–245244

following table where α¼ 13

� �. The full computation of

these coefficients is presented as an Appendix.

n

0 1 2 3 4 5 6 7 8 9

γn¼

9α 17α 18α 25α 26α 27α 33α 34α 35α 36α An¼ 1 �1 �1 1 2 1 �1 �3 �3 �1

6. The correspondence principle

The above development showed that the algorithm hastwo steps:

�

Expand the transfer function in negative powers of sα. � Invert the series term by term.

On the other hand the examples showed that it can beused for both integer or fractional orders. This leads us tothink of the possibility of obtaining the fractional solutionfrom the integer one (and the reverse). In fact this ispossible through what we call the correspondence principlethat we state in the following theorem.

Theorem 5 (Correspondence principle). Let H(s) be theLaplace transform of the impulse response h(t) of a causalcommensurate linear system. The correspondence principlestates the following:The substitution of sα for s in the transfer function

corresponds to the substitution of tnα�1=ΓðnαÞ fortn�1=ΓðnÞ in the series associated with the impulseresponse.

For proof, consider a given integer order transfer function,H(s), of a causal exponential order function, h(t). According tothe development presented in Section 4, we can write

HðsÞ ¼ ∑1

1hðkÞð0þ Þs�k�1

As shown previously the successive coefficients are com-puted using the initial value theorem and hðtÞ is obtainedthrough a term by term inversion leading to (27).

Substitute sα for s in the series for H(s) to obtain theseries of a new transfer function, HαðsÞ. The above state-ment says that the corresponding series for hαðtÞ can beobtained from the series of h(t) using the above substitu-tion. To see whether this is true, write

HαðsÞ ¼ ∑1

1Aks

�αk

Its inversion gives

hα tð Þ ¼ ∑1

1Ak

tnα�1

ΓðnαÞε tð Þ

As known,

Dα tmα�1

ΓðmαÞ

�¼ tðm�1Þα�1

Γððm�1ÞαÞ

The results presented in Section 2.2 show that whencomputing the limit as t goes to zero, all the terms of an

order less than m go to infinite and so they must beremoved as we did in the computation of the series forHðsÞ. The terms with an order greater than m go to zero.We conclude immediately that Am ¼ hðm�1Þ

a ð0þ Þ.A comparison of the two series shows that with the

above substitution we obtain really the correct series.However, the coefficients must be reinterpreted: they areno longer order k derivatives, but order αk derivatives. Thecoefficients hðαkÞð0þ Þ are really independent of α. This is arather surprising result.

Let us see a simple example.The inverse Laplace transform of HðsÞ ¼ 1=ðsþaÞ for

ReðsÞ40 is given by hðtÞ ¼ e�atεðtÞ. Suppose we want tocompute the inverse of HαðsÞ ¼ 1=ðsαþaÞ for ReðsÞ40 usingthe correspondence principle. Begin by rewriting h(t) in aseries format:

h tð Þ ¼ ∑1

0ð�aÞnt

n

n!ε tð Þ ¼ ∑

1

0ð�aÞn tn

Γðnþ1Þε tð Þ

¼ ∑1

1ð�aÞn�1t

ðn�1Þ

ΓðnÞ ε tð Þ

Perform the above substitution to obtain

hα tð Þ ¼ ∑1

1ð�aÞn�1t

ðnα�1Þ

ΓðnαÞ ε tð Þ

The corresponding transform is

Hα sð Þ ¼ ∑1

1ð�aÞn�1s�nα ¼ s�α 1

1þas�α

¼ 1sαþa

¼H sαð Þ

confirming the above statement.

7. Conclusions

We presented a general approach to decompose aLaplace transform into a Laurent like series. From it weobtained a general power series. This result was applied tothe computation of the impulse response of causal linearsystems. In particular, we considered the commensuratecase and proposed a table to obtain the coefficients of theMacLaurin series. Some examples were presented.

Acknowledgements

This work was partially funded by the National Fundsthrough the Foundation for Science and Technology ofPortugal, under the Project PEst-OE/EEI/UI0066/2011 andby project MTM2010-16499 from the Government ofSpain.

Appendix A. Transfer function decomposition forExample 2

Here we deal with the decomposition of the transferfunction HðsÞ ¼ 1=ðs3þs1=3þ1Þ ¼ 1=ðs9αþsαþ1Þjα ¼ 1=3 asseries of the form HðsÞ ¼∑n�1

0 Aks� γk þRnðsÞ. The next table

M.D. Ortigueira et al. / Signal Processing 107 (2015) 238–245 245

presents the computation of the coefficients for the first 10terms of this series:

�1

0 0 0 0 0 0 0 �1 �1 1 0 0 0 0 0 0 0 0 0 γ0¼9α A0¼1

�1

�1 0 0 0 0 0 0 0 0 γ1 ¼ 17α A1 ¼ �1 �1 0 0 0 0 0 0 1 1 0 γ2 ¼ 18α A2 ¼ �1 1 2 1 0 0 0 0 0 0 0 γ3 ¼ 25α A3 ¼ 1 2 1 0 0 0 0 0 �1 �1 0 γ4 ¼ 26α A4 ¼ 2 1 0 0 0 0 0 �1 �3 �2 0 γ5 ¼ 27α A5 ¼ 1

�1

�3 �3 �1 0 0 0 0 0 0 γ6 ¼ 33α A6 ¼ �1 �3 �3 �1 0 0 0 0 1 1 0 γ7 ¼ 34α A7 ¼ �3 �3 �1 0 0 0 0 1 4 3 0 γ8 ¼ 35α A8 ¼ �3 �1 0 0 0 0 1 4 6 3 0 γ9 ¼ 36α A9 ¼ �1

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