UNIVERSITY
OF TRENTO
DEPARTMENT OF INFORMATION AND COMMUNICATION TECHNOLOGY
38050 Povo – Trento (Italy), Via Sommarive 14
http://www.dit.unitn.it
A Criterion of stability for High-Accuracy Digital Oscillators
based on Delta-Sigma Modulators
David Macii, Fernando Pianegiani,
Paolo Carbone e Dario Petri
May 2004
Digital Os illators based on Modulators D. Ma ii 1 , F. Pianegiani 2 , P. Carbone 1 , D. Petri 2 1
Department of Ele troni and Information Engineering,
University of Perugia,
via G.Duranti, 93{ 06125 Perugia, Italy
Phone: +39 075 5853629, Fax: +39 075 5853654, E{mail: arbonediei.unipg.it.
2
Department of Informationand Communi ation Te hnology
University of Trento,
ViaSommarive,14{ 38050 Trento,Italy
Phone: +39 0461 883902, Fax: +39 0461 882093, E{mail: petridit.unitn.it.
Index Terms
Built{InSelf{Test(BIST),digitalresonators,delta{sigma,rootlo us.
I. Introdu tion
The ontinuousevolutionofhighperforman e mixed{signalintegrated ir uitsrequires
to use in reasingly sophisti ated measurement and testing pro edures, whose ost may
urrently over almost 50% of the overall produ tion budget [1℄. In this s enario, a
valuable solution to redu e onsiderably both testing times and instrumentation osts
is provided by Built{In Self{Test (BIST) s hemes. Generally speaking, a BIST s heme
onsists of both stimulus generation and measurement{oriented on{ hip omponents. Of
ourse, inorder tomake the BIST e e tive,su h as heme must beprogrammable,
exi-ble and, above all, inexpensive in terms of integration resour es. As analogue{to{digital
Converters (ADCs) are usually the key devi es of mixed{signal integrated ir uits, the
abilityof hara terizinga urately theirmetrologi alperforman es isoneofthe most
im-portanttasksindesigning aBIST s heme. Tothis purpose, manydi erentsolutionshave
been presented and various kinds of test stimuli have been used [2℄, [4℄. Among them,
high{quality,programmablesinewavesareprobablythemostsuitabletestsignalsbe ause
they are ommonly employed in many standard ADC testing pro edures [5℄. Moreover,
sinusoidal os illators based on a { topology, i.e. digital resonators exploiting 1{bit
delta{sigma properties, are parti ularly suitablefor BIST purposes be ause they an be
implemented withouthardware multipliersandwith a minimum amount ofanalogue
ir- uitry (i.e., a 1{bit DAC followed by a low{order lter). Also, they are able to generate
an output sinusoidal signal of elevated spe tral purity [6℄{[9℄. Unfortunately, most {
{based resonators have proved to work orre tly only under ertain onditions, while
exhibiting serious stability problems when some ir uit parameters are hanged[10℄. Up
tonow,thesephenomenahavenot beenproperlyinvestigated duringthedesignphase. In
fa t, all published results relymostly on rules of thumband extensive omputer
simula-tions. Hen e,the aimofthispaperistoprovideageneralinsightaboutthe stabilityissue
of this kind ofdevi es, thus determininga stabilization riterion. To this purpose, inthe
2
1
+
1
-PSfragrepla ements Refzg Refzg Imfzg z 1 z 1 e s x 1 x 2 x3 x 4 x 5 x 6 k 1 +k 0 k0 H(z) mux y modulator A6 A 5 A4 A 3 B 6 B 5 B4 B 3 z 1 1 z 1 1 1 z 1 b b a a (a) (b) n max(jx 1 j) max(jx 2 j) max(jx 3 j) max(jx 4 j) max(jx 5 j) max(jx 6 j) max(jx 3 j,jx 4 j,jx 5 j,jx 6 j) min() log[max(jx i j)℄ log[max(jx 6 j)℄,log[min()℄Fig.1. Blo kdiagramofageneri {harmoni
res-onator. Theparametersk
0
andk
1
setthefrequen y
ofthe outputsinewave. Theinitial onditionsx
1 [0℄ andx 2 [0℄togetherwithk 0 andk 1
determinethe
am-plitudeoftheoutputtone.
-PSfrag repla ements Refzg Imfzg Refzg Imfzg z 1 e s x1 x2 x 3 x4 x5 x 6 k 1 +k 0 k 0 H(z) mux y modulator A 6 A5 A4 A3 B 6 B 5 B 4 B 3 z 1 1 z 1 z 1 1 z 1 1 1 z 1 1 1 z 1 b b a a (a) (b) n max(jx 1 j) max(jx 2 j) max(jx 3 j) max(jx 4 j) max(jx 5 j) max(jx 6 j) max(jx 3 j,jx 4 j,jx 5 j,jx 6 j) min() log[max(jx i j)℄ log[max(jx 6 j)℄,log[min()℄Fig.2. Blo k diagram of the4 th order loop lter
H(z)insertedinthe{modulator.
II. Des ription of Stability Analysis
In general, harmoni digital resonators are based on the as ade of two dis rete{time
integrators inaloop,withthe signofoneintegratorbeingpositiveandtheothernegative.
In these stru tures, the os illationfrequen y depends onone or two multipli ative
oeÆ- ients[7℄. Inordertoavoidthe useofhardwaremultipliersandmulti{bitD/A onverters,
a 1{bit { modulatoris inserted inthe loopas shown in Fig. 1. The in rement in
ir- uital omplexity due to the introdu tion of this modulator is partially ounterbalan ed
bythebene tofgeneratingahigh{quality,single{bitoutputsignalthat anbemultiplied
bya onstant oeÆ ientsimplybymeansofamultiplexer. Obviously,theimplementation
ost aswell as the spe tral quality of the generated signal depend on the ar hite ture of
the { modulatoremployed. In parti ular, using the well{known additive white noise
model torepresent thebehaviorof the1{bitquantizer, themodulatorshouldbedesigned
so that the Noise Transfer Fun tion (NTF) exhibits an in{band Signal{to{Noise Ratio
(SNR) higher than a given value [9℄. However, the additive noise model is too oarse to
ope with stability issues. Indeed, even if this kind of ir uits seems tobe stable, it has
been shown that, under ertain initial onditions, the amplitude of the internal signals
may diverge suddenly[8℄. Thereasons ofthis behavior an be understoodmore learlyif
ananalysisofthedynami softheresonatorisperformedby modelingthe1{bitquantizer
with a time{varying gain [10℄. This parameter, whi h inthe following willbe referred to
as, resultsfromthe ratiobetween the onstantunit outputamplitudeofthe modulator
and the variable input amplitude of the quantizer. Using this model, it follows that any
nonlinearresonator an beregardedasasequen eofparametri linearsystemsdepending
on di erent values of . Furthermore, the order N of ea h system is equal to the sum
between the orderM of the { modulatorinserted inthe loopand the order K of the
digital resonator. As a typi al harmoni os illator is a se ond{order devi e, it usually
results that N =M+2. Thus, by assumingthat x[℄is a Nx1 state variable ve tor and
that F() is the NxN state transition matrix of ea h linear system, it results that the
1{step updating lawfor the state variablesis:
x[n+1℄=F()x[n℄ (1)
A ordingly, it an be easily shown that the z{transform of the free evolution of the
system isgiven by [11℄: X(z)=z[zI F()℄ 1 x[0℄= N(z;;x[0℄) P(z;) ; (2)
ve torof polynomialsdependingontheinitial onditionsx[0℄. Observe thatboth system
eigenvalues, i.e. the zeros of the hara teristi polynomial, and the mode amplitudes
depend on the parameter . Sin e the system stability is related to the position of the
poles on the omplex plane, the root lo us te hnique isa valid method to investigatethe
degree of stability of parametri { stru tures [12℄. Generally speaking, the positive
rootlo usasso iatedwithaN thorderfeedba ksystem onsistsofN urvesrepresenting
thepositionsofthe systempolesonthe omplexplanefordi erentvaluesoftheloopgain
0. Therefore, even though the sequen e of values, whi h depends on the haoti
dynami s of the internal { modulator, annot be predi ted analyti ally, it is at least
possible to determine whi h modes are liable of possible malfun tions and whether the
existing limit y les are stable. Limit y les o ur when the poles of the system exhibit
a module equal to1, i.e. when the rootlo us interse ts the unit ir le. The fundamental
radian frequen y of ea h limit y le is given by the angle of these interse tion points in
polar oordinates. The stability of ea h limit y les depends instead onhow the module
of the orresponding polevariesfordi erentvalues of. Infa t,if thepolemoduletends
to move out of the unit ir le when de reases (i.e. when the amplitude of the internal
signal in reases) it means that the onsidered limit y le is always riti ally stable, i.e.
the amplitude of the system internal state is destined to diverge. Conversely, if the pole
positiontendstomoveintotheunit ir lewhende reases,the growingamplitudeofthe
statesignalstendstoprodu establemodes(i.e. poleswhosemoduleislowerthan1),thus
leading toa stable os illation. In the nal papera more omplete explanation about the
in uen e of multiple time{varying modes on the overall stability of { resonators will
be provided. A tually,inorder to larifytheproposed approa h,theresults ofastability
analysis arried out ona parti ular kind of resonator ispresented inthe following.
III. Simulation Results
Supposetoinsertinthegeneral{resonators hemeshowninFig. 1,thefourth-order
loop lter H(z) displayed in Fig. 2. The resulting stru ture is a 6 th order system, in
whi heverystate variablex
i
,i=1;:::;6representsthe ontent ofadi erentregister. The
multipli ative oeÆ ients of the lter H(z) have been set toA
3 =2 6 , A 4 =A 5 = 2 4 , A 6 =2 7 ,B 3 =2 14 ,B 4 = 2 10 ,B 5 = 2 9 ,B 6 =2 6
inordertooptimizethespe tralpurity
of the output signal [7℄. By manipulating the algebrai de nition of ea h state variable
of the system, after some algebrai steps, itresults immediatelythat the state transition
matrix is equal to:
F()= 2 6 6 6 6 6 6 6 4 1 k 0 k 1 256k 0 78k 0 32k 0 3=4k 0 1 1 k 0 k 1 256k 0 78k 0 32k 0 3=4k 0 0 0 1 1=16 0 0 0 0 1=64 1021=1024 1=16 1=2048 0 0 0 1=16 1 1=128 0 1 256 78+1=256 32+1=16 3=4+2049=2048 3 7 7 7 7 7 7 7 5 ;
wheretheeigenvaluesareafun tionofanddependontheparametersk
0
andk
1
. InFig.
3(a) and 3(b) the full root lo us of the system and a zoomed part of the plot aroundthe
point(1;0) are shown, by assumingthat k
0 =9:1510 6 and k 1 =2 14
. Observethat the
bran hes of the rootlo us b, b
, ,
interse t the unit ir le in two ouple of symmetri
points. Aspe tral analysis revealedthat the radianfundamentalfrequen y of the output
signal oin ideswiththeabsolutevalueoftheangleasso iatedwiththeinterse tionpoints
between the bran hes ,
and the unit ir le. Further numeri al analyses also proved
that when !0 the poles related to the bran hes and
are slightly inside the unit
ir le, whereas when !1 su h poles lie outside the unit ir le, thus leading to the
on lusion thatthe orresponding limit y le is stable. Nevertheless, repeated simulation
4
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
3.14e8
2.83e8
2.51e8
2.2e8
1.88e8
1.57e8
1.26e8
9.42e7
6.28e7
3.14e7
3.14e8
2.83e8
2.51e8
2.2e8
1.88e8
1.57e8
1.26e8
9.42e7
6.28e7
3.14e7
PSfragrepla ements Refzg Imfzg Refzg Imfzg z 1 e s x1 x2 x3 x4 x5 x6 k 1 +k 0 k 0 H(z) mux y modulator A6 A5 A4 A3 B6 B5 B 4 B 3 z 1 1 z 1 1 1 z 1 b b a a a a (a) (b) n max(jx1j) max(jx 2 j) max(jx 3 j) max(jx 4 j) max(jx 5 j) max(jx 6 j) max(jx 3 j,jx 4 j,jx 5 j,jx 6 j) min() log[max(jx i j)℄ log[max(jx 6 j)℄,log[min()℄ (a) (b)0.94
0.96
0.98
1
1.02
1.04
1.06
−0.25
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25
0.9999
1
1
7.6
7.8
8
8.2
8.4
8.6
x 10
−3
PSfrag repla ements Refzg Refzg Imfzg z 1 e s x1 x 2 x 3 x 4 x 5 x 6 k 1 +k0 k0 H(z) mux y modulator A 6 A 5 A 4 A 3 B 6 B 5 B 4 B3 z 1 1 z 1 1 1 z 1 b b a a (a) (b) n max(jx 1 j) max(jx 2 j) max(jx 3 j) max(jx 4 j) max(jx 5 j) max(jx 6 j) max(jx3j,jx4j,jx5j,jx6j) min() log[max(jx i j)℄ log[max(jx 6 j)℄,log[min()℄ (a) (b)Fig.3. Root lo us ofthe {resonator aftermodeling the1{bitquantizeras atime{varyinggain whenk
0 = 9:1510 6 and k 1 =2 14
. In(a) thefull root lo us is shown, while in(b)a zoomed portion of(a) is plotted
aroundthepoint(1;0)ofthe omplexplane. Thearrowshighlightthedire tionalongwhi hthegaingrows.
state signalsmay suddenlydiverge whendi erentinitialvaluesfor thestate variablesare
set. For instan e, if the initial register values x
1 [0℄=0:0055 and x 2 [0℄=3:5110 6 and x i
[0℄=0,i=3;:::;6 are hosen, the peak{to{peak amplitude of the internalstate signals
tends to in nity after about 810
7
iterations. This is learly re ognizable in Fig. 4(a),
where the urves representing the envelope of absolute values of the time{varying state
variablesare plottedas a fun tion of the number of iterationson a logarithmi s ale. In
this way, the haoti patternofsome state variablesdoesnot hidethe informationabout
the amplitude of the internal signals. A ording to the root lo us analysis, the possible
unstable behavior is due to the riti al modes asso iated with the lo us bran hes b, b
.
Indeed,thelimit y le orrespondingtotheinterse tionpointsbetweensu hbran hesand
the unit ir le is learly unstable be ause when !0 the poles tend to move out of the
unit ir le, thus further stressing the in oming instability. The system will be ertainly
stable only if the amplitude of the quantizer input, namely the sequen e of values, is
maintainedhigherthanthe riti alparameter
orrespondingtothe interse tion points
mentioned above. If this ondition is not met, the behavior of the resonator be omes
unpredi table, thus making the whole devi e unreliable. Of ourse,
depends on the
hosen values for ir uital parameters. In the ase onsidered, for instan e,
is equal
to about 0.47. In order to verify the validity of this assumption, the minimum values of
have been re orded. The olle ted results as well as the maxima of the most riti al
statevariablex
6
, hosenasreferen e, areshowninFig. 4(b). Observethatassoonasthe
sequen e of values be omes lower than the riti al threshold
=0:47, the amplitude
of x
6
in reases abruptly. From this analysisitfollows thaton e
isknown, the stability
of os illatorsbased on a { topology an be obtained simplyby assuring that >
.
However, sin e a simple lipping of the state variables would deteriorate ex essively the
spe tral purity of thegenerated waveform, adi erentapproa h hasto befollowed. Good
results in terms of output a ura y have been a hieved by lipping only minima and
maxima ofthe resonator internalsignals,whi hare re ognizedasthe mainresponsibleof
an in ominginstability. This strategy is justi ed by the fa t that not allsignals in rease
simultaneously at the same rate. In parti ular, the state variable x
6
exhibits an almost
lineartrendwhi hishigherthanthetrendoftheothersstatevariablessothatitseemsto
anti ipate the a tual breakdown. Therefore, a proper monitoring and a areful lipping
5
0
2
4
6
8
10
12
x 10
7
−3
−2
−1
0
1
2
3
4
5
6
PSfragrepla ements Refzg Refzg Imfzg z 1 e s x1 x2 x3 x4 x 5 x 6 k 1 +k 0 k 0 H(z) mux y modulator A6 A5 A4 A3 B 6 B 5 B 4 B 3 z 1 1 z 1 1 1 z 1 b b a a (a) (b) n max(jx1j) max(jx 2 j) max(jx3j) max(jx 4 j) max(jx 5 j) max(jx 6 j) max(jx 3 j,jx 4 j,jx 5 j,jx 6 j) min() log[max( j x i j )℄ log[max(jx 6 j)℄,log[min()℄ (a) (b)0
2
4
6
8
10
12
x 10
7
−6
−4
−2
0
2
4
6
PSfrag repla ements Refzg Refzg Imfzg z 1 e s x1 x2 x3 x4 x 5 x 6 k 1 +k 0 k 0 H(z) mux y modulator A6 A5 A4 A3 B 6 B 5 B 4 B 3 z 1 1 z 1 1 1 z 1 b b a a (a) (b) n max(jx 1 j) max(jx 2 j) max(jx 3 j) max(jx 4 j) max(jx 5 j) max(jx 6 j) max(jx 3 j,jx 4 j,jx 5 j,jx 6 j) min() log[max(jx i j)℄ log[max( j x 6 j )℄, log[min( )℄ (a) (b)Fig.4. In(a)theenvelopeoftheabsolutevaluesofthe{resonatorstatevariableareplottedasafun tionof
thenumberofiterationsonalogarithmi s ale,afterassumingx
1 [0℄=0:0055,x 2 [0℄=3:5110 6 ,k 0 =9:1510 6 and k 1 =2 14
. In (b)the minimum values of are shownand ompared on alogarithmi s ale against the
envelopeoftheabsolutevalueofthestatevariablex
6 .
IV. Novelties
The orre t operation of { harmoni resonators for BIST purposes depends on the
stabilityofthe self{generatedlimit y les. In thispaper, bymodelingthe1{bitquantizer
inside the { modulator asa time{varying gain, the stability of this kindof os illators
is analyzed using the root lo us te hnique. The obtained results not only provide useful
guidelines tounderstandwhether a ertainstru tureis riti ally stable,but alsopromote
the implementation of a exible lipping strategy whi h in reases the stability of the
resonator without deterioratingthe spe tralpurity of the output signal.
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