ΣΑ ADCs: Basic Concepts, Topologies and State of the Art

36 Pages·2017·2.11 MB·English

Chapter 2
(cid:2)(cid:3) ADCs: Basic Concepts, Topologies
and State of the Art
SIGMA-DELTA MODULATION HAS DEMONSTRATED TO BE VERY
SUITED FOR the implementation ofAnalog-to-Digital (A/D) interfaces in many
different electronic systems, covering a large number of applications from instru-
mentationtotelecom[Nors97a, Mede99, Geer02, Schr05b, Ortm06].Thistypeof
A/D converters (ADCs), composed of a low-resolution quantizer embedded in a
negativefeedbackloop,usesoversampling(asamplingfrequencymuchlargerthan
theNyquistfrequency)toreducethequantizationerrorand(cid:2)(cid:3)modulationtopush
thisnoiseoutofthesignalband[Inos62].Thecombineduseofredundanttemporal
data (oversampling) and filtering (noise shaping) results in high-resolution, robust
ADCswhichhavelowersensitivitytocircuitparasiticsandtolerances,andaremore
suited for their implementation in modern standard CMOS technologies [Rodr03,
Schr05b].
Theaimofthischapteristointroducethefoundationsof(cid:2)(cid:3)ADCs.Thefunda-
mentalsofsamplingandquantizationprocesses—inherenttotheA/Dconversion—
arerevisedinSect.2.1forthepurposeofjuxtaposingtheoperationprinciplesof(cid:2)(cid:3)
modulation (oversampling and quantization noise shaping). Sect. 2.2 presents the
basicschemeofa(cid:2)(cid:3)ADC,togetherwithitsidealbehaviorandadefinitionofits
mostimportantperformancemetrics.Aclassificationofpracticalimplementations
of (cid:2)(cid:3) modulators is then presented in Sect. 2.3. Sect. 2.4 describes single-loop
(cid:2)(cid:3)architectures,discussingstabilityissuesandpracticaltopologiestoimplement
high-orderloops. Cascade(cid:2)(cid:3)architecturesarepresentedinSect.2.5asanarchi-
tecturalalternativetoimplementunconditionally-stablehigh-order(cid:2)(cid:3)modulators.
Sect.2.6revises(cid:2)(cid:3)modulatorswithmulti-bitembeddedquantizers,analyzingtheir
prosandprovidingtechniquestocircumventtheirimpactonthemodulatorlinearity,
suchasdual-quantizationordynamicelementmatchingtechniques.Finally,thestate
oftheartin(cid:2)(cid:3)ADCsisreviewedfromintegratedcircuitimplementationsreported
inopenliterature.
A.Morgadoetal.,NanometerCMOSSigma-DeltaModulatorsforSoftwareDefinedRadio, 23
DOI10.1007/978-1-4614-0037-0_2,©SpringerScience+BusinessMedia,LLC2011
24 2 (cid:2)(cid:3)ADCs:BasicConcepts,TopologiesandStateoftheArt
x (t) x [n]
f s
xa(t) S/H Yd[n]
B
f /2 Bbit
s
f
a AAF s Quantizer
xa(t) xa(t) Xa(f)
AAF
t –fs –fs/2 fs/2 fs f
AAF
x(t)
f
X(f)
f
x(t)
f
f S/H t –f –f/2 f /2 f f
x
s
X (f)
x [n] s
s
t –f –f/2 f /2 f f
Quantizer
Y
d Quantization
Yd(f) noise
B
b Yd[n] t –f –f/2 f /2 f f
Fig.2.1 Analog-to-digitalconverter:agenericscheme,bsignalprocessinginvolved
2.1 FundamentalsoftheA/DConversion
AnADCisasystemthattransformssignalswhicharecontinuousintimeandampli-
tude(analogsignals)intosignalswhicharediscreteintimeandamplitude(digital
signals).Fig.2.1ashowsthegenericschemeofanADCintendedfortheconversion
oflow-passsignalsthatincludesthefollowingblocks:ananti-aliasingfilter(AAF),
asampling-and-holdcircuit(S/H)andaquantizer.Thesignalprocessinginvolvedin
theoperationoftheADCblocksisillustratedinFig.2.1bbothintimeandfrequency
domains.First,theanaloginputsignalx (t)oftheADCpassesthroughtheAAFin
a
order to remove spectral components above one half of the sampling frequencyf
s
oftheS/H.Otherwise,accordingtotheNyquisttheorem,out-of-bandcomponents
wouldbefoldedbackintothesignalbandduringthesubsequentsamplingprocess,
thereforecorruptingthesignalinformation.Theresultingband-limitedsignalx[t]is
f
thensampledataratef bytheS/H,yieldingadiscrete-timesignalx [n]=x [nT ],
s s f s
whereT =1/f .Finally,thequantizermapstherangeofamplitudesofx [n]intoa
s s s
discretesetoflevelsusingBbits;i.e.,eachsampleofthecontinuous-valuedinputis
codedontothecloserdiscrete-valuedleveloutofthe2Blevelsthatcoverthevariation
intervaloftheinputsignal.ThisprocessyieldstheconverterdigitaloutputY [n].
d
2.1 FundamentalsoftheA/DConversion 25
The fundamental operations involved in the A/D conversion are sampling and
quantization, as illustrated in Fig. 2.1. On the one hand, the sampling process
performs the continuous-to-discrete conversion of the input signal in the time do-
main. On the other, the quantization process performs the continuous-to-discrete
conversion of the input signal in amplitude. These two transformations inherently
impose limitations to the performance of anADC, even if its implemented using
idealcomponents.
2.1.1 Sampling
Sampling imposes a limit on the bandwidth of the analog input signal.According
to the Nyquist theorem, the minimum frequency f required for sampling a signal
s
with no loss of information is twice the signal bandwidth, BW; i.e., f =2BW,
N
also called the Nyquist frequency. Based on this criterion, theADCs in which the
analoginputsignalissampledatminimumrate(f =f )arecalledNyquistADCs.
s N
Fig.2.1billustratedthesignalprocessinginvolvedassumingaNyquistADC.Since
theinputsignalbandwidthequalsf /2,aliasingwilloccurifx (t)containsfrequency
s a
componentsabovef /2.High-orderAAFsarethereforerequiredinNyquistADCsin
s
ordertoimplementasharptransitionbandandremovetheout-of-bandcomponents
withnosignificantattenuationofthesignalband.
2.1.2 Quantization
QuantizationalsolimitstheperformanceofanidealADC,sincetheprocessitselfof
mappingcontinuous-valuedlevelsintoasetofdiscretelevelsdegradesthequality
oftheinputsignal.Intheprocessanerrorisgenerated,calledquantizationerror.
ThisprocessisillustratedinFig.2.2forthecaseofa3-bitidealquantizer(B=3).
Astheinputsignalchangesfromx tox ,itis‘rounded’tooneoutoftheeight
min max
(2B)discretelevels.ForaB-bitquantizer,theseparationbetweenadjacentlevelsis
definedbythequantizationstep(cid:3)as
X
(cid:3)= FS (2.1)
(2B−1)
withX beingthequantizerfullscale.
FS
The quantizer operation can be therefore described mathematically by a linear
model
y=g x+e(x) (2.2)
q
where g stands for the quantizer gain—the slope of the line intersecting the code
q
transitions—and e(x) stands for the quantization error. This error is a non-linear
26 2 (cid:2)(cid:3)ADCs:BasicConcepts,TopologiesandStateoftheArt
e
g
q
x y x + y
a B bit B b
y
e(x)
X
FS
+ /2
x
max
x x
xmin gq xmax xmin – /2
d
c
Fig.2.2 Idealquantizationprocess:asymbolicrepresentation,blinearmodelofanidealquantizer,
cidealtransfercharacteristic,dquantizationerrorofa3-bitquantizer
function of the input x, as shown in Fig. 2.2d. Note that, if x is confined in the
interval[x ,x ],thequantizationerrorisboundedby±(cid:3)/2.Forinputsoutside
min max
that interval, the absolute value of the quantizer error grows monotonically. This
situationisknownasquantizeroverload.
2.1.3 WhiteNoiseApproximationofQuantizationError
In order to evaluate the performance of an ideal quantizer, some assumptions are
usuallymadeonthestatisticalpropertiesofthequantizationerrorwhicharecollec-
tively called the ‘additive white noise approximation’. As shown in Fig. 2.2d, the
quantizationerrorisstronglydependentontheinputsignalvalue. Nevertheless, if
xisassumedtochangerandomlyfromsampletosampleintheinterval[x ,x ]
min max
andthenumberoflevelsinthequantizerislarge,thequantizationerrorcanbeas-
sumed uncorrelated from sample to sample. Quantization can therefore be viewed
as a random process, with the quantization error exhibiting a uniform probability
density function (PDF) in the range [−(cid:3)/2, +(cid:3)/2] [Benn48, Srip77]. The power
associatedtothequantizationerroristhen
(cid:2)+∞ +(cid:2)(cid:3)/2
1 (cid:3)2
σ2(e)= e2PDF(e)de= e2 de= (2.3)
(cid:3) 12
−∞ −(cid:3)/2
Thispowerwillbeuniformlydistributedintheband[−f /2, +f /2]asthequantized
s s
signalissampledatratef andthequantizationerrorwillthereforeexhibitaconstant
s
2.1 FundamentalsoftheA/DConversion 27
X ( f )
a
AAF
a f
BW f f –BW f
--s- s s
2
2 1
In-band quantization noise power =-----------------
12OSR
S ( f )
E
2
Total quantization noise power =------
12
b f
–f /2 –BW +BW +f /2
s s
Fig.2.3 Benefitsofoversamplingonthe:aAAF,bin-bandquantizationnoise
powerspectraldensity(PSD)inthatfrequencyinterval
σ2(e) (cid:3)2
S (f)= = (2.4)
E
f 12f
s s
ThedegradationintroducedbythequantizerintheperformanceofanADCcanbe
expressedthroughthein-bandquantization‘noise’powerP ,calculatedas
Q
+(cid:2)BW
(cid:3)2
P = S (f)df = (2.5)
Q E
12
−BW
Note that f equals 2BW in a NyquistADC and all the quantization noise power
s
thereforefallsinsidethesignalbandandpassestotheADCoutputasapartofthe
signalitself.
2.1.4 Oversampling
Oversamplingconsistsinsamplingasignalfasterthantheminimumrateimposed
bytheNyquisttheoremtoavoidaliasing.Howmuchfasterthanrequiredthesignal
issampledisexpressedthroughtheoversamplingratio,definedasOSR=f /(2BW).
s
Oversampling has two noticeable effects in an ADC. First, as illustrated in
Fig.2.3a,sincef islargerthantheNyquistrate,theimagesoftheinputcreatedbythe
s
samplingprocessaremoreseparatedthaninaNyquistADC.Spectralcomponentsof
theinputsignalintherange[BW,f −BW]donotaliaswithinthesignalbandand,
s
consequently, the transition band of theAAF can be smoother in an oversampling
ADC,whatgreatlysimplifiesitsdesign.Second,asillustratedinFig.2.3b,whenan
28 2 (cid:2)(cid:3)ADCs:BasicConcepts,TopologiesandStateoftheArt
oversampledsignalisquantized, thequantizationnoiseisuniformlydistributedin
therange[−f /2,+f /2]andonlyafractionofthetotalpowerlayswithinthesignal
s s
band.Thein-bandquantizationnoisepowercanthereforebecalculatedas
+(cid:2)BW +(cid:2)BW
(cid:3)2 (cid:3)2
P = S (f)df = df = (2.6)
Q E
12f 12OSR
s
−BW −BW
anddecreaseswithOSRatarateof3dB/octave.
2.1.5 QuantizationNoiseShaping
Thequantizationnoisepowerwithinthesignalbandcanbefurtherdecreasedthrough
theprocessingofthequantizationerror.Letusconsiderthequantizationofanover-
sampledsignal.IfOSRislarge,theinputsignalvaluewillonlyslightlychangefrom
onesampletoanotherandmostofthechangesinthequantizationerrorwilloccur
athighfrequencies—i.e.,low-frequencycomponentsofconsecutivesamplesofthe
quantization error e[n] will be similar. Hence, low-frequency in-band components
ofthequantizationerrorcanbeattenuatedbysubtractingtheprevioussamplefrom
thecurrentone
e [n]=e[n]−e[n −1] (2.7)
HPF
or further reduced by involving a larger number of previous samples in the error
processing
e [n]=e[n]−e[n −1], 1st−order error processing
HPF,1
e [n]=e[n]−2e[n−1]+e[n −2], 2nd−order error processing
HPF,2
e [n]=e[n]−3e[n −1]+3e[n −2]−e[n −3], 3rd−order error processing
HPF,3
... (2.8)
ThisprocedurecanbeformulatedinanunifiedmannerinZ-domainas
E (z)=(1−z−1)L·E(z) (2.9)
HPF,L
indicating that the processed error is a (high-pass) filtered version of the original.
Thefilteringtransferfunctiononthequantizationerrorduetothisprocessing,called
noisetransferfunction(NTF),isthereforeobtainedas
NTF(z)=(1−z−1)L (2.10)
whereLdenotestheorderofthefiltering.IfOSRislargeenough,NTFtakessmall
valueswithinthesignalbandandcanbeapproximatedto
(cid:3) (cid:4) (cid:5)(cid:3) (cid:3) (cid:3) (cid:6) (cid:7) (cid:6) (cid:7)
(cid:3)(cid:3)NTF ej2πf/fs (cid:3)(cid:3)2 =(cid:3)(cid:3)1−e−j2πf/fs(cid:3)(cid:3)2L =22Lsin2L πf ≈22L πf 2L (2.11)
f f
s s
2.2 Basicsof(cid:2)(cid:3)A/DConverters 29
Fig.2.4 Illustrationofthe
2
S (f )NTF(f )
quantizationnoiseshaping E
No shaping
1
f
–fs/2 –BW 0 +BW +fs/2
Hence,thein-bandpowerofthefilteredquantizationerrorresultsin
+(cid:2)BW
(cid:3)2 π2L
P = S (f)|NTF(f)|2df ≈ · (2.12)
Q E 12 (2L+1)OSR(2L+1)
−BW
thatismuchsmallerthanonlyapplyingoversampling.Theresultingerrorreduction
isillustratedinFig.2.4.
2.2 Basicsof(cid:2)(cid:3)A/DConverters
In contrast to Nyquist ADCs, oversampling sigma-delta ADCs—usually referred
to as (cid:2)(cid:3) ADCs—make use of oversampling and noise shaping to decrease the
quantization noise power within the signal band and increase the accuracy of the
A-to-Dconversion.Fig.2.5illustratesthebasicschemeofa(cid:2)(cid:3)ADC,aswellasthe
signalprocessinginvolved.Asshown,a(cid:2)(cid:3)convertercomprisesthreemainblocks:
• Anti-AliasingFilter(AAF).ItsfunctionisthesameasinNyquistADCs;i.e.,to
bandlimittheinputsignalinordertoavoidaliasingduringsampling.Asstated
above, oversampling considerably relaxes the attenuation requirements for this
analogfilterandsmoothtransitionbandsaresufficient(seeFig.2.3a).
• Sigma-DeltaModulator((cid:2)(cid:3)M).Itsimultaneouslyperformstheoversampling
andquantizationoftheband-limitedinputsignal.Quantizationerrorisalsohigh-
passfilteredbymeansofagivennoise-shapingtechnique.Thisisaccomplished
byplacinganappropriateloopfilterH(z)beforealow-resolutionquantizerand
closinganegativefeedbacklooparoundthem.Thein-bandquantizationnoiseis
thereforegreatlydecreasedincomparisontothatoftheembeddedquantizer.The
outputofthe(cid:2)(cid:3)MisaB-bitdigitalstreamatf samplingrate.
s
• Decimator. Itreducestherateofthe(cid:2)(cid:3)MoutputstreamdowntotheNyquist
rate.Jointly,thewordlengthincreasesfromBtoNinordertopreserveresolution
asthewordratedecreases.Althoughtheblockschemeofadecimatormaydiffer
in practice from that illustrated in Fig. 2.5, it conceptually consists of a high-
selectivitydigitalfilterandadownsampler.Frequencycomponentsofthestream
30 2 (cid:2)(cid:3)ADCs:BasicConcepts,TopologiesandStateoftheArt
xa(t) xf(t) S/Hxs[n] H(z) Y(n) Yf[n] OSR Yd[n]
_
B bit B N N
BW fs- BW fs BW
AAF Digital filter Downsampler
DAC
Decimator
a
Modulator
xa(t) xa(t) Xa(f) AAF
AAF
t BW fs/2 fs-BWfs f
xf(t) xf(t) Xf(f)
R fs S/H
O xs(n)
AT _ t BW fs/2 fs-BWfs f
UL xs
OD AC H(z) Xs(f)
M D
B bit t BW fs/2 fs f
Y Quantization noise
Y(f) shaping
B Y[n]
t BW fs/2 fs f
Digital
OR filter Yf Yf(f) Digital
MAT N Yf[n] filter
CI
DE OSR t BW fs/2 fs f
Yd Yd(f)
N
b Yd[n] t BW fs f
Fig.2.5 (cid:2)(cid:3)ADC:agenericscheme,bsignalprocessinginvolved
above BW are removed1—and, therefore, most part of the shaped quantization
error—toavoidaliasingduringthesubsequentdownsampling,inwhichthestream
rateisdividedbyOSRkeepingonlyoneoutofeveryOSRsamples.
The(cid:2)(cid:3)modulatoristheblockthathasmostinfluenceupontheADCperformance,
basically because it is the responsible of the sampling and quantization processes
and,therefore,ultimatelylimitstheaccuracyoftheA-to-Dconversion.
1Alargesteepnessinthetransitionbandisdemandedusuallyforthefilterinordertoavoiddegrading
thesignalband.However,thisspecificationisimposedonadigitalfilterandisapriorieasierto
fulfillthanfortheanalogAAFofaNyquistADC.
2.2 Basicsof(cid:2)(cid:3)A/DConverters 31
e
g
+ + q
X H(z) Y X H(z) + Y
_ _
B bit
y y
a DAC b
Fig.2.6 (cid:2)(cid:3)Marchitecture:abasicscheme,blinearmodel
2.2.1 SignalProcessingina(cid:2)(cid:3)Modulator
Figure2.6ashowsthebasicschemeofa(cid:2)(cid:3)modulator.Itconsistsofafeed-forward
pathformedbyaloopfilterH(z)andaB-bitquantizerandanegativefeedbackpath
aroundthemusingaB-bitDAC[Inos62].Theoperationofthe(cid:2)(cid:3)Mcanbeexplained
as follows. Assume that H(z) exhibits large gain inside the signal band and small
gainoutsideofit.Duetothenegativefeedback,theerrorsignalx−ywillbecome
practicallynullinthesignalband;i.e.,theinputsignalxandtheanalogversionofthe
outputywillpracticallycoincidewithinthisband.Mostofthedifferencesbetweenx
andywillthereforebeplacedathigherfrequencies,shapingquantizationerrorand
pushingitoutsidethesignalband.
Figure2.6bshowsthelinearmodelofa(cid:2)(cid:3)M,inwhichtheDACisassumedtobe
ideal,thequantizerisreplacedbythemodelinFig.2.2bandtheadditivewhitenoise
approximationisconsideredforthequantizationerror.Thisway,themodulatorcan
beviewedasatwo-inputsystemwhoseoutputisrepresentedinZ-domainas
Y(z)=STF(z)X(z)+NTF(z)E(z) (2.13)
where X(z) and E(z) are the Z-transform of the input signal and the quantization
noise, respectively, and STF(z) and NTF(z) are the respective transfer functions,
givenby
g H(z) 1
STF(z)= q NTF(z)= (2.14)
1+g H(z) 1+g H(z)
q q
Since the signal and the noise pass through different transfer functions, H(z) can
be chosen so that the noise shaping does not affect the signal. Using a loop filter
withlargegainwithinthesignalband,thesignalandnoisetransferfunctionscanbe
approximatedinthatrangeto
1
STF(z)≈1 NTF(z)≈ (cid:5)1 (2.15)
g H(z)
q
Thenoise-shapingfunctionsin(2.10)canbebuiltwithproperselectionofH(z).The
easiest loop filter that exhibits the desired frequency performance is an integrator,
whoseZ-domaintransferfunctionis
z−1
H(z)= (2.16)
1−z−1
32 2 (cid:2)(cid:3)ADCs:BasicConcepts,TopologiesandStateoftheArt
1.0
al 0.8
n
g 0.6
put si 00..24
ut 0.0
o
d –0.2
an–0.4
ut –0.6
p–0.8
n
I–1.0
0 50 100 150 200 250
Clock cycle
Fig.2.7 PDMoutputstreamofa1st-order(cid:2)(cid:3)Mforaninputramp
Assumingthatthequantizergaing equalsunity,the(cid:2)(cid:3)Moutputyields
q
Y(z)=z−1X(z)+(1−z−1)E(z) (2.17)
and the modulator is called a 1st-order (cid:2)(cid:3)M, referring to the order of the noise
shaping.
Figure2.7showstheoutputofa1st-order(cid:2)(cid:3)Mwitha1-bitembeddedquantizer
forarampinputsignal. Duetothecombinedactionofoversamplingandnegative
feedback,the(cid:2)(cid:3)Moutputisapulse-densitymodulated(PDM)signalthatlocally
trackstheinputonaverage:whentheinputlevelislow,the(cid:2)(cid:3)Moutputcontains
more−1’sthan+1’s; whenitishigh, the+1’saredominant; andwhentheinput
signal is close to zero, the density of +1’s and −1’s practically coincides. If the
quantizer resolution is larger, the output tracks the input much closer, since the
separationbetweenthediscretelevelsdecreases.
2.2.2 PerformanceMetricsof(cid:2)(cid:3)Modulators
For the sake of clarity, it is convenient at this time to define the most important
parameterscommonlyusedtoquantifytheperformanceof(cid:2)(cid:3)modulators;namely:
• Signal-to-noiseratio,SNR.Itistheratiooftheoutputpoweratthefrequencyof
aninputsinusoidtotheuncorrelatedin-banderrorpower.Duetonon-idealitiesof
thecircuitrythatimplementsthemodulator, other(linearandnon-linear)errors
apartfromquantizationnoisecontributetothein-banderror.SNRaccountsforthe
linearperformanceofthemodulatorthein-bandpowerassociatedtoharmonicsis
thereforenotincluded.Foranideal(cid:2)(cid:3)modulatorandtakingonlyquantization
errorintoaccount,theSNRcanbeapproximatedto
(cid:3) (cid:6) (cid:7)
(cid:3) A2
SNR(cid:3) =10log (2.18)
dB 10 2PQ
whereAistheamplitudeoftheoutputsinusoid.