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Journal of Advanced Research 25 (2020) 243–255 Contents lists available at ScienceDirect Journal of Advanced Research journal homepage: www.elsevier.com/locate/jare Beyond the particular case of circuits with geometrically distributed components for approximation of fractional order models: Application to a new class of model for power law type long memory behaviour modelling Jocelyn Sabatier IMS Laboratory, Bordeaux University, UMR CNRS 5218, 351 Cours de la liberation, 33400 Talence, France g r a p h i c a l a b s t r a c t Power law type behaviour of a r t i c l e T ð0;sÞ uð0;sÞ. i n f o Article history: Received 20 February 2020 Revised 1 April 2020 Accepted 4 April 2020 Available online 23 April 2020 Keywords: Power law type long memory behaviours Fractional models Cauer networks Foster networks Heat equation Poles and zeros geometric distributions a b s t r a c t In the literature, fractional models are commonly approximated by transfer functions with a geometric distribution of poles and zeros, or equivalently, using electrical Foster or Cauer type networks with components whose values also meet geometric distributions. This paper first shows that this geometric distribution is only a particular distribution case and that many other distributions (an infinity) are in fact possible. From the networks obtained, a class of partial differential equations (heat equation with a spatially variable coefficient) is then deduced. This class of equations is thus another tool for power law type long memory behaviour modelling, that solves the drawback inherent in fractional heat equations that was proposed to model anomalous diffusion phenomena. Ó 2020 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 It is well known that the diffusion equation of the form Peer review under responsibility of Cairo University. E-mail address: Jocelyn.sabatier@u-bordeaux.fr @/ðx; tÞ @ 2 /ðx; t Þ ¼ Df @t @x2 https://doi.org/10.1016/j.jare.2020.04.004 2090-1232/Ó 2020 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/). ð1Þ 244 J. Sabatier / Journal of Advanced Research 25 (2020) 243–255 produces power law type long memory behaviours of order 0.5 (Df is a diffusion coefficient). That is why the Warburg impedance, 1=2 , defined in the frequency domain (variable x) by Z ðjxÞ ¼ ðjxÞ was introduced to model numerous diffusion-controlled processes in many domains such as electrochemistry [44,23,42,36,3], solid-state electronics and ionics [15,41,2]. However, it is also well-known that there are processes whose behaviour cannot be modelled by the Warburg impedance as they exhibit a power law type behaviour of the form Z ðjxÞ ¼ ðjxÞ m 0 <  m 9 > s = þ 1 xh L1 Ima ðsÞ ¼ C 0  m > > s : ; xl þ 1 0  m Z xl B sinðmpÞ ¼ C0 @dðtÞ þ p xh As a first try, the following change of variable is used in relation (14): x ¼ az ¼ ezlnðaÞ 1 xh xb m ðxh  xÞ C dxA ðx  xl Þm ðs þ xÞ ð22Þ m ILb ðsÞ  a0 þ N X s k¼0 xk ak þ1 ð23Þ dx ¼ lnðaÞe thus HðsÞ can be rewritten as: Z H ð sÞ ¼ where dðtÞ is the Dirac impulse function. According to the previous comments, with a 2 Rþ  1 l ezlnðaÞ  zlnðaÞ s þ ezlnðaÞ 1 lnðaÞe Z dz ¼ 1 zlnðaÞ  lnðaÞ l ezlnðaÞ s ezlnðaÞ 1 dz: ð27Þ dz: ð28Þ  þ1 This transfer function can be approximated by: Ha ðsÞ ¼  N X lnðaÞ k¼0 l ezk lnðaÞ s ezk lnðaÞ  þ1 Dz ð29Þ with with m a0 ¼ C 0 ðxxl Þ ; h m sinðmpÞ ðxh xk Þm p ðxk xl Þm xk Dx; h xh xb ak ¼ C 0 ðxxl Þ xk ¼ xh þ kDx; Dx ¼ N : ð24Þ m Fraction expansion of IN ðsÞ in relation (4) can also be written as ImLb ðsÞ ¼ a0 0 þ N X k¼0 a0 k s x þ1 sinðmpÞ p ð25Þ lnðxmax Þ lnðaÞ Dx ¼ Þ  lnlnðxðmin aÞ N zk ¼ lnðxmin Þ þ kDz: lnðaÞ ð30Þ mzk lnðaÞ lnðaÞe Dz C k ¼ sinðmpÞ p 1 lnðaÞe ðmþ1Þzk lnðaÞ Dz ð31Þ and k C kþ1 xkm1 ¼ m1 Ck xkþ1 lnðxmin Þ lnðaÞ Such a discretisation permits the realization of Fig. 1 with: Rk ¼ A comparison of coefficients ak and a0 k is given by Fig. 4. It reveals that, for a large value of N, ak  a0 k and that the two approximations are very close. If the transfer function HðsÞ of relation (12) is considered again (for simplicity but a similar analysis can be done with the transfer functions of Table A1), relation (20) highlights that the capacitors and resistors are linked by the recurrence relations: m1 Rkþ1 xkþ1 ¼ m1 Rk xk z0 ¼ ð26Þ It can be noticed, that unlike relation (10), the ratios linking two resistors or two capacitors are not constant and depend on k. This discretization can be viewed as an alternative solution to algorithm 1, but as parameter N must be very large to have an accurate approximation on a large frequency band, it requires a very large number of components in the network of Fig. 1. Such a defect is due to the fixed step discretization of integral (11). To overcome this defect, it is possible to search for a change of variable that contracts the frequency domain, thus making the fixed step discretization more efficient. xk ¼ 1 ¼ ezk lnðaÞ : Rk C k ð32Þ If ¼ 0:4; a ¼ 10, N ¼ 10, xl ¼ 0:001 rd=s, xh ¼ 1000 rd=s, the Bode diagrams of the approximation Ha ðsÞ in relation (29) with change of variable (27) are shown in Fig. 5. They are very similar to those of Fig. 3 obtained with relation (18) and N ¼ 106 , thus showing the interest of the change of variable (27) in reducing the size of the approximation. Remark 3. Whatever the value of a, and as: zkþ1  zk ¼ lnðxmin Þ lnðxmin Þ þ ð k þ 1 Þ Dz   k Dz ¼ Dz lnðaÞ lnðaÞ ð33Þ It can be noticed that Rkþ1 emzkþ1 lnðaÞ ¼ mz lnðaÞ ¼ emDzlnðaÞ Rk e k C kþ1 1eðmþ1Þzkþ1 lnðaÞ ¼ ðmþ1Þz lnðaÞ ¼ eðmþ1ÞlnðaÞ k Ck e ð34Þ And xkþ1 1 ¼ ¼ eDlnðaÞ : Rk C k xk ð35Þ The previous relation highlights a geometric distribution of the values of resistors, capacitors and corner frequencies, defined by the following ratios: a ¼ emDzlnðaÞ g ¼ eðmþ1ÞlnðaÞ : ð36Þ This geometric distribution generalises the one introduced by Oustaloup [19,20]. The latter is indeed a particular case obtained with a ¼ 10, among the infinite number of distributions obtained for all the other values of a, and for other changes of variable that can be proposed instead of relation (27). Among this infinity, the following one is interesting as it also makes it possible to contract the frequency domain. Using the following change of variable zn ¼ x or z ¼ x1=n with n 2 N thus dx ¼ nzn1 dz ð37Þ relation (14) can be rewritten as: Fig. 4. Comparison of coefficients ak and a0 k , with N ¼2000 and xl ¼ 1 rd=s, xh ¼ 106 rd=s (zoom inside the figure). m ¼ 0:3, H ð sÞ ¼ sinðmpÞ p Z 0 1 zmn sinðmpÞ nzn1 dz ¼ s þ zn p Z 1 n 0 zmn1 dz s þ1 zn ð38Þ 248 J. Sabatier / Journal of Advanced Research 25 (2020) 243–255 Fig. 5. Bode diagram of Ha ðsÞ with change of variable (27) and comparison with the Bode diagrams of approximation (18). and permits the network of Fig. 1 with: Rk ¼ sinðmpÞ p nzkmn1 Dz; C k ¼ sinðmpÞ p 1 nzkmn1þn Dz Extension to Cauer type networks ; ð39Þ and xk ¼ 1 ¼ znk : Rk C k ð40Þ If m ¼ 0:4; n ¼ 60, N ¼ 10, xl ¼ 0:001 rd=s, xh ¼ 1000 rd=s is the Bode diagrams of the approximation Ha ðsÞ obtained by discretisation of integral (38) and change of variable (37) are shown in Fig. 6. They are compared with the Bode diagrams obtained with change of variable (27). The comparison reveals that the two changes of variable are of equivalent quality with the same complexity (N ¼ 10). As an infinity of changes of variable can be proposed, an infinity of Foster type networks can be used to generate a power law behaviour. The following section shows that Cauer networks can also generate this type of behaviour with an infinity of different distributions. The Cauer network of Fig. 7 is considered. For the geometric distribution, such as the one defined by relations (5) and (6), an analytical result can be obtained to show that a Cauer type network generates a power law behaviour. Considering the network in Fig. 7, the following relations hold 1 ðIk1 ðsÞ  Ik ðsÞÞ ¼ U k ðsÞ sC k ð41Þ and U kþ1 ðsÞ  U k ðsÞ ¼ Rk U k ðsÞ: ð42Þ From relations (41) and (42) respectively, it can be written that U k ðsÞ ¼ U k1 ðsÞ 1 þ 1 sC k 1 sC k Ik ðsÞ U k ðsÞ and Fig. 6. Bode diagram of Ha ðsÞ with the change of variable (37) and comparison with change of variable (27). ð43Þ 249 J. Sabatier / Journal of Advanced Research 25 (2020) 243–255 If the network results from an infinitesimal slicing of a continuous medium of abscissa z, the ratio of two consecutive components (capacitor or resistor) denoted Fis given by: F kþ1 F ðkdz þ dzÞdz ¼ F ðkdzÞdz Fk where dz denotes the thickness of the considered slices, with dz ! 0. Given that Fig. 7. Cauer type RC network. Ik ðsÞ ¼ U k ðsÞ 1 þ 1 Rk U iþ1 ðsÞ I i ðsÞ 1 Rk : ð44Þ I 0 ð sÞ ¼ U 0 ð sÞ 1 R1 1þ F ðkdz þ dzÞ  F ðkdzÞ ¼ F 0 ðkdzÞ dz ! 0 dz 1 sC 1 R1 1 sC R 1þ 1 12 sC 2 R2 1þ 1þ : ð45Þ F ðkdz þ dzÞ ¼ F ðkdzÞ þ F 0 ðkdzÞdz F kþ1 F 0 ðkdzÞ dz: ¼1þ F ðkdzÞ Fk 0 Rkþ1 ¼ r and Rk C kþ1 ¼ q; Ck with and if Z ðsÞ ¼ relation (45) becomes I 0 ð sÞ ¼ U 0 ð sÞ 1 þ ð46Þ Z ðsÞ 1þ : ð47Þ 1þ Introducing the function 1þ Z ðsÞ=r Z ðsÞ=rq 1þ Z ðsÞ=r2 q 1þ Z ðsÞ=r2 q2 1þ Z ðsÞ=rN qN 1þZ ðsÞ=rNþ1 qN 1þ After resolution of the differential equation (57), function F ðkdzÞ is given by z 2 ½0; 1½: ð58Þ This shows that the lineic characteristics of the discretized medium that produces the network of Fig. 7 are defined by: 1þZ ðsÞ=rNþ1 qN Z ðsÞ ð57Þ F ðkdzÞ ¼ F 0 ekf kdz Z ðsÞ=r2 q2 Z ðsÞ=rN qN 1þ g ðZ ðsÞ; r; qÞ ¼ ð56Þ F ðkdzÞ ¼ kf F ðkdzÞ Z ðsÞ=r Z ðsÞ=rq Z ðsÞ=r2 q 1þ F ðkdzÞ dz ¼ 1 þ kf dz F ðkdzÞ 0 1 R1 1þ ð55Þ If this ratio, only a function of dz, is assumed constant 8k as in relation (46) and equal to K, using relation (55), K¼1þ HðsÞ ¼ ð54Þ the ratio of relation (52) becomes Suppose that in Fig. 7, the resistors and capacitors are geometrically distributed and linked by the following ratios (as in [19]): 1 sC 1 R1 ð53Þ where F 0 ðzÞ denotes the derivative of F ðzÞ and thus Combining relations (43) and (44), it can be shown that the input admittance of the network in Fig. 7 is defined by the continued fraction: HðsÞ ¼ ð52Þ RðzÞ ¼ R0 ekR z ð48Þ and C ðzÞ ¼ C 0 ekC z z 2 ½0; 1½: ð59Þ The ratio of two consecutive resistors and capacitors is thus defined by: Rkþ1 Rððk þ 1ÞdxÞ ¼ ¼ ekR Dz RðkdxÞ Rk C kþ1 C ððk þ 1ÞdxÞ ¼ ¼ e kC D z : C ðkdxÞ Ck and relation (48) becomes HðsÞ ¼ 1 R1 I 0 ð sÞ ¼ : U 0 ðsÞ 1 þ g ðZ ðsÞ; r; qÞ ð60Þ ð49Þ If N tends towards infinity, function g ðZ ðsÞ; r; qÞ meets the following property: Property 1. Function g ðZ ðsÞ; r; qÞ meets the following relation g ðZ ðsÞ; r; qÞ ¼ 1þgðZZððssÞÞ;q;rÞ 1 R1 K ðr; qÞZ ðsÞm x 2 ½1; 1½ k and C k ¼ C ðkdxÞ ¼ C 0 ðkdxÞ C dx ð62Þ The ratio of two consecutive resistors and capacitors is thus defined by: k ð50Þ ð51Þ ð61Þ With an infinitesimal slicing of the continuous medium, the system can be characterised by the network of Fig. 7 with: kR Rkþ1 R0 ððk þ 1ÞdxÞ R dx ðk þ 1Þ ¼ ¼ k k Rk ðk Þ R R0 ðkdxÞ R dx Using theorem 1, demonstrated in Appendix A.2, for jZ ðsÞj  1, r > 1), admittance HðsÞ meets the relation HðsÞ  and C ðxÞ ¼ C 0 xkC k Theorem 1. With N ! 1 (as RðxÞ ¼ R0 xkR Rk ¼ RðkdxÞ ¼ R0 ðkdxÞ R dx Thanks to property 1, the function g ðZ ðsÞ; r; qÞ can be written under the form of a rational function with descending powers. g ðZ ðsÞ; r; qÞ ¼ Kðr; qÞZ ðsÞm 2  mkþ1  k 3 P1 Z ðsÞ þ C 2k ðr; qÞ ZrðsÞ 6 1 þ k¼1 C 2k1 ðr; qÞ r 7 4 5 P mkþ1 k 1þ 1 C ð q ; r Þ ð Z ð s Þ Þ þ C ð q ; r Þ ð Z ð s Þ Þ 2k k¼1 2k1 Now consider the change of variable z ¼ log ðxÞ, x 2 ½1; 1½, then relation (59) becomes k C kþ1 C 0 ððk þ 1ÞdxÞ C dx ðk þ 1Þ ¼ ¼ k k Ck ðkÞ C C 0 ðkdxÞ C dx and kC ð63Þ These ratios are similar to those given by relation (26) for the Foster circuit of Fig. 1. The following change of variable z ¼ log ðxn Þ, x 2 ½1; 1½, n 2 Nþ is now considered. Relation (62) thus becomes 250 J. Sabatier / Journal of Advanced Research 25 (2020) 243–255 RðxÞ ¼ R0 xnkR and C ðxÞ ¼ C 0 xnkC x 2 ½1; 1½: ð64Þ With an infinitesimal slicing of the continuous medium, the system can be characterised by the network of Fig. 7 with: Rk ¼ RðkdxÞ ¼ R0 ðkdxÞ nkR dx nkC and C k ¼ C ðkdxÞ ¼ C 0 ðkdxÞ dx: Rkþ1 ¼ ekR Dz Rk Rkþ1 ðk þ 1Þ ¼ k Rk ðkÞ R ð65Þ The ratio of two consecutive resistors and capacitors is thus defined by: nk nk and nkC C kþ1 C 0 ððk þ 1ÞdxÞ C dx ðk þ 1Þ ¼ ¼ nk nk Ck ðk Þ C C 0 ðkdxÞ C dx : ð66Þ These ratios are similar to those given by relation (39) for the Foster circuit of Fig. 1. These networks and the associated distributions are used in the next section to introduce a class of heat equation that exhibits a power law type long memory behaviour. Heat equation with spatially variable coefficients for power law type long memory behaviour modelling The following heat equation with spatially dependent parameters is now considered.   @T ðz; t Þ @ @T ðz; tÞ ¼ cðzÞ bðzÞ @t @z @z ð67Þ with z 2 Rþ . This equation is a simplified form of the equation studied in [13]. Let uðz; tÞ ¼ bðzÞ @T ðz; tÞ : @z ð68Þ Discretisation of equation (68) with a discretisation stepDzleads to: ð69Þ and thus: nkR cðzÞ ¼  Dz uðz; tÞ: bðzÞ ð70Þ Using relation (69), relation (67) can be rewritten as: @T ðz; t Þ @ uðz; tÞ ¼ cðzÞ : @t @z Spatial discretisation of Eq. (71) with a discretisation step Dz leads to: cðzÞ ¼  Dz cðkDzÞ ¼ C ðkDzÞDz and Rk ¼  ð75Þ nkC and C kþ1 ðk þ 1Þ ¼ nk Ck ðk Þ C ðrelation ð72ÞÞ 1 C 0 e kC z and bðzÞ ¼  1 R0 ekR z ð76Þ z 1 C 0 zkC bðzÞ ¼  and ð77Þ 1 R0 zkR z 2 ½1; 1½ ðrelation ð60ÞÞ ð78Þ 1 1 and bðzÞ ¼  C 0 znkC R0 znkR 2 ½1; 1½ ðrelation ð63ÞÞ cðzÞ ¼  z ð79Þ Of course, as previously explained, many other spatially varying coefficients can be obtained using other changes of variable than those proposed at the end of Section ‘Extension to Cauer type networks’. Discussions around some other distributions for further Now, among the infinity of distributions that can be obtained using changes of variable as shown in Section ‘Beyond geometric distribution’, the following is studied: or x ¼ z1=m 1 1 thus dx ¼  zm1 dz m ð80Þ Using this change of variable, relation (14) becomes: sinðmpÞ Z 0 z   1 1m1 dz  z 1 p m þ1 s þ z m Z 1 1m sinðmpÞ 1 z ¼ dz p m s þ z1m 0 H ð sÞ ¼ sinðmpÞ ð81Þ mp Z 1 0 s z 1 m 1 dz þ1 ð82Þ and permits the realization of Fig. 1 with: Ha ðsÞ ¼ PN Rk k¼0 Rk C k sþ1 Rk ¼ sinmðpmpÞ Dz ¼ Cte 1 mpzm C k ¼ sinðmpÞk ð72Þ For z ¼ kDz and if the following notations are introduced Ck ¼  ðrelation ð62ÞÞ or after simplification ð71Þ @T ðz; t Þ uðz þ dz; tÞ  uðz; tÞ ¼ cðzÞ @t Dz cðzÞ ¼ ðuðz þ dz; t Þ  uðz; t ÞÞ: Dz ð74Þ kC 2 ½0; 1½: ðrelation ð59ÞÞ H ð sÞ ¼ T ðz; t Þ  T ðz þ dz; tÞ ¼  C kþ1 ðk þ 1Þ ¼ k Ck ðkÞ C and Rkþ1 ðk þ 1Þ ¼ nk Rk ðkÞ R z ¼ x m ; T ðz þ dz; t Þ  T ðz; t Þ uðz; tÞ ¼ bðzÞ Dz kR ðrelation ð46ÞÞ and according to the relations (59), (60) and (63), the heat equation (67) exhibits a power law type long memory behaviour if (as cðzÞ ¼ 1=C ðzÞ and bðzÞ ¼ 1=RðzÞ according to relation (73)) nkR Rkþ1 R0 ððk þ 1ÞdxÞ R dx ðk þ 1Þ ¼ ¼ nk nk Rk ðk Þ R R0 ðkdxÞ R dx C kþ1 ¼ e kC D z Ck and Dz ¼ RðkDzÞDz ð73Þ bðkDzÞ discretisation of Eq. (67) thus leads to the Cauer network of Fig. 8. As C k ¼ C ðkDzÞDz and Rk ¼ RðkDzÞDz, according to relations (46), (62) and (72), the transfer function uð0; sÞ=T ð0; sÞ of the Cauer network of Fig. 8 exhibits a power law type long memory behaviour if p Dz 1 and xk ¼ R 1C ¼ zk m : ð83Þ k k If ¼ 0:4; N ¼ 10; 000, xl ¼ 0:001 rd=s, xh ¼ 1000 rd=s the Bode diagrams of the approximation Ha ðsÞ are shown in Fig. 9. They are compared with the Bode diagrams of approximation (18) and the one obtained with change of variable (37). As for approximation (18), parameterNmust be very large to have an accurate approximation of sm on a large frequency band, but the interest of this change of variable is not there. The distribution of resistors and capacitors of relation (83) is now used to build the Cauer network of Fig. 7, with m ¼ 0:4, N = 1000, Dz ¼ 2 and 251 J. Sabatier / Journal of Advanced Research 25 (2020) 243–255 (0,t) T(0,t) R1 (z,t) C1 (k z,t) Rk ((k+1) z,t) Ck T((z,t) Rk+1 T(k z,t) Ck+1 Fig. 8. Cauer type RC network resulting from the discretization of relation (67). Fig. 9. Bode diagram of Ha ðsÞ of relation (83) with change of variable (80) and comparisons with the Bode diagrams of approximation (18) and the one obtained with change of variable (37). C0 ¼ mp : sinðmpÞ ð84Þ The resulting Bode diagram of the transfer function I0 ðsÞ=U 0 ðsÞ is represented by Fig. 10. This diagram shows yet again that a power law behaviour can be obtained without a geometric distribution of resistors and capacitors. In this circuit, all the resistors have the same values and the capacitors are linked by the following relation 1 1 C kþ1 ððN  k  1ÞDzÞm ðN  k  1Þm ¼ ¼ : 1 1 Ck ððN  kÞDzÞm ðN  kÞm ð85Þ This class of components distribution, that cannot be deduced using a change of variable in relation (59), and the resulting class of spatially varying coefficients in relation (67) will be studied by the author in future work. Fig. 10. Bode diagram of transfer function I0 ðsÞ=U 0 ðsÞ of the Cauer type RC network with distribution of relation (83). 252 J. Sabatier / Journal of Advanced Research 25 (2020) 243–255 Conclusion h1 ðt Þ ¼ This paper shows that an infinity of – pole and zero distributions (frequency modes) in classical integer transfer functions, – passive component value distributions (such as capacitors or resistors) in Foster type networks, can generate power law type long memory behaviours. Hence, the geometric distributions [19,20] often encountered in the literature are a particular case among an infinity of distributions. For the Foster type network the proof is easy to establish using several changes of variables, as this network results directly from the discretisation of a filter transfer function that exhibits a power law behaviour. The proof for the Cauer type network is more tedious and is developed in the paper. Due to the close link between Cauer type networks and heat equations (through discretisation), this paper also shows the ability of heat equations with a spatially variable coefficient to have a power law type long memory behaviour. This class of equation is thus another tool for power law type long memory behaviour modelling that solves the drawback inherent in fractional heat equations. This class of equation will be more deeply studied by the author. Finally, this paper shows, without proof, that other distributions and thus heat equations with spatially variable coefficients also exhibit power law type long memory behaviours. Moreover, by increasing the number of components in each branch of the Cauer network, it is possible to keep a power law behaviour, which suggests that there are a very large number of partial differential equations, other than the heat equations, which can produce a power law type long memory behaviour, some were already proposed in [27]. With reference to other papers recently published by the author [33,35], this work is a new contribution to the dissemination of models not based on fractional differentiation but which exhibit power law type long memory behaviours. Z 1 2pj cþj1 H1 ðsÞets ds with c > xl : For the computation of integral (A1.2), path C ¼ c0 [ ::: [ c7 of Fig. A11 is considered with c > xl . This path bypasses the negative axis around the branching point z ¼ xl and z ¼ xh for t > 0. It thus avoids the complex plane domain for which the transfer function H1 ðsÞ is not defined, i.e. the segment ½xh ; xl . On path C, the radii of sub-path c1 and c7 tend towards infinity, and the radius of sub-path c4 tends towards 0. Using Cauchy’s theorem with c > xl : h1 ðt Þ ¼ þ 1 2pj X Z cþj1 H1 ðsÞets ds ¼  cj1 1 2pj Z H1 ðsÞets ds Cc0 ðA1:3Þ Res H1 ðsÞets : poles in C Since Res½H1 ðsÞets  ¼ 0; ðA1:4Þ operator H1 ðsÞ being strictly proper, by Jordan’s lemma integrals on the large circular arcs of radius R, R ? 1 can be neglected: Z c1 þc7 H1 ðsÞets ds ¼ 0 ðA1:5Þ Let s ¼ xejp , x 21; xh  on c2 and thus ds ¼ ejp dx. Let also s ¼ xejp , x 2 ½xh ; 1½ on c6 and thus ds ¼ ejp dx. Then Z H1 ðsÞets ds ¼ c2þ c6 xl m þ m1 xh Z 1 xh xl m xh m1 Z xh  jp m1 xe þ xh m 1 ðxejp þ xl Þ ext ejp dx m1 ðxejp þ xh Þ xt jp m e e dx ¼ J c2 þc6 ðtÞ ðxejp þ xl Þ Compliance with ethics requirements This article does not contain any studies with human or animal subjects. Declaration of Competing Interest The author has declared no conflict of interest. Appendix A.1. Impulse response of some transfer functions that exhibit power law type long memory behaviours The approximations given in Section ‘Beyond geometric distribution’ are made on the integral form of the impulse response of the transfer function HðsÞ ¼ s1m . The methodology used to derive the approximations and the change of variable used in Sections ‘Beyond geometric distribution’ and ‘Extension to Cauer type networks’ can be extended to other transfer functions. The following one is now considered:  s xh H1 ðsÞ ¼ C 1  þ1 s xl m1 þ1 m   2 1 xl with C 1 ¼   2 1 xh þ1 þ1 The impulse response of H1 ðsÞ is defined by 2m : m1 2 ðA1:2Þ cj1 ðA1:1Þ Fig. A1.1. Integration path considered. ðA1:6Þ