Bilinear Double-Order Filter Designs and Application Examples

Recei ed 7 Janua y 2024, accep ed 17 Janua y 2024, da e o publica ion 22 Janua y 2024, da e o cu en e sion 30 Janua y 2024. Digi al Objec Iden i ie 10.1109/ACCESS.2024.3357092 Bilinea Double-O de Fil e Designs and Applica ion Examples JULIA NAKO 1, (G adua e S uden Membe , IEEE), COSTAS PSYCHALINOS 1, (Senio Membe , IEEE), FABIAN KHATEB 2,3,4, AND AHMED S. ELWAKIL 5,6,7, (Senio Membe , IEEE) 1Depa men o Physics, Elec onics Labo a o y, Uni e si y o Pa as, Rio, 265 04 Pa as, G eece 2Depa men o Mic oelec onics, B no Uni e si y o Technology, 601 90 B no, Czech Republic 3Facul y o Biomedical Enginee ing, Czech Technical Uni e si y in P ague, 272 01 Kladno, Czech Republic 4Depa men o Elec ical Enginee ing, Uni e si y o De ense, 662 10 B no, Czech Republic 5Depa men o Elec ical Enginee ing, Uni e si y o Sha jah, Sha jah, Uni ed A ab Emi a es 6Depa men o Elec ical and So wa e Enginee ing, Uni e si y o Calga y, Albe a, AB T2N 1N4, Canada 7Nanoelec onics In eg a ed Sys ems Cen e (NISC), Nile Uni e si y, Giza 12677, Egyp Co esponding au ho : Cos as Psychalinos ([email p o ec ed]) This wo k was suppo ed by HEAL-Link. ABSTRACT A no el kind o non-in ege o de bilinea il e s, named double-o de bilinea il e s, is in oduced in his wo k. They a e based on he employmen o wo non-in ege o de s, o e ing he maximum design lexibili y in compa ison wi h hei ac ional-o de and powe -law coun e pa s. An a ac i e o e ed bene i is ha his is achie ed wi hou inc easing he ci cui complexi y, since he p oposed s uc u e is capable o ealizing all non-in ege kinds o il e s. Two design examples a e p o ided, whe e i is shown ha lead/lag compensa o s u ilized in con ol applica ions and low/high shel ing il e s employed in acous ic applica ions a e ac ually bilinea il e s wi h sui ably selec ed pole/ze o equencies. Simula ion and expe imen al esul s, using he O CAD PSpice simula o and a Field P og ammable Analog A ay de ice, espec i ely, suppo he indings o his wo k. INDEX TERMS Analog il e s, bilinea il e s, compensa o s, cu e- i ing app oxima ion, ield p og ammable analog a ay, ac ional-o de il e s, powe -law il e s, shel ing il e s. I. INTRODUCTION The e m "bilinea il e " is used o he cha ac e iza ion o il e s which a e exp essed as a a io o wo linea unc ions. The ans e unc ion o a i s -o de bilinea il e is HIO(s)=GL·yτs+1 xτs+1,(1) wi h x,y>0 being dimensionless scaling ac o s, τbeing a ime cons an , and GLbeing he low- equency gain o he il e . Employing ac ional calculus, he ans e unc ion o a ac ional-o de bilinea il e is HFO(s)=GL·y(τs)α+1 x(τs)α+1,(2) The associa e edi o coo dina ing he e iew o his manusc ip and app o ing i o publica ion was And ea De Ma cellis . wi h 0 < α < 1 being he o de o he il e . The ans e unc ion o a powe -law bilinea il e o o de 0< β < 1 is gi en by HPL(s)=GL·yτs+y0 xτs+x0β .(3) Meanwhile, he s anda d i s -o de low-pass and high-pass il e unc ions a e di ec ly de i ed om (1), by se ing y= 0 and x=y, espec i ely. Fo ms o ac ional-o de bilinea il e unc ions ha e been ealized in [1],[2],[3], and [4], while he co esponding ealiza ion o powe -law ones ha e been p esen ed in [3]. Bo h he a o emen ioned kinds o il e s o e imp o ed design lexibili y wi h ega ds o hei in ege -o de coun e - pa s, because o he a iable non-in ege o de o he il e s, which allows he adjus men o he main cha ac e is ics o hei equency beha io . 14040 2024 The Au ho s. This wo k is licensed unde a C ea i e Commons A ibu ion-NonComme cial-NoDe i a i es 4.0 License. Fo mo e in o ma ion, see h ps://c ea i ecommons.o g/licenses/by-nc-nd/4.0/ VOLUME 12, 2024 J. Nako e al.: Bilinea Double-O de Fil e Designs and Applica ion Examples In his wo k, a double-o de bilinea il e unc ion is in oduced whe e wo deg ees o eedom a e o e ed because o he employmen o wo o de s, ins ead o a single o de in ac ional-o de and powe -law il e s. This enables ha ing ull con ol o he cha ac e is ics o he il e . This wo k is an ex ension o he wo k p esen ed in [5]. Two possible implemen a ions a e demons a ed, wi h he i s one based on he employmen o a single Cu en Feedback Ope a ional Ampli ie (CFOA) as he ac i e elemen , which also o e s he capabili y o ealizing all non-in ege o de unc ions by he same RC ne wo k, and jus adjus ing he alues o esis o s and capaci o s. The second implemen a ion is based on he u iliza ion o a Field P og ammable Analog A ay (FPAA) de ice, which o e s design p og ammabili y and e sa ili y in he sense ha all kinds o (non-in ege o de ) bilinea il e s can be implemen ed by e-p og amming he cha ac e is ics o he in e media e s ages. This wo k is o ganized as ollows: a sys ema ic e iew o he bilinea il e ans e unc ions, p esen ed in he li e a- u e, is pe o med in Sec ion II. The p oposed gene alized bilinea il e ans e unc ion is in oduced in Sec ion II, whe e i s possible implemen a ions a e also discussed. Two applica ion examples a e p o ided in Sec ion IV, and he e alua ion o he pe o mance o he esul ing schemes is pe o med h ough simula ion esul s, ob ained wi h he employmen o he O CAD PSpice sui e and h ough expe imen al esul s using he FPAA AN231E04 de ice om Anadigm [6]. II. BILINEAR FILTER TRANSFER FUNCTIONS A. INTEGER-ORDER BILINEAR FILTERS Conside ing he exp ession in (1), he ime cons an is associa ed wi h a cha ac e is ic equency ω0acco ding o he o mula: τ=1/ω0, and he pole and he ze o a e loca ed in he le -hal o he s-plane wi h hei magni udes being ωp=1 xτ=ω0 x, ωz=1 yτ=ω0 y.(4) Acco ding o (4), he pole (ωp) and ze o (ωz) equencies a e no symme ically loca ed a ound he cha ac e is ic equency ha ing (in loga i hmic scale) a dis ance equal o xand y, espec i ely. Using (4), he ans e unc ion in (1) can be al e na i ely w i en as in (5) HIO(s)=GLωp ωzs+ωz s+ωp .(5) The gain a high equencies (GH) ends o GH=GLωp ωz=GLy x.(6) Se ing s=jωin (5), he de i ed gain and phase esponses a e gi en by (7a)–(7b), espec i ely |HIO(jω)| = GL· u u u u 1+ω ωz2 1+ω ωp2,(7a) HIO(jω)= an−1(ω/ωz)− an−1(ω/ωp).(7b) Gene ally, he knee equencies o he il e a e calcula ed om (7a) by se ing he alue o he gain o a desi ed le el. The mos common le el is he ±3dB le el and his will be employed in wha ollows. In o de o simpli y he analysis, i is assumed ha he pole and ze o o he il e a e sepa a ed in such a way ha hey independen ly de e mine he beha io o he il e . The asymp o ic (Bode) beha io o he equency esponse is de e mined by he ela i e sepa a ion be ween he pole and he ze o o he il e . This will be also assumed he eina e o all he ypes o il e s which will be conside ed. Type-I: ωz> ωp(x>yand GL>GH), hen he gain esponse has a cons an alue equal o GLun il he lowe knee equency ωL, which is equal o he pole equency, and s a s mono onically dec easing un il he highe knee equency ωH, which is equal o he ze o equency. A e his equency, i eaches a cons an alue equal o GH, due o he e ec o he ze o. The e o e, ωL=ωp=ω0 x, ωH=ωz=ω0 y.(8) Type-II: ωp> ωz(x<yo GL<GH), hen he gain esponse has a cons an alue equal o GLun il he lowe knee equency ωL, bu now becomes equal o he ze o equency, and hen i mono onically inc eases un il eaching he highe knee equency ωH, which is now equal o he pole equency, eaching a cons an alue equal o GH. Thus, ωL=ωz=ω0 y, ωH=ωp=ω0 x.(9) De ining he geome ic mean o he knee equencies as he mean equency (ωm) ωm≡√ωL·ωH,(10) hen, using (8) o (9) and (10), i is eadily ob ained ha he ela ionship be ween he mean equency and he cha ac e is ic equency ω0is ωm=ω0 √xy.(11) The gain a he mean equency (Gm) is de ined by (12) Gm≡| HIO(jωm)|= pGLGH,(12) which is equal o he geome ic mean o he gains a low and high equencies. I is now clea ha he ela ionship be ween he gains a low (GL) and high equencies (GH), and he gain a he mean equency (Gm), is GL=Gm x y,GH=Gm qx y .(13) The e o e, he gains o he il e GLand GH(in dBs) a e equally spaced a ound Gm. VOLUME 12, 2024 14041 J. Nako e al.: Bilinea Double-O de Fil e Designs and Applica ion Examples FIGURE 1. Bode plo s o he gain esponses o he Type-I (blue) and Type-II ( ed) bilinea il e s, wi h no a ion o hei mos impo an equency cha ac e is ics. The phase a his equency (Φm) eaches i s mini- mum/maximum alue gi en by Φm≡HIO(jωm)=sin−1 1−x y 1+x y!.(14) In o de o acili a e he eade , he Bode plo s associa ed wi h Type-I and II a e demons a ed in Fig.1, whe e he a o emen ioned equency cha ac e is ics a e depic ed. B. FRACTIONAL-ORDER BILINEAR FILTERS The pole and ze o o he ac ional-o de il e unc ion in (2) can be exp essed as ωp=1 x1/ατ=ω0 x1/α , ωz=1 y1/ατ=ω0 y1/α ,(15) whe e he dis ance be ween he pole and he ze o is con olled by he o de o he il e . The ans e unc ion in (2) can be al e na i ely w i en as HFO(s)=GLωp ωzα ·sα+ωα z sα+ωα p ,(16) wi h he gain a high equencies being GH=GLωp ωzα =GLy x.(17) Se ing sα=ωα·[cos(0.5απ)+jsin(0.5απ)]in (16), he gain and phase esponses o he il e a e gi en by |HFO(jω)| = GL· u u u u 1+ω ωz2α +2ω ωzαcos(0.5απ) 1+ω ωp2α +2ω ωpαcos(0.5απ) , (18a) HFO(jω)= an−1sin(0.5απ) ωz ωα+cos(0.5απ) − an−1sin(0.5απ) ωp ωα+cos(0.5απ).(18b) Thus, he asymp o ic beha io will be as ollows: Type-I: ωz> ωp(x>y,GL>GH), hen he il e ’s gain esponse has simila beha io as i s in ege -o de coun e pa . The di e ence is ha he low and high knee equencies a e no equal o he pole/ze o equencies. They depend on he o de o he il e , and hey a e gi en by ωL=ω0 x1/α ·hp1+cos2(0.5απ)−cos(0.5απ)i1/α , (19a) ωH=ω0 y1/α ·hp1+cos2(0.5απ)+cos(0.5απ)i1/α . (19b) Type-II: ωp> ωz(x<yo GL<GH), and he equency beha io is simila o ha o he Type-II in ege -o de il e s. The knee equencies a e gi en by (19a)–(19b) a e xand y in e changing. Using (15), i is eadily ob ained he ela ionship be ween he mean equency (ωm) and he cha ac e is ic equency ω0= 1/τ, gi en by (20) ωm=ω0 (√xy)1/α .(20) The ela ionship be ween he low and he high equency gains wi h he gain a he mean equency is also gi en by (13) making hem equally spaced a ound he gain a he mean equency. The phase a he mean equency eaches i s mini- mum/maximum alue calcula ed by (21) 8m= an−1sin(0.5απ) qx y+cos(0.5απ)− an−1sin(0.5απ) qy x+cos(0.5απ) . (21) C. POWER-LAW BILINEAR FILTERS The pole and ze o loca ions o he il e in (3) a e de e mined by (4). Thus, he ans e unc ion in (3) becomes HPL(s)=GLωp ωzβs+ωz s+ωpβ .(22) A high equencies, he gain ends o he alue GH=GLωp ωzβ =GLy xβ .(23) The gain and phase esponses o he il e a e |HPL(jω)| = GL·   1+ω ωz2 1+ω ωp2   β/2 ,(24a) HPL(jω)=β·h an−1(ω/ωz)− an−1(ω/ωp)i.(24b) The asymp o ic beha io o he il e is as ollows: Type-I: ωz> ωp(x>yand GL>GH), wi h he gain esponse ha ing he same beha io as ha o i s ac ional- o de coun e pa . Again, he knee equencies a e no equal 14042 VOLUME 12, 2024 J. Nako e al.: Bilinea Double-O de Fil e Designs and Applica ion Examples o he pole/ze o equencies and hey depend on he o de o he il e wi h hei associa ed exp essions gi en by ωL=ω0 x·p21/β −1, ωH=ω0 y·1 √21/β −1.(25) Type-II: ωp> ωz(x<yand GL<GH), whe e he knee equencies a e gi en by (26) ωL=ω0 y·p21/β −1, ωH=ω0 x·1 √21/β −1.(26) The exp ession o he mean equency, de i ed using (25)–(26) is he same as ha which co esponds o he in ege - o de case, i.e., (11). The phase, is calcula ed om Φm=β·sin−1 1−x y 1+x y!,(27) and his is he minimum/maximum alue. III. PROPOSED GENERALIZED (DOUBLE-ORDER) BILINEAR FILTERS A. FILTERS CHARACTERISTICS The ans e unc ions o he in ege -o de , ac ional-o de , and powe -law bilinea il e s can be gene alized acco ding o he ollowing o m HDO(s)=GL·y(τs)α+1 x(τs)α+1β ,(28) wi h 0 < α, β ≤1 being he o de s o he il e . Acco ding o (28), he in ege -o de , ac ional-o de , and powe -law bilinea il e s co espond o α=β=1, β=0, and α=0, espec i ely. The pole and ze o o he il e a e gi en by he exp ession in (15) and, he e o e, he ans e unc ion in (28) can be e- o med as HDO(s)=GLωp ωzαβ sα+ωα z sα+ωα p!β .(29) Hence GL GH=ωz ωpαβ =x yβ .(30) The esul ing gain and phase esponses a e desc ibed by (31a)–(31b) |HDO(jω)|=GL    u u u u 1+ω ωz2α +2ω ωzαcos(0.5απ) 1+ω ωp2α +2ω ωpαcos(0.5απ)     β , (31a) HDO(jω)=β· an−1sin(0.5απ) ωz ωα+cos(0.5απ) − an−1sin(0.5απ) ωp ωα+cos(0.5απ).(31b) TABLE 1. F equency cha ac e is ics o double-o de bilinea il e s. The shape o he asymp o ic beha io s o he il e is he same as ha o Type-I and Type-II kinds, wi h he knee equencies gi en by he exp essions in (32a)–(32b) ωL=ω0 x1/α ·hp21/β −1+cos2(0.5απ)−cos(0.5απ)i1/α , (32a) ωH=ω0 y1/α ·hp21/β −1+cos2(0.5απ)−cos(0.5απ) i−1/α , (32b) o he Type-I and wi h he same exp essions o Type II a e in e changing xand y. The exp ession o he mean equency o he il e is also gi en by (20), while he phase eaches i s mini- mum/maximum alue calcula ed by (33) 8m=β· an−1sin(0.5απ) qx y+cos(0.5απ) − an−1sin(0.5απ) qy x+cos(0.5απ).(33) The mos impo an equency cha ac e is ics o he gene alized bilinea il e a e summa ized in Table 1. I mus be men ioned a his poin ha he equency cha ac e is ics o he in ege -o de il e s (α=β=1), ac ional-o de (β=0), and powe -law (α=0) bilinea ile s could be eadily ob ained om his Table. In o de o demons a e he design lexibili y o e ed by he double-o de il e , o a gi en se o alues {x,y,GL, ω0} he con ol o he equency cha ac e is ics o he il e is desc ibed in Table 2. Conside ing he ex a deg ees o eedom {α, β} in he case o he double-o de il e , i is e iden om his Table ha he i e cha ac e is ics a e con olled by i e pa ame e s, o e ing he highes possible eedom o he designe . B. REALIZATION OF THE PROPOSED GENERALIZED BILINEAR FILTER 1) MINIMUM ACTIVE COMPONENT COUNT REALIZATION Le us conside he s uc u e in Fig.2a whe e a CFOA has been chosen as he ac i e elemen [7]. The ealized ans e VOLUME 12, 2024 14043 J. Nako e al.: Bilinea Double-O de Fil e Designs and Applica ion Examples TABLE 2. Con ollabili y o he equency cha ac e is ics o in ege -o de (I-O), ac ional-o de (F-O), powe -law (P-L) and double-o de (D-O) bilinea il e s. FIGURE 2. (a) CFOA based gene alized s uc u e o implemen ing in ege and non-in ege o de bilinea il e unc ions, (b) RC ne wo k o implemen ing Z2 o Type-I o Z1 o Type-II in ege -o de il e s, and (c) Caue -I RC ne wo k o implemen ing Z2 o Type-I o Z1 o Type-II gene alized non-in ege o de bilinea il e s. unc ion is H(s)=Z2 Z1 .(34) Assuming ha he impedance Z2is ealized by he ne wo k in Fig.2b, i s alue is gi en by Z2(s)=R2 R1C1s+1 (R1+R2)C1s+1.(35) Using (34)–(35) and conside ing ha Z1=R3, hen he ollowing ans e unc ion is eadily ob ained HIO−I(s)=R2 R3 R1C1s+1 (R1+R2)C1s+1.(36) Compa ing (1) and (36), i is de i ed ha : GL=R2/R3, and x/y=1+R2/R1>1. The e o e, his opology implemen s he Type-I in ege -o de bilinea ans e unc ion. The alues o xand ydepend on he de e mina ion o he ime cons an . Fo example: •Assuming ha τ=RC1, hen x=(R1+R2)/Rand y=R1/R, wi h Rbeing an a bi a y alue esis o . •Assuming ha τ=τz=R1C1(i.e., equal o he ime cons an associa ed wi h he ze o equency), hen x= 1+R2/R1and y=1. •Assuming ha τ=√τpτz=√R1(R1+R2)C1(i.e., equal o he geome ic mean o he ime cons an s associa ed wi h he pole and ze o equencies), hen x=1/y=√1+R2/R1. This choice es ablishes ha he mean equency will be equal o he cha ac e is ic equency (ωm=ω0) and ha he pole and ze o equencies will be symme ically loca ed a ound he mean equency. The co esponding Type-II il e s a e implemen ed by in e changing he posi ion o he ne wo k ha implemen s Z2 in he p e ious case wi h he posi ion o R3. As a esul , and since (34) is s ill alid, he ans e unc ion becomes HIO−II (s)=R3 R2 (R1+R2)C1s+1 R1C1s+1,(37) whe e i is ob ious ha x<y, and he a o emen ioned possible choices o he cha ac e is ic equency a e s ill alid. The ealiza ion o he co esponding ac ional-o de Type-I and Type-II bilinea il e s could be pe o med by subs i u ing he capaci o in Fig.2b by i s ac ional- o de coun e pa . The app oxima ion o i s impedance Zα=1/Cαsαcan be pe o med by Fos e o Caue RC ne wo ks [8]. Howe e , he ealiza ion o he powe -law and double-o de il e s can no be pe o med by his way, due o he p esence o non-in ege o de s ha a e no di ec ly associa ed wi h Laplacian ope a o s. The e o e, Type-I bilinea non-in ege o de il e (i.e., ac ional-o de , powe -law, and double-o de ) will be eal- ized by assuming ha Z2(s)=RHDO(s)=RGLy(τs)α+1 x(τs)α+1β ,Z1=R,(38) while o he Type-II Z2=R,Z1(s)=R HDO(s)=R GLx(τs)α+1 y(τs)α+1β .(39) Employing a 3 d-o de app oxima ion and using he cu e- i ing based app oxima ion employed also in [9], he e- quency dependen impedances in (38)–(39) a e app oxima ed by he ans e unc ion in (40) Zapp ox (s)≃B3s3+B2s2+B1s+B0 s3+A2s2+A1s+A0 ,(40) wi h Aiand Bj(i=0,1,2,j=0,1,2,3)being posi i e and eal coe icien s. Conside ing, o example, he Caue -I ne wo k demons a ed in Fig.2c, he con inued ac ion expansion o (40) akes he o m Zapp ox (s)=q0+1 q1s+1 q2+1 q3s+1 q4+1 q5s+q6 ,(41) and he design equa ions will be gi en by (42) R0,c=q0Ci,c=qiRj,c=qji=1,3,5. . . j=2,4,6 (42) whe e qi(j)a e he coe icien s o he con inued ac ion expansion in (41). 14044 VOLUME 12, 2024 J. Nako e al.: Bilinea Double-O de Fil e Designs and Applica ion Examples 2) PROGRAMMABLE REALIZATION The p og ammabili y o he p oposed bilinea il e unc ions can be achie ed as ollows: s a ing om he ans e unc ion in (28) and u ilizing he cu e- i ing based app oxima ion (as in he p e ious Sec ion), he esul ing ans e unc ion has he o m Happ ox (s)≃D3s3+D2s2+D1s+D0 s3+C2s2+C1s+C0 ,(43) wi h Ciand Dj(i=0,1,2,j=0,1,2,3)being also posi- i e and eal coe icien s. The ans e unc ion in (43) can be implemen ed by a mul i- eedback s uc u e desc ibed by he ans e unc ion in (44) CFLF (s)= G3s3+G2 τ1s2+G1 τ1τ2s+G0 τ1τ2τ3 s3+1 τ1s2+1 τ1τ2s+1 τ1τ2τ3 .(44) The scaling ac o s and he ime cons an s a e calcula ed by equa ing he coe icien s o (43) and (44). The ans e unc ion in (44) can be implemen ed using Ope a ional T ansconduc ance Ampli ie s (OTAs) as ac i e elemen s, wi h hei small-signal elec onically con olled ansconduc ance pa ame e used o implemen ing he scaling ac o s and ime cons an s [7]. Ano he al e na i e is he u iliza ion o an FPAA de ice such as he Anadigm AN231E04 de ice, whe e he p og ammabili y is achie ed h ough he u iliza ion o he swi ched-capaci o echnique [10],[11]. IV. APPLICATION DESIGN EXAMPLES The ans e unc ions o in ege -o de , ac ional-o de , and powe -law compensa o s a e he ollowing HIO,C(s)=GL·τs+1 xτs+1,(45) HFO,C(s)=GL·(τs)α+1 x(τs)α+1,(46) HPL,C(s)=GL·τs+1 xτs+1β .(47) The e o e, compensa o s a e a special case o bilinea il e s wi h y=1. In he case ha x>1, his is a Type- I compensa o known as lag-compensa o , while o x< 1 he esul ing Type-II compensa o is known as lead- compensa o [1],[2],[12],[13],[14],[15],[16],[17],[18], [19],[20],[21]. The co esponding exp essions o shel ing il e s a e HIO,SF (s)=HSF (s)=GL· τs x+1 xτs+1,(48) HFO,SF (s)=GL· (τs)α x+1 x(τs)α+1,(49) HPL,SF (s)=GL·τs x+1 xτs+1β .(50) Consequen ly, shel ing il e s a e a special case o hei co esponding bilinea il e coun e pa s, wi h x=1/y. Thus, he case o Type-I shel ing il e s co esponds o x>1, and hese a e known as low-shel ing il e s, while o x<1, he esul ing Type-II shel ing il e s a e known as high- shel ing il e s [3],[22],[23],[24]. Concluding, shel ing il e s and compensa o s a e di - e en aspec s o he same co e, which is a bilinea il e . In o he wo ds, ha ing a ailable a bilinea il e s uc u e, i can beha e like a shel ing il e o a compensa o by choosing sui able alues o he pole/ze o equencies. The only di e ence be ween shel ing il e s and compensa o s is ela ed o he conside ed equency ange; shel ing il e s a e employed o applica ions in he acous ic ange (i.e., 20Hz–20kHz), while compensa o s a e employed in con ol applica ions in he ange o Hz. In o he wo ds, he di e ence is in he loca ion o he mean equency o , equi alen ly, in he loca ion o he cha ac e is ic equency ω0. The ans e unc ions o he p oposed compensa o s and shel ing il e s will be HDO,C(s)=GL·(τs)α+1 x(τs)α+1β ,(51) HDO,SF (s)=GL·"(τs)α x+1 x(τs)α+1#β ,(52) espec i ely, o e ing he a o emen ioned bene i s in e ms o design lexibili y and ci cui complexi y. A. DESIGNS OF COMPENSATORS Assuming o ins ance ω0=10 ad/s, he ange o app oxima ion (10−2ω0,10+2ω0), and R=10k, hen he alues o passi e elemen s ( ounded o he E96 se ies de ined in IEC 60063 s anda d) which a e equi ed o implemen ing he conside ed Type-I non-in ege o de compensa o s wi h GL=20dB and x=1/y=3.162, a e summa ized in Table 3a. The co esponding alues in he case o Type-II compensa o s wi h GL=0dB and x=1/y=0.3162, a e p o ided in Table 3b. The alues o he passi e elemen s {R1,R2,R3,C1}, which co espond o he case o in ege - o de compensa o s, a e {10k, 100k, 10k, 3.01µF} and {1.1k, 10k, 10k, 28.7µF} o Types-I and II, espec i ely. The pe o mance o bilinea compensa o s is e alua ed using he O CAD PSpice sui e, wi h he AD844 disc e e componen biased a ±10V employed as CFOA. Using he componen alues in Table 3, he simula ed esponses a e depic ed in Fig.3, whe e he heo e ical plo s a e also p o ided by dashes. The mos impo an pe o mance cha ac e is ics o he non-in ege o de il e s a e sum- ma ized in Table 4, accompanied by he heo e ically p edic ed alues gi en in pa en heses. The co espond- ing esul s in he case o Type-I in ege -o de compen- sa o s a e 3.13(3.19) ad/s, 27.43(31.29) ad/s, 9.3(10) ad/s, 10(10)dB, and −55.9o(−54.9o), while o he Type-II he esul s a e 3.22(3.19) ad/s, 32.98(31.29) ad/s, 10.3(10) ad/s, 10.2(10)dB, and 54.3o(54.9o). VOLUME 12, 2024 14045 J. Nako e al.: Bilinea Double-O de Fil e Designs and Applica ion Examples TABLE 3. Values o he RC-ne wo k in Fig.1c o implemen ing (a) Type-I, and (b) Type-II non-in ege o de compensa o s. FIGURE 3. Simula ed gain and phase esponses o (a) Type-I and (b) Type-II in ege -o de , ac ional-o de , powe -law, and double-o de compensa o s implemen ed using he s uc u e in Fig.2. FIGURE 4. Time-domain beha io o (a) Type-I, and (b) Type-II double-o de compensa o s s imula ed a hei mean equency by a 2Vp-p sinusoidal inpu . The ime-domain beha io is e alua ed in he case o Type-I and II double-o de compensa o s. Fo his pu pose, hey a e s imula ed by a 2Vpeak- o-peak sinusoidal signal a hei mean equency. Acco ding o he wa e o ms in Fig.4a, he gain and he phase di e ence be ween he ou pu and inpu wa e o ms a e equal o {12.1dB, −32.9o} o he Type-I, wi h he associa ed heo e ical alues being {12dB, −33.09o}. The simula ed alues in he case o he Type-II compensa o , de i ed om Fig.4b, a e {8.04dB, 33.6o} close o he heo e ically p edic ed ones {8dB, 33.08o}. TABLE 4. F equency esponse pe o mance cha ac e is ics o (a) Type-I, and (b) Type-II non-in ege o de compensa o s. FIGURE 5. Mon e-Ca lo analysis esul s abou he mean equency o (a) Type-I, and (b) Type-II double-o de compensa o s. TABLE 5. Values o he coe icien s and ime cons an s in (44) o implemen ing (a) Type-I, and (b) Type-II non-in ege o de shel ing il e s. The sensi i i y analysis is pe o med by employing he Mon e-Ca lo analysis, o e ed by he Ad anced Analysis ool o he O CAD PSpice. Fo his pu pose, ±5% de ia ion om he nominal alues o passi e elemen s is assumed and he ob ained his og ams, o N=500 uns, o he mean equency o he Type-I and II compensa o s a e gi en in Fig.5. The alue o he s anda d de ia ion in he case o Type-I compen- sa o is 0.21 ad/s, while o Type-II is 0.24 ad/s, con i ming ha he p esen ed schemes ha e easonable sensi i i y cha ac e is ics. 14046 VOLUME 12, 2024 J. Nako e al.: Bilinea Double-O de Fil e Designs and Applica ion Examples FIGURE 6. Implemen a ion o he p oposed gene alized bilinea il e (a) Design ob ained using he Anadigm Designe ®2EDA so wa e, (b) expe imen al se up, and (c) FPAA boa d wi h an ex a boa d including he inpu and ou pu in e aces o single- o-di e en ial con e sion and ice e sa. FIGURE 7. Expe imen al ime-domain beha io o (a) Type-I, and (b) Type-II double-o de shel ing il e s s imula ed a hei mean equency by a sinusoidal inpu (g een:inpu , magen a:ou pu ). FIGURE 8. Expe imen al demons a ion o he p og ammabili y o he p oposed double-o de il e o (α=0.8, β =0.7) (a) Type-I, and (b) Type-II (g een:inpu , magen a:ou pu ). B. DESIGNS OF SHELVING FILTERS Assuming ha ω0=104 ad/sand a ange o app oxima ion (10−2ω0,10+2ω0), he alues o he coe icien s and ime cons an s in (44) o implemen ing non-in ege o de Types- I and II shel ing il e s wi h low- equency gains and ime cons an scaling ac o s he same as in he p e ious designs, a e p o ided in Tables 5a–b, espec i ely. The u ilized clock equency is equal o 1MHz. Using he Anadigm Designe ®2 EDA so wa e, he esul ing design is demons a ed in Fig.6a, while he expe imen al se up is depic ed in Figs.6b-c. The gain and he inpu -ou pu phase di e ence o he il e s measu ed om he inpu -ou pu wa e o ms in Fig.7, which a e ob ained in he cases o Type-I and Type-II double-o de shel ing il e s s imula ed by a sinusoidal signal a hei mean TABLE 6. Values o he coe icien s and ime cons an s in (44) o implemen ing (a) Type-I, and (b) Type-II double-o de shel ing il e s wi h (α=0.8, β =0.7) and (α=0.9, β =0.8). FIGURE 9. Expe imen al demons a ion o he p og ammabili y o he p oposed double-o de il e o (α=0.9, β =0.8) (a) Type-I, and (b) Type-II (g een:inpu , magen a:ou pu ). equency, a e {11.9dB, −33o} o Type-I, and {8.4dB, 29o} o Type-II. The co esponding heo e ical p edic ed alues a e {12dB, −33.08o} and {8dB, 33.08o}. The p og ammabili y ea u e o he p oposed double-o de il e is demons a ed o (α=0.8, β =0.7) and (α= 0.9, β =0.8). The alues o he scaling ac o s and VOLUME 12, 2024 14047 J. Nako e al.: Bilinea Double-O de Fil e Designs and Applica ion Examples ime cons an s o implemen ing he esul ing app oxima ion ans e unc ions a e gi en in Table 6. The ime-domain beha io o hese il e s is demons a ed in Fig.8 o he i s case, while Fig.9co esponds o he second one. In bo h cases, he il e s a e s imula ed by a sinusoidal signal a hei mean equency. The alues o he gain and phase a e {13.04dB, −29o} o he Type-I and {7.46dB, 26o} o he Type-II il e s wi h (α=0.8, β =0.7), when he co esponding heo e ical alues a e {13dB, -28.95o} and {7dB, 28.95o}. Respec i ely, when (α=0.9, β =0.8), he measu ed alues o gain and phase a e {12.02dB, −39o} o Type-I and {8.76dB, 32o} o Type-II double-o de shel ing il e s, when he co esponding heo e ical p edic ed alues a e equal o {12dB, −38.28o} and {8dB, 38.28o}. V. CONCLUSION Bilinea il e s a e applicable in con ol sys ems as lead/lag- compensa o s and in acous ic sys ems as low/high-shel ing il e s. The employmen o non-in ege o de s in hei ans e unc ions o e s design lexibili y bu , in gene al, su e s om he inc eased ci cui complexi y. The p oposed double-o de bilinea il e unc ion o e s he mos economical solu ion in e ms o he ac i e componen coun , while he p esen ed FPAA based implemen a ion o e s p og ammabili y. All possible non-in ege o de e sions (i.e., ac ional-o de , powe -law, and double-o de ) a e implemen able by he same co e, jus by in e changing he impedances in he case o ac i e RC implemen a ion, o adjus ing he coe icien s o he app oxima ion ans e unc ion. Fu u e esea ch plans include explo ing double-o de PID con olle s, which gene alize ac ional-o de con olle s. ACKNOWLEDGMENT The publica ion o he a icle in OA mode was inancially suppo ed by HEAL-Link. REFERENCES [1] C. Mu niz-Mon e o, L. A. Sánchez-Gaspa iano, C. 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JULIA NAKO (G adua e S uden Membe , IEEE) ecei ed he B.Sc. and M.Sc. deg ees om he Uni e si y o Pa as, G eece, in 2021 and 2023, espec i ely, whe e she is cu en ly pu suing he Ph.D. deg ee wi h he Pos g adua e S udies P o- g am ‘‘Elec onics-Ci cui s and Sys ems,’’ Physics Depa men . She is a membe o he Analog VLSI Design Team, Elec onics Labo a o y, wo king unde he supe ision o P o . Cos as Psychalinos. He cu en esea ch in e es s include he design o analog in eg a ed ci cui s and sys ems o signal p ocessing, including non-in ege o de ci cui s, con ol sys ems, and biomedical ci cui s. 14048 VOLUME 12, 2024