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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 54, NO. 4, APRIL 2007 757

Efficient Multibit Quantization in Continuous-Time�� Modulators

Jeroen De Maeyer, Pieter Rombouts, and Ludo Weyten

Abstract—A drawback of continuous-time �� modulators istheir sensitivity to clock jitter. One way to counteract this is to use amultibit feedback loop which requires a (high resolution) multibitquantizer. However, every extra bit in the quantizer doubles itscomplexity, power consumption and capacitive load for the analogcircuit that needs to drive the quantizer. In this paper a new con-cept for the quantization in sigma delta modulators is proposed. Itallows to significantly reduce the required amount of comparatorsin the multibit quantizer. Three architectures that realize this newconcept are presented and their implementation issues discussed.The architectures’ performance has been compared with a con-ventional modulator through computer simulations. Compared tothe conventional modulator, the proposed architectures achieve thesame performance, with much less comparators in the quantizer.

Index Terms—Analog-to-digital conversion, quanitzation,sigma–delta (��) modulation.

I. INTRODUCTION

I N sigma–delta ( ) modulators, the use of a multibit quan-tizer with many levels is an interesting path. The more levels,

the lower the quantization noise. Firstly, this enhances the dy-namic range. Secondly, the quantizer behaves more linearly,which makes the modulator intrinsically more stable. This al-lows for the implementation of a more aggressive noise transferfunction (i.e., a noise transfer function (NTF) with e.g., a higher

, [1]). This again results in a higher resolution. Thirdly, theslew rate requirements on the integrators of the loop filter aregreatly loosened. Hence, they can operate on a lower bias cur-rent. Last but not least, in the special case of continuous-time

modulators, using many levels in the quantizer might bethe only way to reduce their sensitivity to jitter [2], [3]. Fromthe above reasoning, it is clear that the number of bits in thequantizer should be as high as possible.

However, every extra bit in the quantizer requires a doublingof the number of comparators. Thus, the complexity, power con-sumption, and the silicon area of this part of the system doubleswith every extra bit. Also, the capacitive load formed by thisquantizer increases, leading to a power hungry analog drivingcircuit. In this paper a novel concept to perform the multibit

Manuscript received April 9, 2004; revised February 23, 2005 and June 28,2006. The work of De Maeyer was supported by the Fund for Scientific Re-search—Flanders (F.W.O.-V., Belgium). This paper was recommended by As-sociate Editor Y. Lian.

J. De Maeyer was with the Department of Electronics and Informations Sys-tems (ELIS), Ghent University, Ghent 9000, Belgium. He is now with AMISemiconductor, Oudenaarde B 9700, Belgium (e-mail: [email protected]).

P. Rombouts and L. Weyten are with the Department of Electronics and In-formations Systems (ELIS), Ghent University, Ghent 9000, Belgium.

Digital Object Identifier 10.1109/TCSI.2007.890607

Fig. 1. A conventional continuous-time �� modulator.

quantization in modulators is introduced. Based on this con-cept, the number of comparators in the quantizer can be reducedsignificantly. Three different architectures that implement thisconcept will be presented.

In [4], an alternative way for reducing the number of com-parators in the quantizer was proposed. There a so-callednonuniform quantizer is used, this technique might be morerobust due to the fact that it is less complex than the proposedtechnique. However, three main differences in preformance canbe observed. The first is that in our technique the signal-to-noiseratio (SNR) does not top for higher input amplitudes. Thesecond is that the modulator in the technique of [4], due tostability reasons, needs to be designed in correspondence withthe large quantization step for the higher input signals. Lastbut not least, these larger quantization steps do not decreasethe jitter sensitivity of the modulator which is one of our majormotives.

A. Traditional Modulator

1) General Remarks: A typical continuous-time modu-lator is shown in Fig. 1 [2], [5]. It consists of a loop filter ,a feedback digital–analog converter (DAC) with frequency re-sponse , a main quantizer and a sampler,with sampling period . In the rest of the paper, the samplingperiod is normalized to 1 in order to simplify the equations.Modulators, with a return-to-zero pulse are much more sensitiveto jitter compared to the ones with a nonreturn-to-zero. Hence,we assume a nonreturn-to-zero DAC-pulse

(1)

Furthermore, we will denote the quantization step and the full-scale input range of the quantizer respectively and . Inthe traditional linear model for the modulator, this quantizer ismodeled as an additive source of white noise , with a noisepower of . This models the behaviour of a multibit quan-tizer in a modulator well, as long as the input signals do notexceed the full-scale input range of the quantizer .

Applying, the traditional linear white noise model for thequantizer to Fig. 1, we derive the discrete time output of the

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758 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 54, NO. 4, APRIL 2007

modulator [6], [7] is as follows ( is the NTF and thesignal transfer function (STF) is depicted by ):

(2)

The equivalent discrete time of the continuous-timemodulator can be obtained by using, e.g., the state-spacemethod of [5] or equivalently the impulse-invariant method. InAppendix A we go into more detail about the meaning of theso-called star-operator used in . At this point,it is sufficient to provide the following (intuitive) interpretation:

(3)

This corresponds to the well-known aliasing phenomenon dueto the sampling operation. Finally, in terms of the loop filter, thesignal transfer function equals

(4)

An interesting approximation can be found for signals with afrequency in the (low-pass) signal band of interest ,where the magnitude of the loop filter is high. For these fre-quencies, the STF is approximated by the inverse of the feed-back path

(5)

In this paper, we will assume the is known, it mighthave been designed using any kind of design method, e.g., [1],[8], [9]. To illustrate our ideas, we will investigate four modula-tors. All are third-order modulators with optimized zeros. In twoof the four cases the zeros are optimized for an oversamplingratio (OSR) of 16 in the other two cases this is 128. In all thecases the was designed with the method of [1]. Two ofthe modulators had an of 2, for the other two modulators the

was 3. This way we end up with a discrete time loop filterfor which the continuous-time loop filter was next derivedusing the impulse-invariant method (see, e.g., Appendix A).

2) Input Signal of the Quantizer: In this paper we are espe-cially interested in the sampled input signal of the quantizer, i.e.,

. With the above definitions this signal is given by

(6)

As can be seen from (6) there are two contributors to the signal, both will be analyzed theoretically. The results are shown in

Fig. 2. This figure shows the signal range of the two contributorsto as a function of the input frequency of the input signal.This signal range is expressed in terms of . Hence, this canbe considered as an estimation of the number of comparators

Fig. 2. Contributors to the signal levels of V (z) in a conventional continuous-time �� modulator. (H = 2).

that are addressed. In this specific case was set in corre-spondence to a 6-bit (mid-rise) quantizer. If a full-scale inputsignal with an amplitude of is applied to the quantizer,this signal covers the whole quantizer input range of . In the6-bit case such a full-scale input signal hence corresponds to asignal range of or 63 in the figure.

We first look at the contribution of the term in the quantiza-tion noise. For this contribution an upper limit can be found inthe -norm of cfr. [8]. Of course, this term de-pends on the designed NTF, e.g., for the design with anof 2 (respectively, 3) the -norm is 2.6 (respectively, 3.56).This means that the contribution of the quantization noise hasa signal range of (respectively, ). Hence, about3 (respectively, 4) comparators spread by are required tocover this signal range. The contribution of this term is inputsignal amplitude and signal frequency independent and corre-sponds to the horizontal line in Fig. 2.

The second term in (6), which comes from the input signal,of course depends on the specific type of the input signal. Fur-thermore, for the contribution of this term also the signal transferfunction is important. This is also shown in Fig. 2 wherethe contribution of this term for a sinewave with a full-scale am-plitude is plotted for two modulators with the same NTF but witha different STF. Using the modulator of Fig. 1, the path from theinput of the modulator to the input of the quantizer only affectsthe STF and is optional. If this path is present we will speak ofthe CTI modulator, if it is not this will be denoted by CT0. Thecorresponding signal transfer functions (shown in Fig. 2) arecalled STFI and STF0. As can be seen, in the signal band of in-terest (indicated by the vertical lines) the influence of the feedinpath to the quantizer is negligible. Outside this signal band theeffect is important, see also [6], [7]. This contribution is signalamplitude dependent. In Fig. 1, the signal range is shown for afull-scale (amplitude of ) sine wave.

In order for the modulator not to become unstable and thewhite noise quantization model to be an accurate model for thebehaviour of the multibit quantizer in a modulator, it isimportant that the signal range of does not exceed thefull-scale input range of the quantizer . As a result themodulator has a maximum stable amplitude (MSA) which is

DE MAEYER et al.: EFFICIENT MULTIBIT QUANTIZATION 759

Fig. 3. Conceptual diagram of the proposed modulator.

smaller then . Using the analysis above, this MSA couldbe approximated by

(7)

as part of the input range of the quantizer is occupied by the termin the quantization noise. Most of the times the modulator is de-signed using the gain of the STF in the signal band of interest,where . In a traditional modulator with a multibitquantizer one could say that . This means that thenumber of comparators equals . Thelatter indicates that, given a quantization step set by e.g.,jitter requirements, the number of comparators in the quantizeris proportional to the desired MSA. In the next section, we pro-pose a way to break up this proportional relationship. This willallow to reduce the total number of comparators required in thequantizer.

B. Proposed Concept

A conceptual diagram of the proposed modulator isshown in Fig. 3. Compared to the conventional modulator, acontinuous-time signal is subtracted at the input ofthe quantizer. In the quantizer this signal is sampled. Likebefore the sampled version of is denoted by ,see also Appendix A. Another modification compared to theconventional modulator is the addition of signal to theoutput of the quantizer. The discrete time signals and

are nominally the same and are both “predictions” ofthe signal . In Section II three practical ways to obtainthese predictions are proposed.

For the moment, we assume these predictions are available.Then the output of the modulator is

(8)

As the signals and are nominally identicalthe performance of the modulator is the same as for the tradi-tional modulator (of course as long as the quantizer is not over-loaded). This can also be understood by noting that the block inthe dashed rectangle of Fig. 3 is (in the linear model) function-ally equivalent to the main quantizer of the conventional modu-lator of Fig. 1. This also means that the behaviour of the internalnodes of the modulator is identical to that of the traditional mod-ulator.

Fig. 4. MRIn-realization of the proposed technique, here the prediction is basedon the input signal of the modulator.

Now we will, like we did for the traditional modulator, con-centrate on the input signal of the quantizer, . This signalis given by

(9)

As it is our intention to make identical tois equal to the difference between the original signaland its prediction . The better the prediction,

the smaller the signal range at the input of the quantizer. This isthe basic idea of the proposed concept as a smaller signal rangerequires less comparators in the quantizer.

II. PRACTICAL REALIZATIONS

A. MRIn-Realization

Fig. 4 shows a first possible practical realization of the pro-posed modulator of Fig. 3. Here, the prediction signals

and are derived from the input signal. We will callthis realization the MRIn-realization (modulator with a Reducednumber of comparators based on the Input of the modulator). Asseen in Fig. 3, the input signal of the modulator is sampled onthe falling edge of the clock period. Next, this signal is roughlyquantized with a low resolution auxiliary quantizer .This digital discrete time signal can be used for . The originof the unit delay in the path of is explained in Appendix A.To obtain the continuous-time version of , the latter is sentthrough an auxiliary DAC with transfer function typ-ically equal to . The resulting signal is then sub-tracted from the output of the loopfilter. Analog techniques toperform the auxiliary quantization and digital to analog conver-sion are readily available. The adder in Fig. 3 can be imple-mented using e.g., the analog techniques presented in [10].

To further illustrate the working principle, the third-ordermodulator with optimized zeros described before was used withan OSR of 128 and with an of 2. The number of bits in theauxiliary quantizer was set to 3. The main quantizer had a quan-tization step corresponding to a 6-bit quantizer and as advisedin [11] a small dither signal was applied to it. Furthermore, theinput signal was a sine wave with an amplitude ofand had a frequency of 0.875 times the bandwidth of the mod-ulator.


Fig. 5. Signal levels at the input of the quantizer.

Fig. 6. MRLoop-structure as a possible realization of the proposed modulator.

A simulation result is shown in Fig. 5. Three signals are dis-played. The first is , the input signal of the quantizer for theconventional modulator of Fig. 1. The second signal is , itis a roughly quantized version of the modulator’s input signal.The last signal is , the input signal of the quantizer for theproposed modulator of Fig. 4. In the right part of Fig. 5 a his-togram of a much longer simulation is shown, the width of 1 binwas set to the quantization step . Hence, this histogram cor-responds to the number of times the corresponding comparatoris addressed.

As can be seen from Fig. 5, the signal range of is dic-tated by the input signal. Subtracting the prediction signalfrom results in . The signal range of is much smaller.Here, it is roughly equal to the quantization step of the auxiliaryquantizer, i.e., , meaning that the output value of the outercomparators never change in value. This means that the corre-sponding comparators can be omitted. The signal range of(and hence the required number of comparators) will be furtheranalyzed in the next sections, but first we will propose two otherpractical realizations.

B. MRLoop-Realization

In the structure of Fig. 6, the signals and are de-rived from the output of the loop filter. We will call this theMRLoop-structure (Modulator with a Reduced number of com-parators based on the output of the Loop filter). In this struc-ture the combination of the auxiliary quantizer followed by themain quantizer is somewhat similar but not identical to a twostep flash quantizer [12]. The auxiliary quantizer generates a lowresolution digital approximation of the output of the loop filter,

Fig. 7. MROut-structure as a possible realization of the proposed modulator.

which of course can serve as a good prediction of . The mainquantizer further refines the digital approximation.

C. MROut-Realization

In another possible realization, the output of the modulator isused for the prediction signals and . This structure isshown in Fig. 7 and is called the MROut-structure (Modulatorwith a reduced number of comparators based on the output ofthe modulator). In contrast to the previous structures, the outputsignal is already a discrete time digital signal. Hence, neithersampling nor quantization is needed. As the paths actually forman extra feedback loop (see Fig. 7), a delay of one clock periodis required to make it practically realizable. A somewhat sim-ilar modulator has been proposed in [13] and recently in [14] amodulator using a similar technique was implemented.

III. NOMINAL NUMBER OF COMPARATORS

In this section, we investigate the required number of com-parators for the three structures. First, we start with a theoreticalanalysis of each structure. Next, simulation results are used toillustrate the effectiveness of the proposed techniques and thecorrectness of the theoretical analysis. At the end of this sectiona design example is presented.

A. Theoretical Analysis

1) Expressions for : First we will derive an expressionfor the signal for the three structures, for which we willuse (9). Hence, we need to calculate the signals and

from which we can find as .


Doing so, we find for the MRIn-structure

(10)

Here, corresponds to the quantization noise of the auxil-iary quantizer. Note that the factor comes from the factthat the auxiliary quantizer is sampling on the falling edge (seealso Appendix A). Using , this re-sults in

(11)

Cleaning up the equation, we find for the MRIn-structure

(12)

The same calculations can be performed for the other twostructures. This way we obtain for the MRLoop-structure (formore details see Appendix A)

(13)

For the MROut-structure this becomes

(14)

As can be seen form the above equations, there are, in general,three contributors to , one originating from the input signal,another from and finally one from . We will now discussthe influence of these contributors to the signal levels at the inputof the quantizer.

2) Contribution of : The term in is present in the ex-pression for in the case of the MRIn and MRLoop-struc-ture. It is easily shown that for these structures at leastcomparators are required (see Fig. 5). Remember that the mainquantization step is set by the desired number of bits in themain quantizer. Now, the quantization step of the auxiliaryquantizer can be chosen based on the following trade off. Forsmaller , less comparators in the main quantizer are required,at the expense of more comparators in the auxiliary quantizer.

3) Contribution of : The second contributor to the signalcomes from . This will be investigated in the same way

as for the traditional modulator, it is by means of the -normof the factor of . In Table I, the contribution of this term isdetermined using the theoretical analysis and after simulation.First it should be noted that the simulation results are in goodcorrespondence with the theoretical analysis taking into accountthat the -norm forms an upperlimit for the signal level thatcan result from quanization noise [8]. The longer the simulationthe closer the simulation approximates this upper limit. Next, it

TABLE ICOMPARISON BETWEEN THE THEORETICAL UPPER BOUND AND THE

SIMULATED CONTRIBUTION OF THE TERM IN Q FOR THE DIFFERENT

STRUCTURES

can be concluded that the contribution of depends on thedesigned NTF. Not surprisingly, higher result in a higher

-norm.Next, comparing the different structures, we find that the MR-

Loop-structure has the lowest -norm. This could intuitivelybe explained by the fact that in this structure the estimationsignal is closest to , see (9). For the MRIn-structure the -norm is the same as in the traditional mod-ulator, which is somewhat larger than for the MRLoop-struc-ture. The MROut-structure has the largest -norm. This couldbe explained by the fact that there is a full delay in the predic-tion path in combination with the fact that the quanization noisesignal changes rapidly.

4) Contribution of : The contribution of the term inthe input signal is shown in Fig. 8 for the three differentstructures, on the top for the CTI modulator, on the bottom forthe CT0 modulator . First we concentrate on thebehaviour of the structures for signals with a frequency in thesignal band of interest. Here, the approximation for the STFpresented in (5) comes in hand. If we apply this approximationtogether with the approximation to (12),(13) and (14) it can be found that in the MRIn and MROut-structure is multiplied with . For the MRLoop-structure thisis . This difference in behaviour is also observed in Fig. 8.For large OSR this difference between the structures might notbe so important. However for lower OSR it might be significant,as will be shown in the next section.

Using the (12), (13), and (14) we can simply investigate thebehavior of the modulator for larger signal frequencies, i.e., out-side the signal band of interest. As noted before this behaviourstrongly depends on the STF as is shown by the difference be-tween the curves in the top and bottom figure. Next it also de-pends on the structure. Here, especially the strong increase inthe case of the MROut-structure should be noted. The impor-tance of the behaviour of the modulator for signals outside thesignal band of interest depends on the application of the modu-lator and the presence of pre-filtering [6], [7]. The designer canderive the influence of out of band signals using the above equa-tions.

B. Verificiation by Simulation

Taking the sum of the above three contributions results in aclose upper bound and hence a good approximation of the signallevels of . To illustrate this, we show the results for theMRLoop-structure. Like before the third-order modulator withoptimized zeros and an of 2 was used. The main quan-tizer had a quantization step corresponding to a 6-bit quantizer


Fig. 8. Contribution of the input signal to V (z), on the top for the CTI mod-ulator, on the bottom for the CT0 modulator.

Fig. 9. Comparison between the predicted signal levels for the MRLoop-struc-ture versus simulation results (H = 2). Here, TFI stands for the transferfunc-tion undergone by V (s) in the MRLoop-structure (CTI modulator) [see (13)].

and the auxiliary quantizer was a 3-bit variant. Again, for theinput signal a sinewave with an amplitude of waschosen. The frequency of this sinewave was varied resulting inthe curve of Fig. 9, where TFI stands for the transferfunction un-dergone by in the MRLoop-structure, see (13) in the caseof the CTI modulator. The dotted-dashed lines in the figure showthe calculated values of the three individual contributions to

. The full-line corresponds to the overall theoretical pre-diction. It also shows the simulated signal levels which were ob-tained in a time-domain simulation where the maximum valuesof the sampled input of the quantizer were collected. Basedupon this figure the good correspondence between the theoryand simulation can be observed both for the behaviour for inand out-of-band signals. As expected the proposed theory pro-vides an upperlimit for the signal levels due to the fact that sucha limit was used for the prediction of the contribution of .

Interesting points in this figure are the crosspoints betweenthe curve with the simulated or theoretical values and thevertical lines corresponding to the baseband of the modulator.These crosspoints correspond to the highest signal values forsignals in the baseband. In the case of Fig. 9, these crosspointsare 15.9 and 10.7 for the and case,respectively. This clearly shows the importance of the term inthe input signal for lower OSR compared to the case of largerOSR.

The same calculations and simulations can be performed forthe other structures and for the other modulators having anof 3. The crosspoints that are found in these situations are sum-marized in Table II. In this table first the influence of the OSRand the for a given structure can be clearly identified. Thetable also shows that the signal levels can be decreased to a rea-sonable extent compared to the traditional modulator if the pro-posed technique is used. Finally, it is interesting to compare theresults of the MRIn and MRLoop structure as for these struc-tures the contributions of and are about the same. There-fore, for large OSR the difference between both is small. How-ever, for low OSR where the contribution of the input signal ismore important the fact that the input signal is multiplied with

for the MRLoop structure instead of for the other twostructures can be detected from the figures in the table.

C. Two Design Examples

This section is dedicated to the design of a modulator usingthe proposed technique, more specifically we will show howthe theoretical analysis can be used to determine the requirednumber of comparators in the main quantizer. We assume theother aspects like the OSR, the NTF, the STF, and the ofthe modulator are set by other performance requirements likebandwidth, performance, interference immunity and jitter per-formance respectively. We will use Fig. 10 for the design. Inthis figure the theoretical signal range for all the contributorswas calculated for a sinewave with an amplitude of inthe case of the MRLoop-structure. Based on this analysis theinput range of the main quantizer covers for this sine(16.2 with an amplitude of ) with a frequency at theedge of the bandwidth for the case of an OSR of 16 and anof 2. In the case where the OSR is 128 this input range covers by

, almost independent of the amplitude of the sinewave.Next, full time-domain simulations were performed with an

input sinewave with an amplitude of and a frequencyof , which is close to the worst case situation. Assuggested in [11] we applied a (small) dither signal to the quan-tizer. In the simulations the number of comparators was variedand the performance of the modulator was measured. Table IIIshows, for the design example, two main figures. The first figure


TABLE IISIMULATED SIGNAL LEVELS AND CALCULATED THEORETICAL UPPER BOUND FOR THE SIGNAL LEVELS (BOTH IN 1=� ) FOR A SINEWAVE WITH AN INPUT

AMPLITUDE OF 0:75V =2 AND A FREQUENCY AT THE EDGE OF THE BASEBAND

TABLE IIINUMBER OF TIMES THE MAIN QUANTIZER GOT OVERLOADED AND THE CORRESPONDING SNDR AS A FUNCTION OF THE NUMBER OF COMPARATORLEVELS

(COMPARATORS)

Fig. 10. Theoretical curves used for the design example. TF0 stands for thetransferfunction undergone by V (s) in the MRLoop-structure (CT0 modu-lator), see (13).

is the number of times the main quantizer got overloaded duringthe simulation. The second figure represents the obtained simu-lated performance signal-to-noise distortion ratio (SNDR).

The following things can be concluded from this table. Usingthe number of comparators based on the theoretical analyses in-deed leads to the full performance of the modulator and to a sit-uation where the main quantizer never gets overloaded. On theother hand, the modulator can become unstable if too few com-parators are used in the main quantizer. In between these twoextreme situations, the modulator gets overloaded occasionally.However, if this overloading occurs rarely the modulator is notunstable and the additional quantization noise is limited. There-fore, in this case only a small reduction in performance is ob-served.

Based on these results, we conclude that a design based on thetheoretical upper limit for the number of comparators providedby the theoretical analysis results in a modulator with a fullperformance, where the main quantizer never gets overloaded.

Moreover, compared to the traditional modulators instead of 63comparators the modulator achieved full performance with only15–17 comparators in the case and 9–11 in the

case which means a significant reduction.

IV. EFFECT OF NONIDEALITIES

In the previous sections, we assumed all components used inthe proposed structures to be ideal. In this section we will inves-tigate the influence of most common and possible nonidealitieson the behaviour of the proposed structures. Looking at the anal-ysis, it falls apart into two aspects. The first is the influence withrespect to the performance of the modulator, the other is the ef-fect on the required number of comparators. The purpose of thissection is to derive the requirements that need to be fulfilled in aactual implementation of the auxiliary quantizer, auxiliary DACand analog subtractor.

A. Requirements on the Auxiliary Quantizer

Both the MRIn and MRLoop-structure use an auxiliary quan-tizer. In this section, we investigate the influence of offset onthe individual comparator decission levels (comparator offset), aglobal gain error and a global offset error in this auxiliary quan-tizer. These nonidealities will cause no difference between thetwo prediction signals and . Because these nonideali-ties are situated in the common path of these signals, nothingchanges with respect to the output signal. So, there will be noperformance degradation, provided the main quantizer does notoverload, whether this happens will be investigated next.

1) Comparator Offset in the Auxiliary Quantizer: The anal-ysis of comparator offset in the auxiliary quantizer is very sim-ilar to the analysis of [12]. The comparator offset in the auxiliaryquantizer can cause the main quantizer to become overloaded.A simple way to counteract this is to add some extra compara-tors. However, no extra comparators are required if this offsetremains smaller than about of the quantization step[12]. This condition means that the comparator offset require-ments for the auxiliary quantizer are similar to the offset require-ments of the main quantizer.


2) Gain Error and Overall Offset Error in the Aux-iliary Quantizer: With regard to the contribution of a gainand offset error to the input signal of the main quantizer,we find (15) respectively, (16) for the MRIn respectively, MR-Loop-realization1

(15)

(16)

For the equation above we used (32) from Appendix A. Theapproximation below for baseband signals ( or ) isinteresting

(17)

So, normally this gain and offset error will slightly increase thesignal range of . Again, this can be compensated for by usingsome more comparators. For an input signal in the frequencyband of interest where , this is not necessary if

(18)

B. Requirements on the Auxiliary DAC and the Subtractor

In all the proposed structures, the signal passes throughan auxiliary DAC, while does not. In this section we as-sume the auxiliary DAC and subtractor introduce a gain error

and offset . These parasitic effects cause a differencebetween and and the effect of this parasitic effectcan best be calculated by observation of the structures in Figs. 4,6 and 7.

1) Offset : The offset causes a constant difference be-tween the signals and . At the output this constant ismultiplied by the NTF [see (8)]. Since, the offset is at dc, thisoffset causes no performance degradation, provided the mainquantizer does not overload.

For the MRIn, MRLoop, and MROut structures the offsetresults in an additional contributor to . By inspection of

Figs. 4, 6, and 7, the following equations can be found:

(19)

for the MRIn, MRLoop, and MROut-structure, respectively. So,the MRIn and MROut-structure behave very alike, meaning theydo not suffer from this offset. In the MRLoop-structure the in-fluence of is the same as of , which was also observed inour simulations.

1Note that we indicate the additional contribution to V by�V .

2) Gain Error : Now we will calculate the influence ofthe gain error for the different structures.

• For the MRIn-structure we can calculate the influenceof the gain error directly by using (10) for .The effect of is identical to an additive source of

, which means we can reuse (19). So, wederive

(20)

The influence of the gain error on the signal is verysmall, as both (remember is the quantizationstep of the auxiliary quantiser) and (for signals in the bandof interest) is small.With respect to the output of the modulator the second termis equivalent to a linear gain error, which can be tolerated.However, depending on the properties of there may besome performance degradation caused by the first term in

. If we make the assumption that is white noise, (forlarge values of ) a performance degradation of

(21)

can be expected. In practice, is not white noise, e.g.,for a sine wave it may contain harmonics. Allthough theseharmonics appear shaped at the output of the modulator,harmonics in the frequency band of interest may cause asmall performance degradation.

• The influence of the gain error is most complex in theMRLoop-structure. First of all, the gain error changesthe loop filter such that the equivalent discrete time loop-filter equals

(22)

This means that the NTF is also changed to ,shown in Fig. 11 (for an unrealistic high value of

). The gain of the loopfilter in the signal band of interestis affected by a factor of due to . This will onlyhave a minor effect on the performance of the modulator.For higher frequencies may have a more significantinfluence on the loop filter and in theory it may affect thestability. However, the authors did not observe instabilityin any of the simulations even for unrealistic high valuesof (up to 10 %).Now let us look at the output signal for which we find

(23)

Of course, the output changes to a small extent due to thechange in the NTF, this corresponds to the first two terms.The last term results from the fact that causes a a dif-ference of between and . Thisdifference appears shaped by the new NTF at the output


Fig. 11. Effect of a DAC-gain error on the overall NTF for the case of theMROut and MRLoop structure.

of the modulator [see (8)]. With respect to this additionalthird term, the same remarks mentioned above (21) con-cerning the spectral content of apply. However, thenoisy term caused by in the input signal of the auxil-iary quantizer is a kind of dithering signal to the auxiliaryquantizer. So, in this structure the white noise assumptionis a better approximation for .Finally, we look at the input signal of the main quantizer.The influence of can be calculated exactly, usingequations which are similar to the ones presented inAppendix A. However, the equations are very compli-cated and lack easy interpretation. Therefore, we derivea first-order approximation based on the expression for

in the case where . We assume this signalremains unaltered due to the gain error . Next, we canmodel the effect of as an additive source and use (19).As such, we find

(24)

There are two contributors to namely andand there is also one term in . The term in ispossibly the largest one. It means that part of the inputsignal is leaked to the input of the main quantizer. Thisleakage which was also observed in simulations increasesthe signal range of . Still it is guaranteed that no extracomparators are required if .

• For the MROut-structure the gain error also changes theloop filter this time such that the equivalent discrete timeloopfilter is

(25)

The NTF corresponding to this loopfilter withis also shown in Fig. 11. Again, the NTF is only slightlychanged in the signal band of interest. At higher frequen-cies the extra term in is more important. In theory it

could change the stability, which was not observed by theauthors. For this structure the output signal is given by

(26)

So, causes no performance degradation.For the influence of to we can use the samestrategy as for the MRLoop-structure. Next we can reuse(19) which reveals that in this structure in a first-orderapproximation does not affect the signal range at the inputof the main quantizer.

3) DAC Nonlinearity: The auxiliary DAC is a multibit DAC.Hence, static mismatch between the unit elements will causenonlinearity. This nonlinearity can be modeled as an additivesource. Then, the influence of this source is the same as forthe offset . This means that for all the structures the DACnonlinearity appears shaped at the output of the modulator. Withrespect to the number of comparators, this static mismatch willhave a negligible effect.

C. Design Guideline

The above equations reveal that in every structure (for inputsignals in the band of interest) the effect of somekind of gainerror or offset error is upper bounded by orrespectively. The resulting errors need to be compared with thequantization step . As the gain and offset errors normallyarise from matching errors it is observed that the matching re-quirements on the building blocks correspond to the accuracyrequired in the main quantizer. E.g., for a 6-bit main quantizerit is sufficient that gain and offset are matched to a 6-bit accu-racy.

V. CONCLUSION

In this paper, a new modulator concept is proposed. Ituses a high-resolution quantizer with a reduced number of com-parators. The basic idea of the new modulator is to decrease thesignal range at the input of the quantizer. Compared to the quan-tizer in the traditional modulator, this allows to omit the ma-jority of the comparators in the quantizer as they are never ad-dressed. Three possible practical realizations are discussed andcompared with each other. The strength of the proposed tech-nique and the design procedure were illustrated based on the-oretical analysis and simulation results. For the simulations, athird-order continuous-time modulator was used as a testvehicle. The quantizer had a quantization step correspondingto 6-bit. However, instead of 63 comparators the same perfor-mance was obtained with much less (e.g., as less as 9) compara-tors. Two of the structures require an additional 3-bit quantizerand DAC. The influence of nonidealities in these circuit blocksis discussed and compared for the proposed structures.

APPENDIX A

The purpose of this Appendix is two fold. First, we want togive some more details about the star operator. Next, this oper-ator is used for the derivation of (13), the expression forin the MRLoop-structure.


A. The Star-Operator

The star-operator [15], [16] is used to indicate the effect ofa sampler on a continuous-time signal. As such corre-sponds to sampling the analog signal at thesampling instants and then taking the -transformation

(27)

where the -variable should be replaced by . A more efficientway to calculate this (periodic) discrete transfer function usesresidues [15], [16]. It is also easy to show that (27) correspondsto

(28)

Also commercially available numerical software providesfunctions to perform this conversion (e.g., the c2d-function inmatlab).

Finally, we would like to note that applying the star-operatorto corresponds to finding the equivalent discretetime transfer function of the loop filter of the modulator see(27). As such the star-operator is a very efficient and shorthandnotation to express the effect of the sampler in the frequencydomain.

Interesting mathematical rules are

(29)

(30)

(31)

B. Derivation of (13)

Looking at the MRLoop-structure of Fig. 6 the followingequations can be derived. First, equals (assume for themoment )

(32)

Here, equals for a CT0 modulator and forthe CTI modulator. For the sampled version of the signalwe know

(33)

The derivation of is somewhat more difficult and canmost easily be understood if the sampler which samples at thefalling edge of the clock is conceptually replaced by the cas-cade of a negative delay block (delay -T/2, ) and a sampler

which samples at the rising edge. This way all discrete signalsare synchronized at the same edge. Doing so, we find

(34)

Due to the presence of the auxiliary DAC an epsilon delayshould be taken into account. This leads to a full delay betweenthe sampling operation in the redefined auxiliary and mainquantizer, or

(35)

This corresponds to what is happening intuitively. The implicitpresence of this delay in the analog path also explains the ex-plicit delay in the discrete path to the output of the main quan-tizer in all the structures and more specifically in the one ofFig. 6.

Subtracting the above (33) and (35), we find

(36)

or with (32)

(37)

Next, we can use the mathematical rules (29)–(31) and the factthat

(38)

to find the first term in (13). The second and third term in (13)result from and and can easily be found using the samereplacement for the sampler in the auxiliary quantizer.

REFERENCES

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[2] E. J. van der Zwan and E. C. Dijkmans, “A 0.2-mW CMOS��modu-lator for speech coding with 80-dB dynamic range,” IEEE J. Solid-StateCircuits, vol. 31, no. 12, pp. 1873–1880, Dec. 1996.

[3] J. A. Cherry and W. M. Snelgrove, “Clock jitter and quantizer metasta-bility in continuous-time delta-sigma modulators,” IEEE Trans. Cir-cuits Syst. II, Analog Digit. Signal Process., vol. 46, no. 6, pp. 661–676,Jun. 1999.

[4] Z. Zhang and G. Temes, “Multibit oversampled �� A/D converterwith nonuniform quantization,” Electron. Lett., vol. 27, no. 6, pp.528–529, Mar. 1991.

[5] R. Schreier and B. Zhang, “Delta-Sigma modulators employing con-tinuous-time circuitry,” IEEE Trans. Circuits Syst. I, Fundam. TheoryAppl., vol. 43, no. 4, pp. 324–332, Apr. 1996.

[6] J. De Maeyer, J. Raman, P. Rombouts, and L. Weyten, “Controlledbehavior of the STF in CT �� modulators,” Electron. Lett., vol. 41,no. 16, pp. 896–897, Aug. 2005.


[7] J. De Maeyer, J. Raman, P. Rombouts, and L. Weyten, “STF behaviourin a CT �� modulator,” in Proc. IEEE Int. Conf. on Electr. CircuitsSyst. (ICECS), Gammarth, Tunesié, Dec. 2005, vol. 12, op CD.

[8] J. G. Kenney and L. R. Carley, “Desing of multibit noise-shaping dataconverters,” Anal. Integr. Circuits Signal Process., vol. 3, pp. 259–272,May 1993.

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Jeroen De Maeyer was born in Brugge, Belgium, in1979. He received the Ir. degree in electronics and thePh.D. degree in electronics and information systemsfrom Ghent University, Ghent, Belgium, in 2002 and2006, respectively.

Currently he works as a Design Engineer at AMISemiconductor, Oudenaarde, Belgium. His researchfocuses on analog–digital and digital–analog conver-sion, more specifically continuous-time sigma–deltamodulators.

Dr. De Maeyer received support from the Fund forScientific Research (F.W.O.-Vlaanderen), for his doctoral work.

Pieter Rombouts was born in Leuven, Belgium, in1971. He received the Ir. degree in applied physicsand the Dr. degree in electronics from Ghent Univer-sity, Ghent, Belgium, in 1994 and 2000, respectively.

Since 1994 he has been with the Electronics andInformation Systems Department of the Ghent Uni-versity where he is currently a Professor. In 2002, hewas a Visiting Professor at the University Carlos III,Madrid, Spain. His technical interests are signal pro-cessing, circuits and systems theory, and analog cir-cuit design.

Ludo Weyten was born in Mortsel Antwerpen, Bel-gium, in 1947. He received the Ir. and Dr. degreesfrom Ghent University, Ghent, Belgium, both in elec-trical engineering, in 1970 and 1978, respectively.

From 1970 to 1972, he joined the ElectronicsLaboratory, Ghent University, as a Research As-sistant. From 1972 to 1975, he was teaching atthe National University of Zaïre (now Congo),Lubumbashi, Congo, in a government technicalco-operation project. Since 1975, he has been withthe Engineering Faculty of Ghent University, where

he is currently a Professor. His teaching and research interests are in the fieldof electronics circuits and systems and e-learning.

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