IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 48, NO. 5, OCTOBER 1999 873

Narrow-Band Variable Center FrequencySingle-Loop and Multistage Sigma–Delta

Modulators for Bandpass SignalsMohammed Al-Janabi,Student Member, IEEE, Izzet Kale,Member, IEEE, and Richard C. S. Morling,Member, IEEE

Abstract—Oversampled narrow-band single-loop and multi-stage resonator-based bandpass Sigma–Delta(�–�) modulatorsthat can accommodate different passband center to samplingfrequency ratios are reported in this paper. These tunable band-pass configurations are designed by analytically determining andsubsequently verifying through detailed empirical simulationsthe required compensation hardware to deliver enhanced noise-shaping. It is demonstrated that comparatively superior in-bandsignal-to-noise ratios (SNR’s) and dynamic ranges (DR’s) areattributed to the inclusion of appropriate digital feedforward andfeedback compensators within these structures.

Index Terms— Analog–digital conversion, bandpass sigma–delta modulation, nonlinear feedback systems, quantizationnoise, resonators.

I. INTRODUCTION

BANDPASS Sigma–Delta – modulators combineoversampling and noise-shaping to perform highly ac-

curate analog-to-digital (A/D) conversion for high frequency,narrow-band signals, that are centered at an arbitrary frequencyaway from base-band. The bulk of the quantization noise inthese bandpass– modulators is shifted to either side awayfrom the region of interest leaving the original signal virtuallyintact [1]. Bandpass – modulators are suitable for a broadrange of applications such as AM digital radios, receivers fordigital cellular mobile radios, direct A/D conversion for IFsignals [2] and in phased-array ultrasound imaging [3]. Most ofthe work conducted thus far on resonator-based bandpass–modulators has involved using a convenient center frequency

that is one quarter of the sampling frequency. This paperreports different single-loop and multistage structures thatcan cater for a variety of center frequencies overcomingthe popular restriction. The following sections containan explanation of the design criteria, a description of theinternal structure and function for each modulator as well asa comparison of all configurations based on Matlab systemlevel simulations (in Simulink) of in-band SNR’s and DR’s.

II. DESIGN CONSIDERATIONS

In bandpass – modulators, the noise-shaped band centeris positioned at a fixed frequency which is a fraction of the

Manuscript received June 23, 1999.The authors are with the Department of Electronic Systems,

University of Westminster, London W1M 8JS, U.K. (e-mail:aljanabi@cmsa.westminster.as.uk).

Publisher Item Identifier S 0018-9456(99)08382-5.

Fig. 1. Second-order variable center frequency bandpass�–� modulator.

sampling frequency. The choice of a particular band centeris based upon a compromise of fundamental specificationparameters such as the sampling frequency, oversampling ratio(OSR) and antialiasing filter requirements [4]. The placementof the bandwidth close to dc increases the OSR leadingto better in-band SNR performance. In addition, the designrequirements of the antialiasing filter may be substantiallyrelaxed since the image frequencies that may fold back intothe signal region will be positioned further away from it[4]. However, extremely small center frequencies can lead toexcessively high and impractical clock rates for the modulatorloop-filter and antialiasing filter [4]. On the other hand, shiftingthe band center closer to Nyquist results in moderate clockrates, but simultaneously imposes more rigid constraints on thenoise-shaping loop and antialiasing filters [4]. The placementof the resonator poles exactly on the unit circle is highlydesirable, as this will result in a very deep noise notch. If theloop-filter poles are positioned at the wrong frequency, then thenoise notch will be improperly centered [5]. More seriously, ifthe poles move-off the unit-circle toward the origin, then thenotch will be shallower resulting in a poorer noise-shapingresponse [5]. Also, a resonator gain less than unity will resultin a smaller in-band SNR and hence inferior resolution. Thedesign of the noise transfer function must satisfy twomore design requirements. The first is the causality criterion,which dictates that , implying that the

modulator feedback loop can not be delayless [1]. The secondis the stability criterion by Lee, which empirically stipulatesthat the peak magnitude gain of should not exceedtwo for the modulator to remain stable, i.e.,

, [6].

0018–9456/99$10.00 1999 IEEE

874 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 48, NO. 5, OCTOBER 1999

III. FOUR BANDPASS – MODULATORS

This section presents the internal structure for four variablecenter frequency resonator-based bandpass– modulators.These are the second-order single-loop, fourth-order double-loop, double-stage (1-1) MASH, and double-stage (2-1)MASH – modulators where the sequence of numbersused for a given MASH modulator topology corresponds tothe number of loops in each stage. For example, 2-1 refersto a double-stage modulator whose first and second stagesconstitute double and single loops respectively. A conventionalapproach of analyzing – modulators involves modeling the1-bit nonlinear quantizer by an equivalent additive white noisesource, which is statistically independent of the input signal,to enable the application of linear theory. This approachdelivers results which are representative for dithered single-loop structures and reasonably accurate in the case of ditheredMASH structures [7].

A. Second-Order Single-Loop

A second-order bandpass– modulator consists of adiscrete-time variable center frequency resonator and a1-bit quantizer in the feedforward path as well as a suitablecompensator in the feedback path as shown in Fig. 1.In addition, random dither , of amplitude 0.1, is addedprior to the quantizer input to whiten the quantization noiseand reduce spurious tones.

Bandpass – modulators are designed by specifying therequired to provide maximum in-band attenuation. Thezeros of are located at where is chosen to centerthe signal band at any frequency given by [8].It is algebraically shown that

(1)

(2)

where is the resonator gain. This analysis reveals that a dou-ble delayer is insufficient to cause complete cancellation of thedenominator of and hence deliver acceptable results.Improved in-band SNR’s can be achieved by incorporatingthe summation of a double delayer and a single delayer whoseassociated coefficient depends on. Thus can be seento be a compensating structure in the feedback path to providebetter in-band SNR’s and DR’s. A closer inspection revealsthat degenerates to the double delayer of the conventionalcase when . It must be emphasized that any othercenter to sampling frequency ratio will inevitably require theinclusion of the extra term in the compensatingstructure. Therefore, the modulator center frequency tunabilityis obtained by changing the zero locations of . Thiscorresponds to the movement of the poles of along theunit circle to the selected frequency locations.

B. Fourth-Order Single-Loop

The same principles are extended for the analysis of fourth-order resonator-based structures where performance improve-ments in terms of in-band SNR’s, DR’s and tones reductions

Fig. 2. Fourth-order variable center frequency bandpass�–� modulator.

are accomplished. This structure consists of two discrete-timevariable center frequency resonators and a 1-bit quantizer inthe feedforward path and a compensator in the feedback pathas shown below in Fig. 2.

Analysis shows that and are identicalto the expressions already derived for the second-order case.Rigorous system level simulations, however, demonstrate thatthe 1-bit quantizer is prevented from overloading by makingthe gain of the first resonator equal to or less than 1/8in an attempt to ensure stability [8], [9]. This gain reductionfixes the stability problem, but unfortunately increases thequantization noise level resulting in degraded in-band SNRvalues. One suitable solution to this problem is to compensatethe output of the modulator using a second-order digital finiteimpulse response (FIR) filter in the feedforward path as shownin Fig. 2. This digitally adjustable feedforward compensator

is given by

(3)

where . Note that the final output of the modulatorin the time-domain is no longer a combination of . Thestream of values enters the feedforward compensator andis scaled by its coefficients to achieve one of the followingeight values,and where .

C. Double Stage 1-1 MASH

Higher-order single-loop structures beyond fourth-order– modulators are highly prone to instability. A well

known approach with guaranteed stability is the MASH– modulator [10] where the newly introduced variable

center frequency resonators coupled with the necessary digitalcompensation hardware are embedded in each section. Thus,these bandpass– modulators are a cascade of independentsecond-order or fourth-order – modulators where thequantization noise in each stage is modified by feeding theerror of each section into the input of the following stage [11].

The 1-1 MASH is a cascade of two second-order resonator-based bandpass– modulators as shown above in Fig. 3.The notch filter is given by

(4)

The correctness of this 1-1 MASH modulator can be verifiedby checking its output in the time-domain. The output signal

AL-JANABI et al.: MODULATORS FOR BANDPASS SIGNALS 875

Fig. 3. Double-stage 1-1 MASH bandpass�–� modulator.

of the first stage has values of1. The output signal of thesecond stage which has values of1 after the notch filterbecomes a signal which consists of samples having one of sixamplitudes and . All combinationsfrom both stages are added to produce a cumulative multibitoutput signal having one of the following values,

and .

D. Double Stage 2-1 MASH

This is a cascade of a fourth-order followed by a second-order resonator-based bandpass– modulators [11] asshown in Fig. 4.

The validity of the modulator can be verified by observingits output in the time–domain. The output signal of thefirst stage which has values of 1 after the feedforwardcompensator becomes a signal which consists of

and .The output signal of the second stage which has values of1after the double notch filter becomes a signal which consistsof samples having one of the amplitudes

and . Outputcombinations from both stages are added to produce the finaloutput signal having one of the following values:

(5)

where is an even integer having one of three values 0,4 and is an odd integer having one of six values,1,3, and 5. Note that the presence of an extra notch filter

as well as this novel feedforward compensator haveresulted in more cumulative multibit output combinations. Themain advantage of the 2-1 structure is that the quantizationnoise of the first stage output produces fourth-order noise-shaping which is much smaller in magnitude and less tonal.Consequently, the effects of component mismatches betweenfirst and second stages are less significant [8] resulting in betterperformance.

IV. SIMULATION RESULTS

System level Matlab simulations (in Simulink) were con-ducted by injecting single tone sinusoids at the inputs of the

Fig. 4. Double-stage 2-1 MASH bandpass�–� modulator.

Fig. 5. SNR plots of four�–� modulators for OSR= 128.

Fig. 6. SNR comparison of different OSR’s for fourth-order�–� modu-lator.

modulators for a block length of 262 144 samples followedby a Hanning Windowed FFT on the modulator output for allfour modulators for different center frequencies. The first setof simulation results in Fig. 5. illustrate the in-band SNR’s forall four structures for an OSR of 128.

The next set of simulations (Fig. 6) shows the in-bandSNR’s for the fourth-order modulator for five OSR’s of 256,

876 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 48, NO. 5, OCTOBER 1999

TABLE IPEAK SNR RESULTS OF ALL FOUR STRUCTURES

TABLE IIDR RESULTS OF ALL FOUR STRUCTURES

128, 64, 32, and 16. Note that the in-band SNR is improvedby an average of 15.6 dB for every doubling of the OSR.

Tables I and II contain a summary of the main performanceparameters for a selection of resonator center frequencies.

In addition, these simulations revealed that the distribu-tion of quantization noise became asymmetrical when

. Detailed empirical evaluation demonstrated that noise-shaping symmetry may be significantly improved by selectingappropriate feedback coefficients. For example, making thefeedback factor inversely proportional to in the caseof the fourth-order modulator delivers enhanced quantizationnoise-shaping distribution. Note that the addition of varioussingle-bit intermediate outputs yields a multibit final output,which results in a more complicated decimation filter.

V. CONCLUSION

The variable center frequency resonator-based bandpass– modulators considered in this paper could simultaneously

serve the roles of direct A/D conversion and tuning fornarrow-band input signals. Early conversion to digital atthe IF stage leads to more robust communication systemsas well as facilitating IF filter programmability and easiersystem testing. Third-band and two-third-band resonators arecheaper to implement in the loop-filter of these modulatorsbecause they are multiplier-free structures since .The tabulated results show that an average of 9.3 bits, 14.7bits, 15.6 bits, and 19.7 bits of resolution are achieved withthe second-order, 1-1 MASH, fourth-order and 2-1 MASHstructures respectively. Detailed simulations demonstrated thatthe inclusion of feedforward and feedback digitalcompensators within these modulators result in enhanced in-band SNR’s and DR’s.

REFERENCES

[1] S. A. Jantzi, M. Snelgrove, and P. F. Ferguson, “A fourth-order bandpasssigma–delta modulator,”IEEE J. Solid-State Circuits, vol. 28, pp.282–291, Mar. 1993.

[2] F. Gourgue, M. Bellanger, S. Azrouf, and V. Bruneau, “A bandpasssubsampled delta–sigma modulator for narrowband cellular mobilecommunications,” inProc. IEEE ISCAS, 1994, vol. 5, pp. 353–356.

[3] O. Norman, “A band-pass delta–sigma modulator for ultrasound imagingat 160 MHz clock rate,”IEEE J. Solid-State Circuits, vol. 31, pp.2036–2041, Dec. 1996.

[4] S. R. Norsworthy, R. Schreier, and G. C. Temes,Delta Sigma DataConverters: Theory, Design and Simulation. New York: IEEE Press,1996, ch. 9, pp. 282–286.

[5] F. W. Singor and W. M. Snelgrove, “Switched-capacitor bandpassdelta–sigma A/D modulation at 10.7 MHz,”IEEE J. Solid-State Circuits,vol. 30, pp. 184–192, Mar. 1995.

[6] W. L. Lee, “A novel higher-order interpolative modulator topology forhigh resolution oversampling A/D converters,” M.S. thesis, Mass. Inst.Technol., Cambridge, June 1987.

[7] W. Chou, P. W. Wong, and R. M. Gray, “Multi-stage sigma–deltamodulation,”IEEE Trans. Inform. Theory, vol. 35, pp. 784–796, 1989.

[8] M. Al-Janabi, I. Kale, and R. C. S. Morling, “Variable centre frequencyresonator based bandpass�–� modulator,”Electron. Lett., vol. 33, pp.2008–2009, Nov. 1997.

[9] B. E. Boser and B. A. Wooley, “The design of sigma–delta modulationanalog-to-digital converters,”IEEE J. Solid State Circuits, vol. 28, pp.1298–1308, Dec. 1988.

[10] D. B. Ribner, “A Comparison of modulator networks for high-orderoversampled sigma–delta analog-to-digital converters,”IEEE Trans.Circuits Syst., vol. 38, pp. 145–159, Feb. 1991.

[11] D. B. Ribner, “Multistage bandpass delta sigma modulators,”IEEETrans. Circuits Syst. II, vol. 41, pp. 402–405, June 1994.

Mohammed Al-Janabi (S’99) received theB.Eng.(Hons.) degree in electronic engineeringfrom the University of Westminster, London, U.K.,in 1993. He is currently pursuing the Ph.D. degreein bandpass sigma–delta modulation at the sameuniversity.

He joined the academic staff of the Universityof Westminster in September 1993, where he iscurrently a Lecturer teaching analog and digitalelectronics. His research interests include digitalsignal processing, high-resolution analog-to-digital

converters, and nonlinear feedback control systems. His current researchactivities involve the theory, the system level design and performanceevaluation of variable-center frequency resonator-based bandpass sigma–deltamodulators.

Mr. Al-Janabi is the vice-chairman of the London Younger MembersCommittee of the IEE.

Izzet Kale (M’88) was born in Akincilar, Cyprus.He received the B.Sc.(Hon.) in electrical and elec-tronic engineering from the Polytechnic of CentralLondon, London, U.K., the M.Sc. in the designand manufacture of microelectronic systems, fromEdinburgh University, Edinburgh, U.K., and thePh.D. degree in digital signal processing, from theUniversity of Westminster, London.

He joined the academic staff of the Universityof Westminster in 1984, where he is currently aReader in advanced DSP systems. He has under-

taken industrial consultancy, training and contract work for a number ofEuropean, U.S. and British microelectronics and communications companiesin the areas DSP and VLSI architectures. His research, teaching and industrialconsultancy activities include digital and analog signal processing circuits andsystems and their mixed signal silicon implementation, as well as reducedcomplexity digital filter design and implementation. He is currently workingon efficient DSP algorithm development and custom silicon implementationof high-fidelity and bandwidth sigma–delta based A/D and D/A converters,adaptive and fixed signal processing algorithms and software radio techniquesfor telecommunication applications.

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Richard C. S. Morling (M’79) was born inRochford, U.K. He received the B.Sc.(Hons.) degreein physics from the Polytechnic of Central London,London, U.K., in 1971, and the Ph.D. degree ininformation engineering from the City University,London, in 1989.

He worked in the field of television for DeccaRecords, Ltd., and, during a period at the ImperialCollege of Science and Technology, London,was engaged in the design of electro-myographicequipment for the Migraine Trust. In 1971, he

joined the staff of the University of Westminster, where he is currentlyDirector of the Division of Electronic Systems. His research activities haveincluded packet-switched local-area networks, discrete-time signal processingstructures, aircraft braking systems, design methodologies for integratedcircuit design, sigma–delta conversion techniques, and teaching methods inelectronic engineering. He is currently working on the theory and practicalrealization of high-fidelity sigma–delta A/D and D/A converters.