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
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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.
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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
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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).
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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.
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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