Spectral Logic and Its Applications for the Design of Digital Devices PDF
Preview Spectral Logic and Its Applications for the Design of Digital Devices
SPECTRAL LOGIC AND ITS APPLICATIONS FOR THE DESIGN OF DIGITAL DEVICES www.it-ebooks.info SPECTRAL LOGIC AND ITS APPLICATIONS FOR THE DESIGN OF DIGITAL DEVICES MARK G. KARPOVSKY RADOMIR S. STANKOVIC´ JAAKKO T. ASTOLA AJOHNWILEY&SONS,INC.,PUBLICATION www.it-ebooks.info Copyright©2008byJohnWiley&Sons,Inc.Allrightsreserved. PublishedbyJohnWiley&Sons,Inc.,Hoboken,NewJersey PublishedsimultaneouslyinCanada Nopartofthispublicationmaybereproduced,storedinaretrievalsystem,ortransmittedinanyformor byanymeans—electronic,mechanical,photocopying,recording,scanning,orotherwise—exceptas permittedunderSection107or108ofthe1976UnitedStatesCopyrightAct,withouteithertheprior writtenpermissionofthePublisherorauthorizationthroughpaymentoftheappropriateper-copyfeeto theCopyrightClearanceCenter,Inc.,222RosewoodDrive,Danvers,MA01923,(978)750-8400,fax (978)750-4470,oronthewebatwww.copyright.com.RequeststothePublisherforpermissionshould beaddressedtothePermissionsDepartment,JohnWiley&Sons,Inc.,111RiverStreet,Hoboken,NJ 07030,(201)748-6011,fax(201)748-6008,oronlineathttp://www.wiley.com/go/permission. LimitofLiability/DisclaimerofWarranty:Whilethepublisherandauthorhaveusedtheirbesteffortsin preparingthisbook,theymakenorepresentationsorwarrantieswithrespecttotheaccuracyor completenessofthecontentsofthisbookandspecificallydisclaimanyimpliedwarrantiesof merchantabilityorfitnessforaparticularpurpose.Nowarrantymaybecreatedorextendedbysales representativesorwrittensalesmaterials.Theadviceandstrategiescontainedhereinmaynotbesuitable foryoursituation.Youshouldconsultwithaprofessionalwhereappropriate.Neitherthepublishernor theauthorshallbeliableforanylossofprofitoranyothercommercialdamages,includingbutnot limitedtospecial,incidental,consequential,orotherdamages. Forgeneralinformationonourotherproductsandservicesorfortechnicalsupport,pleasecontactour CustomerCareDepartmentwithintheUnitedStatesat(800)762-2974,outsidetheUnitedStatesat(317) 572-3993,orfax(317)572-4002. Wileyalsopublishesitsbooksinavarietyofelectronicformats.Somecontentthatappearsinprintmay notbeavailableinelectronicformats.FormoreinformationaboutWileyproducts,visitourWebsiteat www.wiley.com. LibraryofCongressCataloging-in-PublicationData Karpovsky,MarkG. Spectrallogicanditsapplicationsforthedesignofdigitaldevices/MarkG.Karpovsky, RadomirS.Stankovic,JaakkoT.Astola. p.cm. Includesbibliographicalreferences. ISBN978-0-471-73188-7(cloth) 1.Logicdesign–Methodology. 2.Spectrumanalysis. 3.Digitalelectronics–Mathematics. 4.Signalprocessing–Mathematics. 5.Spectraltheory(Mathematics) I.Stankovic,RadomirS.,1952– II.Astola,Jaakko. III.Title. TK7868.L6K3752008 621.39(cid:1)5–dc22 2008002714 PrintedintheUnitedStatesofAmerica 10987654321 www.it-ebooks.info CONTENTS PREFACE xv ACKNOWLEDGMENTS xxv LISTOFFIGURES xxvii LISTOFTABLES xxxiii ACRONYMS xxxix 1. LOGICFUNCTIONS 1 1.1 DiscreteFunctions 2 1.2 TabularRepresentationsofDiscreteFunctions 3 1.3 FunctionalExpressions 6 1.4 DecisionDiagramsforDiscreteFunctions 10 1.4.1 DecisionTrees 11 1.4.2 DecisionDiagrams 13 1.4.3 DecisionDiagramsforMultiple-ValuedFunctions 16 1.5 SpectralRepresentationsofLogicFunctions 16 1.6 Fixed-polarityReed–MullerExpressionsofLogic Functions 23 1.7 KroneckerExpressionsofLogicFunctions 25 1.8 CircuitImplementationofLogicFunctions 27 v www.it-ebooks.info vi CONTENTS 2. SPECTRALTRANSFORMSFORLOGICFUNCTIONS 31 2.1 AlgebraicStructuresforSpectralTransforms 32 2.2 FourierSeries 34 2.3 BasesforSystemsofBooleanFunctions 35 2.3.1 BasisFunctions 35 2.3.2 WalshFunctions 36 2.3.2.1 OrderingofWalshFunctions 40 2.3.2.2 PropertiesofWalshFunctions 43 2.3.2.3 HardwareImplementationsofWalshFunctions 47 2.3.3 HaarFunctions 50 2.3.3.1 OrderingofHaarFunctions 51 2.3.3.2 PropertiesofHaarFunctions 55 2.3.3.3 HardwareImplementationofHaarFunctions 56 2.3.3.4 HardwareImplementationoftheInverse HaarTransform 58 2.4 WalshRelatedTransforms 60 2.4.1 ArithmeticTransform 61 2.4.2 ArithmeticExpressionsfromWalshExpansions 62 2.5 BasesforSystemsofMultiple-ValuedFunctions 65 2.5.1 Vilenkin–ChrestensonFunctionsandTheirProperties 66 2.5.2 GeneralizedHaarFunctions 70 2.6 PropertiesofDiscreteWalshandVilenkin–ChrestensonTransforms 71 2.7 AutocorrelationandCross-CorrelationFunctions 79 2.7.1 DefinitionsofAutocorrelationandCross-Correlation Functions 79 2.7.2 RelationshipstotheWalshandVilenkin–Chrestenson Transforms,theWiener-KhinchinTheorem 80 2.7.3 PropertiesofCorrelationFunctions 82 2.7.4 GeneralizedAutocorrelationFunctions 84 2.8 HarmonicAnalysisoveranArbitraryFiniteAbelianGroup 85 2.8.1 DefinitionandPropertiesoftheFourierTransformon FiniteAbelianGroups 85 2.8.2 ConstructionofGroupCharacters 89 2.8.3 Fourier–GaloisTransforms 94 2.9 FourierTransformonFiniteNon–AbelianGroups 97 2.9.1 RepresentationofFiniteGroups 98 2.9.2 FourierTransformonFiniteNon-AbelianGroups 101 www.it-ebooks.info CONTENTS vii 3. CALCULATIONOFSPECTRALTRANSFORMS 106 3.1 CalculationofWalshSpectra 106 3.1.1 MatrixInterpretationoftheFastWalshTransform 109 3.1.2 DecisionDiagramMethodsforCalculationofSpectral Transforms 114 3.1.3 CalculationoftheWalshSpectrumThroughBDD 115 3.2 CalculationoftheHaarSpectrum 118 3.2.1 FFT-LikeAlgorithmsfortheHaarTransform 118 3.2.2 MatrixInterpretationoftheFastHaarTransform 121 3.2.3 CalculationoftheHaarSpectrumThroughBDD 126 3.3 CalculationoftheVilenkin–ChrestensonSpectrum 135 3.3.1 MatrixInterpretationoftheFast Vilenkin–ChrestensonTransform 136 3.3.2 CalculationoftheVilenkin–ChrestensonTransform ThroughDecisionDiagrams 140 3.4 CalculationoftheGeneralizedHaarSpectrum 141 3.5 CalculationofAutocorrelationFunctions 142 3.5.1 MatrixNotationfortheWiener–KhinchinTheorem 143 3.5.2 Wiener–KhinchinTheoremOverDecisionDiagrams 143 3.5.3 In-placeCalculationofAutocorrelationCoefficients byDecisionDiagrams 148 4. SPECTRALMETHODSINOPTIMIZATIONOFDECISION DIAGRAMS 154 4.1 ReductionofSizesofDecisionDiagrams 155 4.1.1 K-ProcedureforReductionofSizesofDecisionDiagrams 156 4.1.2 PropertiesoftheK-Procedure 164 4.2 ConstructionofLinearlyTransformedBinaryDecisionDiagrams 169 4.2.1 ProcedureforConstructionofLinearlyTransformed BinaryDecisionDiagrams 171 4.2.2 ModifiedK-Procedure 172 4.2.3 ComputingAutocorrelationbySymbolicManipulations 172 4.2.4 ExperimentalResultsontheComplexityofLinearly TransformedBinaryDecisionDiagrams 173 4.3 ConstructionofLinearlyTransformedPlanarBDD 177 4.3.1 PlanarDecisionDiagrams 178 4.3.2 ConstructionofPlanarLT-BDDbyWalshCoefficients 181 4.3.3 UpperBoundsontheNumberofNodesinPlanarBDDs 185 www.it-ebooks.info viii CONTENTS 4.3.4 ExperimentalResultsforComplexityofPlanarLT-BDDs 187 4.4 SpectralInterpretationofDecisionDiagrams 188 4.4.1 HaarSpectralTransformDecisionDiagrams 192 4.4.2 HaarTransformRelatedDecisionDiagrams 197 5. ANALYSISANDOPTIMIZATIONOFLOGICFUNCTIONS 200 5.1 SpectralAnalysisofBooleanFunctions 200 5.1.1 LinearFunctions 201 5.1.2 Self-DualandAnti-Self-DualFunctions 203 5.1.3 PartiallySelf-DualandPartiallyAnti-Self-Dual Functions 204 5.1.4 QuadraticForms,FunctionswithFlatAutocorrelation 207 5.2 AnalysisandSynthesisofThresholdElementNetworks 212 5.2.1 ThresholdElements 212 5.2.2 IdentificationofSingleThresholdFunctions 214 5.3 ComplexityofLogicFunctions 222 5.3.1 DefinitionofComplexityofSystemsofSwitching Functions 222 5.3.2 ComplexityandtheNumberofPairsofNeighboring Minterms 225 5.3.3 ComplexityCriteriaforMultiple-ValuedFunctions 227 5.4 SerialDecompositionofSystemsofSwitchingFunctions 227 5.4.1 SpectralMethodsandComplexity 227 5.4.2 LinearizationRelativetotheNumberofEssential Variables 228 5.4.3 LinearizationRelativetotheEntropy-Based ComplexityCriteria 231 5.4.4 LinearizationRelativetotheNumbersofNeighboring PairsofMinterms 233 5.4.5 ClassificationofSwitchingFunctionsbyLinearization 237 5.4.6 LinearizationofMultiple-ValuedFunctionsRelativeto theNumberofEssentialVariables 239 5.4.7 LinearizationforMultiple-ValuedFunctionsRelativeto theEntropy-BasedComplexityCriteria 242 5.5 ParallelDecompositionofSystemsofSwitching Functions 244 5.5.1 PolynomialApproximationofCompletelySpecified Functions 244 5.5.2 AdditiveApproximationProcedure 249 www.it-ebooks.info CONTENTS ix 5.5.3 ComplexityAnalysisofPolynomialApproximations 250 5.5.4 ApproximationMethodsforMultiple-ValuedFunctions 251 5.5.5 EstimationoftheNumberofNonzeroCoefficients 255 6. SPECTRALMETHODSINSYNTHESISOFLOGIC NETWORKS 261 6.1 SpectralMethodsofSynthesisofCombinatorialDevices 262 6.1.1 SpectralRepresentationsofSystemsofLogicFunctions 262 6.1.2 SpectralMethodsfortheDesignofCombinatorialDevices 264 6.1.3 AsymptoticallyOptimalImplementationofSystems ofLinearFunctions 266 6.1.4 WalshandVilenkin–ChrestensonBasesfortheDesign ofCombinatorialNetworks 270 6.1.5 LinearTransformsofVariablesinHaarExpressions 272 6.1.6 SynthesiswithHaarFunctions 274 6.1.6.1 MinimizationoftheNumberofNonzeroHaar Coefficients 274 6.1.6.2 DeterminationofOptimalLinearTransformof Variables 275 6.1.6.3 EfficiencyoftheLinearizationMethod 283 6.2 SpectralMethodsforSynthesisofIncompletelySpecified Functions 286 6.2.1 SynthesisofIncompletelySpecifiedSwitchingFunctions 286 6.2.2 SynthesisofIncompletelySpecifiedFunctionsbyHaar Expressions 286 6.3 SpectralMethodsofSynthesisofMultiple-ValuedFunctions 292 6.3.1 Multiple-ValuedFunctions 292 6.3.2 NetworkImplementationsofMultiple-ValuedFunctions 292 6.3.3 CompletionofMultiple-ValuedFunctions 293 6.3.4 ComplexityofLinearMultiple-ValuedNetworks 293 6.3.5 MinimizationofNumbersofNonzeroCoefficientsinthe GeneralizedHaarSpectrumforMultiple-Valued Functions 295 6.4 SpectralSynthesisofDigitalFunctionsandSequencesGenerators 298 6.4.1 FunctionGenerators 298 6.4.2 DesignCriteriaforDigitalFunctionGenerators 299 6.4.3 HardwareComplexityofDigitalFunctionGenerators 300 6.4.4 BoundsfortheNumberofCoefficientsinWalsh ExpansionsofAnalyticalFunctions 302 www.it-ebooks.info x CONTENTS 6.4.5 ImplementationofSwitchingFunctionsRepresented byHaarSeries 303 6.4.6 SpectralMethodsforSynthesisofSequenceGenerators 304 7. SPECTRALMETHODSOFSYNTHESISOFSEQUENTIAL MACHINES 308 7.1 RealizationofFiniteAutomatabySpectralMethods 308 7.1.1 FiniteStructuralAutomata 308 7.1.2 SpectralImplementationofExcitationFunctions 311 7.2 AssignmentofStatesandInputsforCompletelySpecified Automata 313 7.2.1 OptimizationoftheAssignmentsforImplementation oftheCombinationalPartbyUsingtheHaarBasis 315 7.2.2 MinimizationoftheNumberofHighestOrderNonzero Coefficients 320 7.2.3 MinimizationoftheNumberofLowestOrderNonzero Coefficients 322 7.3 StateAssignmentforIncompletelySpecifiedAutomata 333 7.3.1 MinimizationofHigherOrderNonzeroCoefficientsin RepresentationofIncompletelySpecifiedAutomata 333 7.3.2 MinimizationofLowerOrderNonzeroCoefficientsin SpectralRepresentationofIncompletelySpecified Automata 338 7.4 SomeSpecialCasesoftheAssignmentProblem 342 7.4.1 PreliminaryRemarks 342 7.4.2 AutonomousAutomata 342 7.4.3 AssignmentProblemforAutomatawithFixedEncoding ofInputsorInternalStates 344 8. HARDWAREIMPLEMENTATIONOFSPECTRALMETHODS 348 8.1 SpectralMethodsofSynthesiswithROM 349 8.2 SerialImplementationofSpectralMethods 349 8.3 SequentialHaarNetworks 350 8.4 ComplexityofSerialRealizationbyHaarSeries 352 8.4.1 OptimizationofSequentialSpectralNetworks 356 8.5 ParallelRealizationofSpectralMethodsofSynthesis 358 8.6 ComplexityofParallelRealization 359 8.7 RealizationbyExpansionsoverFiniteFields 362 www.it-ebooks.info CONTENTS xi 9. SPECTRALMETHODSOFANALYSISANDSYNTHESIS OFRELIABLEDEVICES 370 9.1 SpectralMethodsforAnalysisofErrorCorrectingCapabilities 370 9.1.1 ErrorsinCombinatorialDevices 370 9.1.2 AnalysisofError-CorrectingCapabilities 371 9.1.3 CorrectionofArithmeticErrors 381 9.2 SpectralMethodsforSynthesisofReliableDigitalDevices 386 9.2.1 ReliableSystemsforTransmissionandLogic Processing 386 9.2.2 CorrectionofSingleErrors 388 9.2.3 CorrectionofBurstErrors 391 9.2.4 CorrectionofErrorswithDifferentCosts 393 9.2.5 CorrectionofMultipleErrors 396 9.3 CorrectingCapabilityofSequentialMachines 399 9.3.1 ErrorModelsforFiniteAutomata 399 9.3.2 ComputinganExpectedNumberofCorrectedErrors 400 9.3.2.1 SimplifiedCalculationofCharacteristic Functions 400 9.3.2.2 CalculationofTwo-Dimensional AutocorrelationFunctions 404 9.3.3 Error-CorrectingCapabilitiesofLinearAutomata 408 9.3.4 Error-CorrectingCapabilityofGroupAutomata 410 9.3.5 Error-CorrectingCapabilitiesofCountingAutomata 411 9.4 SynthesisofFault-TolerantAutomatawithSelf-ErrorCorrection 414 9.4.1 Fault-TolerantDevices 414 9.4.2 SpectralImplementationofFault-TolerantAutomata 415 9.4.3 RealizationofSequentialNetworkswithSelf-Error Correction 416 9.5 ComparisonofSpectralandClassicalMethods 419 10. SPECTRALMETHODSFORTESTINGOFDIGITALSYSTEMS 422 10.1 TestingandDiagnosisbyVerificationofWalshCoefficients 423 10.1.1 FaultModels 423 10.1.2 ConditionsforTestability 426 10.1.3 ConditionsforFaultDiagnosis 428 10.2 FunctionalTesting,ErrorDetection,andCorrectionby LinearChecks 430 10.2.1 IntroductiontoLinearChecks 430 www.it-ebooks.info
The list of books you might like
