IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 55, NO. 9, OCTOBER 2008 2525
Ma ix Me hods o he Dynamic Range Op imiza ion
o Con inuous-Time
Gm
-
C
Fil e s
Juan F. Fe nández-Boo ello, Manuel Delgado-Res i u o, Membe , IEEE, and
Angel Rod íguez-Vázquez, Fellow, IEEE
Abs ac —This pape p esen s a syn hesis p ocedu e o he op-
imiza ion o he dynamic ange o con inuous- ime ully di e en-
ial - il e s. Such p ocedu e builds up on a gene al ex ended
s a e-space sys em ep esen a ion which p o ides simple ma ix
algeb a mechanisms o e alua e he noise and dis o ion pe o -
mances o il e s, as well as, he e ec o ampli ude and impedance
scaling ope a ions. Using hese me hods, an analy ical echnique
o he dynamic ange op imiza ion o weakly nonlinea - il-
e s unde powe dissipa ion cons ain s is p esen ed. The p oce-
du e is i s explained o gene al il e s uc u es and hen illus-
a ed wi h a simple biquad a ic sec ion.
Index Te ms— - il e s, il e syn hesis, dynamic ange op-
imiza ion, noise analysis, dis o ion analysis, low powe .
I. INTRODUCTION
FULLY DIFFERENTIAL - ci cui s and echniques
a e widely employed o design in eg a ed con inuous- ime
il e s. O e he yea s signi ican con ibu ions ha e been made
ega ding he p oposal o ansconduc o opologies wi h
enhanced noise and dis o ion pe o mance. Based on such
ansconduc o s, il e s wi h enla ged dynamic ange (DR)1can
be buil o di e en p ac ical applica ions. Howe e , despi e
he a ailabili y o high-pe o mance ansconduc o s, dynamic
ange op imiza ion may be hampe ed due o inaccu a e e alu-
a ion o he impac o noise and dis o ion on he o e all il e
pe o mance.
Rega ding he impac o noise, one o he mos signi ican
ea ly con ibu ions was due o G oenewold [1]. He employed
Manusc ip ecei ed Oc obe 22, 2007; e ised Janua y 11, 2008. Fi s
published Ap il 18, 2008; cu en e sion published Oc obe 29, 2008. This
wo k was suppo ed by he Spanish Minis y o Educa ion & Science unde
G an TEC2006-03022, and he Jun a de Andalucía unde G an TIC-02818.
This pape was ecommended by Associa e Edi o T. B. Ta im.
J. F. Fe nández-Boo ello and A. Rod íguez-Vázquez a e wi h AnaFocus,
Se illa 41092, Spain.
M. Delgado-Res i u o is wi h he Ins i u o de Mic oelec Ónica de Se illa,
Cen o Nacional de Mic oelec ónica, Consejo Supe io de In es igaciones
Cien í icas (CSIC), and Uni e sidad de Se illa, 41012 Se illa, Spain (e-mail:
[email p o ec ed]).
Digi al Objec Iden i ie 10.1109/TCSI.2008.921048
1DR is de ined as he a io be ween he maximum and minimum signal ha
can be p ocessed by he il e . The la e is limi ed by he o al in eg a ed ou pu -
e e ed noise powe o he il e ,
U
, and he o me is limi ed by he maximum
ou pu dis o ion le el which can be ole a ed. Assuming a single- one exci a ion
and le ing be he maximum signal le el a he ou pu o he il e , he dynamic
ange can be exp essed as
DR
=(1
=
2)
1
(
 =U
)

P =U
s a e-space desc ip ions o e alua e noise pe o mance h ough
simple ma ix manipula ions. Simila app oaches o noise e al-
ua ion ha e been used in [2], [3]. In his pape , we adop hese
noise e alua ion echniques. Also, we adop he gene al -
s uc u e p oposed in [3].
Rega ding he impac o dis o ion, di e en echniques
ha e been epo ed; o ins ance, hose p esen ed in [4]–[9].
P oposals in [4], [5] a e based on equency-domain calcula ion
using Vol e a se ies which, al hough powe ul, end o be e y
cumbe some as he il e o de inc eases. In [6], [7] nonlinea i-
ies o indi idual ansconduc o s a e p opaga ed by means o
pa ial ans e unc ions o he ou pu , whe e he con ibu ions
a e summed o es ima e he o e all dis o ion beha io o he
il e . Finally, [8], [9] use ime-domain analysis and s a e-space
modelling o he e alua ion o ha monic and in e modula ion
dis o ion in il e s wi h no loa ing capaci o s. None o hese
app oaches p o ides he simple, gene al and sys ema ic ma ix
manipula ion echniques which a e a ailable o noise. Conse-
quen ly, no sys ema ic me hod is ye a ailable o e alua e he
impac o bo h noise and dis o ion on a gene al - il e
s uc u e, he eby limi ing he abili y o designe s o maximize
he dynamic ange o p ac ical il e implemen a ions.
This pape p esen s me hods and echniques o e alua e he
impac o bo h noise and dis o ion on he - il e s uc u e
o [3] h ough simple and sys ema ic ma ix algeb a. Fo dis-
o ion e alua ion we combine s a e-space and Vol e a Se ies
ep esen a ions [10] in such a way ha hey can be easily em-
bedded in ma ix o m.
The pape also add esses he issue o scaling. The op i-
miza ion o DR h ough modi ica ions o he ampli ude and
impedance le els a he in e nal il e nodes is co e ed o bo h
biquad a ic sec ions and gene al - il e s. In he case o
biquad a ic sec ions, he pape epo s compac DR exp essions
which p o ide mo e accu a e es ima ions o he in luence o
he quali y ac o , , han hose in [1]. Fo gene al il e s,
he echniques p esen ed in his pape o e come he lack o
uni ocal solu ions obse ed in he app oach p esen ed in [6].
The pape is s uc u ed as ollows. Sec ion II desc ibes he
s a e-space ep esen a ion o gene al - il e s, and p o ides
ela ionships among he s a e-space ma ix and a se o in e nal
ans e unc ions which a e essen ial o he o egoing analysis.
Sec ions III and IV p esen he sys ema ic echniques o he
e alua ion o noise and dis o ion, espec i ely. Sec ion V deals
wi h he scaling o - il e s. Sec ion VI add esses he dy-
namic ange op imiza ion o gene ic - il e s and he p o-
cedu e is illus a ed in Sec ion VII o biquad s uc u es. Finally,
Sec ion VIII concludes he pape .
1549-8328/$25.00 © 2008 IEEE
2526 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 55, NO. 9, OCTOBER 2008
Fig. 1. Simpli ied schema ic o a ully balanced
G
-
C
il e .
II. SPACE-STATE REPRESENTATION OF - FILTERS
Fig. 1 shows he concep ual schema ic o a gene ic con in-
uous- ime ully balanced - il e wi h in eg a ion nodes.
We assume ha each in eg a ion node can be connec ed o he
inpu and o all he emaining in eg a ion nodes; we u he as-
sume ha hese connec ions can be ealized wi h ei he capac-
i o s o ansconduc o s. I is illus a ed in Fig. 1 o a gene ic
in eg a ion node . Capaci i e
connec ions o o he in eg a ion nodes a e ealized wi h capaci-
o s ; ansconduc ance connec ions o o he
in eg a ion nodes a e ealized wi h ansconduc o s
; connec ions o he inpu a e ealized wi h capaci o s
and ansconduc o s , espec i ely. This gene al diag am
con empla es also he gene al case whe e he il e ou pu is ob-
ained as he linea combina ion o he inpu and an a bi a y
numbe o in e nal node ol ages.Using an ex ended s a e-space
no a ion, he il e is desc ibed as [3]
(1)
which is ac ually a gene aliza ion o he ep esen a ion gi en in
[1]. In he exp ession abo e:
•and a e, espec i ely, he inpu and ou pu ol ages o
he il e and is a s a e ec o which
ga he s all he in eg a ion node ol ages. Ope a o
deno es ma ix anspose.
•is an ma ix whose elemen ep esen s he
ansconduc ance o he ansconduc o connec ed om
node o node .
•is an ec o gi en by
(2)
whe e and ep esen , espec i ely, he ansconduc-
ance and capaci ance be ween he il e inpu node and he
in eg a ion node ; is o med by ansconduc ances
and is composed by capaci ances.
•in (1) is a ec o whose elemen deno es he
ol age ampli ica ion be ween node and he ou pu ; e-
alized by a ansconduc o wi h inpu and ansconduc-
ance , loaded wi h a esis o o esis ance
implemen ed by a eedback ansconduc o [see Fig. 1(a)].
Pa ame e is he ol age gain o he o wa d pa h om he
inpu o he ou pu o he il e ; ealized by a ansconduc o
wi h gain loaded by he same eedback anscon-
duc o as be o e.2
• Finally, is an ma ix composed by capaci ances
wi h he ollowing s uc u e:
(3)
whe e he nega i e and posi i e componen s o he in e-
g a ion nodes a e chosen such ha all he ou -o -diagonal
en ies a e nega i e.
In con en ional - s uc u es, he an i-diagonal en ies o
, namely and , coincide as hey ep esen he same ca-
paci o . Howe e , he e a e p ac ical opologies which lead o
non-symme ical ma ices. This happens, o ins ance, in he
so-called - opamp s uc u es whe e he in eg a ing capac-
i o is connec ed in eedback con igu a ion a ound one opamp
[11].
F om he ep esen a ion in (1) he inpu –ou pu ans e unc-
ion o he il e can be calcula ed as
(4)
In addi ion o his o e all ans e unc ion, o he unc ions need
o be de ined. On he one hand, le be he ans e unc ion
om o he in eg a ion node . I can be shown ha
(5)
whe e
(6)
2Fo he sake o gene ali y he ma hema ical o mula ion in he pape con-
side s he gene al ep esen a ion abo e. In many p ac ical si ua ions, he e is no
o wa d pa h om he inpu so ha
d
=0
. Also, in many cases he ou pu o
he il e is aken om a single in e nal node so ha only one en y o ec o
C
is non-ze o (i.e.,
C
=[0
;
...
;c ;
...
;
0]
). Mo eo e , i no ou pu ol age
ampli ica ion is equi ed, hen ec o
C
is uni a y ( he only non-ze o en y has
uni y alue) and he ou pu summing ne wo k can be supp essed.
FERNÁNDEZ-BOOTELLO e al.: MATRIX METHODS FOR THE DYNAMIC RANGE OPTIMIZATION OF CONTINUOUS-TIME -FILTERS 2527
On he o he hand, le be he ans e unc ion om a cu en
sou ce connec ed o in eg a ion node (be ween he ou pu
e minals o ansconduc o s wi h gain o ) o
he il e ou pu o . I all he ans e unc ions
, a e collec ed in o a ec o
(7)
i is easy o show ha
(8)
As will be shown a e wa ds, ec o s and signi ican ly sim-
pli y he e alua ion o il e dis o ion. Also, hey a e use ul o
calcula e he ela i e sensi i i ies o he il e ans e unc ion
wi h espec o he ma ices in he ep esen a ion (1). They
can be easily ob ained using he Hadama d p oduc as3
(9)
whe e he sensi i i ies a e de ined as .
Up o now, i has been assumed ha ansconduc ances
emain cons an wi h equency. In a mo e gene al case, he
ep esen a ion in (1) mus be ex ended o also cope wi h e-
quency-dependen ansconduc ances [7]. This can be done a
he p ice o a mo e complex s a e-space ma ix desc ip ion. As
an example, le us assume ha ansconduc ance exhibi s a
one-pole oll-o wi h ime cons an . This can be modelled
by eplacing ansconduc o by he ne wo k a he igh
o Fig. 2(a), whe e and he low- equency
ansconduc ance is gi en by .
No e ha his model adds a new node o he opology (la-
belled ) pe ansconduc o . Hence, ma ices o he ex ended
s a e-space ep esen a ion mus be ebuil . This is indica ed in
Fig. 2(b) which shows he ows and columns ha mus be added
o accoun o he new il e node, . Fo consis ency, he low
equency beha io mus be made equal o he o iginal e-
quency-independen en y in (1).
III. NOISE IN - FILTERS
We assume noise is only con ibu ed by ansconduc o s and
ha equency is high enough so ha noise beha io is dom-
ina ed by he mal con ibu ions. The mal noise con ibu ions
om di e en ansconduc o s a e assumed un-co ela ed ( his
is suppo ed by he ac ha ansconduc o s a e di e en phys-
ical en i ies), and a e modelled by ou pu cu en noise sou ces
wi h double-sided powe spec al densi y (PSD)
whe e is he ansconduc ance, is Bol zmann’s cons an ,
3The Hadama d p oduc o wo
m
2
n
ma ices
A
and
B
, deno ed by
A

B
,
is an
m
2
n
ma ix gi en by
(
A

B
) =
a b
[4].
Fig. 2. (a) Ci cui eplacemen o adding one-pole oll-o ansconduc ance
cha ac e is ics o he gene ic schema ic o Fig. 1(a). (b) Requi ed modi ica ions
on he s a e-space ma ices.
is he absolu e empe a u e and is he noise excess ac o o
he ansconduc o — he alue o his la e pa ame e depends
on he ac ual ansconduc o implemen a ion [12].
Using he ans e unc ions de ined in he p e ious sec ion
[see (7) and (8)], he o al ou pu - e e ed noise ol age PSD o
he il e can be exp essed as
(10)
He e, he i s e m accoun s o he noise con ibu ions o all
ansconduc o s in he il e co e. The second e m co esponds
o he ou pu summing s uc u e. F om now on we will assume
ha his second e m is ei he negligible (which occu s o la ge
enough alues o as i happens in p ac ice), o null (which
co esponds o he case whe e he il e ou pu is simply aken
om a single in e nal node). Wi h his assump ion, he o al
ou pu noise o he il e is app oxima ely gi en by
(11)
which ep esen s an uppe -limi alue.
In o de o e alua e he in eg al in (11) i is wo h no ing ha
ma ix , de ined as [1], [13]4
w(12)
can be algeb aically ob ained om he ollowing gene alized
Lyapuno equa ion [14]:
(13)
4
W
is ela ed o he obse abili y g ammian o he sys em,
W
,as
W
=
EWE
[15].
2528 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 55, NO. 9, OCTOBER 2008
Fig. 3. Concep ual schema ic a he
i
h in eg a ion node o he il e including
nonlinea i ies.
and, he e o e, he o al noise o he il e can be w i en as
(14)
whe e w, ep esen he elemen s a he diag-
onal o ma ix is he sum o all he
ansconduc ances d i ing node . In he igh -hand side o his
equa ion, ep esen s he noise con ibu ed o he ou pu
om he h in eg a ion node.
IV. DISTORTION IN - FILTERS
We ocus he e on he impac o nonlinea i ies on he
inpu –ou pu ol age- o-cu en ans o ma ion. Assuming
weakly nonlinea ope a ion condi ions and ha ansconduc o s
a e ully balanced; hei inpu –ou pu cha ac e is ic can be
app oxima ed by
(15)
whe e is he hi d-o de nonlinea i y coe icien and is he
inpu ol age. This simpli ied model, whe e e en-o de non-
linea coe icien s a e null because o he balanced s uc u e,
su ices o mos p ac ical ansconduc o s [11], [12].
Le us i s conside ha he ou pu summing s uc u e o
Fig. 1(a) does no gene a e dis o ion; he in luence o his s uc-
u e will be compu ed in la e on. Using (15), he ex ended
s a e-space ep esen a ion in (1) becomes
(16)
whe e is he Hadama d cube o x(i
is ob ained by h ee consecu i e Hadama d p oduc s—see oo -
no e 3). Fo illus a ion pu poses, Fig. 3 shows a concep ual
schema ic (single-ended o simplici y o he d awing) which
displays he componen s which con ibu e o he h in eg a ion
node acco ding o (16). In his igu e, he linea and nonlinea
componen s a e clea ly sepa a ed.
Se e al app oaches a e ound in li e a u e o he analysis o
he nonlinea beha io desc ibed by (16), [4]–[9]. He e we use
Vol e a’s se ies expansions [10]. This me hod consis s in de-
composing he in e nal nodes a iables in ope a o s, acco ding
o [16]
(17)
whe e is he inpu signal o he sys em, is an a bi a y
ampli ude scaling ac o
(18)
is e e ed o as he h o de Vol e a ope a o and is he
h Vol e a ke nel [10]. The Laplace ans o m o his mul idi-
mensional ke nel is de ined as
(19)
whe e is he -dimensional Laplace a iable. Func ion
desc ibes in equency-domain he h o de
dis o ion pe o mance o he sys em. Hence, desc ibes
he linea beha io o he sys em, accoun s o
he hi d-o de nonlinea beha io , and so on.
The ele an ea u e o Vol e a’s se ies expansions app oach
is ha , o weakly nonlinea sys ems and low alues o , se ies
(17) apidly con e ges and i can be app oxima ed by he i s
ew e ms. The e o e, i he dis o ion beha io o a ully bal-
anced - il e is domina ed by he hi d-o de nonlinea i-
ies o he ansconduc o s, can be simply app oxima ed
by . Replacing his exp ession
in (16) and g ouping e ms wi h he same powe o , he ol-
lowing wo linea sys ems in and a e ob ained
(20)
and
(21)
whe e . The i s equa ions o hese wo sys ems
can be mapped in o he i s - and hi d-o de ci cui s shown in
Fig. 4(a) and (b), espec i ely. Sol ing bo h ci cui s, he i s -
and hi d-o de ans o med ke nels o he il e a e espec i ely
gi en by
(22)
and
(23)
F om hese exp essions, he hi d-o de ha monic dis o ion
o he il e and i s in e modula ion pe o mance can be es i-
ma ed, wi h no ansien analysis needed, by [16]
(24)
whe e is he ampli ude o he inpu ones applied o he sys em.
FERNÁNDEZ-BOOTELLO e al.: MATRIX METHODS FOR THE DYNAMIC RANGE OPTIMIZATION OF CONTINUOUS-TIME -FILTERS 2529
Fig. 4. (a) Fi s -o de and (b) hi d-o de ci cui s a node
i
.
Equa ion (24) can be ela ed o ec o s and , de ined in
(6) and (7), espec i ely, as shown in (25) a he bo om o he
page. I e eals ha he dis o ion e alua ion o a - il e
can be easily accomplished by simple ma ix algeb a. These
closed- o m exp essions compa e a ou ably o o he s in he li -
e a u e, demanding a much mo e cumbe some o mula ion [4],
[5].
Le us now conside he case whe e he ou pu summing s uc-
u e o Fig. 1(a) also con ibu es dis o ion. In his case, he con-
cep ual schema ic o he il e ou pu akes he o m in Fig. 5,
simila o ha in Fig. 3 o he il e co e. As abo e, he analysis
o his ci cui encompasses decomposi ion in o a i s - and a
hi d-o de schema ics (shown, espec i ely, in Fig. 6(a) and (b))
which a e sol ed one a e he o he o gi e
(26)
Fig. 5. Concep ual schema ic o he ou pu s uc u e including nonlinea i ies.
Fig. 6. Fi s -o de (a) and (b) hi d-o de (b) ci cui s o e alua e he dis o ion
o he ou pu s age.
whe e and a e ob ained om (22) and (23). In hese
ep esen s he linea pa o he il e esponse, and
he hi d-o de nonlinea con ibu ion. A e some algeb a, he
hi d-o de ha monic dis o ion o he il e and i s in e modula-
ion pe o mance ake he o m in (27), shown a he bo om o
he page, whe e e ms and , de ined in
(25), a e due o he co e o he il e .
As an illus a ion o he p oposed dis o ion e alua ion
me hod, Fig. 7 shows in solid lines he calcula ed hi d-o de
in e modula ion and ha monic dis o ion componen s o a ully
balanced se en h-o de low-pass Chebyshe il e wi h 10 MHz
cu -o equency. The il e uses a s anda d leap- og s uc u e
(25)
(27)

2530 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 55, NO. 9, OCTOBER 2008
Fig. 7. Es ima ions o
IM
(2
!
0
!
)
and
HD
(
!
)
. using ansien analysis
and he p oposed me hod.
and i is assumed ha all ansconduc o s exhibi a hi d-o de
nonlinea e m .
Calcula ions a e made assuming ha he il e is d i en by
inpu ones o 125 mV a di e en equencies. The hi d-o de
in e modula ion es ima ion assumes a 100 kHz o se equency
be ween he inpu ones. Fo compa ison pu poses, Fig. 7 in-
cludes also en ies o he hi d-o de in e modula ion (as e -
isks) and ha monic dis o ion (ci cles) calcula ed a di e en
inpu equencies by means o con en ional Fou ie analysis o
a ansien esponse. As can be seen, bo h me hods gi e app ox-
ima ely he same esul s, howe e , whe eas he compu a ional
cos o he ansien app oach is 135 s cpu ime, ha o he
p oposed me hod is o only 0.6 s (da a using a 1.8 GHz Pen-
ium Mobile p ocesso ), i.e., mo e han wo o de s o magni ude
lowe .
V. SCALING OF - FILTERS
This sec ion p esen s analy ic exp essions o accoun o he
impac o scaling on il e noise and dis o ion. This is wo h
doing because scaling is cus oma ily employed o il e de-
sign [6], [12], [17], [18]. We conside only scaling ope a ions
which e ain he equency esponse, , equi ed by he ap-
plica ion. This excludes equency scaling ope a ions which a e
easily implemen ed by mul iplying all il e capaci ances by he
same ac o [ his makes and,
he ea e , ] o by scaling all il e anscon-
duc ances by [ his makes and,
he ea e , ].
In he o egoing analysis, il e s a e g ouped in o wo ca e-
go ies. On he one hand, il e s wi h all capaci o s g ounded ex-
cep ing, pe haps, hose connec ing he in eg a ion nodes wi h
he il e inpu ( hey a e cha ac e ized by a diagonal ma ix ).
On he o he hand, il e s which include loa ing capaci o s be-
ween in eg a ion nodes.
Unless o he wise s a ed, i deno es a gi en a iable in he
p o o ype sys em, ep esen s he co esponding ans o med
a iable.
A. Fil e s Wi hou Floa ing Capaci o s
Table I summa izes esul s o he h ee ypes o scaling con-
side ed he ein. The second column shows he basic and de i ed
ma ix equa ions o each ype o scaling. The hi d column in-
cludes commen s ega ding he e ec s o co esponding ans-
o ma ion on he noise and dis o ion pe o mance o he il e .
A i s scaling app oach consis s o mul iplying each ow o
he s a e equa ion in (1) by a co esponding posi i e numbe , .
This ans o ma ion, deno ed as noise scaling in Table I, mod-
i ies he local impedance a each node o he il e wi hou al-
e ing hei ol age swings. Hence, i does no a ec he dis o -
ion beha io o he il e -in e es ing p ope y ha will be ex-
ploi ed la e on.
The noise con ibu ed o he ou pu om he h in eg a ion
node in he scaled il e becomes , whe eas he
sum o all ansconduc ances d i ing he h node o he il e
scales as . I means ha o
educe he noise con ibu ion a node by a ac o ,
he o al ansconduc ance mus be inc eased by he same
ac o . This inc eases he a ea occupa ion o he il e as well,
because capaci ances a e also scaled by .
Fo a gi en alue o he o al ansconduc ance,
, he e is an op imum se o scaling co-
e icien s , which minimizes he o al noise
con ibu ed by he il e . A e some calcula ions (de ailed in
Appendix I) i is ound ha such op imum se is ob ained when
all he diagonal elemen s o he ans o med ma ix a e iden-
ical, i.e.,
w w w (28)
which gi es
w
w
(29)
In his case, he o al noise o he il e can be exp essed as
(30)
A special case o noise scaling is powe scaling in which all
he mul iplying ac o s ake on he same alue and, hence,
all capaci o s and ansconduc o s o he il e co e a e scaled
by . In his case, he o al ou pu noise alue o he il e is
ans o med acco ding o , wi hou a ec ing he
dis o ion beha io . This ac will be used in Sec ion VI o ela e
he o al noise o he il e wi h i s powe consump ion.
Conside now ha scaling is made by mul iplying column
en ies ins ead o ow en ies. This ans o ma ion is labelled
dis o ion scaling in Table I whe e scaling ac o s a e called
. I a ec s he ampli ude le el o he ol ages
a he in e nal nodes o he il e and, he e o e, modi ies i s dis-
o ion beha io . Ac ually, dis o ion imp o es o scaling ac-
o s . Howe e , his ope a ion a ec s he noise beha io
and de ines a ade-o be ween noise and dis o ion. I is wo h
no ing ha ma ix emains unal e ed a e dis o ion scaling
and so w w . This ac will be exploi ed in he nex sec ion.
FERNÁNDEZ-BOOTELLO e al.: MATRIX METHODS FOR THE DYNAMIC RANGE OPTIMIZATION OF CONTINUOUS-TIME -FILTERS 2531
TABLE I
BASIC SCALING TRANSFORMATIONS
A las op ion o il e scaling consis s o applying a sim-
ila i y ans o ma ion on he s a e-space ep esen a ion in
(1). A ypical case, illus a ed in Table I, co esponds o a
diagonal ans o ming ma ix wi h coe icien s .Asin
he p e ious case, his mapping ca ies on bo h noise and
dis o ion modi ica ions on he p o o ype il e , and a ade-o
can be es ablished as well: scaling ac o s dec ease
he noise con ibu ion bu wo sen dis o ion. This ype o
ans o ma ion is ypically used o equalize and maximize
he peak ol ages a he inpu o he ansconduc o s in he
il e — his is done by means o a diagonal ma ix wi h
coe icien s , whe e
is he maximum inpu signal ange o ansconduc o s and
a e he peak alues o each de ined
in (5) ( hey usually occu nea he passband o he il e ) [17].
B. Fil e s Wi h Floa ing Capaci o s
The p esence o loa ing capaci o s be ween he in eg a ion
nodes makes he ma ix non-diagonal. As a consequence,
p e ious scaling ope a ions canno be applied in s and-alone
manne . O he wise he symme y o [see (3)] would be los ,
he eedback and o wa d pa hs o he loa ing connec ions
would be di e en and, consequen ly, hey would be un ealiz-
able wi h simple capaci o s. To o e come his si ua ion, scaling
ope a ions should be pai -wise applied so ha symme y o
is always es o ed a e scaling.5
As an example, le us assume ha a p o o ype il e wi h
loa ing capaci o s is scaled by a diagonal simila i y ans o -
ma ion . The non-diagonal componen s o ma ix
change as , hus, b eaking he symme y o
. In o de o es o e his p ope y and, hence, allow he
use o loa ing capaci o s as in he p o o ype il e , one possi-
bili y is o noise scale he il e (see Table I), in such a way ha
he ela ionship is me o ,
wha gua an ees ha . A simila p ocedu e can be en i-
5Rigo ously speaking, scaling could be pe o med in a single s ep by means
o a simila i y ans o ma ion wi h an o hogonal
T
ma ix
(
T
=
T
)
.
Howe e , om a syn hesis pe spec i e, i is mo e simple and in ui i e o use
wo consecu i e scaling ope a ions.
2532 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 55, NO. 9, OCTOBER 2008
sioned i , ins ead o noise scaling, a dis o ion scaling is applied
o es o ing symme y.
VI. DYNAMIC RANGE OPTIMIZATION OF
GENERAL - FILTERS
We ha e seen ha o a gi en o al ansconduc ance he noise
o a - il e can be minimized by making all diagonal
e ms o ma ix iden ical. We ha e also seen ha , by ap-
plying dis o ion scaling, he ha monic dis o ion and in e mod-
ula ion pe o mance can be imp o ed while keeping he e ms
unal e ed. Hence, an op imum noise-scaled il e will e-
ain his ea u e a e dis o ion scaling. This obse a ion is a
he co e o a ecen ly p oposed p ocedu e o dynamic ange
op imiza ion o weakly nonlinea - il e s unde powe
dissipa ion cons ain s [6]. I consis s o he consecu i e appli-
ca ion o noise and dis o ion op imiza ions on he p o o ype
il e . Howe e , his p ocedu e p esen s wo limi a ions. Fi s ,
he noise and dis o ion con ibu ions, as well as powe con-
sump ion, o ansconduc o s no accoun ed in ma ix a e ne-
glec ed. Indeed, ansconduc o s assessed in ma ix a e e-
placed by ideal cu en sou ces. This simpli ica ion p ecludes
ob aining uni ocal solu ions om he op imiza ion p ocedu e,
as will be illus a ed in he nex sec ion by means o an example.
Second, dis o ion op imiza ion is cons ained by he condi ion
ha he noise gene a ed a he il e co e keeps unal e ed. This
condi ion educes he design space o il e pa ame e s and, con-
sequen ly, may p eclude ha a global op imum is eached.
These d awbacks a e o e come by he p ocedu e p esen ed
in his sec ion. Wi hou loss o gene ali y and o keep ma he-
ma ics as simple as possible, le us assume ha all ansconduc-
o s in he il e sha e he same opology and linea ange, and
exhibi he same cu en e iciency, ; whe e his la e pa am-
e e is de ined as he a io be ween he ansconduc ance and
he biasing cu en o he cell [18], [19]. In his case, he o al
powe consump ion o he il e , , is p opo ional o he sum
o all i s ansconduc ances, ,as ,
whe e is he powe supply ol age o he il e [18]. A p ac-
ical way o ul illing his assump ion is by implemen ing all
ansconduc ance alues h ough he pa allel connec ion o uni-
a y ansconduc o s.6
Le us assume ha a gene ic ully balanced - il e
wi h no ou pu ne wo k (see Fig. 1(a)) has been, i s , op imally
noise scaled and, hen, dis o ion and powe scaled. In his case,
aking in o accoun Table I and applying a powe scaling ac o
, he o al noise o he il e o a o al
powe consump ion becomes
(31)
6This is, indeed, a common p ac ice among in eg a ed ci cui designe s. By
using mul iple ins ances o a gi en ansconduc o , he design complexi y is
no ably educed and he obus ness o he il e agains a ia ions o he ech-
nological p ocess is imp o ed. Using his s a egy in combina ion wi h p ope
layou echniques, ma ching be ween ansconduc o s is la gely a ou ed and
he unabili y o he il e , simpli ied [20], [21].
whe e and a e he coe icien s o he ma ices and
a e op imum noise scaling, and is he sum o all he
ansconduc ances o he il e a e dis o ion scaling. Use ul
o he o egoing analysis, can be also exp essed as
(32)
whe e ( espec i ely, ) is he sum o he ansconduc-
ances o all he ansconduc o s wi h inpu a he h node ( e-
spec i ely, il e inpu ) o he op imum noise-scaled il e .7
Le us u he assume, wi hou loss o gene ali y, ha he dis-
o ion pe o mance o he il e is e alua ed by he hi d-o de
in e modula ion. F om Table I and assuming ha inpu ones
a e close oge he , a can be
exp essed as
(33)
whe e , see de ini ion in (7), is ob ained a e op imum noise
scaling. Equa ion (33) can be also w i en as
(34)
whe e coe icien s and , bo h independen o , a e de-
ined as
(35)
The maximum powe a he ou pu o he il e , , o a peak
in e modula ion dis o ion alue, , can be ob ained
om (34) as
(36)
om whe e he dynamic ange o he il e (see oo no e 1) can
be calcula ed, using (31), as
(37)
This exp ession can be ecas as
(38)
whe e is an adimensional numbe , which
depends on he pa icula ansconduc o implemen a ion used
in he il e , and depends on
he il e s uc u e. Pa ame e gi es a measu e on how la ge
he dynamic ange o a il e can be o a gi en powe dissipa ion
7In his sec ion and he ollowing, i
x
deno es a a iable o coe icien o he
o iginal p o o ype,
~
x
e e s o he co esponding a iable o coe icien o he
op imally noise scaled il e .
FERNÁNDEZ-BOOTELLO e al.: MATRIX METHODS FOR THE DYNAMIC RANGE OPTIMIZATION OF CONTINUOUS-TIME -FILTERS 2533
and dis o ion pe o mance. Hence, i can be used as a igu e o
me i o compa ing ansconduc o implemen a ions.
Equa ion (38) also shows ha o a gi en powe consump-
ion , maximum ole a ed dis o ion and
selec ed ansconduc o opology , he op imiza ion o he
il e dynamic ange implies maximizing .As e-
mains unal e ed a e scaling, he only scaling-dependen ac o
in is , which we de ine as a cos unc ion in
he dynamic ange op imiza ion p ocedu e
(39)
Clea ly, o op imize he dynamic ange o a il e , mus
be minimized. The minimum o is calcula ed by se ing
, which a e some algeb a ob ains he se o equa-
ions
(40)
whe e is he conjuga e o and ex ac s he eal
pa o he a gumen . Summing he le -hand e ms o he abo e
equa ions, on he one hand, and he igh -hand e ms, on he
o he , he ollowing wo ela ionships a e ob ained:
(41)
om whe e, equa ing he esul s and using (40), he op imum
dis o ion-scaling coe icien s can be calcula ed by ecu -
si ely sol ing he exp ession
(42)
using he de ini ion o in (34). Wi h hese alues, he ab-
solu e minimum o becomes
(43)
No e ha i inpu ansconduc o s a e no accoun ed o in he
abo e analysis, which is he si ua ion conside ed in [6], coe -
icien s and a e null and (43) becomes inde e mina e.8
8By emo ing he e ec s o inpu ansconduc o s on dis o ion and noise,
hey a e assumed o pe o m as pe ec ol age- o-cu en con e e s. In essence,
hese ideal con e e s play he same ole as he inpu cu en -con olled cu en
sou ces used in [6], i.e., o injec signal in o a cu en -inpu il e . Assuming ha
B
has no capaci i e componen s, i.e.,
B
is a null ec o , he d i ing signal o
he cu en -inpu il e akes in ou model he o m
B
. In he case o [6], such
d i ing signal is simply gi en by
Bi
. Ob iously, he dimensionali y o ma ix
B
in bo h p ocedu es is di e en bu , om he poin o iew o op imizing he
dynamic ange o he cu en -inpu il e , de ined by ma ix
A
and
E
, his is
i ele an .
Fig. 8. Fully di e en ial
G
-
C
biquad a ic sec ion.
This e eals ha he e is an in ini e numbe o solu ions ha
achie e he same dynamic ange o a gi en powe dissipa ion,
and no a single solu ion as s a ed in [6]. I is o be unde s ood
ha by including he e ec s o inpu ansconduc o s on dis o -
ion and noise, he op imiza ion algo i hm inds he necessa y
cons ain o achie e a single and uni ocal solu ion.
I is also wo h men ioning ha he dynamic ange op imiza-
ion may esul in a non-uni y dis o ion-scaling coe icien o
he ou pu node o he il e , i.e., i he ou pu is aken om node
, coe icien will mo e likely be . This implies ha
ec o becomes non-uni a y, howe e , he e is no need o add
an ampli ica ion ou pu s age o he il e (see Fig. 1) o gua -
an ee an op imum dynamic ange. Ins ead, he ou pu o he il e
could be di ec ly aken om node . No e ha any po en ial
ou pu ne wo k will ideally scale he ou pu noise and desi ed
signal powe by he same ac o while e aining he dis o ion
pe o mance o he il e co e. The e o e, he dynamic ange will
emain unal e ed a e ampli ica ion.
Finally, no e ha he op imiza ion p ocess desc ibed abo e is
es ic ed o a single ope a ing equency. The e o e, such e-
quency mus be ca e ully chosen so ha i co esponds o he
wo s -case dynamic ange o he il e . This can be done by a
p e ious analysis on he noise and dis o ion dependence wi h
equency h ough (10) and (25).
VII. CASE STUDY:BIQUADRATIC SECTIONS
As a case o s udy, he p ocedu e in he p e ious sec ion
is he ein applied o he biquad a ic sec ion o Fig. 8. In he
ollowing analysis, ansconduc o -dependen pa ame e s ha e
been de i ed om a simple olded-cascode opology, designed
in a 0.13 m CMOS echnology a a powe supply o 3.3 V.
They a e and gi ing
. Using hese ansconduc o s, a low-pass il e
wi h cu -o equency a MHz has been designed.
The powe consump ion o his il e is mW.
Using he ex ended s a e-space ep esen a ion o Sec ion II,
he biquad in Fig. 8 can be desc ibed by he ma ices
(44)