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IEEE TRANSACTIONS COMMUNICATIONS, ON VOL.NO. COM-33, 3, 249 MARCH 1985 A Use of Double Integration in Sigma Delta Modulation JAMES C . C A h Y , FELLOW,IEEE 1 ' Abstract-Sigmadeltamodulation is viewedas a technique that employs integration andfeedbacktomove quantizationnoise outof baseband. This techniquemaybeiterated by placing feedback loop around feedback loop, but when three or more loops are used the circuit canlatch into undesirable overloading modes. In the, desired mode, a simple linear theorygives a good description the of modulation even when the quantization has only two levels. A modulator that employs double integration andtwo-level quantizationis easy to implement andtolerant is of parameter variation. Atsampling ratesof 1 MHz it provides resolution are equivalent to 16 bit PCM for voiceband signals. Digital filters that suitable for converting the modulation to PCM are also described. multipleintegration,andcalculatethesignal-to-noiseratios. Theresultsareconfirmedbysimulation,andexperimental measurements. We show that when more than two integrators are used, the circuit can latch into undesirable modes where its performance is ruined. Finally, we give a design for a digital processor for constructing PCM from this modulation. 11. QUANTIZATIONWITH FEEDBACK Fig. 1 (a) shows the circuit of a differential quantizer which is a form of the well-known delta modulator; we will use this circuit t o explain our view of feedback modulation. Fig. l(b) shows a sampled data model of the circuit; it assumes that the A/D ahd D/A conversion are ideal and that signals are random, I. INTRODUCTION so that the quantization may be represented by added noise e and linear gain G (level-spacing/threshold spacing). AccumulaHIS paper describes the design of a digital modulator that isinteti'ded for use in oversampled PCM encoders. These tion A representstheintegration.Mathematicaldescriptions of related circuits have been presented in several places [ 11, encodersmodulatetheiranaloginputsintoasimpledigital [ 3 1 , [ 41, [ 121, [ 141 -[ 161. They show that the presence of format highspeed;thendigitalprocessingtransformsthe feedback around the quantizer has three uses, which are summodulationto PCM sampledattheNyquistrate [ 11-[9]. marized below. Propertiesofthepreliminarymodulation have stronginPrediction and Preemphasis: The modulated signal A4 comfluence on the design-of the entire encoder. For example, the isp.roporresolution of the PCM can be no better than that. of the modu- prisesanoisecomponentandacomponentthat tional to the rate of change of input amplitude. Modulating a lation, and the complexity and speed of the digital processor rateofchangecanbemoreefficientthanmodulatingthe depends on the kind of modulation used and its resolution. amplitudedirectly,particularlyforvideoandaudio signals The tolerance of the analog circuits employed in the modulator can determine the suitability of the design for integrated cir- whosespectraldensitiesfallwithincreasingfrequencyand cuit implementation and the power consumed by these circuitswhosesamplevaluesarehighlycorrelated [ 151, [ 161. This can be a large part of the power used b y the entire encoder. improved efficiency can result in decreased sampling rate or a There is, therefore, much incentive to find an efficient modureduction in the number of quantization levelsneeded for a lator,onethatprovideshighresolution(idlechannelnoise given resolution. morethan 80 dBbelowpeaksignal)atmoderatesampling Control of Overloading: Ordinary PCM quantization overrates (less than 1 MHz for 4 kHz telephone signals) yet employs loads by clippingsignal amplitudes directly. When this happens simple robust circuits (tolerances no tighter than +3 percent). t o t h e signal applied to the A/D in Fig. 1, it is t h e derivative Early work on oversampled encoders [ 11 , [ 21 was mostly of the input signal that is clipped, resulting in slope overloadtheoreticalandbasedondeltamodulation.Later,practical ing of the output signal. Distorting the slope of video [ 171 and realization preferred sigma delta modulation but modified it to audio signals can be less disturbing than clipping their amplilower t h e samplingrateandsimplifythedigitalprocessing. tudes directly. This also can lead to improved efficiency of the For a video application, multilevel quantization was used [3] modulation. toreducethemodulationrate.Fortelephoneapplications, NoiseShaping: Placing thequantizerinafeedbackloop some modulators [4], [5] achieve highresolutionbybiasing with a fiiter shapes the spectrum of the modulation noise[ 3 1 , the modulator to an especially quiet state for idle channel op[ 131, and at the same time it can decorrelate the noise from eration. One [ 71 employed triple integration in the sigma delta t h e signal. If we assume that the quantization noise e in Fig. modulator;another [ 81 employeddigitalaccumulationand l ( b ) is white,thenthespectraldensityofthenoiseinthe companded quantizationlevels in the feedback path. modulated signalrises withfrequency;butafterintegration Recentadvancesindigitalintegratedcircuittechnology t h e noise is white again at the output. We willsee that other have greatly reduced the need to have simple digital processciicuit configurations [ 131 -[ 181 can shape the output noise ing; indeed, now it is feasible [ 101 to have digital line equalispectrum to suit particular applications. on zation,echocanceling,digitalhybrids,andconferencing When higher order filters are used in place of simple integrathechipwiththecodec.Theseapplications,however,place tion [ l l ]-[ 141, the propertiesof the modulation are modified. to be Themodulated signalincludescomponentsthatareproporstringentdemandsonresolutionanddynamicrange provided by the modulator. tional to high-order derivatives of the signal. Overloading limits The present work explores the advantages of having double not only the slope but also the rate of change of slope of the integration in a sigma delta modulator. We demonstrate that signal, and modulation noiserises more steeply with frequency. a particular class of circuits can provide high resolution and be The restriction on the design of these high-order filters is the toleranttoimperfection. We explainthereasonsforusing need to keep the feedback stable. All these properties of feedback quantization can influence t h e design of modulators for oversampled codecs, and in some Paper approved by the Editor for Signal Processing and Communication Electronics of the IEEE Communications Society for publication without oral applications the requirements are in conflict. For example, oppresentation. Manuscript received March28, 1984; revised October 16,1984. timum design of predictors usually calls for leaky integration, The author is with AT&T Bell Laboratories, Holmdel, NJ 07733. but optimum noise shaping calls for long-time constant inte- T 0090-6778/85/0300-0249$Ol.OO 0 1985 IEEE 250 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-33, NO. DEMODULATOR 3, MARCH 1985 MODULATOR r------1 I QUANTIZER I 1'" I Lc7-' DELAY DELAY "n-l+""-l I A I ACCUMULATOR (b) Fig. 1. (a) An exampleof a differential modulator and demodulator. @) A sampled data representation of the differential modulation. MODULATOR INTEGRATOR A , DEMODULATOR LOW PASS AID FILTER 1 1 lxh+en-e.-~) (b) Fig. 2. (a) A sigma delta modulator and demodulator. (b) A sampleddata representation of sigma delta modulation. . . gration [ 15 1 . It is the different empGasis given to these separ- has gained favor bcause it is 'very tolerant of imperfection and mismatch of the two D/A circuits. [3]1 The structure.of Fig. 1 ate prop.erties that accounts for the different filters that have is not so tolerant ,because its D/A imperfections are multiplied beenproposedforfeedbackquantizers. Our design forthe the integrating filter at the remodulator will be based on the requirements of telephone toll-by the large baseband gain of ceiver. networks. The next section of this work will be directed atthe, task of 111. REQUIREMENT O F GENERAL PURPOSE MODULATORS generalizing the filter A used in this modulator for the purpose FOR TOLL NETWORK USE of moving quantization noise out of t h e signal band. Applicasignal prediction and special overload Digital ,codecs used in the telephone toll-network must ac- tions that want to use cept a wide range of signals, and their design may not r,ely o n characteristics could do so by providing preemphasis and deemphasis filters external to the modulator[ 151 . properties of restricted classes of signals nor.properties of special receivers. We may not rely on there'being high correlation between Nyquist samples, nor assume t,hat slope overloading IV. SIGMADELTA MODULATION isanymoreacceptablethanclippingamplitudesorthata The modulator shown in Fig. 2 generates a quantized signal colored noise is less objectionable than white noise. We may take advantage of the fact that the signal is band that oscillates between levels, keeping its average equal to the average input. It is easy to show [ 31 that for active inputs, the limited, however, by moving quantization noise out of band spectral density of the noise in the quantizedsignal is given by whereitcanberemovedbyappropqatefilters.Thesigma delta modulator shown in Fig. 2 does this without differentiating the signal: it eliminates the need for integration at the receiverbecauselow-frequencycomponents of themodulated signal representtheinputamplitudedirectly.Thisstructure 25 1 CANDY: DOUBLE INTEGRATION IN SIGMA DELTA MODULATION QUANTIZER A2 DELAY INNER LOOP , SECOND LOOP Fig. 3. A quantizerwith two feedbackloopsaround elements are nominally unity. 20 kHz it. Thegainsofall 40 MHz 4 MHz 400 kHz SAMPLINGRATE (4lr) Fig. 4. Quantizationnoiseplottedagainstsamplingfrequencyforvarious numbers of feedback loops. Baseband is 3.5 kHz and the noise is referred to the step size: the noise of ordinary PCM is - 10.8 dB on this scale. The circuits that can be derived by reiterating simple feedwhere a is the quantization step size and 7 the sampling period. class of feedback 0 < f < fo is then given approxi- backloopsofthiskindareaveryuseful The net noise in baseband mately by quantizers with high-order filters [ 141, [ 181. Ritchie points out the penalties that must be paid for using feedback, and they are summarized in the next section. providedthat f0.r Q 1and eo = a/& is t h e averagenoise generated by the quantizer alone. Thus, the resolution can be greatly increased by oversampling and feedback. For example, quantizinga 3.5 kHzsignal at16 MHz withthisfeedback quantizer reduces the noise eo by 96 dB, which is equivalent to a 2 reduction of the step size. This procedure for increasing the resolution with feedback can be reiterated as illustrated in Fig.3 . Here we have included gains G, in cascade with each integrator in order to describe circuit imperfections that cause the loop gains to be other than unity. Appendix A shows that the spectral densityof the noise in the modulated signalgeneratedbytwofeedbackloops is given by ~ ~ (= (1 f -) z - ~ ) ~ E ( = z >2 e 0 f i ( 1 - cos ( 2 n f ~ ) ) ( 3 ) when G, = 1. The in-band noise is given approximately by This double feedback increases the resolution by 95 dB for a samplingrateofonly1 MHz and 3.5 kHzbaseband.Fig. 4 compares the resolution of PCM to that obtainable with one, two, and three feedback loops. Measurements on circuits with active input signals agree with these calculated values. V. PENALTIES FOR USINGFEEDBACK In Fig. 2 t h e signal applied to the quantizer can be expressed as the input less the noise from the previous cycle: w, = x, e,andthisnoiseusesupsomeofthedynamicrangeof the quantizer. Overloading may be avoided by adding one extra level to the quantizer because e spans + a / 2 . In a similar manner it may be shown that, when the number of feedback loops is L > 0, the range of signalsapplied to the quantizer is increased by 2L- a. If this signal exceeds the range of the quantizer, the modulation noise increases, as is illustrated by Fig. 5 . For many applications, the increase in noise for large signal valuescanbetolerated,providedanadequatesignal-to-noise ratio is maintained. Besidesrequiringadditionalquantizationrange,feedback demands increased precision in the gains of the circuits. Gains in the range +lo percent are usually acceptable for quantizers having a single feedback loop, but more precise gains are needed when additional feedback loops are used. In Fig. 6 we plot the calculated and measured signal-to-noise ratio against values of gains G, placed in series with each integrator for one, two, and three feedback loops. The measured change of noise with gain is larger and more variable than predicted by calculation. This isbecausethenoise iscorrelatedwithsignalamplitudeina way that depends on the gains G,, and this correlation is ignored in the calculations. The third penalty for using feedback concerns the depend- ' 252 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-33, NO. 3, MARCH 1985 POSITION OF QUANTIZATION LEVELS -30 I fi.5 20.5 t2.5 6 Z -40 -30 -40 '0 W -0 -50 -50 -60 i.o 0.5 0 -70 2.5 2.0 i.5 -80 AMPLITUDED OF SINEWAVE Fig. 5 . The noise introduced into sinusoidal signals of various amplitudes by modulation with two feedback loops placed around a quantizer having the stated number of levels. Sampling is at 128 kHz. -30 -40 m 9 W -50 -60 z W- s 8 z 6 0 -70 -80 - - U 0 4 \! W m z 1.1 4 I u 0.5 0.6 0.7 0:8 0,'9 . 4:O 1.2 \ GAIN 1 ' I I 1 I 0 0.2 0.5 0.8 i.o 01AS. a (C) Fig. 7. Graphs of idle channel noise plotted against input bias. (a) Ordinary PCM. (b) Simple sigma delta modulation. (c) Two-level quantization with double feedback. Noise is referred to the step size. The sampling rate is 256 kHz. tx: . t 0.5 0.6 0.7 0.8 0.9 GAIN , i,'O\ 4 ence of modulation noise on signal amplitude. A determination of this dependence for ordinary sigma delta modulation in [ 191 shows that it can have an important influence on the performance of oversampled codecs. Fig. 7 is a graph of noise plotted against the dc bias, x, applied to the modulator. Fig. 7(a) is the quantization error without feedback; the error is zero for x = 0 or 1 which corresponds to the position of t w o adjacent levels, and elsewhere the error is proportional to t h e distance to a level. Fig. 7(b) shows the idle channel noise of a quantizer with a single feedback loop. The noise is gathered into a series of narrow peaks, and t h e large peaks can be an embarrassment in the designof oversampled codecs. VI. TWO-LEVEL QUANTIZATION We will demonstrate how the three penalties, described in the previous section, can be avoided by using just two feedback loops and degrading the quantizer to a single threshold circuit that generates a two-level output. With a singlethresholdtheinconvenience of establishing threshold spacing is removed, and the concept of gain of t h e quantizer becomes unreal unless other circuit properties provide a calibration for the amplitudes applied to the threshold. We find by experiment that signal levels adjust themselves so that the effectivegainof the quantizer compensates for changes O f I I 1I in the values of circuit gains G,. The measured noise shown in -2 0.5 0.6 0.7 0.8 0 Fig. 8 is almost independent of these gains and corresponds t o GAIN -4 thecalculatedvalueforloopgains of unity.Likewise,the (C) change in the attenuation of the signal Y ( o ) / X ( o is ) less t h a n Fig. 6. The change in noise with gain placed in cascade with each integrator 50.05 dB for values of gain 0.5 < G, < 2. We find that the for multilevel quantization. The reference noise is that measured at unity penalty of having to establish quantizer gains in one and two gain value. The curve gives calculated values that apply for sampling rates feedback loops is substantially eliminated by the use of twothat are at least eight times the Nyquist rate. (a), (b), and (c) are for one, level quantization: signal levels in the circuit automatically adtwo,andthreefeedback loops, respectively.Thesecircuitsbecome just themselves to make the effective loop gains unity. With unstable at gains 2, 1.236, and 1.087. - L 253 CANDY: DOUBLE INTEGRATION IN SIGMA DELTA MODULATION 0.5 I I I I.o 1.5 2.0 GAINS G i AND G2 Fig. 8. The noise indoublefeedback,two-levelmodulationplottedagainst gains G, at various sampling rates. The input was excited with a random signalspanning *O.la. Thehorizontallines mark calculatedvalues of noise for unity gain and multilevel quantization with the same step size. more than two feedback loops, undesirable oscillations spoil the functioning of the circuit when the quantizer has only two levels. We discuss these oscillations in the next section. The second penalty concerns the loss of quantization range because feedback increases the amplitudes that are applied to the quantizer. With a single threshold there is no limitation on the range of its input amplitude, only the output levels are defined, but we see in Fig. 5 that for two-level quantization the noiseincreasesrapidlywithsignalamplitude.Fig. 9 presents similar data plotted on logarithmicscales. The third penalty concerned the correlation of noise with input level. Fig. 7(c) shows a graph of modulation noise plotted against input bias for a modulator utilizing two feedback loops. Comparing it to the graphs in Fig. 7(b), we see that use of two loops substantially decorrelates the noise except at the ends of the range, where the modulation noise peaks in the L same fashion that it does in modulators that utilize only one - 35 feedback loop. Reference [ 191 shows that the amplitude of these peaks of noise is given by Nmax =fi(fo+ which can be large compared to the calculated noise their width u is narrow: (5) (4). But 50 40 30 20 p io I - 30 I I I I 1 -25 -20 -15 -10 -5 INPUTAMPLITUDE /STEPSIZE. dB Fig. 9. The dependenceof noise on the amplitudeof a sinusoidal input signal for two-level modulation with double feedback. The noise is referred to its small signal value. This result applies to sampling rates that are eight or more times the Nyquist rate. of the quantizer. In particular, when the quantizer has only t w o levels, the undesirable modes are easily excited and difficult t o extinguish. The susceptibility of the circuit to enter unIt appears that the use of two-level quantization and double desirable modes has prevented measurement of the resolution integration could be the basis of a useful modulator. The next of circuits that have triple integration and two-level quantizasection explains why it is wise to use no more than two intetion except for gains G 2 , G 3 less than 0.55. Even for gains as grators. low as 0.1, theunwantedmodeswereself-sustainingafter beingexcited.Whentwofeedbackloopsareused,however, VII. LIMIT-CYCLES THAT OVERLOADTHE QUANTIZER the contents of the integrators always decay to a small value gains G , The feedback quantizers that we have described cause their whenexcitationsareremoved,providedthatthe are less than 1.23 for multilevel quantization and less than 2 outputs to oscillate between levels in a way that keeps their for two-level quantization. average value equal t o t h e average input. In the desired mode of operation the signalsheld in the integrators are comparaWhen modulators function in an undesirable mode, the sigtively small, but when three or more feedback loops are present, nalsintheintegratorsare so large,compared t o t h e largest quantization levels, thattheinnerfeedbacksareineffective. other modes can be excited [ 141. These modes are characterThebehavior of thecircuit is dominatedbytheoutermost ized by being very noisy and having large-amplitude, low-frefeedback, which, if it contains more than two integrations, is quency oscillations in the integrators, which exceed the range u = fora. (6 1 254 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-33, NO. 3, MARCH 1 9 8 5 unstable.Stabilityisregainedwhentheinnerfeedbacksare made effective by clipping the amplitudes ofsignalsheldin the integrators, or by nonlinear feedback [ 141. But it is questionable whether the extra resolution obtained in practice can justifytheuse of thesemorecomplicatedcircuitsandthe tighter tolerances that they demand. VIII. IMPLEMENTATION OF THE MODULATORSUSING Two INTEGRATORS , x(tl The circuit shown in Fig. 3 is a sampled data model of t h e modulator; its signals are represented by impulse sample values. This circuit could well be implemented using switched capacitorsforaccumulation.Implementationsinabipolartechnology, however, would prefer t o use continuous signals such as those in the circuit of Fig. 10. The analysis of the switched and the continuous circuits in Appendix C shows that their operation is equivalent when the feedback signal y ( t ) is held constantthroughoutthesampleinterval,thetimeconstant R C equals 1.57, and the two inputs are related by the expression Xn7 -InT ‘- THRESHOLD ~ ~ _ _ _ _ y(tl Fig. 10. Ananalogversion of thedoubleintegratingmodulator. 30 50 (2x(t) - x(t - 7)) dt. (7) (n-1)r The analog circuit in Fig. 10 is relatively easy to construct becausethereareonlytwomainconstraintsonitsdesign. There is the need to keep signals small in order to conserve power yet have signal levels large enough to swamp noise and imperfectioninthethresholdcircuit,andthere is theneed t o set the time constant R C with sufficient precision. Fig. 1 1 showsgraphs of themodulator’sresolutionplottedagainst input amplitude for three values of the time constant. Time constants changing in the range 1.27 to 1.87 give less than 1 dBvariationinnoiselevel;thisshouldsatisfymostapplications. Equation (7) can be used to define the frequency response of the filter that should be placed in cascade with the input 3 in order to make its reof the sampled data circuit in Fig, sponse identical to that of the analog circuit in Fig. 8 . That frequency response is given by ’ ( 0 ) = ( 2 - f - w ) smc G(0)=X . (f7). X(w> (8) This low-pass filter, inherent to the circuit of Fig. 10, is usefulforreducingaliasingdistortion.Forexample,when 3.5 kHzsignalsarebeingmodulatedat 512 kHz, anyspurious signals in the range 508-5 16 kHz alias into band. But the distortion is small because I G(508) I < -40, dB. Table I lists the attenuation of t h e signalsaliasedintobandforvarioussampling frequencies; also listed is gain introduced into baseband. This gain could be equalized in the digital processor. Simulations show that signal amplitudes in the integrators of these two-level feedback modulators can be very large, and realimplementationsneedtolimittheirsize.Clippingtheir amplitudesspeedstherecoveryfromoverloadbutmayincrease the noise. Fig. 12 shows that essentially full resolution of two-level quantization is obtained by allowing the integrated signals t o swing through at least k l . 0 step sizes. Variation of themodulator’snetgainwithinputamplitudewas less than +0.025 dB. Fig. 13 shows that leakage in the integration has negligible effect on noise, provided its time constant exceeds 1/2fo seconds. The data in this figure agree with calculations that represent quantization by added noise, and assumes unity gain in the feedback loops. IX. DESIGNO F THE DIGITAL PROCESSOR A digital processor will convert the output of the modulator into PCM by smoothing the signal with a digital low-pass filter 40 % - 9 5 Y - u) 30 5 z 0 - -70 -60 -50 -40 -20 -30 SIGNALAMPLITUDE, - 10 - 0 0 -10 dB Fig. 11. Thesigna-to-nolse ratio plotted against the amplitude of a sinusoidal input signal for the circuit in Fig. 10. The sampling rate is 256 kHz with 3.5 kHz baseband; the time constantRC, is (a) 1.5 7;(b) 27; (c) 7. TABLE I GAINS OF THEFILTER (2 - exp (7 j 2 x f 7 ) ) sinc SAMPLINGFREQUENCY G A I N A T 3.5 k H z I/TkHz dB cf7) GAIN AT (Ih-3.5) kHz. de 64 0.87 -23 9 I28 0 24 -30.8 256 0.06 -37.1 512 0.01 5 -43 2 1024 0.004 -49.3 2048 0.001 -55.3 4096 0 00 -6 I .J 8192 0.00 -67 4 255 CANDY: DOUBLE INTEGRATION IN SIGMA DELTA MODULATION quency response (1 0) and impulse response n(n h , =- + 1) for 1 < n < N 2 0 forN<n<2N I 0.1 I I 1 I I 0.2 0.3 0.4 0.5 0.6 h, = AMPLITUDE OF INPUTISTEPSIZE 250 150 500 6400 0 ps 80 25 50 I I I I I I I I o 0.5 ?.a i5 2.0 2.5 3.0 35 'INTEGRATIONLOWERCUTOFF-FREQUENCY, - 1)(3N - n ) for2N<n<3N , l L Fig. 12. The effect on noise of clipping the amplitude of signals in the two integrators for various amplitudes of a sinusoidal input. W is the ratio of the clipping level to the step size. Quantization levels are at k0.5. I N T E G R A T I O N TIME-CONSTANT, (3N - n I I 4.0 kHz Fig. 13. Thedependence of noise on leakage in bothintegratorsfortwolevel quantization and double feedback. Basebandfo = 3.5 H z . andresamplingitattheNyquistrate.Thefilterattenuates spurioushigh-frequencycomponentsofthe signal andthe high-frequency components of the modulation noise, so that resampling does not alias significant noise into baseband. Reference 191 describes t h e design of adigitalprocessor that is suitable for use with ordinary sigma delta modulation. It reduces the sampling rate in stages, employing 32 kHz as anintermediatefrequencyforreachingthe 8 kHzNyquist rate of telephone signals. This technique leads to simple and efficient circuits. Use of double integration in the modulator does not influence the design of the filter with respect to outof-band signal components, but it does influence the requirementsforattenuatingmodulationnoise,becausethenoise now rises more rapidly with increasing frequency. The filter used in [91 for lowering the sampling rate to 3 2 kHz has a triangular-shaped impulse response, and frequency response given by sin fNr sin fr Calculations of noise, in Appendix B, show that such filtering does not provide sufficient attenuation for modulations generated by means of doubleintegration.It is shown that the modulation frequency would need to be raised from 1 to 2.5 MHz in order to make up for use of such inadequate filtering. Afilterthatdoesprovideadequateattenuationhasfre- where N is the number of input sample values that occur in one period of the resampling. The duration of the filter impulseresponse is threeresamplingperiods.Itsfrequency response has triple zeros at the 32 kHz resampling rate and all harmonics of it. When this filter is used, the noise, aliased into band, resu1t.s in less than 0.5 dB loss of resolution. This filter can be implemented as a n FIR structure that does not require full multipliers because the modulated signal is a 1 bit code. In t h e finalstageofdigitalprocessingasharpcutofffilteris needed. Analysis in Appendix C shows that at least 25 dB of attenuation is needed,which is provided bythefilterdesecscribedin [9] ; itcomprisestwosecond-orderlow-pass tions followed by an accumulation and dump. X. CONCLUSION We have demonstrated that a sigma delta modulator with double integration can be designed t o provide resolution equivalent to that of 16 bit PCM when modulating 4 kHz signals a t 1 MHz, with simple robust circuits. A somewhat higher modulation rate, 2.5 MHz, would permit use of-simpler digital processing given by (9) for converting the modulation to PCM. It is instructive to compare this modulator to the interpolating modulator described in [ S I . Theyprovidecomparable 9 bit resolution when the interpolating modulator generates words at 256 kHz and the sigma delta modulator generates 1 1 MHz. Theinterpolativemodulatorhasmore bitwordsat complex circuits, but the tolerances of the two modulators to imperfection are roughly equivalent. The overriding adva,ntage of the modulator described here is the fact that its quantization is uniform. To obtain comparable resolution from the interpolatingmodulatorwithoutcompandingitsquantization levels would require that it generate 4 bit words at 2MHz. There isneed t o have uniform quantization in order that the, encoder can accept the sum ofseveral independent signals without having interaction between them. For example, when digital hybrids and digital conferencing are to be provided, the sumofthreeormorespeech signals maybepresentinthe modulator at one time; then we require that the quantization noisebeindependentofthe signal amplitude.Thepossible use of triple integration has been rejected because it can latch into noisy modes of operation. APPENDtX A MODULATION NOISE QuantizationinthemodulatorshowninFig. 3 is represented by additive noise e. We assume it is white with spectral power density 2reO2, where eo is the noise power in the band of frequencies below t h e half sampling rate. When the gains G,, areunity, we candescribethemodulated signal bythe 256 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-33, NO. 3, MARCH 1985 Thus, the noise (13) is increased by the root of t h e decimation ratio N . This loss of resolution could be made up for by increasing the input modulation rate from / r1 t o N 1 / 4 / ~ . A better decimating filteris z-transform expression Y'(Z) =Z-lX'(z) -t(1 - z-')~E(z). 1)(1 The noise in this signal has spectral density N M ( ~ = ) 2(1 - cos ( 0 7 ) ) e o f i . (12) If NMOis the component of noise in baseband, 0 < f < f o and w0 = 27rf0, then - 8 sin ( + ~ ~ 7sin ) (2w07)]. ) be approximated When f07 is small enough that sin( 2 ~ 0 7 can by the first five terms in its Taylor expansion, This noise can be made very small by using high modulation rates. Subsequent digital processing, called decimation, lowers the rate without increasing the noise. N L(=N)3 3 1 -Z-l It modifies the modulation noise to be The spectral density of the noise following resaypling can be obtained by reverting to the time domain. Let E (z) represent the accumulated noise which is equivalent to N A ~ P E N D IBX DESIGNOF THE DECIMATOR The decimator uses 1,ow-pass,filters to attenuate out-of-band components of the modulation that, will be aliased into band by the resampling. TO simplify the design of these filters, there is much advantage in lowering the sampling rate in stages. In theinitialstageswherethesamplingrate isstilllargecompared to the Nyquist rate, it is wise to place zeros of the filters atthenewsamplingrateandharmonics of it.Suitable sequences of,evenly spaced zeros occur in trigonometric functions, and a particularly easy spectral response to implement is (sin wNT/sin cj7). In the time domain it is an averaging o f N samples with ,T being the peripd of the input samples and NT the period of the output samples. In [ 91 a cascade of two such filters having response When this is resampled with period N T , i.e., every Nth sample is retained, there will b e n o cGrrelation between the samples, if t h e original samples are uncorrelated. The resampled noise e'(iN7) will be white with spectral density N e o f i . It follows that the noise at the decimator output after resampling can be described as and its baseband component forN7fo 2- 1 is approximated by eo n2 N~~ = -(2f07)5/2. 6 (14) There is no change from (1 3). All of the penalty for decimating is represented by the in-band attenuation of the decimating filter which, when equalized, increases the noise. was used to lower sampling rates of sigma delta modulation to This decimating filter can be used successfully to lowerSam'32 kHz. This filter is inadequate for use with modulators em- pling rates to about four times the Nyquist rate, and its in-band ploying double integration, as the following analysis willdeattenuation will be less than 3 dB. Filters with much sharper monstrate.Afterfilteringby (14), themodulationnoisein cutoff characteristics are needed in the final stage of decima(1 1) becomes R mustbesufficienttomakethe tion.Theirattenuation modulation noise that is aliased into band small with respect t o N M Oin (13). That is, Resampling with period NT results in eo (16) (1 N D ( 2 ) = ___ - Z-')2 N2 where Z - = z - represents ~ aone-perioddelayatthenew rate.Followingargumentssimilarto(1 1)-( 13),thein-band noise can be approximated by which is satisfied if 257 CANDY: DOUBLE INTEGRATION IN SIGMA DELTA MODULATION When 1/Nr is 32 kHz and f o is 4 kHz, this requires R < -25 way in the sample time( n dB. This is easier than the requirement for antialiasing filters in D-channel banks, R < -32 dB. The low-pass filter described in [ 91 is adequate for use with double integrating modulators. APPENDIXC COMPARISONOF A MODULATOR THAT INTEGRATES ANALOGSIGNALS WITH ITS SAMPLED DATA - EQUIVALENT +i)~ we get , y(.- (. +)+ 7 ) + y n 7 . (33) n=O When the modylator in Fig. 3 has input samples x , ’ and output samples y n and the signal applied to the quantizer is w,- 1 ‘, a relationship between their values can be expressedas We areprimarilyinterestedinmodulatorsthataresampled at very high frequency compared to baseband so that the signal varies little during the sampling interval. To simplify the analysis, we shall now assume that the center of area of the inputwaveformforeachsample liesmidwayintheinterval t n = (n *)T. Then + or N-1 N- 1 N-1 provided all signals are initially zero. Ananalog.circuitthatcan haveequivalentresponse is shown in Fig. 10. Here x ( t ) is a continuous signal and the feed- Yn=O (N-n+(ficI!-))ytlT. back signal y ( t ) is held constant throughout each sample in(34) terval. To analyze this circuit we make use of the following relationshipsbetweenintegrationandsummation of sample Comparing this result to (27) we see that they can be equivavalues. lent, with the right initial conditions and Lemmas: F o r f ( t )= 0 when t < 0 and r 2 C 1 C 2= r 2 , R C , = 1.57, Yn’=Yn7 and This leads t o t h e final result (29) I = 2xn7 - x(i2-1)7 and which has spectral equivalent REFERENCES where t , is the position, in time, of the center of area of the signal waveform during the nth sample interval, i.e., tf(t)dt. When we assume that all signals in the circuit are zero for t < 0 , wecanexpresstheamplitudethat isappliedtothe quantizer in Fig. 3 at the Nth sample time as y ( t ) is Applyingresults(13)-(15)andnotingthatbecause heldconstant,thecenter of area of itswaveformliesmid- [I] D. J. Goodman,“Theapplication of deltamodulationtoanalog-todigital PCM encoding,” Bell Syst. Tech. J., vol.48,pp.321-343, Feb.1969. [2] L. D. J. 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