Table Of Content

Look-Ahead Based Sigma-Delta Modulation ANALOGCIRCUITSANDSIGNALPROCESSING SeriesEditors: Mohammed Ismail.TheOhioStateUniversity Mohamad Sawan.ÉcolePolytechniquedeMontréal Forothertitlespublishedinthisseries,goto www.springer.com/series/7381 Erwin Janssen (cid:2) Arthur van Roermund Look-Ahead Based Sigma-Delta Modulation ErwinJanssen ArthurvanRoermund NXPSemiconductors,CentralR&D ElectricalEngineering MixedSignalCircuits&Systems Mixed-signalMicroelectronicsGroup HighTechCampus,Building32 TechnicalUniversityEindhoven Eindhoven5656AE Eindhoven5600MB TheNetherlands TheNetherlands [email protected] [email protected] SeriesEditors: MohammedIsmail MohamadSawan 205DreeseLaboratory ElectricalEngineeringDepartment DepartmentofElectricalEngineering ÉcolePolytechniquedeMontréal TheOhioStateUniversity Montréal,QC 2015NeilAvenue Canada Columbus,OH43210 USA ISBN978-94-007-1386-4 e-ISBN978-94-007-1387-1 DOI10.1007/978-94-007-1387-1 SpringerDordrechtHeidelbergLondonNewYork LibraryofCongressControlNumber:2011926230 ©SpringerScience+BusinessMediaB.V.2011 Nopartofthisworkmaybereproduced,storedinaretrievalsystem,ortransmittedinanyformorby anymeans,electronic,mechanical,photocopying,microfilming,recordingorotherwise,withoutwritten permissionfromthePublisher,withtheexceptionofanymaterialsuppliedspecificallyforthepurpose ofbeingenteredandexecutedonacomputersystem,forexclusiveusebythepurchaserofthework. Coverdesign:VTeXUAB,Lithuania Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 BasicsofSigma-DeltaModulation . . . . . . . . . . . . . . . . . . . 5 2.1 AD,DD,andDASigma-DeltaConversion . . . . . . . . . . . . . 8 2.1.1 ADConversion . . . . . . . . . . . . . . . . . . . . . . . 8 2.1.2 DDConversion . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.3 DAConversion . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Sigma-DeltaStructures . . . . . . . . . . . . . . . . . . . . . . . 10 2.3 LinearModelingofanSDM . . . . . . . . . . . . . . . . . . . . . 12 2.4 Sigma-DeltaModulatorPerformanceIndicators . . . . . . . . . . 17 2.4.1 GenericConverterPerformance . . . . . . . . . . . . . . . 17 2.4.2 SDMSpecificFunctionalPerformance . . . . . . . . . . . 22 2.4.3 SDMSpecificImplementationCosts . . . . . . . . . . . . 26 2.4.4 Figure-of-MeritofanSDM . . . . . . . . . . . . . . . . . 27 3 TransientSDMPerformance . . . . . . . . . . . . . . . . . . . . . . 29 3.1 MeasuringSignalConversionQuality . . . . . . . . . . . . . . . . 29 3.1.1 Steady-State . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.1.2 Non-steady-State. . . . . . . . . . . . . . . . . . . . . . . 30 3.2 TimeDomainSINADMeasurement . . . . . . . . . . . . . . . . 31 3.3 Steady-StateSINADMeasurementAnalysis . . . . . . . . . . . . 33 3.3.1 ObtainingtheLinearizedSTF . . . . . . . . . . . . . . . . 34 3.3.2 TimeDomainSINADMeasurement . . . . . . . . . . . . 37 3.4 Non-steady-StateSINADMeasurementAnalysis . . . . . . . . . . 37 3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4 Noise-ShapingQuantizerModel. . . . . . . . . . . . . . . . . . . . . 43 4.1 GenericQuantizer . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.2 Noise-ShapingQuantizer . . . . . . . . . . . . . . . . . . . . . . 44 4.3 Noise-ShapingQuantizerwithMultipleCostFunctions . . . . . . 46 4.4 SpecificRealizationStructures . . . . . . . . . . . . . . . . . . . 47 v vi Contents 5 Look-AheadSigma-DeltaModulation . . . . . . . . . . . . . . . . . 49 5.1 Noise-ShapingQuantizerwithLook-Ahead. . . . . . . . . . . . . 49 5.2 Look-AheadEnabledSDMModel . . . . . . . . . . . . . . . . . 51 5.3 Look-AheadPrinciple . . . . . . . . . . . . . . . . . . . . . . . . 52 5.3.1 QuantizerCostFunction . . . . . . . . . . . . . . . . . . . 54 5.4 ObtainingInformationAbouttheFuture . . . . . . . . . . . . . . 55 5.4.1 ApproximatedFutureInput . . . . . . . . . . . . . . . . . 55 5.4.2 ActualFutureInput . . . . . . . . . . . . . . . . . . . . . 56 5.5 FullLook-AheadAlgorithm . . . . . . . . . . . . . . . . . . . . . 56 5.6 LinearModelingofaLook-AheadSDM . . . . . . . . . . . . . . 59 5.6.1 BoundaryConditionsandAssumptions . . . . . . . . . . . 59 5.6.2 Feed-ForwardLook-AheadSDM . . . . . . . . . . . . . . 60 5.6.3 Feed-BackLook-AheadSDM . . . . . . . . . . . . . . . . 62 5.7 BenefitsandDisadvantagesofLook-Ahead . . . . . . . . . . . . . 64 5.7.1 Benefits . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 5.7.2 Disadvantages . . . . . . . . . . . . . . . . . . . . . . . . 67 5.8 Look-AheadADConversion . . . . . . . . . . . . . . . . . . . . 68 5.8.1 PotentialBenefitsandDisadvantagesofLook-Aheadin ADConversion . . . . . . . . . . . . . . . . . . . . . . . 68 5.8.2 FeasibilityofaLook-AheadADC. . . . . . . . . . . . . . 69 5.8.3 HybridLook-AheadADC . . . . . . . . . . . . . . . . . . 71 5.8.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 72 5.9 Look-AheadDDConversion . . . . . . . . . . . . . . . . . . . . 72 5.10 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 6 Reducing the Computational Complexity of Look-Ahead DD Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 6.1 FullLook-Ahead . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 6.1.1 Complete Response Calculation with Reuse of IntermediateResults . . . . . . . . . . . . . . . . . . . . . 78 6.1.2 SelectandContinuewithHalfoftheSolutions . . . . . . . 78 6.1.3 LinearDecompositionoftheFilterResponse . . . . . . . . 79 6.1.4 ConditionalComputationoftheSolutions . . . . . . . . . 80 6.1.5 CalculatingMultipleOutputSymbolsperStep . . . . . . . 80 6.1.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 6.2 PrunedLook-Ahead . . . . . . . . . . . . . . . . . . . . . . . . . 82 6.2.1 MotivationforPruning . . . . . . . . . . . . . . . . . . . 83 6.2.2 BasicPrunedLook-AheadModulation . . . . . . . . . . . 84 6.2.3 PrunedLook-AheadModulationwithReuseofResults . . 86 6.2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 6.3 PrunedLook-AheadModulatorRealizations . . . . . . . . . . . . 95 6.3.1 TrellisSigma-DeltaModulation . . . . . . . . . . . . . . . 96 6.3.2 EfficientTrellisSigma-DeltaModulation . . . . . . . . . . 97 6.3.3 PrunedTreeSigma-DeltaModulation . . . . . . . . . . . . 98 6.3.4 PrunedTreeSigma-DeltaModulationforSA-CD . . . . . 100 6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Contents vii 7 TrellisSigma-DeltaModulation . . . . . . . . . . . . . . . . . . . . . 103 7.1 Algorithm—KatoModel . . . . . . . . . . . . . . . . . . . . . . 104 7.1.1 HiddenMarkovModel. . . . . . . . . . . . . . . . . . . . 104 7.1.2 AlgorithmSteps . . . . . . . . . . . . . . . . . . . . . . . 106 7.2 Algorithm—PrunedLook-AheadModel . . . . . . . . . . . . . . 109 7.3 VerificationoftheLinearizedNTFandSTF . . . . . . . . . . . . 110 7.3.1 NTF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 7.3.2 STF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 7.4 RelationTrellisOrderandTrellisDepth . . . . . . . . . . . . . . 113 7.4.1 SimulationSetup. . . . . . . . . . . . . . . . . . . . . . . 114 7.4.2 TrellisDepthasaFunctionoftheTrellisOrderandthe SignalAmplitude . . . . . . . . . . . . . . . . . . . . . . 114 7.4.3 TrellisDepthasaFunctionoftheSignalFrequency . . . . 116 7.4.4 TrellisDepthasaFunctionoftheLoop-FilterConfiguration 117 7.4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 7.5 FunctionalPerformance . . . . . . . . . . . . . . . . . . . . . . . 119 7.5.1 SNR,SINAD,THDandSFDR . . . . . . . . . . . . . . . 119 7.5.2 ConverterStability . . . . . . . . . . . . . . . . . . . . . . 124 7.5.3 NoiseModulation . . . . . . . . . . . . . . . . . . . . . . 128 7.5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 7.6 ImplementationAspects . . . . . . . . . . . . . . . . . . . . . . . 131 7.6.1 RequiredComputationalResources . . . . . . . . . . . . . 131 7.6.2 Look-AheadFilterUnit . . . . . . . . . . . . . . . . . . . 132 7.6.3 OutputSymbolSelection . . . . . . . . . . . . . . . . . . 134 7.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 8 EfficientTrellisSigma-DeltaModulation . . . . . . . . . . . . . . . . 137 8.1 ReducingtheNumberofParallelPaths . . . . . . . . . . . . . . . 137 8.2 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 8.3 RelationBetweenN andM . . . . . . . . . . . . . . . . . . . . . 141 8.4 RequiredHistoryLength. . . . . . . . . . . . . . . . . . . . . . . 143 8.5 FunctionalPerformance . . . . . . . . . . . . . . . . . . . . . . . 145 8.5.1 SNR,SINAD,THDandSFDR . . . . . . . . . . . . . . . 145 8.5.2 ConverterStability . . . . . . . . . . . . . . . . . . . . . . 149 8.5.3 NoiseModulation . . . . . . . . . . . . . . . . . . . . . . 150 8.5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 8.6 ImplementationAspects . . . . . . . . . . . . . . . . . . . . . . . 153 8.6.1 SelectionStep . . . . . . . . . . . . . . . . . . . . . . . . 154 8.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 9 PrunedTreeSigma-DeltaModulation . . . . . . . . . . . . . . . . . 159 9.1 RemovingtheTestforUniqueness . . . . . . . . . . . . . . . . . 159 9.2 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 9.2.1 InitializationPhase. . . . . . . . . . . . . . . . . . . . . . 162 9.2.2 OperationPhase . . . . . . . . . . . . . . . . . . . . . . . 162 9.3 RequiredHistoryLength. . . . . . . . . . . . . . . . . . . . . . . 163 viii Contents 9.4 FunctionalPerformance . . . . . . . . . . . . . . . . . . . . . . . 165 9.4.1 SNR,SINAD,THDandSFDR . . . . . . . . . . . . . . . 165 9.4.2 ConverterStability . . . . . . . . . . . . . . . . . . . . . . 168 9.4.3 NoiseModulation . . . . . . . . . . . . . . . . . . . . . . 170 9.4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 9.5 ImplementationAspects . . . . . . . . . . . . . . . . . . . . . . . 174 9.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 10 PrunedTreeSigma-DeltaModulationforSA-CD . . . . . . . . . . . 179 10.1 RequirementsofanSA-CDModulator . . . . . . . . . . . . . . . 179 10.2 SA-CDLosslessDataCompression . . . . . . . . . . . . . . . . . 181 10.3 DualOptimization . . . . . . . . . . . . . . . . . . . . . . . . . . 184 10.3.1 PredictorCostFunction . . . . . . . . . . . . . . . . . . . 185 10.3.2 CombiningtheCostFunctions . . . . . . . . . . . . . . . 187 10.3.3 SpectralShaping . . . . . . . . . . . . . . . . . . . . . . . 188 10.4 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 10.5 FunctionalPerformance . . . . . . . . . . . . . . . . . . . . . . . 193 10.5.1 LosslessDataCompression . . . . . . . . . . . . . . . . . 193 10.5.2 SNR,SINAD,THDandSFDR . . . . . . . . . . . . . . . 194 10.5.3 ConverterStability . . . . . . . . . . . . . . . . . . . . . . 196 10.5.4 NoiseModulation . . . . . . . . . . . . . . . . . . . . . . 197 10.5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 10.6 ImplementationAspects . . . . . . . . . . . . . . . . . . . . . . . 201 10.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 11 ComparisonofLook-AheadSDMTechniques . . . . . . . . . . . . . 205 11.1 AlternativeLook-AheadTechniques . . . . . . . . . . . . . . . . 205 11.2 AlgorithmComparison . . . . . . . . . . . . . . . . . . . . . . . 206 11.3 FunctionalPerformanceComparison . . . . . . . . . . . . . . . . 209 11.3.1 SNR,SINAD,THDandSFDR . . . . . . . . . . . . . . . 209 11.3.2 ConverterStability . . . . . . . . . . . . . . . . . . . . . . 213 11.3.3 NoiseModulation . . . . . . . . . . . . . . . . . . . . . . 216 11.3.4 LosslessDataCompression . . . . . . . . . . . . . . . . . 218 11.3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 11.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 12 MaximumSNRAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . 225 12.1 Experiment1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 12.2 Experiment2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 12.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 12.3.1 SecondOrderFilterStability . . . . . . . . . . . . . . . . 229 12.3.2 HighOrderFilterStability . . . . . . . . . . . . . . . . . . 231 12.4 ObtainingtheMaximumSNR . . . . . . . . . . . . . . . . . . . . 233 12.5 TheoreticalMaximumSNR . . . . . . . . . . . . . . . . . . . . . 235 12.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 13 GeneralConclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 Contents ix AppendixA FFTCalculations—CoherentandPowerAveraging. . . . 241 AppendixB DescriptionoftheUsedSigma-DeltaModulators . . . . . . 243 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

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High resolution analog-to-digital conversion and digital-to-analog conversion is often realized with the use of a Sigma-Delta Modulator (SDM). The sigma-delta modulation process, although very non-linear when a 1-bit quantizer is used, is nowadays well understood and can be made to deliver a very hi
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High resolution analog-to-digital conversion and digital-to-analog conversion is often realized with the use of a Sigma-Delta Modulator (SDM). The sigma-delta modulation process, although very non-linear when a 1-bit quantizer is used, is nowadays well understood and can be made to deliver a very hi

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