* [email p o ec ed]; phone +34955056666; ax +34955056686; www.imse.cnm.es
Con inuous-Time Cascaded Σ∆Modula o s o VDSL:
A Compa a i e S udy
Ramón To osa, José M. de la Rosa*, Angel Rod íguez-Vázquez and F ancisco V. Fe nández
Ins i u o de Mic oelec ónica de Se illa, IMSE-CNM (CSIC)
Edi icio CICA-CNM, A da Reina Me cedes s/n, 41012-Se illa, SPAIN
ABSTRACT
This pape desc ibes new cascaded con inuous- ime Σ∆ modula o s in ended o cope wi h e y high- a e digi al subsc ibe
line speci ica ions, i.e 12-bi esolu ion wi hin a 20-MHz signal bandwid h. These modula o s ha e been syn hesized using
a new me hodology ha is based on he di ec syn hesis o he whole cascaded a chi ec u e in he con inuous- ime domain
ins ead o using a disc e e- o-con inuous ime ans o ma ion as has been done in p e ious app oaches. This me hod
allows o place he ze oes/poles o he loop- il e ans e unc ion in an op imal way and o educe he numbe o analog
componen s, namely: ansconduc o s and/o ampli ie s, esis o s, capaci o s and digi al- o-analog con e e s. This leads
o mo e e icien opologies in e ms o ci cui y complexi y, powe consump ion and obus ness wi h espec o ci cui
non-ideali ies. A compa ison s udy o he syn hesized a chi ec u es is done conside ing hei sensi i i y o mos c i ical
ci cui e o mechanisms. Time-domain beha io al simula ions a e shown o alida e he p esen ed app oach.
Keywo ds: Analog- o-digi al con e e s, sigma-del a modula o s, con inuous- ime ci cui s.
1. INTRODUCTION
Con inuous-Time (CT) Sigma-Del a Modula o s (Σ∆Μs)ha e demons a ed o be an a ac i e solu ion o he imple-
men a ion o Analog- o-Digi al (A/D) in e aces in sys ems-on-chip in eg a ed in deep-submic on s anda d CMOS ech-
nologies 1. Al hough mos epo ed Σ∆Ms ha e been implemen ed using Disc e e-Time (DT) ci cui s, he inc easing
demand o b oadband da a communica ion sys ems has mo i a ed he use o CT ci cui echniques. In addi ion o show
an in insic an ialiasing il e ing, CT Σ∆Ms p o ide po en ially as e ope a ion wi h lowe powe consump ion han hei
DT coun e pa s 2,3.
In spi e o hei men ioned ad an ages, CT Σ∆Ms a e mo e sensi i e han DT Σ∆Ms o some ci cui e o s, namely:
clock ji e , excess loop delay and echnology pa ame e a ia ions 2,3. The la e a e specially c i ical o he ealiza ion
o cascaded a chi ec u es. This has o ced he use o single-loop opologies in mos epo ed silicon p o o ypes e en
hough low o e sampling a ios ( ) a e needed 4,5, whe eas e y ew cascaded CT Σ∆M In eg a ed Ci cui s (ICs) ha e
been epo ed 6.
Howe e , he need o achie e medium-high esolu ions ( ) wi hin high signal bandwid hs ( ) while
gua an eeing s abili y, has p omp ed he in e es in p ope me hods o he syn hesis o high-o de cascaded CT Σ∆Ms 7-
9. These me hods a e based on applying a DT- o-CT ans o ma ion o an equi alen DT opology ha ul ils he equi ed
speci ica ions. In mos cases, he use o such a ans o ma ion is no mally ansla ed in o an inc ease o he analog ci cui
complexi y wi h he subsequen penal y in silicon a ea, powe consump ion and sensi i i y o pa ame e ole ances.
This pape p esen s a di ec syn hesis me hod o cascaded CT Σ∆Ms which, dispensing wi h he DT- o-CT equi a-
lence, make i possible o educe he analog ci cui y complexi y and place he ze oes/poles o he quan iza ion noise ans-
e unc ion in an op imal way, hus yielding o mo e obus a chi ec u es han using a DT- o-CT ans o ma ion. As an
applica ion, he p oposed me hodology is used o ind op imum CT Σ∆Ms o Ve y high- a e Digi al Subsc ibe Line
(VDSL). Th ee i h-o de cascaded opologies a e syn hesized: 2-1-1-1, 2-2 and 3-2. These modula o s a e designed o
12-bi @20-MHz speci ica ions and hei pe o mances a e compa ed in e ms o ime-domain simula ions ha ake in o
accoun c i ical e o mechanisms like misma ch and clock ji e e o .
12<
12bi s>20MHz>
VLSI Ci cui s and Sys ems II, edi ed by José Fco. López, F ancisco V. Fe nández,
José Ma ía López-Villegas, José M. de la Rosa, P oceedings o SPIE Vol. 5837
(SPIE, Bellingham, WA, 2005) 0277-786X/05/$15 · doi: 10.1117/12.607923
59
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2. CASCADED CONTINUOUS-TIME Σ∆ MODULATORS
Fig.1 shows he concep ual block diag am o a cascaded CT Σ∆M. Each s age, consis ing o a single-quan-
ize CT Σ∆M, e-modula es a signal con aining he quan iza ion e o gene a ed in he p e ious s age. Once in he digi al
domain, he ou pu s, , o he s ages a e p ope ly p ocessed and combined (by he cancella ion logic) in o de o cancel
ou he quan iza ion e o s o all he s ages, bu he las one in he cascade. This la e e o appea s a he o e all modula o
ou pu shaped by a unc ion o o de equal o he summa ion o he o de s o all he s ages.
Cascaded CT Σ∆Ms a e no mally syn hesized om equi alen (well-known) DT sys ems and use he same digi al can-
cella ion logic 8. This DT/CT equi alence can be gua an eed because he o e all open loop ans e unc ion o each s age
in Fig.1 is in ac a DT sys em 2. Thus, in he case o a ec angula impulsi e esponse o he Digi al- o-Analog Con e e
(DAC), i can be shown ha he equi alen DT loop il e ans e unc ion is gi en by 10,11:
(1)
whe e is he sampling equency; ; ; and a e espec i ely he ime
delay and pulse wid h o he DAC wa e o m; a e he poles o and s ands o he esidue o .
In o de o ge a unc ional CT Σ∆M while keeping he cancella ion logic o he o iginal DT Σ∆M, e e y s a e a iable
and DAC ou pu mus be connec ed o he in eg a o inpu o la e s ages 8. This inc eases he numbe o analog compo-
nen s, i.e ansconduc o s, ampli ie s and DACs. As an illus a ion, Fig.2(a) shows a cascaded CT Σ∆M ob ained
om an exis ing DT Σ∆M12. No e ha a leas eigh scaling coe icien s ( ) and hei co esponding signal pa hs a e
needed o connec he di e en s ages o he modula o . The numbe o in eg a ing pa hs can be educed −as shown in
Fig.2(b) − i he whole cascaded Σ∆M is di ec ly syn hesized in he CT domain as p oposed in he nex sec ion.
3. PROPOSED SYNTHESIS METHODOLOGY
The idea o dispensing wi h he DT- o-CT ans o ma ion was p e iously epo ed in 3 o single-loop a chi ec u es.
Howe e , in he case o cascaded a chi ec u es, he cancella ion logic unc ions (no p esen in single-loop Σ∆Ms) mus
be included in he syn hesis p ocedu e o ge an op imum a chi ec u e.
Le us conside he mo e gene al case o he cascaded CT Σ∆M shown in Fig.1. The o e all ou pu , , is
gi en by:
(2)
Figu e 1. Concep ual block diag am o a cascaded CT Σ∆M.
CT Σ∆M
1
CT Σ∆M
2
CL
1
CL
2
CL
m
.
.
.
.
.
..
.
.
x(s)
y
2
(z)
y
1
(z)
y
m
(z)
y
o
(z)
E
1
(z)
E
2
(z)
E
m
(z)
Cancella ion
Logic
CT Σ∆M
m
+
DAC
To nex s age
E
2
(z)
y
2
(s)
x
2
(s)F(s)
y
2
(z)
m-s age
yi
Fz() Re Fs()
s
----------- em1TSs⋅
ze
TSs⋅
–
--------------------
⋅



pi
∑Re Fs()
s
----------- em2TSs⋅
ze
TSs⋅
–
--------------------
⋅



pi
∑
–=
s1T
s
⁄
=
m11 dTs
⁄
–=
m21 dτ
+
()Ts
⁄
–=
dτ
piFs()s⁄Re x() x
2-1-1
kg29–
m-s age yo
yoz() ykz()CLkz()
k1=
m
∑
=
60 P oc. o SPIE Vol. 5837
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whe e and ep esen espec i ely he ou pu and pa ial cancella ion logic ans e unc ion o he
s age.
I he modula o inpu , , is se o ze o, he ou pu o each s age can be w i en as:
(3)
whe e s ands o he , is he in e se Laplace ans o m, is he ans e unc ion o he
DAC, and
(4)
ep esen s he ans e unc ion om o he inpu o quan ize .
Using he ollowing no a ion
(5)
he ou pu o each s age is gi en by:
(6)
and he ou pu o he modula o can be w i en as:
(7)
Figu e 2. Cascaded 2-1-1 CT Σ∆M a chi ec u e ob ained (a) om an equi alen DT Σ∆M (b) using he p oposed me hod.
DAC
CL
1
CL
3
1
T
s
s
DAC
CL
2
k
g1
1
T
s
s
k
b2
1
T
s
s
k
g5
k
g9
k
g8
k
g7
k
g6
k
b4
x(s)
y
o
(z)
(a)
DAC
DAC
CL
1
CL
3
DAC
CL
2
E
2
(z)
E
1
(z)
E
3
(z)
y
1
(z)
y
2
(z)
y
3
(z)y
3
(s)
y
2
(s)
y
1
(s)
y
o
(z)
x(s)
(b)
E
1
(z)
E
2
(z)
E
3
(z)
y
1
(s)
y
2
(s)
y
3
(s)
y
1
(z)
y
2
(z)
y
3
(z)
k
in1
1
T
s
s
k
b1
DAC
k
g2
k
g3
k
g4
k
b3
k
in1
1
T
s
s
k
b1
k
in3
1
T
s
s
k
b4
k
g1
1
T
s
s
k
b2
k
in2
1
T
s
s
k
b3
ykz() CLkz() k- h
x ()
ykz()
Ekz() ZL
1–HDFik
[]
nTs



yiz()
i1=
k1
–
∑
+
1ZL
1–HDFkk
[]
nTs



–
-------------------------------------------------------------------------------------------
=
ZZ- ans o m L1
–
HDHDAC s()≡
Fij Fij s()≡Inpu Quan ize j
yis()
-----------------------------------------
=
yis() j- h
Z
km ZL
1–HDFkm
()
nTs


≡
ykz() Ekz()
1Zkk
–
-----------------Zik yiz()
1Zkk
–
--------------------
i1=
k1–
∑
+=
yoykCLk
k1=
m
∑Ek
1Zkk
–
-----------------1
1Zkk
–
-----------------Zik yi
i1=
k1–
∑
+



CLk
k1=
m
∑
==
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The pa ial cancella ion logic ans e unc ions ( ) can be calcula ed by imposing he cancella ion o he ans e
unc ion o he i s quan iza ion e o s in (7). This gi es:
(8)
whe e he pa ial cancella ion logic ans e unc ion o he las s age, , can be chosen o be he simples o m ha
p ese es he equi ed noise shaping.
No e ha he design equa ions (2)-(8) do no only ake in o accoun he single-s age loop il e ans e unc ions ( ),
bu also he in e -s age loop il e ans e unc ions ( ). The la e a e con inuous- ime in eg a ing pa hs appea ing
only when he modula o s ages a e connec ed o o m he cascaded Σ∆M and mus be included in he syn hesis me hod-
ology o ob ain a unc ional modula o wi h minimum numbe o in e -s age pa hs.
The e o e, he ollowing p ocedu e can be used in a sys ema ic me hodology o he syn hesis o cascaded CT Σ∆Ms††:
• Fi s , he poles o single-s age ans e unc ions ( ) a e op imally placed in he signal bandwid h o gi en
speci ica ions. This p ocess is ca ied ou en i ely in he CT domain and no equi alence o an exis ing DT modu-
la o needs o be imposed.
• Second, once he indi idual s ages a e designed and op imized, cancella ion logics a e calcula ed using (8).
Fo illus a i e pu poses, he 2-1-1 CT Σ∆M o Fig.2(b) was syn hesized using (2)-(8) o achie e esolu ion in
a bandwid h, wi h a sampling equency o 48MHz (o e sampling a io, ) 12.Fo simplici y, in o de
o acili a e he compa ison o he pe o mance o bo h modula o s in Fig.2, he coe icien s o he i s s age
( ) a e aken o be equal in bo h sys ems and a e ob ained om a DT- o-CT ans o ma ion o he i s
s age o a DT Σ∆Min12. The es o coe icien s in Fig.2(b) a e aken such ha he ime cons an o he in eg a o s is he
in e se o he sampling equency ( ):
(9)
Hence, he single-loop and in e -s age ans e unc ions a e gi en by:
(10)
and he pa ial cancella ion logic ans e unc ions can be calcula ed using (8)-(10). Conside ing a Non-Re u n- o-Ze o
(NRZ) DAC, he ollowing cancella ion logics a e de i ed:
(11)
†† In his p ocedu e, he modula o o de , o e sampling a io and numbe o bi s o in e nal quan ize s a e as-
sumed o be de e mined o gi en speci ica ions om well-known exp essions 1.
CLk
m1
–
Ekz()
C
Lkz() ZkmCLm
–
1Zmm
–
------------------------
ZL
1–HDFkm
[]
nTs



CLmz(
)
–
1ZL
1–HDFmm
[]
nTs



–
------------------------------------------------------------------------
-
==
CLmz()
Fii
F
ij ij≠,
Fii s()
16-bi
750-kHz M32
=
kin1kg1k b1k b2
,, ,
T
s1
s
⁄
=
kin1k b1
–
14⁄
;==
k b238⁄
–=
kg1kin2k– b3kin3k– b41== == =
F11
3Ts
8
---------s1
4
---
+


–
sTs
()
2
------------------------------
=F22 F33
1–
sTs
--------
==
F13
3Ts
8
---------s1
4
---
+


–
sTs
()
4
------------------------------
=F23
1–
sTs
()
2
---------------
=
CL1
z1–
48
-------729+z1–7z2–
–5z3–
–()=
CL2z1–1z1–
+()1z1–
–()
2
=
CL321 z1–
–()
3
=
62 P oc. o SPIE Vol. 5837
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whe e is chosen o ha e h ee ze oes a DC, co esponding o he ze oes con ibu ed by he i s h ee in eg a o s.
In o de o compa e he obus ness o bo h modula o s in Fig.2, he e ec o misma ch on he Signal- o-Noise Ra io
( ) was also simula ed using SIMSIDES, a SIMULINK-based ime-domain beha io al simula o o Σ∆Ms 13.Fo
his pu pose, maximum alues o misma ch we e es ima ed o a 0.13 µm CMOS echnology and bo h modula o s in Fig.2
we e simula ed conside ing a Gm-C implemen a ion. The esul s a e shown in Fig.3, whe e he loss is ep esen ed
as a unc ion o he s anda d de ia ion o he ansconduc ances ( ) and capaci ances ( ). Fo each poin o hese
su aces, 150 simula ions we e ca ied ou using andom a ia ions wi h he s anda d de ia ion gi en in he diag ams. The
alue o loss ep esen ed in Fig.3 s ands o he di e ence be ween he ideal , i.e wi h no pa ame e a ia ion,
and he wi h 90% o he 150 simula ions abo e i . I is shown ha he lowe analog componen coun in Fig.2(b) is
e lec ed in a lowe a iance o he modula o coe icien s, leading o a be e beha io in e ms o sensi i i y o misma ch.
4. APPLICATION TO VDSL
As an applica ion o he p oposed me hodology, h ee 5 h-o de cascaded CT Σ∆Ms, shown in Fig.4, we e syn hesized
o cope wi h VDSL speci ica ions: 12-bi esolu ion wi hin a 20-MHz signal bandwid h. In o de o ul il hese speci ica-
ions wi hou being limi ed by he clock ji e e o , he sampling equency, , and he numbe o bi s o he in e nal quan-
ize s (and DACs), , mus be p ope ly chosen. In he case o a 5 h-o de modula o s like hose shown in Fig.4, he in-
band ji e noise powe is minimized o and 14.
Ano he c i ical sou ce o e o in CT Σ∆Ms is he excess loop delay. As shown in 15 his e o can be compensa ed by
adding an ex a eedback b anch be ween he ou pu and he inpu o he quan ize (DAC2in Fig.4) and wo D-la ches. By
adding his ex a b anch wi h he app op ia e gain, he loop impulse esponse is exac ly he same as ha o he o iginal.
This ex a eedback e m can be easily included in he calcula ion o he cancella ion logic. In a p ac ical implemen a ion
i could be ad an ageous o make DAC2p og ammable 5.
Conside ing he ac o s abo e, he CT Σ∆Ms in Fig.4 we e syn hesized using he me hodology desc ibed in Sec ion 3,
aking in o accoun he ollowing conside a ions:
• The i s s age o he 2-1-1-1 a chi ec u e (Fig.4(a)) is o med by a esona o which has i s poles placed a
, in o de o minimize he quan iza ion Noise T ans e Func ion (NTF) in he signal bandwid h,
. Resis o a ia ions can be uned ou using a combina ion o a disc e e ough uning o he esis o s ( ,
and ) and a con inuous ine uning o he ansconduc o s and . This uning can be also used o cancel
he e ec o ini e Gain-Bandwid h p oduc ( ) o he on -end opamp, due o he ac ha his e o can be mod-
elled as an in eg a o gain e o 5. All he o he ansconduc o s could be uned in o de o keep he ime cons an
unchanged o e a ia ions.
• An addi ional esona o has been used in he 2-2-1 a chi ec u e (Fig.4(b)), in o de o op imally dis ibu e he poles
o NTF 16.
CL3
SNR
Figu e 3. E ec o misma ch on he SNR o a cascaded 2-1-1 CT Σ∆M ob ained om: (a) an equi alen DT Σ∆M; (b) p opose
d
me hod.
(b)
SNR Loss (dB)
SNR Loss (dB)
(a)
SNR
σgm σC
SNR SNR
SNR
s
B
s240MHz
=
B4
=
ωp2π79⁄Bw
=
BwRin R b
R
k
kg1
GB
Cg
m
⁄C
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Figu e 4. Cascaded CT Σ∆Ms syn hesized o VDSL: (a) 2-1-1-1; (b) 2-2-1; (c) 3-2.
+
−
D
La ch
DAC
2
D
La ch
DAC
1
DAC
3
DAC
4
DAC
5
CLK CLK
CLK
CLK
CLK
CLK
CL
1
CL
2
CL
4
CL
3
C
1
C
2
C
3
C
4
C
5
R
in
k
R
b
R
k
g1
k
in3
k
in4
k
g5
k
g2
k
in2
k
g3
k
g4
k
b2
k
b3
k
b4
y
o
(z)
x(s)
(b)(a)
(c)
+
−
D
La ch
DAC
2
D
La ch
DAC
1
DAC
4
CLK CLK
CLK
CLK
CLK
CL
1
CL
2
CL
3
C
1
C
2
C
4
C
5
R
in
k
1
R
b
R
k
g1
k
in3
k
g2
k
in2
k
g4
k
g5
k
b2
k
b3
DAC
2
D
La ch
DAC
1
CLK CLK
D
La ch
k
2
k
2
k
g3
C
3
y
o
(z)
x(s)
+
−
D
La ch
DAC
2
D
La ch
DAC
1
CLK CLK
CLK
CLK
CL
1
CL
2
C
1
C
5
R
in
k
2
R
b
k
g2
k
g3
k
in2
k
g5
k
b2
DAC
2
D
La ch
DAC
1
CLK CLK
D
La ch
k
3
k
2
k
g4
C
4
C
2
k
1
k
1
k
g1
C
3
64 P oc. o SPIE Vol. 5837
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• The 3-2 modula o shown in Fig.4(c) includes a i s s age which consis s o an in eg a o and a esona o . This
opology allows he same op imum pole posi ioning as in he 2-2-1 modula o wi h one less s age. Howe e , s a-
bili y p oblems migh a ise ha comp omise he modula o pe o mance.
Table 1 shows he single-loop and in e -s age ans e unc ions ( ) o he di e en a chi ec u es in Fig.4 as a unc-
ion o he loop il e coe icien s . The exp essions o , ob ained om (8) and (11), a e also shown. I is impo an
o no e ha a e ound om an i e a i e simula ion-based p ocess ha op imizes he i s s age o he modula o in o de
o maximize while keeping s abili y.
The ou come o he op imiza ion p ocess −en i ely done in he CT domain −is summa ized in Table 2.This able
includes he alues o loop il e coe icien s, (implemen ed as ansconduc ances) as well as he capaci ances, , and
esis ances, ob ained om he op imiza ion p ocess.
The modula o s in Fig.4 we e simula ed using SIMSIDES 13. Fig.5 shows he ideal ou pu spec a o he modula o s
when clocked a . I can be obse ed he e ec o he esona o s poles dis ibu ed wi hin he signal band-
wid h. The impac on he in-band noise powe is be e app ecia ed in Fig.6 ha ep esen s he Signal- o-(Noise+Dis o -
ion) Ra io (SNDR) s inpu ampli ude. No e ha , al hough bo h he 2-2-1 and 3-2 a chi ec u es ha e he same loca ion o
he ze oes o he NTF, he 3-2 modula o achie es a wo se esolu ion. This is due o he ac ha he op imiza ion p ocess
applied o ha a chi ec u e was mo e conse a i e as a consequence o he s abili y cons ains imposed by he 3 d-o de
s age.
In addi ion o he ideal pe o mance desc ibed abo e, he e ec o mos c i ical limi ing ac o s has been aken in o
accoun in he high-le el design. Fig.7 shows he SNR loss caused by clock ji e e o . No e ha he 2-1-1-1 a chi ec u e
seems o be less sensi i e o his e o han he o he a chi ec u es. Howe e , i is impo an o no e ha he ideal SNR o
his modula o is lowe han he o he s. The e o e, he e is a highe componen o quan iza ion noise masking he e ec
o clock ji e .
Two c i ical limi ing ac o s in cascaded Σ∆Ms, and pa icula ly in hei CT implemen a ion, a e ci cui ole ances and
componen misma ch. The i s one can be con olled by using uning o ime cons an s 4,5 o digi al calib a ion 6. Howe e
misma ch e o s ill emains. In o de o e alua e he impac o his e o on he pe o mance o he modula o s in Fig.4,
maximum alues o misma ch we e es ima ed o a 0.13 µm CMOS echnology conside ing a Gm-C implemen a ion. The
Fij
bij CLi
bij
SNR
kiCi
Ri
Figu e 5. Ideal ou pu spec a o he cascaded CT Σ∆Ms in Fig.4: (a) 2-1-1-1. (b) 2-2-1. (c) 3-2.
(a) (b)
(c)
s240 MHz=
P oc. o SPIE Vol. 5837 65
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Table 1: T ans e unc ions and cancella ion logic unc ions o he modula o s in Fig.4
Modula o T ans e unc ions Cancella ion Logic
2-1-1-1
2-2-1
3-2
Cancella ion Logic Coe icien s
2-1-1-1
2-2-1
F14
b10
s3s2ωp
2
+()
---------------------------
=
F
24
1–
Ts
3s3
-----------
=F34
1–
Ts
2s2
-----------
=
F44
1–
Tss
--------
=
C
L1z1
–
n14 n+13z1
–
n12z2
–
n11z3
–
n10z4
–
+++()=
C
L2
1
6
---z1–14z1–z2–
–+()12 Tsωp
()cos z1–
–z2–
+()=
C
L3
1
2
---z1–1z1–
+()1z1–
–()12 Tsωp
()cos z1–
–z2–
+()=
C
L41z1–
–()
212 Tsωp
()cos z1–
–z2–
+()=
F13
b10
ss
2ωp1
2
+()s2ωp2
2
+()
-----------------------------------------------------
=
F
23
b20
ss
2ωp2
2
+()
----------------------------
=F44
1–
Tss
--------
=
C
L1z1–n14 n+13z1
–
n12z2
–
n11z3–n10z4–
+++()=
C
L2z1–n22 n21z1–n20z2–
++()12 Tsωp1
()cos z1–
–z2–
+()=
C
L312 Tsωp1
()cos z1–
–z2–
+()12 Tsωp2
()cos z1–
–z2–
+(
)
=
F12
b11sb
10
+
ss
2ωp1
2
+()s2ωp2
2
+()
-----------------------------------------------------
=
F
22
b21sb
20
+
s2ωp2
2
+
----------------------- eTSs–kceTSs–2⁄
+=
C
L1z2–n14 n+13z1
–
n12z2
–
n11z3
–
n10z4
–
+++()=
C
L212 Tsωp1
()cos z1–
–z2–
+()1z1–
–()⋅=
12 Tsωp2
()kc
–cos()z1–
–1n21 kc2Tsωp2
()cos–+()z2–n20 k+c
()z3–
++
()
n
10 n14
b
–
10
Ts
3ωp
5
------------- Tsωp
()sin Tsωp
–1
6
---Tsωp
()
3
+==
n
11 n13
b–10
Ts
3ωp
5
------------- [Tsωp
()
32Tsωp
()cos–
3
----------------------------------- 4Tsωp
()2TsωpTsωp
()1+cos()+sin–==
n
12
b–10
Ts
3ωp
5
------------- [Tsωp
()
314 Tsωp
()cos–
3
---------------------------------------6Tsωp
()sin 2Tsωp12 Tsωp
()cos+()]–+=
n
10 n14
b
–
10
ωp1
3ωp2
3ωp2
2ωp1
2
–()
------------------------------------------------ Tsωp1ωp2
3ωp1
3ωp2
–()ω
p1
3Tsωp2
()ω
p2
3Tsωp1
()sin–sin+[==
n
11 n13
2b10
–
ωp1
3ωp2
3ωp2
2ωp1
2
–()
------------------------------------------------ [Tsωp2ωp1
3ωp2
3ωp1
–()Tsωp1
() Tsωp2
()cos+cos()()+==
ωp2
3Tsωp1
()1Tsωp2
()cos+()ω
p1
3Tsωp2
()1Tsωp1
()cos+()]sin–sin+
n
12
2b10
–
ωp1
3ωp2
3ωp2
2ωp1
2
–()
------------------------------------------------ [Tsωp1ωp2
3ωp1
3ωp2
–()12 Tsωp1
()Tsωp2
()coscos+()+=
ωp1
3Tsωp2
()12 Tsωp1
()cos+()ω
p2
3Tsωp1
()12 Tsωp2
()cos+()]sin–sin+
n
20 n22
b20
–
ωp2
3
---------- Tsωp2Tsωp2
()sin–[]== n21
2b20
–
ωp2
3
--------------Tsωp2
()Tsωp2Tsωp2
()cos–sin[
]
=
66 P oc. o SPIE Vol. 5837
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esul s o his analysis a e shown in Fig.8 whe e he SNR is ep esen ed as a unc ion o he s anda d de ia ion o he
ansconduc ances ( ) and capaci ances ( ). Fo each poin o hese su aces, a Mon eCa lo analysis o 150 simula-
ions was ca ied ou . The alue o he SNR ep esen ed in he e ical axis o Fig.8 is ob ained by 90% o he simula ions
o each case o and . No e ha e en in he wo s -case misma ch, he esolu ion is abo e he speci ied ( ).
Finally, he modula o s in Fig.4 we e high-le el sized, i.e, he sys em-le el speci ica ions ( ) we e
mapped on o building-block speci ica ion using s a is ical op imiza ion o design pa ame e selec ion, and beha io al
simula ion o e alua ion. The esul s o his sizing p ocess a e summa ized in Table 3 and Table 4 showing he maximum
(minimum) alues o he ci cui e o mechanisms ha can be ole a ed in o de o ul il he equi ed modula o pe o m-
ance. As an illus a ion, Fig.9 shows he ou pu spec a o he modula o s aking in o accoun he non-ideali ies lis ed in
hese ables. The e ec i e esolu ion is and o he modula o s in Fig.4(b) and (c), espec i ely.
Table 1: T ans e unc ions and cancella ion logic unc ions o he modula o s in Fig.4.(Con .)
Mod. Cancella ion Logic Coe icien s
3-2
10
b–10 Tsωp1ωp2ωp2
2ωp1
2
–()ω
p1
3Tsωp2
()ω
p2
3Tsωp1
()sin–sin+()
ωp1
3ωp2
3ωp2
2ωp1
2
–()
-------------------------------------------------------------------------------------------------------------------------------------------------------------------- +=
b–11 ωp1
21Tsωp2
()cos–()ω
p2
21Tsωp1
()cos–()–()
ωp1
2ωp2
2ωp2
2ωp1
2
–()
----------------------------------------------------------------------------------------------------------------------------------
+
11
2b10
–Tsωp1ωp2ωp1
2ωp2
2
–()Tsωp1
()Tsωp2
()cos+cos()ω
p2
3Tsωp1
()1Tsωp2
()cos+()sin+( )
ωp1
3ωp2
3ωp2
2ωp1
2
–()
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+=
2b10 ωp1
3Tsωp2
()sin 1 Tsωp2
()cos+()()
ωp1
3ωp2
3ωp2
2ωp1
2
–()
----------------------------------------------------------------------------------------------------2b11 ωp2
2Tsωp2
()cos 1 Tsωp1
()cos–()ω
p1
2Tsωp1
()cos 1 Tsωp2
()cos–(
)
–(
ωp1
2ωp2
2ωp2
2ωp1
2
–()
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
--
+
12
2b10
–Tsωp1ωp2ωp2
2ωp1
2
–()12 Tsωp1
()cos Tsωp2
()cos+()ω
p1
3Tsωp2
()12 Tsωp1
()cos+()sin+( )
ωp1
3ωp2
3ωp2
2ωp1
2
–()
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+=
2b10 ωp2
3Tsωp1
()sin 1 2 Tsωp2
()cos+()()
ωp1
3ωp2
3ωp2
2ωp1
2
–()
-------------------------------------------------------------------------------------------------------
13
2b10
–Tsωp1ωp2ωp1
2ωp2
2
–()Tsωp1
()Tsωp2
()cos+cos()ω
p2
3Tsωp1
()1Tsωp2
()cos+()sin+( )
ωp1
3ωp2
3ωp2
2ωp1
2
–()
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+=
2b10 ωp1
3Tsωp2
()sin 1 Tsωp2
()cos+()()
ωp1
3ωp2
3ωp2
2ωp1
2
–()
----------------------------------------------------------------------------------------------------+2b11 ωp1
2Tsωp1
()cos 1 Tsωp2
()cos–()ω
p2
2Tsωp2
()cos 1 Tsωp1
()cos–()–(
)
ωp1
2ωp2
2ωp2
2ωp1
2
–()
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
-
14
b–10 Tsωp1ωp2ωp2
2ωp1
2
–()ω
p1
3Tsωp2
()ω
p2
3Tsωp1
()sin–sin+()
ωp1
3ωp2
3ωp2
2ωp1
2
–()
-------------------------------------------------------------------------------------------------------------------------------------------------------------------- –=
b
11 ωp2
21Tsωp1
()cos–()ω
p1
21Tsωp2
()cos–()–()
ωp1
2ωp2
2ωp2
2ωp1
2
–()
-
------------------------------------------------------------------------------------------------------------------------------
–
2
0
b21 Tsωp2
()sin
ωp2
-------------------------------------b20 1Tsωp2
()cos–()
ωp2
2
----------------------------------------------------
–= n21
b–21 Tsωp2
()sin
ωp2
---------------------------------------- b20 1Tsωp2
()cos–()
ωp2
2
----------------------------------------------------
–=
σgm σC
σgm σC72-dB
12-bi @20-MHz
13.4 bi s≅13.2bi s≅
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