On the Use of FIR Feedback in Bandpass Delta-Sigma Modulators

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1
On he use o FIR Feedback in Bandpass
Del a-Sigma Modula o s
Ja ad Go ji, S uden Membe , IEEE, Shan hi Pa an, Fellow, IEEE, and Jos´
e M. de la Rosa, Fellow, IEEE
Abs ac —This pape p esen s a new a chi ec u e o band-
pass del a-sigma modula o s (BP∆ΣMs) ea u ing ini e impulse
esponse (FIR) il e s in he eedback pa h. The e ec i eness
o FIR eedback in lowpass del a-sigma modula o s (LP∆ΣMs)
has been well-es ablished in imp o ing loop- il e linea i y and
obus ness o clock ji e . Building upon hese indings, we explo e
he applica ion o bandpass FIR il e s in single-bi BP∆ΣMs.
By con as o con en ional BP∆ΣMs, he p oposed echnique
signi ican ly educes ou -o -band quan iza ion e o con en s in
he eedback signal. This app oach is applicable o bo h disc e e-
ime and con inuous- ime implemen a ions. Fu he , we show ha
pe o mance does no imp o e by inc easing he numbe o FIR
aps beyond a ce ain poin . Howe e , we can enhance il e ing
pe o mance by employing non-equal coe icien s wi hin he il e .
To alida e he e icacy o he p esen ed app oach, he pape
includes elec ical simula ion o a 4 h-o de ac i e-RC BP∆ΣM.
Index Te ms—Analog- o-digi al con e sion, del a-sigma mod-
ula ion, con inuous- ime ci cui s, FIR DAC, clock ji e .
I. INTRODUCTION
BANDPASS ∆ΣMs digi ize signals placed in an a bi a y
band by applying band-s op il e ing o quan iza ion
noise. Consequen ly, he noise is pushed ou o a bandwid h
(BW) cen e ed a ound he equency n(also called he
no ch equency) [1]. Thei main applica ion is o digi ize
in e media e- equency (IF) o adio- equency (RF) signals in
wi eless ecei e s, hus placing he analog- o-digi al con e e
(ADC) close o he an enna. This way, mos o he signal
p ocessing can be mo ed o he digi al domain and bene i
om echnology downscaling and highe p og ammabili y [2]–
[8].
As RF ADCs need sampling equencies in he GHz ange,
s a e-o - he-a BP∆ΣMs a e domina ed by con inuous- ime
(CT) ci cui s [6]–[8]. CT∆ΣMs can ope a e a highe speeds
when compa ed o hei disc e e- ime (DT) coun e pa s while
consuming less powe . Fu he mo e, hey a e easy o d i e
and possess he p ope y o inhe en an i-aliasing [9], [10].
The choice o he numbe o quan ize le els is pe haps he
Manusc ip ecei ed July 31, 2023; e ised Sep embe 25, 2023.
This wo k was suppo ed in pa by G an s PID2019-103876RB-I00,
PID2022-138078OB-I00, unded by MCIN/AEI/10.13039/501100011033, by
he Eu opean Union ESF In es ing in you u u e, by ERDF A way o making
Eu ope, and in pa by he Cen e o Excellence in RF, Analog and Mixed-
Signal ICs (CERAMIC), IIT Mad as.
Ja ad Go ji and Jos´
e M. de la Rosa a e wi h he Ins i u e o Mic oelec onics
o Se ille, IMSE-CNM (CSIC/Uni e si y o Se ille), C/ Am´
e ico Vespuc-
cio 28, 41092 Se illa, Spain (e-mail: [email p o ec ed]; j osa@imse-
cnm.csic.es).
Shan hi Pa an is wi h he Depa men o Elec ical Enginee ing, Indian
Ins i u e o Technology Mad as (IIT Mad as), Chennai 600036, India (e-mail:
[email p o ec ed]).
i s choice ha needs o be made in any ∆ΣM design. A
mul i-bi quan ize enables he use o a noise ans e unc ion
(NTF) wi h la ge ou -o -band gain (OBG). Howe e , a mul i-
bi ADC is di icul o d i e, pa icula ly a he high sampling
a es needed in an RF ADC. A mul i-bi DAC has p oblems
wi h elemen misma ch, which needs o be add essed wi h
echniques such as dynamic elemen ma ching. This inc eases
he ha dwa e complexi y and makes i di icul and cos ly o
each he equi ed pe o mance in GHz- ange applica ions.
A single-bi quan ize , on he o he hand, has he ad an age
o an ADC ha is easy o d i e and an inhe en ly linea
eedback DAC [11], [12]. Un o una ely, howe e , he ail- o-
ail eedback DAC wa e o m necessi a es inc eased loop- il e
powe dissipa ion o achie e he desi ed linea i y. Fu he mo e,
he modula o ’s sensi i i y o clock ji e is g ea ly inc eased.
This wo k ollows up on he app oach p esen ed in [13] and
p oposes o add ess hese p oblems in a single-bi BP∆ΣM
using bandpass FIR (BP-FIR) eedback. The mo i a ion is
simila o ha in LP∆ΣMs [14], [15]. Like in he lowpass
case, BP-FIR eedback in a BP∆ΣM educes he heigh o
he s eps in he eedback DAC wa e o m, he eby educing
he magni ude o he e o signal ha d i es he loop- il e .
Fu he mo e, FIR eedback implemen ed using a semi-digi al
FIR DAC main ains he inhe en linea i y o he single-bi
quan ize . Finally, sensi i i y o clock ji e and compa a o
me as abili y a e educed.
Al e na i ely, in ini e impulse esponse (IIR) il e ing has
been applied o he eedback pa h o a BP∆ΣM be o e
[16]. Apa om needing a mul i-bi eedback DAC whose
pe o mance is deg aded due o elemen misma ch, he in e nal
eedback needed o implemen an IIR il e also appea s
challenging. An FIRDAC a oids hese p oblems.
Following his in oduc ion, he a icle is o ganized as
ollows: Sec ion II desc ibes he p oposed BP∆ΣM wi h
FIR il e ing, conside ing a DT implemen a ion. Sec ion III
ex ends his app oach o CT BP∆ΣMs and compa es hem
wi h con en ional implemen a ions. We conside wo case
s udies: a second- and a ou h-o de loop- il e , and in his
con ex , we ha e discussed he calcula ion o coe icien s using
wo di e en me hods. In Sec ion IV, a compa ison is made
be ween he use o FIR il e in LP∆ΣMs and BP∆ΣMs.
We will show how, in bo h cases, using p ope FIR il e ing
will emula e mul i-bi ope a ion. The conside a ions o he
op imum design o he FIR il e a e add essed in Sec ion
V. Finally, mapping hese sys em-le el concep s o ci cui
implemen a ion is discussed in Sec ion VI, and conclusions
a e d awn in Sec ion VII.
2
(a)
u[n] n[ ]
-1+-2
z
-2
z
-
e n[]
1
F eq.
s4
Magni ude
1
(c)
-z-2
n[ ]
[n]
1
a0a1a
N i -1
-z-2 -z-2
(b)
u[n] n[ ]
--
F( )z
e n[]
2
n[]
1
1+-2
z
-2
z
-
F ( )z
c
Fig. 1. Block diag am o a DT BP∆ΣM. (a) Con en ional. (b) P oposed
wi h BP-FIR eedback. (c) BP-FIR il e s uc u e and equency esponse.
II. DISCRETE-TIME BP∆ΣMS WITH BP-FIR FEEDBACK
To build insigh , we i s conside a second-o de disc e e-
ime BP∆ΣM wi h a cen e equency s/4as shown in
Fig. 1(a). The NTF o he modula o is NTF(z) = 1+z−2[9],
[10]. The objec i e is o add an FIR il e o he eedback [see
Fig. 1(b)] and ebuild he loop in such a way ha he NTF
is es o ed. The inpu signal is a n= s/4 a he han DC,
hence a BP-FIR il e cen e ed a n[see Fig. 1(c)] mus be
used o F(z) o il e he 2-le el quan ize ou pu sequence
be o e d i ing he eedback in he main pa h.
We can s a by conside ing he gene al o m o a lowpass
FIR (LP-FIR) il e wi h he ollowing ans e unc ion
FIRLP(z) =a0+a1·(z−1) + a2·(z−1)2+···+
aN i −1·(z−1)(N i −1) (1)
whe e N i is he numbe o aps in he FIR il e and ai
a e he il e coe icien s. The equi alen BP-FIR il e wi h a
passband cen e ed a s/4, is ob ained by applying a z→ −z2
ans o ma ion o a LP-FIR il e (1), yielding
FIRBP(z) =a0+a1·(−z−2) + a2·(−z−2)2+···+
aN i −1·(−z−2)(N i −1).(2)
The gain in he passband o he il e is de e mined by
summing all he coe icien alues oge he . Fo simplici y,
he ap weigh s o he FIR il e F(z)a e conside ed o be
he same and equal o 1/N i , so ha he gain o he signal
ans e unc ion (STF) a s/4is uni y. Based on (2), we can
w i e he ans e unc ion o his il e as ollows
F(z) = 1
N i ·
N i −1
X
n=0
(−z−2)n. (3)
The delay in oduced by he FIR il e would ende he loop
uns able wi hou a compensa ion pa h a ound he quan ize
[15]. This is shown in Fig. 1(b). Fo he modula o s in
-1
0
1u[n]
-1
0
1 [n]
w w o& / FIR
-2
-1
0
1
2
e1[n]
w/o FIR
-2
-1
0
1
2
e [n]2
2Tap-FIR
-2
-1
0
1
2
0 8 16 24 32 40 48 56 64
e [n]2
4Tap-FIR
Sample Index
Ampli ude (V)
(a)
(b)
(d)
(e)
c)(
Fig. 2. Impac o BP-FIR eedback in DT BP∆ΣM sequences. (a) Inpu o he
modula o , u[n]( in = s/4+BW/3). (b) Ou pu , [n]. (c) Resona o inpu
o a con en ional BP∆ΣM, e1[n]. (d), (e) Resona o inpu o he p oposed
BP∆ΣM, e2[n] o N i = 2 and N i = 4.
Figs. 1(a) and (b) o ha e iden ical NTFs, we see ha he
ollowing has o be sa is ied
F(z)·−z−2
1 + z−2+Fc(z) = −z−2
1 + z−2(4)
hence, Fc(z)is calcula ed as
Fc(z) = [1 −F(z)] ·−z−2
1 + z−2(5)
since F(z)has a uni y gain a z=j(co esponding o =
s/4), [1 −F(z)] has ze os a z=±j. Consequen ly, we can
w i e i as
[1 −F(z)] = (1 + z−2)·P(z)(6)
whe e P(z)is a polynomial. Using (6) in (5), Fc(z)is seen
o be
Fc(z)=(−z−2)·P(z). (7)
F om he abo e discussion, i can be concluded ha Fc(z)is
also an FIR il e . This indica es ha he o iginal NTF can
be es o ed exac ly. Fu he mo e, he aps o Fc(z)a e no
iden ical in gene al.
Fo a 2- ap FIR, whe e F(z)=(−z−2)[0.5+0.5(−z−2)],
Fc(z) u ns ou o be (−z−2)[1 + 0.5(−z−2)] which demon-
s a es a BP-FIR ollowed by wo delays in each pa h. The
simula ion esul s o Fig. 1(a) and (b) a e shown in Fig. 2.
The ou pu o he modula o emains iden ical in bo h cases,
as he NTFs a e he same. The main di e ence lies in he e o
signal ed in o he esona o . Thanks o FIR eedback, e2[n]
is conside ably smalle wi h 2- and 4- ap FIR eedback when
compa ed o he case wi hou an FIR il e (i.e., e1[n]). The
easoning can be ex ended o a bi a y NTFs and o he CT
domain.
3
z-2
u () n[ ]
s
--
RZ
DAC
ωs
s2+2
ω
k3
NRZ
DAC
e( )
1( )
(a)
y[ ]
HRZ
DAC k1
k2
n[ ]
s
-
ωs
s2+2
ω
k3
NRZ
DAC
(b)
y[ ]
z-1
z-1
-1/2
z
NRZ
DAC
u ()
-
e( )
1( )
k1
k2
NRZ
DAC
Fig. 3. Con en ional 2nd-o de CT BP∆ΣMs wi h a mul i-pa h eedback
loop based on: (a) Di e en DAC (RZ/HRZ) wa e o ms [17]. (b) Iden ical
DAC (NRZ) wi h di e en delays [18].
200 210 220 230 240 250 260 270 280 290 300
Ampli ude (V)
0
1
-1
Inpu (u)
DAC ( 1)
200 210 220 230 240 250 260 270 280 290 300
Ampli ude (V)
Time (nTs)
0
-1
1E o Signal
Fig. 4. Inpu signal ( in = s/4+BW/3), DAC ou pu signal, and modula o
inpu e o signal in a con en ional 2nd-o de single-bi CT BP∆ΣM.
III. CONTINUOUS-TIME BP∆ΣMS
Fig. 3 shows he block diag am o he wo common
app oaches o implemen ing CT BP∆ΣMs wi h mul i-pa h
eedback, u ilizing an ideal esona o wi h a ans e unc ion
HRes(s) = ωos/(s2+ω2
o). He e, ωo= 2π nis he cen e
equency. A 2nd-o de loop- il e made up o a esona o
wi h n= s/4is conside ed. The eedback loop can be
implemen ed in mul iple ways. Fo example, Fig. 3(a) [17]
uses wo di e en DAC pulse shapes, e u n- o-ze o (RZ)
and hal -delayed e u n- o-ze o (HRZ), while Fig. 3(b) [18]
includes a non- e u n- o-ze o (NRZ) DAC wi h a di e en
delay o each pa h. In bo h cases, mul iple eedback pa hs
a e equi ed o implemen he desi ed NTF.
In he main pa h o a con en ional CT BP∆ΣM (Fig. 3),
he eedback signal allows passing bo h he digi ized signal
(placed a ound n= s/4) as well as mos o he ou -o -
band equency componen s o he shaped quan iza ion e o .
As a consequence, he DAC ou pu 1( )does no app oach
he inpu signal u( ). Acco dingly, he e o signal a he inpu
node o he modula o e( ), which coincides wi h he inpu o
he esona o , displays conside able changes. This is illus a ed
in Fig. 4 o a 2nd-o de BP∆ΣM wi h a maximally la NTF
wi h an OBG o 1.5.
As p e iously s a ed, his phenomenon imposes limi a ions
on he signal- o-noise a io (SNR) in BP∆ΣMs, pa icula ly
u () n[ ]
s
z-1
--
z-1/2
-z-2 -z -2
NRZ
DAC
1/N i
z-1
ωs
s2+2
ωk0
1/N i k2k1
NRZ
DAC
e( )
1( )
(a)
(b)
(c)
Compensa ion ( as ) pa h:
NRZ
DAC
z-1
zz-1
z
(d)
P ecise a h :p by me ged blocks
z-1
zk0
P ecise a h:p
z-1/2
z-1
z
ωs
s2+2
ω
NRZ
DAC k0
F( )z
F( )z
F( )zH ( )s
Aux
[ ]n
[ ]n
[ ]n
y[ ]
Fig. 5. (a) P oposed 2nd-o de CT BP∆ΣM wi h 2- ap BP-FIR DAC. (b)
Compensa ion pa h a ound he quan ize . (c) P ecise pa h. (d) P ecise pa h
ede ined by me ging hal delay, CT esona o , and he NRZ DAC in o an
auxilia y block.
when op ing o single-bi quan iza ion o simpli y he sys em
in high-speed design. Howe e , his simpli ica ion is no
es ic ed o high speeds; e en a low-speed design bene i s.
In such scena ios, employing FIR eedback p o es highly
bene icial, as i e ec i ely il e s he quan iza ion e o and
con e s a high-speed 1-bi s eam in o a mul i-le el wa e o m.
A. 2nd-o de CT BP∆ΣM wi h 2- ap BP-FIR DAC
Le us conside he simples case o a 2nd-o de BP∆ΣM as
shown in Fig. 5(a). The o wa d pa h is made up o a esona o
and a 1-bi quan ize , and he eedback pa hs a e implemen ed
using delayed 2- ap BP-FIR DAC. While in DT BP∆ΣM wi h
FIR eedback we had he same delay o 2, be o e F(z)and
Fc(z), i u ns ou ha ano he deg ee o eedom is needed
in he CT domain o ma ch he loop- il e ans e unc ion
wi h an equi alen DT BP∆ΣM. This is accomplished by
applying di e en delays o he main eedback pa h (1.5-cycle
delay) and he compensa ion pa h a ound he quan ize (2-
cycle delay).
The objec i e is o ind he coe icien s ha es o e he
co ec NTF. Sch eie ’s oolbox [19] is used o ge he de-
si ed NTF(z)wi h an OBG o 1.5. The 2nd-o de DT loop-
il e ans e unc ion can be easily ob ained om Ld(z) =
1/NTF(z)−1as ollows
Ld(z) = −0.6667
(z2+ 1) . (8)
The equi alen DT loop- il e ans e unc ion o he desi ed
CT BP∆ΣM, deno ed as Lc(z), is de i ed using a me hod
simila o ha ound in [20]. The loop- il e o he modula o
consis s o a as pa h a ound he quan ize Fig. 5(b), which
is en i ely in he disc e e- ime domain, and a p ecise pa h
ha is esponsible o he noise shaping in he modula o
Fig. 5(c). The p ecise pa h consis s o bo h disc e e- ime
and con inuous- ime componen s. Fo simpli ica ion pu poses,
we can agg ega e he hal delay, he CT esona o , and he
4
NRZ DAC in o an auxilia y block, deno ed as HAux(s), and
ede ine he p ecise pa h as shown in Fig. 5(d). Based on hese
assump ions, he loop- il e o he modula o in Fig. 5(a) can
be w i en in he z-domain:
Lc(z) = k0·Z{HAux(s)}·F(z)·z−1
|{z }
P ecise pa h
+Fc(z)·z−2
| {z }
Fas pa h
(9)
whe e k0 ep esen s he scaling gain o he esona o ou pu .
Based on (9), o de e mine Lc(z)all we need o do is calcula e
HAux(z), which can be exp essed as ollows
HAux(z) = Z[HRes(s)·HDAC(s)] (10)
whe e HRes(s) ep esen s he esona o ans e unc ion and
HDAC(s)deno es he ans e unc ion o a hal -delayed NRZ
DAC pulse shape. In he s-domain, we de ine an a bi a y
ec angula DAC pulse as
HDAC(s) = e− ds−e−( d+τ)s
s(11)
he e τdeno es he pulse wid h, and dsigni ies he applied
delay o he pulse. In he case o an NRZ DAC pulse delayed
by hal , τand d ake alues o 1and 0.5, espec i ely.
Subs i u ing (11) in o (10), we ob ain
HAux(z) = ZHRes(s)·e−0.5s
s−ZHRes(s)·e−1.5s
s.
(12)
In (12), we ha e delays ha a e no in ege mul iples o he
sampling pe iod. The e o e, he use o a modi ied z- ans o m
is equi ed [21]. Hence, we can ew i e (12) as
HAux(z) = Zm1 HRes(s)
s−Zm2 HRes(s)
s(13)
whe e mi= 1 − dis he ac ional delay pa ame e and is
alid o 0< mi<1. In he case o a hal -delayed NRZ DAC,
m1= 0.5and m2=−0.5. He e we can elimina e a comple e
delay om m2and conside ha m2is a ac ional delay
in he i s sampling phase, so m2= 0.5. We will ake in o
accoun he impac o his delay in ou u he calcula ions.
Gi en hese, (13) can be de i ed using he esidue heo em
[22] as ollows
HAux(z) = X
poles o HRes(s)
s
Residues HRes(s)
s·em1s
z−es
−

X
poles o HRes(s)
s
Residues HRes(s)
s·em2s
z−es

·(z−1)
(14)
he e he delay ha is mul iplied in he second pa o (14) is
due o he assump ion we made o m2. The e o e, HAux(z)
is de e mined as
HAux(z) = 1
√2·z−1(1 −z−2)
1 + z−2.(15)
By subs i u ing (15) in (9) and equa ing Lc(z) = Ld(z), he
alues o k0,k1, and k2a e ob ained as −0.47,−0.5, and
−0.17, espec i ely. By ollowing a simila p ocedu e, he
200 210 220 230 240 250 260 270 280 290 300
Ampli ude (V)
0
1
-1
Inpu (u)
FIR+DAC ( 1)
200 210 220 230 240 250 260 270 280 290 300
Ampli ude (V)
Time (nTs)
0
1
-1
E o Signal
( )b
0 0.10 0.15 0.25 0.30 0.35 0.40
-80
-40
0
40
80
0.450.20 0.500.05
NTF
2-Tap FIR
4-Tap FIR Resona o
6-Tap FIR
8-Tap FIR
Magni ude (dB)
Rela i e F equency
(a)
Fig. 6. The impac o he p oposed BP-FIR DAC in a 2nd-o de single-bi CT
BP∆ΣM: (a) Inpu signal ( in = s/4 + BW/3), FIR+DAC ou pu signal,
and modula o inpu e o signal o N i = 8. (b) NTF and FIR il e ans e
unc ion o di e en numbe o aps.
0
10
20
30
40
50
60
70
SNR (dB)
-120
-100
-80
-60
-40
-20
0
0.24 0.245 0.25 0.255 0.26
Rela i e F equency
-70 -60 -50 -40 -30 -20 -10 0
Inpu Ampli ude (dBFS)
Ideal Disc e e-Time
P oposed Con inuous-Time
Ideal Disc e e-Time
P oposed Con inuous-Time
OSR = 256
OSR = 64
(a)
(b)
PSD (dBFS)
Fig. 7. Compa ison o he p oposed 2nd-o de BP∆ΣM wi h N i = 8 and
i s DT coun e pa : (a) Ou pu PSD. (b) SNR e sus inpu ampli ude.
coe icien s o CT BP∆ΣM assis ed wi h BP-FIR wi h mo e
aps can be de e mined.
As illus a ed in Fig. 6(a), he e o p ocessed by he loop-
il e , which is e( ) = u( )− 1( ), is signi ican ly educed wi h
espec o con en ional BP∆ΣMs wi h single-bi quan ize
(see Fig. 4). As a esul , he linea i y equi emen o he
esona o in he loop- il e is elaxed, jus like in CT∆ΣMs
wi h mul i-bi quan iza ion. The p esence o F(z)accoun s o
his unc ionali y. As depic ed in Fig. 6(b), F(z)is a bandpass
il e wi h a passband equency cen e ed a n= s/4. Thus,
he inpu signal componen o [n]is no a ec ed, hough he
powe o he shaped noise is diminished. The e o e, he DAC
ou pu , 1( ), has educed ou -o -band (quan iza ion e o )
con en and only con ains equency componen s close o he

5
(b)
(a)
u () n[ ]
--
k2
z-1/2
k0
-
ωs
s2+2
ω
ωs
s2+2
ω
z-1
z-1
NRZ
DAC
e( )
1( ) s
k1
-z-2
1/N i 1/N i
1/N i
-z-2
-z-2
-z-2
k3
k4
km
F( )z
F( )z
F( )z
F ( )z
c
y[ ]
u () n[ ]
--
k2
z-1/2
k0
-
ωs
s2+2
ω
ωs
s2+2
ω
z-1
z-1
k1NRZ
DAC
e( )
1( ) s
-z-2
1/N i 1/N i
1/N i
-z-2
-z-2
-z-2
k3
k4
km
F( )z
F( )z
F( )z
F ( )z
c
y[ ]
(c)
z-1.5 ωs
s2+2
ω
NRZ
DAC
F( )z
Main Pa h1:
ωs
s2+2
ω
1
l ( )
1
~
ωs
s2+2
ω
NRZ
DAC
F( )z
Main Pa h2:
z-1.5
z-2
z
F( )zωs
s2+2
ω
NRZ
DAC
Main Pa h3:
Compensa ion (Fas ) Pa h:
NRZ
DAC
z-2
z
l ( )
0l[n]
0
[ ]n
l ( )
2
~
l ( )
3
~
l ( )
0
~
l ( )
3l[n]
3
l ( )
2l[n]
2
l ( )
1l[n]
1
k2
k1
k0
Fig. 8. P oposed 4 h-o de CT BP∆ΣM wi h N i - ap BP-FIR DAC in: (a) CRFB s uc u e, and (b) CRFF-B s uc u e. (c) Open-loop impulse esponse
analysis o he modula o pa hs.
inpu u( )a ound s/4.
Fig. 7(a) and (b) illus a e he powe spec al densi y (PSD)
o he ou pu and he SNR e sus inpu ampli ude o he
modula o in Fig. 5, espec i ely. In bo h cases, excellen
alignmen is obse ed be ween he CT∆ΣM and i s DT
coun e pa . Achie ing his ema kable ag eemen in ol es
compensa ing o he deg ada ion induced by sinc(1/4), as
is he case in CT BP∆ΣM wi h NRZ DACs and n= s/4
sampling. To achie e his, a deg ada ion (≈0.9) is applied o
he inpu be o e eaching he summa ion node. Th ough his
app oach, he maximum s able ampli ude (MSA) is ma ched
o ha o he DT BP∆ΣM.
B. 4 h-o de CT BP∆ΣM wi h N i - ap BP-FIR DAC
The p esen ed app oach can be applied o high-o de
BP∆ΣMs wi h N i - ap BP-FIR il e ed DACs. This is illus-
a ed o a 4 h-o de BP∆ΣM wi h wo di e en s uc u es:
a cascade o esona o s wi h eedback (CRFB) shown in
Fig. 8(a), and a cascade o esona o s wi h eed o wa d and
eedback (CRFF-B) shown in Fig. 8(b).
The p ocess s a s by ob aining a 4 h-o de DT loop- il e
ans e unc ion. Fo a maximally la NTF wi h an OBG o
1.5, we ha e
Ld(z) = −0.7749z2−0.5585
(z2+ 1)2. (16)
Rega dless o he s uc u e ype, he coe icien s can be de e -
mined using he echnique discussed in he p e ious sec ion.
Howe e , pe o ming ma hema ical calcula ions o highe -
o de NTFs can be edious and imp ac ical in eal designs.
This is due o he complexi y o he ans e unc ion, which
includes addi ional poles and ze os caused by ini e gain-
bandwid h (GBW) in he ac i e blocks (such as op-amps o
ansconduc o s) and pa asi ics o he ci cui . De e mining he
exac loca ions o hese addi ional poles and ze os om ci cui
simula ions is a challenging ask ha is no s aigh o wa d.
The e o e, in his sec ion, we ollow a obus nume ical
me hod known as “closed-loop i ing” [23] o de e mine he
coe icien s and aps o he compensa ion FIRDAC. The main
idea behind his me hod is based on ew i ing he ela ionship
be ween NTF and he loop- il e ans e unc ion as
Ld(z)×NTF(z) = 1 −NTF(z). (17)
Le us deno e he impulse esponse co esponding o Ld(z)
and NTF(z)by l[n]and h[n], espec i ely. We can exp ess
equa ion (17) in ime domain as ollows
l[n]∗h[n] = δ[n]−h[n]. (18)
Acco ding o Fig. 8(c), depic ing he a ious pa hs o
he 4 h-o de modula o , we can de e mine ha l[n] =
[˜
l1[n]˜
l2[n]˜
l3[n]˜
l0[n]] K
Fc. Fu he , le h0[n] = ˜
l0[n]∗h[n],
h1[n] = ˜
l1[n]∗h[n],h2[n] = ˜
l2[n]∗h[n], and h3[n] =
˜
l3[n]∗h[n]. Subsequen ly, (18) can be w i en as











h1[1] h2[1] h3[1] h0[1] 0 0 ···
h1[2] h2[2] h3[2] h0[2] 0 0 ···
h1[3] h2[3] h3[3] h0[3] h0[1] 0 ···
h1[4] h2[4] h3[4] h0[4] h0[2] 0 ···
h1[5] h2[5] h3[5] h0[5] h0[3] h0[1] ···
h1[6] h2[6] h3[6] h0[6] h0[4] h0[2] ···
.
.
..
.
..
.
..
.
..
.
..
.
....















k0
k1
k2
Fc




=











1
0
0
0
0
0
.
.
.











−











h[1]
h[2]
h[3]
h[4]
h[5]
h[6]
.
.
.











.
(19)
6
TABLE I
TABLE OF CALCULATED COEFFICIENTS FOR A 4TH-ORDER CT BP∆ΣM
WITH OBG = 1.5AND EQUAL TAPS (1/NFIR )IN F(z)
2-Tap 3-Tap 4-Tap 5-Tap 6-Tap 7-Tap 8-Tap
k0−0.097 −0.097 −0.097 −0.097 −0.097 −0.097 −0.097
k1−0.662 −0.739 −0.815 −0.892 −0.968 −1.045 −1.121
k20.054 0.054 0.054 0.054 0.054 0.054 0.054
k3−0.527 −0.591 −0.623 −0.643 −0.656 −0.665 −0.672
k4−0.193 −0.405 −0.511 −0.574 −0.617 −0.647 −0.670
k5-−0.147 −0.344 −0.463 −0.542 −0.598 −0.640
k6- - −0.123 −0.308 −0.431 −0.518 −0.584
k7---−0.109 −0.283 −0.408 −0.501
k8- - - - −0.1−0.266 −0.391
k9-----−0.094 −0.253
k10 ------−0.089
In a p ac ical implemen a ion, hese impulse esponses can be
easily ob ained h ough simula ion, which akes in o accoun
he impac o all non-ideali ies. By sol ing he equa ion
men ioned abo e, we can de e mine he weigh ing coe icien s.
Table I summa ises he calcula ed coe icien s o a 4 h-o de
CT BP∆ΣM wi h 2- ap o 8- ap FIR il e .
Analyzing he impulse esponse o he CT sys em and
compa ing i wi h he DT sys em is essen ial o iden i ying
he expec ed ma ching c ossing poin s. Fig. 9(a) illus a es his
concep in he con ex o a 4 h-o de CT BP∆ΣM wi h a 4-
ap FIR il e . The addi ional delay in oduced by F(z)a ec s
samples 2,4,6, and 8. Howe e , as discussed in Sec ion II, by
employing a BP-FIR il e wi h an equal numbe o aps Fc(z),
we can e ec i ely compensa e o his dis o ion. Fig. 9(b)
demons a es he con o mi y o he CT NTF wi h he DT NTF
a e his compensa ion p ocess.
The i s ap in he FIR il e plays a c i ical ole, as
i is he closes s age o he quan ize and is pa icula ly
suscep ible o da a-dependen ji e esul ing om compa a o
me as abili y. Howe e , in all he cases discussed, he p oposed
s uc u e inco po a es a su icien delay be ween he ou pu
o he modula o and he i s ap in he FIR il e . This
cha ac e is ic alle ia es he equi emen s o quan ize in e ms
o me as abili y.
The decision be ween CRFB and CRFF-B s uc u es de-
pends on he in ended applica ion o he ADC. CRFF-B
is p e e ed o wo easons. Fi s ly, his s uc u e o e s
he ad an age o emo ing one o he FIRDACs om he
eedback, esul ing in powe and a ea sa ings. Secondly, he
inco po a ion o a o wa d pa h a ound he second esona o
con ibu es o be e educing he inpu swing o he second
esona o . The e o e, we can highligh ha in he p oposed
a chi ec u e, he bene i s o he BP-FIR DAC in educing e o
signal swings ex end beyond jus he inpu o he i s esona o
and also apply o he inpu o he second esona o . This is
depic ed in Fig. 10, whe e he inpu swing o he second
esona o exhibi s a ema kable educ ion when compa ed
o he con en ional case wi hou FIR il e ing. Howe e , he
main limi a ion o he eed o wa d app oach, in gene al, is
i s inabili y o e ec i ely handle close o in-band in e e e .
As demons a ed in Fig. 11, he p oposed CRFF-B exhibi s
STF peaking nea s/4. This peaking can po en ially lead he
modula o o sa u a e when he ADC needs o ope a e in he
p esence o s ong blocke s close o in-band, as is he case in
02 4 6 8 10 12 14 16 18 20
Time (Second)
-3
-2
-1
0
1
2
3
y( )
Wi hou
Compensa ion Pa h
Wi h
Compensa ion Pa h
Impulse
Response ( )l n[ ]
(a)
0
-100
-80
-60
-40
-20
0
20
/4
s /2
s
F equency
Magni ude (dB)
DT NTF
CT NTF
OBG=1.5
3.54
3.52
3.50
(dB)
(b)
Fig. 9. 4 h-o de CT BP∆ΣM wi h 4- ap BP-FIR DAC: (a) Impulse esponse
o y( )and he in luence o Fc(z)on i s es o a ion. (b) Res o ed la CT
NTF wi h OBG = 1.5, in compa ison o he DT NTF.
Numbe o ccu encesO
P oposed CRFB
P oposed CRFF-B
Con en ional
-1 -0.8 -0.6 -0.4 -0.2 0 1
0.80.60.40.2
Ampli ude (V)
10K
8K
6K
4K
2K
0
Inpu Swing
2 Resona o
nd
Fig. 10. Compa ison o he p oposed 4 h-o de CT BP∆ΣM wi h 6- ap BP-
FIR DAC wi h con en ional implemen a ion in e ms o inpu swing in he
second esona o .
0 0.10 0.15 0.25 0.30 0.35 0.40 0.450.20 0.500.05
Rela i e F equency
-70
|STF| (dB)
- 06
- 05
- 04
- 03
- 02
-10
10
0
STF CRFB (w & w/o FIR)
STF CRFF-B (w FIR 4 6 8 Tap), , -
0-dB a ound
s4
mo e aps
Fig. 11. STF in CRFB and CRFF-B s uc u es o a 4 h-o de CT BP∆ΣM.
ce ain adio applica ions.
Conside ing ha we a e es o ing he o iginal NTF wi h
his app oach, i does no esul in an imp o emen in SNR.
Howe e , we can apply he same app oach o syn hesize a
mo e agg essi e NTF wi h a highe -o de and/o a highe OBG
while conside ing he s abili y o he sys em. A highe OBG
co esponds o lowe in-band gain o he NTF, which can lead
o an enhanced SNR. This e ec is demons a ed in Fig. 12
o he PSD o a 6 h-o de CT BP∆ΣM wi h OBGs o 1.5
and 1.7, using 8- ap BP-FIR eedback.
7
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
-200
-160
-120
-80
-40
0
0.245 0.25 0.255 0.26 0.265 0.27 0.275
-200
-160
-120
-80
-40
0
Rela i e F equency
PSD (dBFS)
Rela i e F equency
PSD (dBFS)
4.61 dB
NTF wi h OBG = 1.7
SNR = 8 dB2.3
@ OSR = 64
OBG = 1.7
SNR = 7 dB9.4
@ OSR = 64
OBG = 1.5
Smoo hed FFT
Inpu Tone
-6 dBFS
(b)
(a)
Fig. 12. (a) PSD and NTF o a 6 h-o de CT BP∆ΣM wi h 8- ap BP-FIR
DAC ( la NTF wi h OBG = 1.7). (b) PSDs o he same modula o wi h OBGs
o 1.5and 1.7.
IV. COMPARISON BETWEEN LP∆ΣMS AND BP∆ΣMS
WITH FIR FEEDBACK
In gene al, o a single-bi ope a ion, assuming ha he
modula o ou pu bi s eam is [n] = ±1, he magni ude o
ansi ions in [n]is 2. In a LP∆ΣM assis ed wi h FIRDAC,
his is dec eased o 2/N i hanks o he ac ion o F(z). As
a esul , he s ep magni udes a e N i imes smalle , and in
he inpu summa ion node he signals a e simila o a mul i-
bi LP∆ΣM [see Fig. 13(a)]. The same condi ion applies o
BP∆ΣM. As shown in he p e ious sec ions, we can emula e
mul i-bi ope a ion by using a single-bi DAC and assis ing i
wi h a BP-FIR il e . In his case, he magni ude o ansi ions
dec eased, bu no in a manne simila o he LP∆ΣM. Due
o he limi a ion imposed by n= s/4, he sampling occu s
only ou imes du ing one pe iod o he inpu signal. As a
esul , he signal can a y a lo om sample o sample. In
such a si ua ion, al hough we ha e le els in be ween, hese
le els a e sepa a ed in he ime domain [see Fig. 13(b)].
These discussions hold signi icance when aiming o explo e
he ji e pe o mance o he modula o , since clock ji e is
p opo ional o he heigh o he ansi ions in he DAC ou pu
[24]. Gene ally, he ji e is composed o an inpu ela ed
componen and a quan iza ion noise ela ed componen . In
bo h cases o LP/BP∆ΣMs, no ma e whe he using mul i-
bi ope a ion wi hou FIRDAC o single-bi wi h FIRDAC,
he ji e componen associa ed wi h he quan iza ion noise
is educed since his pa o he ji e is p opo ional o he
heigh o he ansi ions in he DAC ou pu , while he ji e
componen ela ed o he inpu signal is no changed. Bea ing
in mind ha he a ia ion in he inpu signal o a BP∆ΣM is
much mo e han ha o a LP∆ΣM. Recall ha e en mul i-bi
BP∆ΣM has he same limi a ion.
(a) CT ΔΣLP Ms
1bi DAC + w/o FIR
1bi DAC + 3- ap FIR
2bi DAC + w/o FIR
3bi DAC + w/o FIR
1/3
-1/3
-1
1
5/7
1
3/7
1/7
-5/7
-1
-3/7
-1/7
1bi DAC + 7- ap FIR
1/3
-1/3
-1
1
1
-1
5/7
1
3/7
1/7
-5/7
-1
-3/7
-1/7
Time
Time
Time
Time
Time
M-Bi w/o FIR 2M
1-Bi + N-Tap FIR N+1
All ansi ion
le els occu
du ing one
pe iod.
Num. Le els
in ( )
1
(b) CT ΔΣBP Ms
1bi DAC + w/o FIR
1bi DAC + 7- ap FIR
1bi DAC + 3- ap FIR
2bi DAC + w/o FIR
3bi DAC + w/o FIR
1/3
-1/3
-1
1
1/3
-1/3
-1
1
1
-1
5/7
1
3/7
1/7
-5/7
-1
-3/7
-1/7
5/7
1
3/7
1/7
-5/7
-1
-3/7
-1/7
Time
Time
Time
Time
Time
M-Bi w/o FIR 2M
1-Bi + N-Tap FIR N+1
Max. 4 ansi ion
le els occu
du ing one
pe iod.
Num. Le els
in ( )
1
Inpu u( ) Feedback ( )
1
Fig. 13. Feedback signal o he modula o [ 1( )] in single-bi ope a ion,
mul i-bi ope a ion, and single-bi assis ed wi h FIRDAC in: (a) LP∆ΣMs.
(b) BP∆ΣMs.
V. BP-FIR FILTER CONSIDERATIONS
A. Numbe o FIR aps
The ji e pe o mance o a 4 h-o de BP∆ΣM wi h FIR-
DAC is shown in Fig. 14(a) o a −7dBFS inpu . The
esul ma ches ou expec a ions based on he discussion in
he p e ious sec ion. The pe o mance o he sys em will no
imp o e much a e 6- ap, while in a simila case o LP∆ΣM
wi h FIRDAC, ji e immuni y imp o es by inc easing numbe
o FIR aps up o 10-12 aps. Ano he me ic in he decision
o he numbe o FIR aps is he modula o inpu e o signal
swing, e( ), ha is shown in Fig. 14(b) o a −5dBFS inpu .
No e ha he signal swing is educed wi h N i . Howe e , his
educ ion becomes less e ec i e again a e 6- ap. In addi ion,
as seen om Table I, k1inc eases wi h N i and i u ns ou ha
he passband gain o he calcula ed Fc(z)inc eases wi h N i
as shown in Fig. 14(c). This phenomenon inc eases he second
8
-1.5 -1 -0.5 0 0.5 1 1.5
0
2.5K
5K
7.5K
10K
12.5K
15K
Numbe o ccu encesO
8-Tap FIR
6-Tap FIR
4-Tap FIR
2-Tap FIR
E o Signal Ampli ude (V)
(b)
35
40
45
50
55
60
SNR (dB)
w/o FIR
2-Tap FIR
4-Tap FIR
6-Tap FIR
8-Tap FIR
0.01 0.02 0.03 0.04 0.05
RMS Ji e Rela i e o Ts
(a)
OSR = 128
-10
-5
0
5
10
0 0.10 0.15 0.25 0.30 0.35 0.40 0.450.20 0.500.05
Rela i e F equency
Magni ude (dB)
8-Tap FIR
6-Tap FIR
4-Tap FIR
2-Tap FIR
c)(
F ( )z
c
Fig. 14. Impac o numbe o FIR aps (N i ) on: (a) Ji e pe o mance. (b)
His og am o e( ). (c) T ans e unc ion o Fc(z).
esona o ou pu swing. This ex a swing can be compensa ed
by adding a eed o wa d pa h a ound he esona o s, al hough
his would lead o a highe STF peak. The e o e, he e will be
a ade-o be ween ha dwa e complexi y, STF peaking, and
signal swing.
B. Non-equal FIR aps
To his poin , all he discussion was based on conside ing
equal aps (1/N i ) o he F(z) il e . We can imp o e he
pe o mance o he il e by using non-equal coe icien s. This
echnique was used be o e in LP∆ΣMs (see [14] and [25]), bu
like o he concep s in his pape , we can apply i o BP∆ΣMs.
All we need o do is keep he sum o all he il e coe icien s
equal o 1 o main ain he uni y gain in he passband. As
an example, a 7- ap FIR wi h equal coe icien s is compa ed
wi h an op imized non-equal e sion o i in Fig. 15(a). The
op imized F(z)shows be e ou -o -band ejec ion – simila
o an FIR wi h mo e aps. Al hough he coe icien s (a1−7)
a e no equal, we can s ill implemen he il e by using uni
elemen s. Conside ing he alue o a uni elemen 1/36, he
coe icien s a e a1,7= 2uni , a2,6= 5uni , a3,5= 7uni ,
and a4= 8uni . Wi h hese changes in mind, a new se o
coe icien s should be calcula ed o Fc(z) o keep he NTF
co ec , as depic ed in Fig. 15(b). In he nex sec ion, we will
use his op imized il e o ou ci cui implemen a ion.
C. Misma ch be ween FIR aps
A i s glance, i migh seem ha by applying complica ed
eedback wi h a lo o weigh s ( he FIR il e ), he obus ness
-10
-8
-6
-4
-2
0
2
4
6
8
10
12
0 0.10 0.15 0.25 0.30 0.35 0.40 0.450.20 0.500.05
Rela i e F equency
Magni ude (dB)
(b)
7-Tap Equal F( )z
7-Tap Non-Equal F( )z
calcula ed based on:
F ( )z
c
F ( )z
c
-80
-70
-60
-50
-40
-30
-20
-10
0
Magni ude (dB)
0 0.10 0.15 0.25 0.30 0.35 0.40 0.450.20 0.500.05
Rela i e F equency
7 Tap Equal-
7 Tap NonEqual-
1/7 1/7 1/7 1/7 1/7 1/7 1/7
2/36 5/36 7/36 8/36 7/36 5/36 2/36
Tap wigh :s
(a)
7 Tap Equal-
7 Tap Non-Equal-
F( )z
Fig. 15. Compa ison be ween FIR wi h equal aps and FIR wi h non-equal
aps o op imal ou -o -band ejec ion: (a) F equency esponse o F(z). (b)
F equency esponse o co esponding calcula ed Fc(z).
0
50
100
150
Numbe o ccu encesO
71 72 73 74 75 76 77 78
SNR (dB)
2-Tap FIR
OSR = 128
71 72 73 74 75 76 77 78
SNR (dB)
0
50
100
150
Numbe o ccu encesO
OSR = 128 4-Tap FIR
0
50
100
150
Numbe o ccu encesO
71 72 73 74 75 76 77 78
SNR (dB)
OSR = 128 8-Tap FIR
0
50
100
150
Numbe o ccu encesO
71 72 73 74 75 76 77 78
SNR (dB)
OSR = 128 6-Tap FIR
Fig. 16. Mon e Ca lo SNR simula ion esul s o a 4 h-o de CT BP∆ΣM
(CRFF-B) wi h BP-FIR DAC and 10% andom misma ch in all ap weigh s.
o he sys em will deg ade. Howe e , his is no a co ec
assump ion; andom misma ch only al e he il e esponse.
We pe o med Mon e Ca lo simula ion by adding a ia ion o
he coe icien s in all h ee FIR il e s o a CRFF-B s uc u e
[see Fig. 8(b)] and de e mined he a ia ion in ADC SNR.
Fig. 16 shows he esul o 1000 uns, conside ing di e en
numbe o FIR aps. The igh dis ibu ion demons a es he
design’s s u diness.
VI. CIRCUIT IMPLEMENTATION
In his sec ion, we will show how his sys em-le el concep
can easily be applied o well-known BP∆ΣM ci cui s uc-
u es. We will p esen a case s udy o a single-bi CT BP∆ΣM
wi h a 7- ap BP-FIR DAC ea u ing non-equal coe icien s
[F(z) = (2/36)−(5/36)z(−2) +(7/36)z(−4) −(8/36)z(−6) +
(7/36)z(−8) −(5/36)z(−10) + (2/36)z(−12)] using a CRFF-
B s uc u e. Fig. 17(a) illus a es he schema ic o he ci cui .
The calcula ed coe icien s (k0 o k9) o such a sys em a e
−0.097,−1.045,0.054,−0.732,−0.793,−0.727,−0.561,
−0.350,−0.158, and −0.036. The modula o is uned o
n= 200 MHz when clocked a s= 800 MHz. A ully-
di e en ial ci cui – no shown in Fig. 17(a) o simplici y