Extending the Concept of Analog Butterworth Filter for Fractional Order Systems

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1 Extending the Conept of Analog Butterworth Filter for Frational Order Sytem Anih Aharya a, Saptarhi Da b, Indranil Pan, and Shantanu Da d a) Department of Intrumentation and Eletroni Engineering, Jadavpur Univerity, Salt Lake Campu, LB-8, Setor 3, Kolkata , India. b) Communiation, Signal Proeing and Control Group, Shool of Eletroni and Computer Siene, Univerity of Southampton, Southampton SO17 1BJ, United Kingdom. ) MERG, Energy, Environment, Modelling and Mineral (E M ) Reearh Setion, Department of Earth Siene and Engineering, Imperial College London, Exhibition Road, London SW7 AZ, United Kingdom. d) Reator Control Diviion, Bhabha Atomi Reearh Centre, Mumbai , India. Author anihaharya91@gmail.om (A. Aharya) aptarhi@pe.jul.a.in,.da@oton.a.uk (S. Da*) indranil.jj@tudent.iitd.a.in, i.pan11@imperial.a.uk (I. Pan) hantanu@magnum.bar.gov.in (Sh. Da) Abtrat Thi paper propoe the deign of Frational Order (FO) Butterworth filter in omplex w-plane (w= ; being any real number) onidering the preene of under-damped, hyper-damped, ultra-damped pole. Thi i the firt attempt to deign uh frational Butterworth filter in omplex w-plane intead of omplex -plane, a onventionally done for integer order filter. Firtly, the onept of frational derivative and w-plane tability of linear frational order ytem are diued. Detailed mathematial formulation for the deign of frational Butterworth-like filter (FBWF) in w-plane i then preented. Simulation example are given along with a pratial example to deign the FO Butterworth filter with given peifiation in freueny domain to how the pratiability of the propoed formulation. Keyword: Butterworth filter; frational order filter; frational order linear ytem; w-plane tability; analog filter deign 1. Introdution In reent year, frational alulu i being widely ued in modeling the dynami of many real life phenomena due to the fat that it ha higher apability of providing aurate 1

2 deription than the integer dynamial ytem. Thi added flexibility of FO ytem i mainly due to the fat that pole in higher Riemann heet whih often ontribute to the important dynamial feature of phyial ytem an eaily be analyzed and modeled uing thi tehniue e.g. the onideration of hyper-damped and ultra-damped pole et. Thu due to greater flexibility in modeling, more aurate analyi and ontrol deign, frational derivative have been ueful in many field like vio-elatiity, aouti, rheology, polymeri hemitry, biophyi et. [1]. The appliation of frational order ytem a deign element have motly been ued in ontrol and ignal proeing ommunity a FO ontroller and FO filter [], [3]. Though different frational order ontroller like CRONE, PI λ D μ, FO lead-lag ompenator have already been famou, there are very few attempt on ytemati deign of FO filter. Previou attempt on FO analog filter deign have motly been retrited in the generalization of the firt order and eond order analog Infinite Impule Repone (IIR) tranfer funtion template a frational order tranfer funtion by Radwan et al. [4], [5]. But the author in [4], [5] did not onider the pole loation and the nature of the pole in higher Riemann heet whih i inevitable while dealing with frational order linear dynamial ytem. Other relevant approahe of deigning frational eond order filter an be found in Li et al. [6]. Initial effort in thi diretion an be een in Soltan et al. [7] with frational apaitor and indutor baed Butterworth filter deign. Our preent paper i the firt attempt to extend the onept of aigning pole of a frational order Butterworth filter lying on a irle in omplex w-plane with onideration of it tability uing well known Matignon theorem [8]. A detailed analyi regarding the tability of different frational order ytem an be found in Radwan et al. [9]. The pole of uh a filter will omprie of everal ae like o-exitene of under-damped, hyper-damped and ultra-damped pole whih ha not been invetigated yet. Several other approahe in frational filter deign may inlude tep filter [10], [11], different inuoidal oillator [1], [13], multi-vibrator iruit [14] et. The Butterworth filter i one of the mot popular analog filter deign paradigm, firt deribed in 1930 by Stephen Butterworth [15]. The bai philoophy of the onventional or integer order analog Butterworth filter i well pratied in variou appliation. It i deigned to have a freueny repone a flat a it i poible. The freueny repone of thee filter i monotoni and the harpne of the roll-off from pa band to top band i determined by the order of the filter. For onventional Butterworth filter the pole aoiated with the magnitude uared funtion are eually ditributed in angle on a irle in the omplex -plane around the origin and having radiu eual to the ut-off freueny ( ). When the ut-off freueny and the filter order are peified, the pole an be obtained readily and from the pole poition the tranfer funtion of the filter an eaily be obtained. Now, while deigning a Butterworth filter we generally have four peifiation, the pa band freueny ( p ), top band freueny ( ), maximum allowable pa band and top band attenuation ( αp, α ). The motivation of deigning a frational order Butterworth filter intead of the onventional integer order Butterworth filter i highlighted in thi paper in the light of optimally finding the filter order to meet thee deign peifiation rather than deigning an over-peified filter. A diued earlier, given thee peifiation when a Butterworth filter i to be deigned, the optimum order whih i apable of meeting thee peifiation may be a fration whih i onventionally raied to the nearet integer value due to unavailability of filter order being a fration. Thi i jut imilar to adding a bit more fator of afety in the deign or over-

3 atifying the deign peifiation a the tandard pratie wa to opt for a filter with next integer value. Hene it i obviouly deirable neither to under atify nor to over atify the deign peifiation whih an eaily be met with the onept of frational order filter with tunable parameter. With further enhanement of the realization tehniue for frational order filter, it will be poible to atify the exat reuirement aording to the given peifiation. The other reaon behind opting for frational order Butterworth filter over the integer order one i the flexibility of deign with the pole in w-plane, a it i well known that the -plane deign i a ub-et of the frational order deign onidering the poibility of exiting filter pole in different Riemann heet. Although there are everal work on Butterworth filter deign in pat but it modifiation with the onept of frational order derivative i new. By doing o, the onventional -plane tability need to be modified firt for frational ytem and we here ytematially propoed the way of deigning uh filter by onidering the w-plane tability given by Matignon theorem [8] for frational order ytem. In a reent paper Soltan et al. [7], the deign of the frational order Butterworth filter ha been reported. However the approah adopted in [7] ha been uite different from the one propoed in our preent paper. In [7] the frational Butterworth filter deign ha been attempted from a iruit theory point of view, where two paive element have been taken a Z1 = β Land Z = 1 C α and thu deigned the Butterworth filter following the onventional iruit deign priniple. However, the number of table pole differ from the expeted one. For example, in the pole loation data provided in [7], a 0.7 order filter having two pole on the irle ha been demontrated. However, it i undertandable from [16], [17] that the filter order being 0.7 we would expet 7 pole ditributed over 10 Riemann heet. Thu the formulation of FO Butterworth filter ha been preented in a different way in order to generalize the deign tehniue with tate of the art analyi methodologie onerning the tability of frational order linear ytem. It i to be noted that in [7], the number of pole ha been onidered to be four whih i a peial ae of frational eond order Butterworth filter wherea many other variant may alo exit and ha been generalized and illutrated with deign example in the preent paper. In thi paper the frational Butterworth filter ha been formulated in the w-plane ( w=, + ). Thu all poible type of pole (under-damped, hyper-damped, ultra-damped) have been taken into aount. An analytial formulation of the filter i done in the w-plane and ertain ondition for the loation of the table and the untable pole are derived. The eene of laial Butterworth filter i kept unhanged and it i found that even in ae of the frational order formulation, the pole are plaed along the irumferene of a irle whoe radiu i eual to the ut-off freueny. Thu, by the propoed formulation all the table pole have been taken into aount. The orreponding tranfer funtion are obtained in w-plane and are then mapped 1 bak into the -plane uing the relation = w. The orreponding repone urve are obtained to onfirm the maximally flat Butterworth nature. Sine thi i one of the firt approahe toward filter deign in the w-plane, ome generi iue related to frational order filter deign need to be diued. It i known from the elementary knowledge of frational order linear ytem that a filter having an order PQ implie that P number of pole are ditributed over Q Riemann heet. However, thi reult i a eriou iue related to the order of the filter. Say the order of 0.9 ha been wrongly etimated a 3

4 0.99 or or that a firt order ytem ha been wrongly etimated a So, it i worth onidering whether inreae in the number of pole and number of Riemann heet for apturing deliate freueny domain information i atually reuired, ine thi would lead to higher deign omplexity [16], [17]. For a 0.9 order filter, the propoition will reuire 9 pole ditributed over 10 Riemann heet, however if the order i etimated to be 0.99, 99 pole ditributed over 100 Riemann heet are to be handled,whih will be umberome to realize in hardware. Thi reulting omplexity would make it diffiult to generalize filter in frational domain and the advantage of greater flexibility will be outweighed by fabriation ot. So, a deign heme whih onidered the relative trade-off between omplexity and auray ha been adopted. Thu for a filter whoe overall order i a fration, it i better to realize it only up to the firt deimal plae (by trunating the trailing deimal) and then inreaing the auray by intuitive judgment for improving the freueny domain harateriti. There are many different onnotation related to the term filtering. In the laial ene, filter deign refer to FIR or IIR truture whih are deigned to atify ome freueny domain peifiation like ut-off freuenie for the pa-band or top band, maximum allowable ripple in magnitude repone et. Thi notion of filtering i onidered in the preent paper. The other notion of filtering may be from the point of view of the theory of linear etimation [18]. In uh ae the filter deign reult in ome FIR/IIR truture whih minimize ome performane objetive in time domain, e.g. Wiener filter. The third paradigm of filtering i a model baed tate etimation or parameter etimation tehniue like the Kalman filter, extended and unented Kalman filter et. Thi onept ha been augmented to obtain different enhaned performane objetive and for varied omplex ytem with H filtering and partile filtering [19]. In [0], diue the latet development in thi ategory uing fuzzy logi. Thi ha been an ative reearh topi for the lat few deade. But in reent year, with the advent of frational alulu in ytem deign, our notion of filtering for all the three above mentioned paradigm need to be modified in thi light. Our preent work an be een a a firt tep for ytemati frational filter deign with the onideration of tability for frational order ytem. Time domain filter like fuzzy [0] or H filtering [1] reuire ome ort of a ytem model. However, in ae of onventional IIR/FIR filtering, the tehniue i applied on the freueny petrum of the ignal diretly. Hene the need of a ytem model i eliminated. In ae where the ignal i orrupted by random diturbane or noie, linear etimation baed filtering tehniue work well [18]. But in many indutrial proee there i a learly identified freueny band of interet (or pa band), and ret of the freueny petrum i not important. In uh ae, imple FIR/IIR filtering tehniue are preferred over model baed filter to redue deign omplexity. All thee tohati filtering tehniue reuire additional parameter for their deign unlike laial FIR/IIR filter deign. For example, mixing propertie (additive/multipliative nature), tatitial propertie of the noie before mixing, parameterized linear/nonlinear model of the ytem generating the ignal, upper bound of the diturbane petra et. are reuired depending on the peifi deign tehniue. We propoe here a new paradigm of deigning frational Butterworth filter to meet exat deign reuirement whih i not poible in onventional integer order filter deign. We have alo hown that integer order Butterworth filter will alway under-atify or over-atify the deign peifiation. The propoed tehniue i a new way of looking at analog filter deign with frational alulu baed tehniue. The magnitude roll-off an be adjuted to any deired 4

5 lope, whih an only be ahieved uing frational order ytem. While dealing with frational order tranfer funtion, the w-plane tability onept or Matignon theorem hould be ued intead of the onventional -plane tability. It i to be noted that in reent year everal effort have been there to propoe new automati ontroller deign uing the onept of w-plane tability for example []-[4]. Thi i the firt attempt to extend the onept of tability in automati ontrol deign to analog Butterworth filter deign uing w-plane tability notion for FO ytem. Exiting tate of the art deign method would inreae the order of the IIR filter to meet deign peifiation or to get a maller tranition band. Sine frational order ytem are inherently infinite dimenional, they naturally model the exat magnitude roll-off. The order of realization an be hoen by the uer after the exat order of the filter i alulated from the freueny domain peifiation. The ret of the paper i organized a follow. Setion review the fundamental of frational alulu reuired to undertand the mathematial formulation of the frational order Butterworth filter. In Setion 3 the mathematial formulation for the realization of frational Butterworth filter i introdued. In Setion 4 imulation reult are preented. Setion 5 illutrate a realiti example of deigning a Frational order Butterworth Filter from given peifiation. The paper end in Setion 6 with the onluion and ome omment about the future ope in thi field.. Bai of frational alulu baed linear dynamial ytem The bai of frational alulu that ha been ued in thi paper, are diued here a the bakground of the filter deign tehniue..1 Funtion ued in frational alulu Some important funtion in relation to frational alulu are a follow: i. Gamma funtion u z 1 ( ) Γ z = e u du, z (1) For omplex z the real part ha to be finite to get a finite value of the gamma funtion. 0 ii. Beta funtion (, ) (1 ) p,, Bp = u u du p + () 0 The relation between the gamma funtion and the beta funtion i Γ( p) Γ( ) Bp (, ) =. Γ ( p + ). Definition of frational differ-integral The two popular definition of frational differ-integration are a follow [1] 5

6 i. Grunwald-Letnikov (G-L) Definition: Thi i baially an extenion of the bakward finite differene formula for ueive differentiation. Thi formula i ued widely for the numerial olution of frational differentiation or integration of a funtion. Aording to Grunwald-Letnikov definition theα th order differ-integration of a funtion f ( t ) i defined a α 1 j Γ ( α + 1) Dt f( t) = lim ( 1) f( t jh) h 0 α (3) h Γ ( j+ 1) Γ( α j+ 1) ii. j= 0 Riemann-Liouville (R-L) Definition: Thi i an extenion of n-fold ueive integration. Theα th order integration of a f t i defined a funtion ( ) and imilarly the a t f τ τ α α (4) α + 1 t τ a α 1 ( ) It f( t) = d for a,, < 0 Γ ( α) ( ) th α order differentiation of a funtion f ( t ) i defined a n t α 1 d f( τ ) a t n α n+ 1 Γ( n α) dt ( t τ) a D f( t) = dτ for n 1 < α < n (5).3 Stability region in w-plane 1 Q Generally, for the multi-valued funtion defined a w= = where Q N there are Q heet in the Riemann urfae. The etor π / Q< arg( wk ) < π / Qorrepond to the firt Riemann heet and k i the number of pole. Thu when pole of any frational order ytem in -plane are mapped into w-plane, all the Riemann heet are taken into aount. Hene all the different kind of pole howing different dynamial behavior are onidered, epeially the ultra-damped and hyper-damped pole [5]. The table region i now ontrained within the region [8] arg ( wk ) > π (6) Table 1 how the different pole type and their orreponding region in the w -plane. Table 1: Type of pole depending on their poition in the w-plane Pole poition Comment on pole nature arg( w) < π / Untable π / < arg( w) < π Stable under-damped π < arg( w) < π Stable hyper-damped arg( w) = π Stable ultra-damped 6

7 3. Propoal for a new frational order Butterworth (FBW)-like filter In thi etion, a frational Butterworth like filter ha been propoed uing the onept of w-plane tability. The pole are loated on the w-plane whih onit of all the pole lying on different higher Riemann heet. Then the tability riterion i taken under onideration and all the untable pole are rejeted and a frational order BW like filter an be obtained. The onept i omewhat imilar to the onventional notion of finding integer BW filter pole lying on a irle and then diarding the right-half plane pole in order to avoid intability. The FBW filter i developed on the line of the integer order Butterworth filter with the onideration of the tability of frational linear ytem [6]. Theorem 3.1: The pole loation of a FO Butterworth low-pa filter of any real order PQ, { PQ, } + j(k 1) π /P given by wk =± j Ce, where k = 1,,, P and the ondition for tability of the root i π given by arg ( wk ) >, where = 1 Q i the ommenurate order and P i the number of pole ditributed over Q Riemann heet. Proof: The uared magnitude funtion of an analog Butterworth filter i given by the form are H( j ) = 1 1+ N (7) where, N i the order of the filter. Now onidering = j = j, the uared magnitude ytem funtion i of the form HH () ( ) = 1 1+ j N (8) For frational order filter the magnitude uared funtion would be written in term of frational power of Laplae variable = w. Then we have the magnitude uared funtion in w-plane a So if PQ, { PQ, } + H( wh ) ( w) = 1 w 1+ C be the order of the filter where = 1 Q, +, where i the ommenurate order hene we now have P number of pole ditributed over Q Riemann heet. Therefore the deign of PQ order FBW like filter i imilar to deign a P th order integer P (9) 7

8 Butterworth filter with the pole eually ditributed on the irumferene of a irle in the omplex w-plane while only diarding the untable pole uing the relation (6). So in w -plane we find the pole loation from the harateriti polynomial P w 1+ = 0 C (10) Thi i being propoed a the Butterworth like polynomial whih ha been expreed a a π j(k 1) P funtion of w intead of. Hene the pole loation are found to be wk =± j Ce, where k = 1,,, P. and the riterion for table pole i the ame a (6). The reult are illutrated and the nature of the pole are invetigated for variou ae. Let P = 1 then the pole are loated at wk = ± j Ce = ± C. The pole loated within the region ± π are untable, hene it i obviou that w = C orrepond to untable ytem. th Therefore, we onider only w = C and thu the tranfer funtion of ( 1 Q) order filter and it mapping bak to the -plane i found to be C C H( w) = H() = (11) w Therefore, here we got a ingle ultra-damped pole for the FBW filter. C 4 Now when P = i onidered, the pole loation are found to be w=± j e, 3π 4 ± jce whih may be table depending on the ommenurate order. Therefore, the tranfer funtion beome 4 C H() = (1) ( )( + ) π j C C C C It i intereting to note that for laial eond order Butterworth filter, eem to have the ame truture a in (1), exept that the untable pole are in the right half w-plane (yielding negative oeffiient of tranfer funtion). For imilar laial deign i.e., = 1, the term ( C C ) ued to be diarded a they repreent untable omplex onjugate in - plane. But for frational order ytem, thee pole may be table and thu mut be inluded in the deign depending on the repetive tability riteria. Therefore frational order tranfer funtion with negative oeffiient with pole lying in the right half part of w-plane, but outide the intability region, i an effetive way for the development of filter having any arbitrary real order. Now the nature of the pole i dependent on the ommenurate order a diued in etion.3. On analyzing the nature of the root for different ombination of P and, i.e. for different order of the filter, the nature of the pole need to be judged for poible effet in the filtering ation uing the notion given in Table 1. For impliity, the pole loation of the C C π j 8

9 frational filter are deigned for overall order le than unity. In other word, for a frational Butterworth like filter with order greater than unity, the integer and the frational part are to be deigned eparately and then need to be aaded together. Thi i important due to the fat that P for order greater than unity (e.g. order N+, N +, Q> P, { PQ, } + ), the number of pole Q NQ + P number of pole ditributed over Q Riemann heet). in w-plane inreae (i.e. ( ) th Therefore, it i expedient to deign the N order BW filter uing onventional notion of integer order BW filter and then deigning the ( PQ ) th order filter uing the propoed tehniue. It i important to mention that there hould not be any ommon fator in{ PQ, }, in order to avoid unneeary inreae in the number of pole and number of Riemann heet. The deign approah preented here i valid for frational filter having any arbitrary real order, repreented by a rational fration le than unity with no ommon fator among the numerator and denominator. Thi i beaue the generalization of finding untable pole among the ( NQ + P) number of pole i umberome. In Table the pole loation onidering = 1 rad/e and Q> P, Q= [ 1,,, P 1] ha been reported. A argued above, the real part of the pole are poitive in few ae, though they may repreent table ytem depending on the value of Q whih determine the angle of the untable region. Therefore it an be een that for a frational filter of order ( PQ ) < 1, P repreent the number of pole in the omplex w-plane irrepetive of the value of Q and the Q play the role of hooing only the table pole by defining the intability region uing relation (6). From Table it an be een that the filter will have ombined dynami of under-damped, hyper-damped pole and ultra-damped pole. Table : Pole loation for different frational order Butterworth polynomial in w-plane Q Untable region P Stable pole loation π 1-1 arg ( w k ) > arg ( ) w k π > i, i, i, i 4 arg ( ) w k π > i, i, i, i 9

10 i, -1, i, i, i i, i, i, i w k 5 arg ( ) π > i, -1, i, i, i i, i, i, i, i, i, i, i In Table 3 the table frational order tranfer funtion of the Butterworth like filter in w- plane i hown in term of the ut-off freueny. For frational order filter deign, the peifiation on hould be uitably tranformed o a to math the w tranformation. If it be onidered that the irle having a radiu a the ut off freueny in -plane, then uing the relation w= the tranformed radiu ( = ) in w-plane beome < for > 1 +, ( 0,1) (13) > for < 1 Table 3: Stable frational order tranfer funtion uing the onept of Butterworth filter in w-plane P H() P =1 P = P = 3 ( )( ) ( )( )( )

11 P=4 P=5 P=6 P=7 P=8 P=9 ( )( )( )( ) ( )( )( )( )( ) ( 1.93 )( )( ) ( )( 1.93 )( ) ( 1.80 )( 1.47 )( )( 1.80 ) ( 1.47 )( )( ) ( 1.96 )( 1.96 )( )( )( ) ( )( )( ) ( )( )( 1.53 )( 1.53 ) ( )( )( )( )( ) In other word, the radiu of the irle on whih the pole of the Butterworth filter lie, get redued or amplified depending on the utoff freueny being higher or lower than unity repetively. In Table 3 the tranfer funtion have been expreed in term of the tranformed radiu a the pole are thoe belonging to the w-plane, inluding the higher Riemann heet. th Similar to the onventional integer P order Butterworth filter tranfer funtion, P i varied from 1 to 9. Thee tranfer funtion have a imilar template to the original Butterworth filter, but have the following differene: i. i replaed by w= ii. i replaed by = iii. Integer BW pole are loated only in the left half -plane. For frational BW like filter, ame et of pole ome in the right half w-plane a a mirror image with repet to the imaginary w-axi. 3.1 Remark Though the above mentioned formula i expeted to be uite generalized however there i a mall iue to be taken into aount. Firtly, thi i a trade-off deign, i.e. trunation of the higher deimal plae i needed to redue omplexity of frational filter realization beaue otherwie plaement of a large number of pole and deriving their loation in different Riemann heet annot be handled with eae. Therefore, we trunated the filter order to make a deign 11

12 with only up to the firt deimal plae and then improving the auray by intuitive judgment for improving freueny domain harateriti. Seondly, another thing that i to be noted i the fat that the above mentioned theorem will not hold true in ae of filter order being an irrational fration. For example an overall order of 3 annot be realized uing the above mentioned theorem. In uh ae a trunated verion of the reurring deimal need to be onidered. 4. Simulation and reult Here different ommenurate order have been hoen and orrepondingly the number of pole ditributed over different number of Riemann heet ha alo been onidered to be different. The nature of the pole for all thoe ae i invetigated. Initially the ommenurate order i hoen to be Q = 10 and the order of the repetive filter ha been obtained by varying P and thu onidering all poible filter order from 0.1 to 0.9 while the loation and the nature of the pole are alo tudied aordingly. By nature of the pole, it i implied how many pole are table, among the table pole how many lie in the under-damped, hyper-damped or ultradamped region. Then imulation are added in order to verify the behavior of the FBWF deigned uing the propoed mathematial formulation. Similarly all poible ae for different ommenurate order Q = to 9 have alo been taken into aount and orrepondingly all poible different frational order of the Butterworth filter are tudied and then imulated to get an overall idea about their repone to unit tep exitation. Uing theorem 3.1 the imulation for time and freueny repone of different filter have been done and reported here, after diarding the untable pole and therefore onidering only the table one in the filter tranfer funtion. 4.1 Comparion of freueny repone of frational Butterworth filter Figure 1 how the freueny repone of the FBWF for variou ombination of PQ, P= { 1,,, Q 1}. From thee four et of freueny repone it an be oberved that the magnitude and the phae roll-off dereae with the inreae in the value ofq. Similarly a P inreae (and hene the order of the FBWF), the magnitude and phae roll-off inreae, for ame value of Q. However the magnitude and phae roll off are not a fat a the integer order filter. Thi typial phenomenon help to meet deign peifiation preiely and obtain an aurate filtering ation a reuired. In the imulation example, the term order of the FBWF repreent the value P, determining the number of pole in the w-plane, whih i independent of the ommenurate order Q. 1

13 Figure 1: Bode diagram of variou order of Frational Butterworth like Filter with unit ut-off freueny for different value of Q. 4. Step Repone omparion Figure how the time repone of the frational order BW filter whoe Bode diagram are hown in Figure 1. In eond order FO BW filter due to the o-exitene of under-damped and hyper-damped pole made the urve to have oillatory behavior for Q = 3. Wherea, in third order FBWF the o-exitene of under-damped, hyper-damped and 1 ritially damped pole made the tep repone to have loal oillation (due to the under-damped pole) but not let the tep repone magnitude exeed unity at all time. A we know that uh ultra-damped pole aue extreme luggihne in the tep repone [5] but exhibit loal oillation intead of a monotoni tep repone due to having two under-damped pole for Q = 3. Similarly for the fourth order FBWF, o-exitene of two under-damped pole and many hyper-damped pole make the ytem oillatory for Q = 3 to 7. For further higher value of Q, ine the intability region and hene the under-damped region of the primary Riemann heet beome more narrow, a a reult all the pole lie in the hyper-damped region indiating a luggih tep repone. 13

14 Figure : Step repone of variou order of frational Butterworth like filter with unit ut-off freueny for different value of Q. 4.3 Pole loation of the FBW filter in omplex w-plane Figure 3 how the pole loation of the FBW filter in the w-plane for different ommenurate order. The firt order ( P = 1) FBWF ha one ultra-damped pole and i alway table irrepetive of the value of Q. The preene of the ultra-damped pole implie a very luggih time repone when the filter i ubjeted to a unit tep input. A eond order ( P = ) FBWF ha two under-damped and two hyper-damped pole whih are alway table for all integer value of Q. It an be een that two of the pole lie on the right half w-plane, but they are till outide the angular etor of intability ± π. The third order FBWF filter ha two underdamped, two hyper-damped and one ultra-damped pole. Similar to the previou ae, it i alway table for all integer value of Q. The fourth order FBWF ha four under-damped and four hyper-damped pole. However unlike the previou ae, the tranfer funtion given in Table 3 i only table for Q> 3. For Q= and Q= 3, one untable pair of pole hould be exluded to obtain the filter tranfer funtion. The fifth order ae ha four under-damped, four hyper-damped pole and one ultra-damped pole. It i table for Q>. For Q=, the one untable pair of pole mut be exluded to obtain a table filter tranfer funtion. For the ixth order ae ix under-damped and ix hyper-damped pole are there. The filter i table for Q> 5. For Q= to 5, the one untable pair of pole mut be exluded to obtain the filter tranfer funtion. The eventh order ae ha ix under-damped, ix hyper-damped pole and one ultra-damped pole. The filter i table for Q> 3. For Q= and Q= 3, the one untable pair of pole mut be exluded to obtain the filter tranfer funtion. In the eighth order ae, there are eight underdamped and eight hyper-damped pole. The filter i table for Q> 7. For Q= to 5, the one untable pair of pole mut be exluded to obtain the filter tranfer funtion. It i alo oberved 14

15 that another pair of pole fall within the intability region for Q=. Thu for the eighth order FBWF with Q=, two pair of untable pole have to be exluded and the reulting tranfer funtion ha to be obtained. For ninth order BW filter a muh impler ae an be oberved where only one et of untable pole have to be exluded for Q= to 4. Thi filter ha the ame number of under-damped and hyper-damped pole for Q= a in the eighth order ae, in addition with an ultra-damped pole. It i alway table for Q> 4. It i to be noted that odd order of FBWF yield an ultra-damped pole whih produe high luggihne in the tep repone, wherea, with the even order uh oillation are avoided. A mentioned above, it an be een that for higher value of P (higher than 3), ome of the onjugate pair are untable for low Q but table for other higher value of Q. In uh ae the tability hek uing (6) mut be done and the untable pair of pole need to be diarded. Figure 3: Pole zero map of variou order of Butterworth filter in omplex w-plane for different Q value. It i alo intereting to note that the pole of the frational order filter are eui-paed in omplex w-plane and alo table whih ha not been done by ontemporary reearher. A onventionally followed in the frational order ontrol ommunity, we mut onider the w-plane tability notion or Matignon theorem to verify that the pole of the filter fall in the table region and alo are eually ditributed in w-plane. 5. Deign advantage of frational Butterworth filter Generally, when a filter i to be deigned it i expeted to meet ertain freueny domain deign peifiation. There are four major peifiation to be dealt with, while deigning an analog Butterworth filter. The freueny for the end of the pa band ( p ) and the tart of the top band ( ) are peified and hene the tarting and ending freuenie of the tranition band 15

16 are alo peified beforehand. Then the maximum pa band attenuation ( α ) and minimum top band attenuation ( α ) are alo peified. In order to deign a BW filter whih meet all thee peifiation, the reuired order of the filter need to be hoen firt [6]. It an eaily be obtained that given thee peifiation the exat order of the filter in term of thee peifiation i given a euation (14) p 0.1α 10 1 log 0.1α 10 p N = 1 log p (14) Alo, the ut-off freueny ( ) an be alulated from the peifiation of the topband ( ) uing the following relation. = ( α 1 ) 1 N (15) Obviouly it an be dedued that the value of N annot alway take integer value. In mot ae it i more likely to take frational value. So in the traditional integer order ae, we had to over atify the deign need. For example, if a filter of order 1.5 wa reuired in order to exatly meet the given peifiation, a eond order filter had to be deigned. Thi an be overome by frational filter of an order exatly eual to 1.5. A mentioned before in order to deign a filter of order 1.5, a 0.5 order filter an be aaded with a firt order filter. One realiti deign example i illutrated here and upported with MATLAB baed imulation tudie. 5.1 Realiti Deign example Suppoe given the peifiation that the maximum pa band attenuation be 0.5dB and minimum top band attenuation be 0 db. The pa band and top band edge freuenie are given a rad/e and 3 rad/e. It i deired to realize a frational order Butterworth like filter uing the propoed tehniue. Hene a from the given peifiationα p = 6 db, α = 0 db, p = rad e, = 3 rad e [6]. Thu the deired order of the filter reuired i N = Trunating up to one deimal plae the order of the filter an be approximated a N = 4.3. But from the onvention of integer order BW filter, we had to over atify the reuirement and deign a filter of order 5. But uing the propoed onept of FBW filter, we an now deign a BW filter having an integer and a frational part, having an exat order of 4.3. In the deign proe, the nominal frational part ha been repreented a a ratio of two integer with no ommon fator i.e. PQ= 3 10 for our ae. Here it i worth notiing that a filter of order 4.3 ould have been deigned whih ha 43 pole ditributed over 10 Riemann heet. But in uh ae the number of untable pole would have inreaed and a uh no generalization an be poible like that reported in Table and Table 3. Therefore in order to keep impliity, it i preferable that the nominal integer and frational part are deigned eparately uing the repetive onept of 16

17 deigning BW polynomial in orreponding -plane or w-plane and ontrut filter uing table pole only and then aade thee two part. Here, we have taken a imilar approah and put a FBWF of order 0.3 in aaded with a 4 th order onventional BW filter. N = 4 For the integer order part ( N = 4 ) it i found uing (15) that = rad/e. For onventional integer order deign the next integer value i ued i.e. N = 5and in that ae the N = 5 ut-off freueny = rad/e. For integer order deign the onventional 4 th and 5 th order Butterworth filter tranfer funtion are HN ( ) HN ( ) = 4 = = 5 = (16) (17) For the frational order part, the ut-off freueny get modified due to the mapping of the radiu from -plane to w-plane a =, where, i the obtained from onventional third order P Butterworth filter deign a C = with P = 3. The 1 obtained here orrepond 0.1α P P (10 1) to the -plane. Thi ha to be tranformed into the w-plane in order to deign frational filter and will thu beome =. It ha been found that for P = 3, i greater than 1. Thu the irle on whih the pole lie, will get ueezed on tranformation from the -plane to the w-plane uing E. (13). The nominal frational part of the BW like filter i found out to be of the form a given in E. (18) uing the tandard notion of deigning a third order BW filter but in the tranformed w- plane. HN ( ) = 0.3 = (18) The aimed filter deign of 4.3 order would be the aaded verion of two filter derived in E. (16) and E. (18). It i intereting to note that all the pole of the frational and integer part in E. (16) and E. (18) are deigned in uh a way o a to enure the ame ut off freueny in - plane and w-plane by meeting the ame et of deign riterion. The upper and lower bound (i.e. the integer order) are plotted next in Figure 4 along with the exat FBWF deign of order 4.3. The amplitude and phae of the propoed BW like filter aymptotially lie between that orreponding to the deign with the upper and lower bound in the high freueny region. Figure 4 how that the propoed FBWF aymptotially fall between the nearet upper and lower integer order Butterworth filter deign. Though at lower freuenie the magnitude and phae repone of the FBWF of order 4.3 i low ompared to the repetive integer order ae but for higher freuenie they follow a trajetory whih lie between 4 th and 5 th order laial BW filter. A oberved in the magnitude uared repone in Figure 5, the FBWF d gain eem to be different from the integer order filter. Thi peuliar 17

18 behavior i oberved due to the preene of ultra-damped and hyper-damped pole in the filter tranfer funtion where the time repone traking of an input tep ommand take infinite time, even though the frational order tranfer funtion ha a d gain of unity a reported in Table 3. Thi heavily luggih time repone, in pite of mathematially yielding zero teady-tate error (from the final value theorem) appear a a relatively low gain at low freuenie. Figure 4: Comparion of the Bode diagram for the integer order BW filter and the FBW like filter. 5. Few diuion on the new deign tehniue Our imulation reult i optimum in the ene that we deigned a filter of the deired order whih exatly atifie the deign peifiation and neither under-atifie nor overatifie it. In order to how the effet of rounding off the reommended order for Butterworth filter deign by the loet integer value jut higher and lower than the exat one i hown in the Bode magnitude and magnitude uared plot for the filter freueny repone. Thi paper propoe a new type of frational order deign and hene advane theoretial apet of olving the problem. The hoie of the filter order an be done in future work by minimizing ome performane objetive and uing intelligent optimization algorithm a done in [7]. Alo, after obtaining the FBWF, eah FO operator an be approximated uing welletablihed analog or digital realization tehniue []. Sine the main goal of the FBWF deign i to math the magnitude repone, the well-known Charef method [8] may be eaily applied ine thi tehniue mathe the magnitude roll-off with mall pieewie linear operator. For digital realization the FO filter may be expanded uing Continued Fration Expanion (CFE) and Power Serie Expanion (PSE) after diretizing the filter uing Euler, Tutin or Al-Alaoui rule [9]. In the preent paper, the attempt wa to have analog domain realization of FBWF. We adopted the approah of deigning in w-plane and then retrofitting into -plane. The z-plane diretization i a further development, where the frational order theory of tability ha to be 18

19 extended, in a new way. Thi diretization, in z-plane, will not be o diret-vi-à-vi w-plane and need further exploration. Alo, a an extenion of the preent work, the analog FBWF an be tranformed to deign high-pa, band-pa and band-top filter from the low-pa deign, a done onventionally, ine the frational order operator are linear in nature. Figure 5: Magnitude uared funtion for the IO/BW filter deign example. 6. Conluion A new kind of frational Butterworth like filter ha been deigned uing the onideration of pole lying on a irle in the tranformed w-plane and alo uing onept of tability in omplex w-plane. The obtained FBWF are analyzed from the nature of pole, a well a in the time and freueny domain. The new frational filter deign tehniue i apable of meeting the deign peifiation in an exat manner rather than under or over atifying it. Thi ha been poible by making the filter order exatly a per the deign reuirement without rounding-off to the nearet upper or lower integer. A pratial deign example ha been given to how the pratiability of the propoed deign tehniue. Future ope of reearh an be direted toward extending the preent onept for direte domain filtering tehniue whih ha higher flexibility of realization uing digital ignal proeor. Alo, the onept an be extended for other la of FO IIR filter and optimization of the filter order with optimization algorithm a done in [7] an alo be done in future. Referene: [1] Shantanu Da, Funtional frational alulu, Springer, 011. [] Saptarhi Da and Indranil Pan, Frational order ignal proeing: introdutory onept and appliation, Springer,

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