Quaternion-Valued Adaptive Signal Processing and Its Applications to Adaptive Beamforming and Wind Profile Prediction Mengdi Jiang Supervisors: Dr. Wei Liu and Dr. Yi Li Thesis submitted in candidature for graduating with degree of doctor of philosophyAugust 2016 ⃝c MengdiJiang2016 Abstract Quaternion-valuedsignalprocessinghasreceivedmoreandmoreattentionsinthepasttenyears due to the increasing need to process three or four-dimensional signals, such as colour images, vector-sensor arrays, three-phase power systems, dual-polarisation based wireless communica- tion systems, and wind profile prediction. One key operation involved in the derivation of all kindsofadaptivesignalprocessingalgorithmsisthegradientoperator. Althoughtherearesome derivationsofthisoperatorinliteraturewithdifferentlevelofdetailsinthequaterniondomain, it is still not fully clear how this operator can be derived in the most general case and how it can be applied to various signal processing problems. In this study, we will give a detailed derivationofthequaternion-valuedgradientoperatorwithassociatedpropertiesandthenapply it to different areas. In particular, it will be employed to derive the quaternion-valued LMS (QLMS) algorithm and its sparse versions for adaptive beamforming for vector sensor arrays, and another one is its application to wind profile prediction in combination with the classic computationalfluiddynamics(CFD)approach. For the adaptive beamforming problem for vector sensor arrays, we consider the crossed- dipole array and the problem of how to reduce the number of sensors involved in the adap- tive beamforming process, so that reduced system complexity and energy consumption can be achieved,whereasanacceptableperformancecanstillbemaintained,whichisparticularlyuse- ful for large array systems. The quaternion-valued steering vector model for crossed-dipole arrays will be employed, and a reweighted zero attracting (RZA) QLMS algorithm is then pro- posed by introducing a RZA term to the cost function of the original QLMS algorithm. The RZA term aims to have a closer approximation to the l norm so that the number of non-zero 0 valuedcoefficientscanbereducedmoreeffectivelyintheadaptivebeamformingprocess. For wind profile prediction, it can be considered as a signal processing problem and we can solve it using traditional linear and non-linear prediction techniques, such as the proposed QLMSalgorithmanditsenhancedfrequency-domainmulti-channelversion. Ontheotherhand, wind flow analysis is also a classical problem in the CFD field, which employs various simu- lation methods and models to calculate the speed of wind flow at different time. It is accurate buttime-consumingwithhighcomputationalcost. Totackletheproblem,acombinedapproach based on synergies between the statistical signal processing approach and the CFD approach is proposed. There are different ways of combining the signal processing approach and the CFD approach to obtain a more effective and efficient method for wind profile prediction. In the combinedmethod,thesignalprocessingpartemploystheQLMSalgorithm,whilefortheCFD part,largeeddysimulation(LES)basedontheSmagorinskysubgrid-scale(SGS)modelwillbe employedsothatmoreefficientwindprofilepredictioncanbeachieved. 2 Contents ListofFigures 4 ListofTables 6 ListofPublications 8 ListofAbbreviations 10 Acknowledgements 11 1 Introduction 12 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.2 OriginalContributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.2.1 AgeneralHRgradientoperatoranditsapplications . . . . . . . . . . . 13 1.2.2 Applicationtoadaptivebeamforming . . . . . . . . . . . . . . . . . . 14 1.2.3 Applicationtowindprofileprediction . . . . . . . . . . . . . . . . . . 15 1.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2 Quaternion-valuedAdaptiveSignalProcessing 18 2.1 IntroductiontoQuaternionAlgebra . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2 DifferentiationwithRespecttoaVectorinComplexDomain . . . . . . . . . . 20 2.3 Complex-valuedLeastMeanSquareAlgorithm . . . . . . . . . . . . . . . . . 23 2.4 Quaternion-valuedGradientOperatorandtheCorrespondingLMSAlgorithm . 26 2.5 Quaternion-valuedDiscreteFourierTransform . . . . . . . . . . . . . . . . . . 30 1 2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3 GeneralQuaternion-valuedGradientOperationandItsApplications 33 3.1 TherestrictedHRgradientoperator . . . . . . . . . . . . . . . . . . . . . . . 34 3.2 TherightrestrictedHRgradientoperator . . . . . . . . . . . . . . . . . . . . . 39 3.3 PropertiesandRulesoftheOperator . . . . . . . . . . . . . . . . . . . . . . . 40 3.4 RestrictedHRDerivativesforaClassofRegularFunctions . . . . . . . . . . . 44 3.5 TheRightRestrictedHRGradients . . . . . . . . . . . . . . . . . . . . . . . . 53 3.6 Quaternion-valuedAlgorithms . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.6.1 Theincrementofaquaternionfunction . . . . . . . . . . . . . . . . . 56 3.6.2 TheQLMSalgorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.6.3 TheAQLMSalgorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.6.4 TheQuaternion-valuednonlinearadaptivealgorithm . . . . . . . . . . 61 3.7 AZero-attractingQLMSAlgorithmforSparseSystemIdentification . . . . . . 65 3.7.1 Thezero-attractingQLMS(ZA-QLMS)algorithm . . . . . . . . . . . 65 3.7.2 Simulationresults . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4 ApplicationtoAdaptiveBeamforming 72 4.1 IntroductiontoAdaptiveBeamforming . . . . . . . . . . . . . . . . . . . . . . 74 4.1.1 Arraymodelandbeamformingstructure . . . . . . . . . . . . . . . . . 74 4.1.2 LMS-basedRSBbeamformer . . . . . . . . . . . . . . . . . . . . . . 76 4.2 AdaptiveBeamformingBasedonVectorSensorArrays . . . . . . . . . . . . . 78 4.2.1 Quaternionicarraysignalmodel . . . . . . . . . . . . . . . . . . . . . 78 4.2.2 Referencesignalbasedquaternion-valuedadaptivebeamforming . . . . 80 4.3 TheRZA-QLMSAlgorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.4 SimulationResults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 2 5 ApplicationtoWindProfilePrediction 93 5.1 CorrelationAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.1.1 TheWienersolution . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.1.2 Correlationswithdifferentsamplingfrequencies . . . . . . . . . . . . 96 5.1.3 Simulationresults . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5.2 Transform-domain Quaternion-valued Adaptive Filtering and Its Application forWindProfilePrediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.3 SimulationResults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.3.1 ResultsbasedonCFDgenerateddata . . . . . . . . . . . . . . . . . . 106 5.3.2 Resultsbasedonrealdata . . . . . . . . . . . . . . . . . . . . . . . . 107 5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 6 CombinedApproachtoWindProfilePrediction 115 6.1 ComputationalFluidDynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 116 6.1.1 Thefluiddynamicsequations . . . . . . . . . . . . . . . . . . . . . . 116 6.1.2 Reynoldsnumber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 6.1.3 TurbulenceorLaminar . . . . . . . . . . . . . . . . . . . . . . . . . . 118 6.1.4 Discretisationmethods . . . . . . . . . . . . . . . . . . . . . . . . . . 118 6.1.5 Simulationsmethodsandmodelsforturbulence . . . . . . . . . . . . . 121 6.2 TheCombinedApproach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 6.2.1 Alternating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 6.2.2 Weighting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 6.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 7 ConclusionsandFuturePlan 138 7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 7.2 FutureWork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 Bibliography 142 3 List of Figures 2.1 Astandardadaptivefilterstructure. . . . . . . . . . . . . . . . . . . . . . . . 24 3.1 Magnitudeoftheimpulseresponseofthesparsesystem. . . . . . . . . . . . . 69 3.2 Learningcurvesforthefirstscenario. . . . . . . . . . . . . . . . . . . . . . . 70 3.3 LearningCurvesforthesecondscenario. . . . . . . . . . . . . . . . . . . . . 71 4.1 AULAwithMomnidirectionalsensors. . . . . . . . . . . . . . . . . . . . . 74 4.2 Referencesignalbasedbeamformingstructure. . . . . . . . . . . . . . . . . . 77 4.3 AULAwithcrossed-dipoles. . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.4 Referencesignalbasedadaptivebeamforming. . . . . . . . . . . . . . . . . . 80 4.5 Learningcurvesofthethreealgorithms. . . . . . . . . . . . . . . . . . . . . . 87 4.6 Beampatternsofthethreealgorithmswith0◦ desiredsignal. . . . . . . . . . . 88 4.7 Amplitudesofthesteadystateweightcoefficients. . . . . . . . . . . . . . . . 89 4.8 Beampatternofthetwoarrays. . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.9 Beampatternsofthethreealgorithmswith15◦ desiredsignal. . . . . . . . . . 92 5.1 Normalisederrorofwindpredictionatdifferentsampleintervals. . . . . . . . 100 5.2 Normalisederrorofwindpredictionatdifferentfrequencies. . . . . . . . . . . 101 5.3 Generalstructureofamulti-channeladaptivefilter. . . . . . . . . . . . . . . . 102 5.4 Generalstructureofamulti-channelfrequency-domainquaternion-valuedadap- tivefilter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 5.5 PredictionresultsusingtheQLMSalgorithm. . . . . . . . . . . . . . . . . . . 106 5.6 PredictionresultsusingtheAQLMSalgorithm. . . . . . . . . . . . . . . . . . 107 4 5.7 Thesingle-channeltime-domainpredictionresult. . . . . . . . . . . . . . . . 109 5.8 Thesingle-channeltransform-domainpredictionresult. . . . . . . . . . . . . . 110 5.9 Thesingle-channellearningcurveforthemagnitudeoftheerrorsignal. . . . . 110 5.10 Themulti-channeltime-domainpredictionresult. . . . . . . . . . . . . . . . . 111 5.11 Themulti-channeltransform-domainpredictionresult. . . . . . . . . . . . . . 111 5.12 Themulti-channellearningcurveforthemagnitudeoftheerrorsignal. . . . . 112 5.13 ThelearningcurveforthemagnitudeoftheerrorsignalwithL=16. . . . . . . 112 5.14 ThelearningcurveforthemagnitudeoftheerrorsignalwithL=32. . . . . . . 113 5.15 ThelearningcurveforthemagnitudeoftheerrorsignalwithL=64. . . . . . . 113 6.1 Thealternatingprogress. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 6.2 HistogramsfornormalisederrorwiththeQLMSalgorithm. . . . . . . . . . . 132 6.3 HistogramsfornormalisederrorwiththeLESmethod. . . . . . . . . . . . . . 133 6.4 Errorsignalwithcombinedmethodat α =0.3. . . . . . . . . . . . . . . . . . 133 6.5 Errorsignalwithcombinedmethodat α =0.5. . . . . . . . . . . . . . . . . . 134 6.6 Errorsignalwithcombinedmethodat α =0.7. . . . . . . . . . . . . . . . . . 134 6.7 Thespectrumoferrorsignalwithcombinedmethodatα =0.3. . . . . . . . . 135 6.8 Thespectrumoferrorsignalwithcombinedmethodatα =0.5. . . . . . . . . 135 6.9 Thespectrumoferrorsignalwithcombinedmethodatα =0.7. . . . . . . . . 136 5 List of Tables 4.1 Comparisonofcomputationalcomplexity. . . . . . . . . . . . . . . . . . . . . 85 4.2 Comparisonofcomputationalcomplexitywithsensornumbers . . . . . . . . . 90 5.1 Normalised prediction error at sampling frequency f = 5Hz with prediction s stepP andnumberofsamplesinvolvedN (PartI). . . . . . . . . . . . . . . . 97 5.2 Normalised prediction error at sampling frequency f = 5Hz with prediction s stepP andnumberofsamplesinvolvedN (PartII). . . . . . . . . . . . . . . . 98 5.3 Normalised prediction error at sampling frequency f = 2Hz with prediction s stepP andnumberofsamplesinvolvedN (PartI). . . . . . . . . . . . . . . . 98 5.4 Normalised prediction error at sampling frequency f = 2Hz with prediction s stepP andnumberofsamplesinvolvedN (PartII). . . . . . . . . . . . . . . . 98 5.5 Normalised prediction error at sampling frequency f = 1Hz with prediction s stepP andnumberofsamplesinvolvedN (PartI). . . . . . . . . . . . . . . . 99 5.6 Normalised prediction error at sampling frequency f = 1Hz with prediction s stepP andnumberofsamplesinvolvedN (PartII) . . . . . . . . . . . . . . . 99 6.1 Normalisedpredictionerrorfortheproposedalternatingmethodwithprediction advancevalueP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 6.2 RunningtimewithpredictionadvancevalueP (seconds). . . . . . . . . . . . . 128 6.3 Normalised prediction error in (dB) by data sequence with sampling frequency f (Hz)andpredictiontimeP (hours)(PartI). . . . . . . . . . . . . . . . . . . 129 s t 6 6.4 Normalised prediction error in (dB) by data sequence with sampling frequency f (Hz)andpredictiontimeP (hours)(PartII). . . . . . . . . . . . . . . . . . 130 s t 6.5 Normalised prediction error in (dB) by the power of data sequence with sam- plingfrequencyf (Hz)andpredictiontimeP (hours)(PartI). . . . . . . . . . 130 s t 6.6 Normalised prediction error in (dB) by the power of data sequence with sam- plingfrequencyf (Hz)andpredictiontimeP (hours)(PartII) . . . . . . . . . 131 s t 7