Signal processing of random physiological signals

232 Pages·2006·4.139 MB·English

by  Charles Lessard, John Enderle

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SIGNAL PROCESSING
OF RANDOM
PHYSIOLOGICAL
SIGNALS
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Copyright©2006byMorgan&Claypool
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or
transmitted in any form or by any means—electronic, mechanical, photocopy, recording, or any other
exceptforbriefquotationsinprintedreviews,withoutthepriorpermissionofthepublisher.
SignalProcessingofRandomPhysiologicalSignals
CharlesS.Lessard
www.morganclaypool.com
159829038X paper Lessard
1598290398 ebook Lessard
DOI10.2200/S00012ED1V01Y200602BME001
APublicationintheMorgan&ClaypoolPublishersseries
SYNTHESISLECTURESONBIOMEDICALENGINEERING
Lecture#1
LibraryofCongressCataloging-in-PublicationData
FirstEdition
PrintedintheUnitedStatesofAmerica
ii
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SIGNAL PROCESSING
OF RANDOM
PHYSIOLOGICAL
SIGNALS
CharlesS.Lessard,Ph.D.
TexasA&MUniversity,CollegeStation,USA
SYNTHESISLECTURESONBIOMEDICALENGINEERING#1
M
&C &
Morgan Claypool Publishers
iii
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iv
ABSTRACT
This lecture book is intended to be an accessible and comprehensive introduction to
randomsignalprocessingwithanemphasisonthereal-worldapplicationsofbiosignals.
Althoughthematerialhasbeenwrittenanddevelopedprimarilyforadvancedundergrad-
uatebiomedicalengineeringstudentsitwillalsobeofinteresttoengineersandinterested
biomedical professionals of any discipline seeking an introduction to the field. Within
education, most biomedical engineering programs are aimed to provide the knowledge
required of a graduate student while undergraduate programs are geared toward de-
signingcircuitsandofevaluatingonlythecardiacsignals.Veryfewprogramsteachthe
processeswithwhichtoevaluatebrainwave,sleep,respiratorysounds,heartvalvesounds,
electromyograms, electro-oculograms, or random signals acquired from the body. The
primarygoalofthislecturebookistohelpthereaderunderstandthetimeandfrequency
domain processes which may be used and to evaluate random physiological signals. A
secondary goal is to learn the evaluation of actual mammalian data without spending
mostthetimewritingsoftwareprograms.Thispublicationutilizes“DADiSP”,adigital
signalprocessingsoftware,fromtheDSPDevelopmentCorporation.
KEYWORDS
Signals,Processes,TimeDomain,FrequencyDomain,RandomData
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Contents
1. BiomedicalEngineeringSignalAnalysis.....................................1
1.1. Introduction.........................................................1
1.2. GeneralizedSystemsEngineeringApproach............................1
2. SystemClassification ...................................................... 5
2.1. Order...............................................................5
2.2. Causalvs.Noncausal.................................................6
2.3. Linearvs.Nonlinear..................................................6
2.4. Fixedvs.Time-Varying...............................................8
2.5. LumpedParametervs.DistributedParameter...........................8
2.6. ContinuousTimevs.DiscreteTime ................................... 9
2.7. Instantaneousvs.Dynamic............................................9
3. ClassificationofSignals...................................................11
3.1. HowareSignalsClassified...........................................12
3.1.1. Periodicvs.NonperiodicSignals...............................13
3.1.2. Randomvs.NonrandomSignals...............................13
3.1.3. EnergySignalsvs.PowerSignals.............................. 14
3.2. SignalCharacterization(MeasurableParameters) ...................... 15
3.2.1. Bandwidth..................................................16
3.2.2. Time–BandwidthProduct .................................... 17
3.2.3. Dimensionality .............................................. 17
3.3. Reference .......................................................... 18
4. BasisFunctionsandSignalRepresentation..................................19
4.1. IntroductiontoBasisFunctions ...................................... 19
4.2. DesirablePropertiesofBasisFunctions................................19
4.3. EvaluationofCoefficients............................................20
4.4. SignalRepresentations .............................................. 23
4.4.1. FourierTransforms...........................................23
4.4.2. LegendreFunctions..........................................27
4.4.3. LaguerreFunctions...........................................27
4.4.4. HermiteFunctions...........................................28
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5. DataAcquisitionProcess..................................................31
5.1. DataCollection.....................................................31
5.2. DataRecording/Transmission........................................32
5.3. DataPreparation....................................................33
5.4. RandomDataQualification .........................................34
5.5. RandomDataAnalysis..............................................35
6. SamplingTheoryandAnalog-to-DigitalConversion ........................ 37
6.1. BasicConcepts ..................................................... 37
6.2. SamplingTheory ................................................... 39
6.3. Quantization .......................................................43
6.3.1. ResolutionorQuantizationStep...............................43
6.3.2. Signal-to-NoiseRatioofanA/DConverter....................45
6.3.3. DynamicRangeofanA/DConverter..........................45
6.4. DigitalFormat......................................................46
6.4.1. FloatingPointFormat........................................46
6.4.2. BlockFloatingPointFormat..................................47
6.5. SignalReconstruction—Digital-to-AnalogConversion(DAC)..........47
7. StationarityandErgodicRandomProcesses.................................49
7.1. Introduction........................................................49
7.2. EnsembleMethodtoTestforStationarity.............................50
7.2.1. StatisticsandStationarity.....................................52
7.2.2. EnsembleMethod ........................................... 53
7.3. Extension-in-TimeMethod..........................................54
7.3.1. ErgodicRandomProcess ..................................... 54
7.4. ReviewofBasicParametricStatisticsandMoment-Generating
Functions .......................................................... 55
7.4.1. Moment-GeneratingFunction................................55
7.5. Summary...........................................................56
7.6. SuggestedReading..................................................57
8. NonparametricStatisticandtheRunsTestforStationarity...................59
8.1. Introduction........................................................59
8.1.1. ConditionsforaRandomProcesstoBeConsidered
asStationary ................................................59
8.1.2. AlternativeMethodstoTestforStationarity....................60
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8.2. ReviewofNonparametricStatisticsUsedforTesting
Stationarity ...................................................... 67
8.2.1. Kolmogorov–SmirnovOne-SampleTest....................67
8.2.2. One-SampleRunsTest ................................... 68
8.2.3. TestingforErgodicity.....................................68
8.3. References.......................................................70
9. CorrelationFunctions....................................................71
9.1. TheCorrelationProcess...........................................73
9.1.1. AutocorrelationFunction..................................74
9.2. PropertiesoftheAutocorrelationFunction..........................75
9.3. StepsintheAutocorrelationProcess................................75
9.4. NumericalCorrelation:DirectCalculation..........................77
9.4.1. StripsofPaperMethod ................................... 77
9.4.2. PolynomialMultiplicationMethod.........................78
9.5. Cross-CorrelationFunction........................................80
9.5.1. PropertiesoftheCross-CorrelationFunction................83
9.5.2. ApplicationsoftheCross-CorrelationFunction..............83
10. Convolution............................................................85
10.1. ConvolutionEvaluation...........................................88
10.1.1. RealTranslationTheorem.................................91
10.1.2. StepsintheCompleteConvolutionProcess.................93
10.1.3. ConvolutionasaSummation .............................. 99
10.2. NumericalConvolution ........................................... 99
10.2.1. DirectCalculation.......................................100
10.3. ConvolutionAlgebra.............................................104
10.3.1. Deconvolution .......................................... 108
11. DigitalFilters..........................................................111
11.1. ClassificationofFilters...........................................111
11.1.1. Low-PassButterworthFilter.............................112
11.1.2. Chebyshevlow-passFilter................................113
11.1.3. ButterworthHigh-PassFilter.............................115
11.1.4. 2nd-OrderButterworthHigh-PassFilter..................116
11.1.5. Band-PassFilters........................................116
11.1.6. Band-Rejector“Notch”Filters............................117
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11.2. DigitalFilters...................................................118
11.2.1. ClassificationofDigitalFilters............................118
11.2.2. InfiniteImpulseResponse(IIR)DigitalFilters.............120
11.2.3. FiniteImpulseResponse(FIR)DigitalFilters..............121
12. FourierSeries:Trigonometric............................................125
12.1. FourierAnalysis.................................................125
12.2. EvaluationoftheFourierCoefficients ............................. 127
12.3. EquivalentFormoftheFourierTrigonometricSeries................128
12.4. SymmetricalPropertiesasRelatedtoFourier
Coefficients.....................................................128
12.4.1. EvenWaveformSymmetries..............................128
12.4.2. ODDWaveformSymmetries.............................129
12.5. EulerExpansion.................................................133
12.6. Limitations.....................................................134
12.7. LimitingProcess ................................................ 135
12.8. InverseFourierTransform........................................137
12.9. Summary.......................................................138
12.10. SummaryofSymmetryTables....................................139
13. FastFourierTransform ................................................. 141
13.1. ContinuousFourierTransform....................................141
13.2. DiscreteFourierTransform.......................................141
13.3. DefinitionofSamplingRate(orSamplingFrequency)...............142
13.3.1. Short-HandNotationoftheDiscrete
FourierTransform(DFT)................................143
13.4. Cooley-TukeyFFT(DecimimationinTime).......................145
13.4.1. DerivationofFFTAlgorithm ............................ 146
13.5. TheFFT“Butterfly”SignalFlowDiagram ........................ 148
13.6. Decimation-in-Frequency........................................149
13.6.1. ButterflyFirst-StageCalculations.........................151
13.6.2. Second-StageCalculations ............................... 152
13.6.3. Third-StageCalculations.................................153
13.6.4. TheDFTbyDirectedMethod...........................154
13.6.5. TheDFTbyFFTMethod...............................155
13.7. Summary.......................................................155
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14. TruncationoftheInfiniteFourierTransform..............................157
14.1. PracticalApplications............................................160
14.2. Summary.......................................................161
15. SpectralAnalysis.......................................................163
15.1. Introduction .................................................... 163
15.2. SpectralDensityEstimation......................................165
15.2.1. PowerSpectraEstimationViathe
AutocorrelationFunction.................................166
15.2.2. PowerSpectraEstimationDirectlyfrom
theFourierTransform....................................168
15.3. Cross-SpectralAnalysis..........................................169
15.4. PropertiesofSpectralDensityFunctions...........................170
15.5. FactorsAffectingtheSpectralDensityFunctionEstimation......... 170
15.6. AdvancedTopic.................................................172
15.6.1. BriefDescriptionofAutoregressiveSpectralEstimation.....172
15.6.2. Summary...............................................173
15.7. SuggestedReading...............................................174
16. WindowFunctionsandSpectralLeakage.................................175
16.1. GeneralitiesaboutWindows......................................176
16.2. PerformanceMeasures...........................................178
16.2.1. HighestSide-LobeLevel.................................178
16.2.2. SidelobeFall-OffRatio..................................178
16.2.3. FrequencyStraddleLoss.................................179
16.2.4. CoherentIntegrationGain...............................179
16.2.5. EquivalentNoiseBandwidth ............................. 183
16.2.6. Three-DecibelMain-LobeBandwidth ....................183
16.3. WindowFunctionsandWeightingEquations......................183
16.3.1. TheRectangleWindow..................................183
16.3.2. TheRectangularWindowandtheGibbsPhenomenon......184
16.3.3. Parzen/RietzWindow ................................... 186
16.3.4. TheTukeyFamilyof Windows...........................186
16.3.5. HanningWindows......................................187
16.3.6. WelchWindow ......................................... 191
16.3.7. BlackmanWindow......................................191
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16.4. Summary.......................................................192
16.5. References......................................................193
16.6. SuggestedReading...............................................193
17. TransferFunctionViaSpectralAnalysis..................................195
17.1. Introduction .................................................... 195
17.2. Methods........................................................196
17.3. Autocorrelation..................................................197
17.4. TheCross-CorrelationFunction..................................198
17.5. SpectralDensityFunctions.......................................199
17.6. TheCoherenceFunction.........................................203
17.7. Summary.......................................................204
17.8. References......................................................204
18. CoherenceFunctionfromSpectralAnalysis...............................205
18.1. Introduction .................................................... 205
18.2. DescriptionoftheCoherenceFunction............................207
18.3. MisapplicationoftheCoherenceFunction.........................209
18.4. ExamplesoftheUseofCoherenceFunctions.......................210
18.5. ProblemsThatLimittheApplicationofCoherence.................211
18.6. Conclusion......................................................212
19. ErrorinRandomDataEstimateAnalysis
(InformationIsParaphrasedfromBendat&Piersol).......................215
19.1. Cross-SpectralDensityFunctionEstimates........................215
19.2. Summary.......................................................219