Fractional Processes and Fractional-Order Signal Processing: Techniques and Applications

309 Pages·2012·16.499 MB·English

by  Hu Sheng, YangQuan Chen, TianShuang Qiu (auth.)

SIGNALSANDCOMMUNICATIONTECHNOLOGY
Forfurthervolumes:
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Hu Sheng (cid:2) YangQuan Chen (cid:2) TianShuang Qiu
Fractional
Processes and
Fractional-Order
Signal Processing
Techniques and Applications
HuSheng TianShuangQiu
SchoolofElectronicandInformation SchoolofElectronicandInformation
Engineering Engineering
DalianJiaotongUniversity DalianUniversityofTechnology
Dalian Dalian
People’sRepublicofChina People’sRepublicofChina
[email protected] [email protected]
YangQuanChen
DepartmentofElectricalandComputer
Engineering
CSOIS
UtahStateUniversity
OldMainHill4160
Logan,UT84322-4120
USA
[email protected]
ISSN1860-4862 SignalsandCommunicationTechnology
ISBN978-1-4471-2232-6 e-ISBN978-1-4471-2233-3
DOI10.1007/978-1-4471-2233-3
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Mywife,ShuyuanZhao, and
Myparents, YumingShengandGuiyunWang
–HuSheng
ThememoryofmyfatherHanlinChen,and
Myfamily,HuifangDou,Duyun,David,andDaniel
–YangQuanChen
ThememoryofmyfatherZhuxianQiu,and
Myfamily,LianWang andYumeng
– TianShuang Qiu
Foreword
Ifanengineerwishestooptimizetheoperationofanelectricorhybridvehicle,the
firstthingsheneedsisamathematicalmodeltocontrolthecharging/dischargepro-
cess.Itisanaxiomofcontroltheorythattheorderofthemodelshouldmatchthat
of the system to be controlled. So, it is not surprising that fractional (non-integer)
ordercontrolsystemsareneededtoregulatethefractional-orderdiffusionkinetics
inherentinelectrochemicalcells.Whatistheoriginofsuchanomalous,andinsome
cases,non-Gaussianbehavior?Howdowedescribetheelectricalsignalsthatarise?
And,whattoolsareavailabletodayfordesignengineerstouseforfractional-order
signalprocessing?Inthisnewbook(FractionalProcessesandFractional-OrderSig-
nalProcessing)theauthorsanswerthesequestionsinamannerthatiseasilyunder-
stoodbystudents,researchersandbypracticingengineers.
Inordertoputthiscontributionintoitspropercontext,weshouldfirstrealizethat
while the tools of fractional calculus are well known—some key elements having
beendevelopedbymathematiciansinthe18thand19thcenturies—theapplication
offractionalcalculusinphysics,chemistry,biologyandengineeringarose onlyin
thelatterhalfofthe20thcentury.Second,unlikethematurefieldof“analysis”that
forms the foundation for the conventional integral and differential calculus, frac-
tional calculus is still undergoing rapid growth and expansion; driven in part by
the same forces that are developing new mathematical theories of chaos, fractals
andcomplexsystems.Unfortunately,thisdynamicofgrowthsometimeshindersthe
transfer and assimilation of new knowledge into the scientific and technical disci-
plines.
Theproblemissomewhatakintothedevelopmentofnewschoolsofphilosoph-
ical inquiry in ancient Greece: Stoicism, Epicureanism, and Skepticism. The cer-
taintyoftheSophiststhathumanknowledgecouldbeacquiredbyobservationand
analysis, was offset by the Epicureans who felt that all such knowledge was use-
ful,whiletheSkeptics—whosawonlythelimitsofnewknowledge—counseledall
to suspend judgment. This type of philosophical argument is unfortunately being
playedouttodaybetweentheappliedmathematicscommunity,thefractionalcalcu-
lusclan,andadvocatesofnonlinear,chaoticandcomplexprocesses(i.e.,processes
assumedtobebeyondourabilitytomodel).Thechallengethereforeforfractional
vii
viii Foreword
Epicureans is to seek aid from mathematical Sophists to place fractional calculus
withinthecanonofcollegecalculus,whilehelping‘complexitytheory’Skepticsto,
inthewordsofColeridge,“suspendtheirdisbelief”.
Giventhissituation,itisnaturalthattheauthorsplaytoallpartsofthehousein
theirpresentation.Thisisevidentinthefourpartsofthemonograph,whichaddress:
basictheory,thetheoryoffractional-orderprocesses,theanalysisofsignalsgener-
ated by fractional-order processes, and the applications of such models to a wide
range of physical, mechanical and biological systems. The first part thus provides
students with a review (and other readers with a refreshing summary) of random
signalsandstochasticprocesseswhileoutliningthebasicsoffractional-ordermod-
els, experimentally observed fractional-order behavior and signal processing. The
second part focuses on constant-order fractional processes and on multifractional
processes.Hence,thepurposeofthissectionistoconnectthereaderwithfractional-
orderprocessesononehandandwiththemorefundamentalconceptsoffractional
Browningmotionandfractalmeasuresofstability(viatheHurstestimators)onthe
other.
The third section of the book provides the core of the topical analysis with
constant-,variable-anddistributed-orderfractionalsignalprocessingdescribed.The
presentationspanstheoryandapplication(analogueanddigital)fortheimplemen-
tationofcontinuous-timeanddiscrete-timeoperators.Canonicalfractionalsecond
orderfiltermodelsareusedtoillustratetheexpectedbehaviorfortheconstant-order
case.Thesynthesisofvariable-orderfractionalsignalprocessingisnextdescribed
for cases of locally stable long memory with an example of the analogue realiza-
tioninthecaseofatemperature-dependentvariableorderfractionalintegratorand
othersforvariable-orderfractionalsystems.Finally,thissectioncloseswiththede-
scriptionofdistributed-orderintegratorsanddifferentiatorsandadistributed-order
lowpassfilter.Partfourofthebookpresentssixseparateexamplesofapplications
offractional-ordersignalprocessingtechniques:twobiomedicalexamples(latency
of evoked potentials, multifractional properties of human sleep EEG), one model
ofmolecularmotion,anelaborationofoptimalfractional-orderdampingstrategies
for a mass-spring viscoelastic vibration damper, analysis of electrical noise using
fractional signal processing, and a variable-order fractional analysis of changes in
waterlevelintheGreatSaltLake.
Given the breadth and depth of the analysis in this monograph the fractional
EpicureanandperhapseventhemathematicalStoicsshouldbesatedwithitsrange
oftheoryandapplications.However,heretheSkepticmightrecallthephilosophical
paradoxofdescribingthe“state”ofariver:thestateisneverthesameasitalways
varies in time and place, and never repeats. And so it is with fractional calculus,
it seems, with its bewildering variety of definitions for fractional-order operators
(Riemann-Liouville, Caputo, Weyl, Grunwald-Letnikov, Riesz, etc.). It is notable
then that for this text, only the Riemann-Liouville and the Caputo definitions are
employed.Theemphasishereisnotondifferentobserversontheshoreoftheriver,
butondifferentwaysforoneobservertoviewthe“state”,e.g.,asafractionalorder
processwithconstant-order,variable-order,ordistributed-orderdynamicmeasures.
JustasthemeasurementsbyHurstoftheflowoftheNileriverinEgyptprovided
thefirstearlyevidencefortheutilityoffractalmodelsof“real-world”phenomena,
Foreword ix
the fractional-order models of biological, hydrological, electrical and mechanical
phenomena presented here give the strongest evidence of the utility of fractional-
order analysis. These models are robust, resilient, and adaptive. As physics, engi-
neering,chemistryandbiologymovetowardmorecompleterepresentationsofcom-
plexmaterialsandprocesses—intissueengineering,forexample,theregeneration
andregrowthofneworgans—scientistsandengineersneedanexpandedsetoftools
to deal with processes that clearly change, like a river, with time and place. This
book provides one clear view of how to develop and apply fractional-order mod-
els to contemporaryproblemsin polymerformation,protein folding,and even the
optimalchargingoflithium-ionbatteriesforthenextgenerationofelectricvehicles.
DistinguishedProfessorofBioengineering RichardL.Magin
DepartmentofBioengineering
UniversityofIllinoisatChicago,IL
Preface
Inancienttimes,insteadoffractions,thenaturalnumberswereused.However,inte-
gerscannotalwaysresultwhenmeasuringorequallydividingthings.Astimewent
by,fractionsandthennon-integersweregraduallyunderstoodandapplied.Withthe
introductionoffractionsandmoregenerallynonintegers,peoplewereabletohave
a closer look at the beauty of nature around them. For example, people long ago
realizedthatarectangleofthe‘goldenratio’1.618:1ismostpleasing.Thenatural
exponentiale=2.71828...,andtheratioofanycircle’scircumferencetoitsdiam-
eter,π =3.14159...arewidelyusedinmathematicsandengineering.The‘beauty’
ofthefractionwasrecognizedandpeoplecametousethe‘fractionalview’toob-
serve the world, to use ‘fractional thinking’ to understand the natural phenomena,
andtouse‘fractionaltechniques’tosolvetheproblemsathand.
Theterm‘fractal’wasintroducedbyMandelbrotin1975[192].Fractalrefersto
theself-similargeometricshape,thatis,ashapeinwhichisalmostidenticaltothe
entire shape except in size [91, 102]. Many objects manifest themselves in fractal
shape, such as clouds, coastlines and snow flakes. In fractal theory, the fractal di-
mensionwasusedtocharacterizethestateofnature.Differentfromtheconventional
integerdimension,thefractaldimensioncanbefractionaloranynon-integernum-
ber.Basedonthefractaltheory,thetraditionalconceptofthree-dimensionalspace
canbeextendedtothefractal(fractional)dimension(FD)whichcanbeappliedto
characterizecomplexobjects.
Likewise,(integer-order)calculuscanbeextendedtofractionalornonintegeror-
der calculus. It should be remarked at this point that due to historical reasons, the
term ‘fractional’ we use here and throughout this monograph should actually be
understoodas‘non-integer’or‘arbitraryrealnumber’tobeprecise.Fractionalcal-
culus,i.e.,fractional-orderdifferentiationandintegration,isapartofmathematics
dealingwithderivativesofarbitraryorder[139,203,209,218,237].Leibnizraised
the question about the possibility of generalizing the operation of differentiation
to non-integer-orders in 1695 [237]. Fractional calculus, developed from the field
ofpuremathematics,hasbeenstudiedincreasinglyinvariousfields[64,142,311,
315,323].Nowadays,fractionalcalculusisbeingappliedtomanyfieldsofscience,
engineering,andmathematics[49,74,78,135,290].Fractionalcalculusprovidesa
xi
xii Preface
betterdescriptionofvariousnaturalphenomena,suchasviscoelasticityandanoma-
lousdiffusion,thaninteger-ordercalculuscanprovide.Themostfundamentalrea-
sonforthesuperiorityoftheuseisthatfractionalcalculusbasedmodelscancapture
thememoryandtheheredityoftheprocess.Itissafetosaythatfractionalcalculus
providesaparticularlyusefulandeffectivetoolforrevealingphenomenainnature
becausenaturehasmemory.
This “fractionalization” idea can go on. Taking the Fourier transform as an ex-
ample,wecannaturallytalkaboutthefractionalFouriertransform(FrFT),whichis
alineartransformationgeneralizingtheFouriertransform,firstintroducedbyVictor
Namiasin1980[213].FrFTcantransformasignalfromtimedomainintoadomain
betweenthetimeandfrequencydomains.So,theFrFTdiffersfromtheconventional
Fourier transform by the rotation of fractional times of the π/2 angle in the time-
frequency plane. FrFT is widely used in filter design, signal detection and image
recovery[32].
Anotherimportant“fractionalization”isfractionallow-ordermoments(FLOM).
FLOMisbasedonthenon-Gaussian α-stabledistribution,apowerfultoolforim-
pulsiverandomsignals.Thedensityfunctionsofα-stabledistributionwilldecayin
thetailslessrapidlythantheGaussiandensityfunctiondoes.Soα-stabledistribu-
tion can be used to characterize signals which exhibit sharp spikes or occasional
bursts of outlying observations more frequently than normally distributed signals
do.Theα-stabledistributionbasedtechniqueshavebeenappliedtodescribemany
naturalorman-madephenomenainvariousfields,suchasphysics,hydrology,biol-
ogy,financialandnetworktraffic[43,70,253,270].Theα-stabledistributionhasa
characteristicexponentparameterα(0<α≤2),whichcontrolstheheavinessofits
tails.Foranon-Gaussianstabledistributionwithcharacteristicexponentα<2,its
second-ordermomentdoesnotexist.α-stabledistributionhasonlyfinitemoments
oforderlessthan α.SotheFLOMplaysanimportantroleinimpulsiveprocesses
liketheroleofmeanandvarianceintheGaussianprocesses.
In this monograph, we will introduce some complex random signals which are
characterizedbythepresenceofheavy-taileddistributionornon-negligibledepen-
dence between distant observations, from the ‘fractional’ point of view. Further-
more, the analysis techniques for these fractional processes are investigated us-
ing‘fractionalthinking’.Theterm‘fractionalprocess’inthismonographrefersto
somerandomsignalswhichmanifestthemselvesbyheavy-taileddistribution,long
rangedependence(LRD)/longmemory,orlocalmemory.Fractionalprocessesare
widelyfoundinscience,technologyandengineeringsystems.Typicalheavy-tailed
distributedsignalsincludeunderwateracousticsignals,low-frequencyatmospheric
noises,manytypesofman-madenoises,andsoon.TypicalLRD/longmemorypro-
cessesandlocalmemoryprocessescanbeobservedinfinancialdata,communica-
tions networks data and biological data. These properties, i.e., heavy-tailed distri-
bution,LRD/longmemory,andlocalmemory,alwaysleadtodifficultyincorrectly
obtainingthestatisticalcharacteristicsandextractingthedesiredinformationfrom
thesefractionalprocesses.Thesepropertiescannotbeneglectedintimeseriesanal-
ysis,becausethetailthicknessofthedistribution,LRD,orlocalmemoryproperties
of the time series are critical in characterizing the essence of the resulting natural
Preface xiii
orman-madephenomenaofthesignals.Therefore,somevaluablefractional-order
signalprocessing(FOSP)techniqueswereprovidedtoanalyzethesefractionalpro-
cesses. FOSP techniques, which are based on the fractional calculus, FLOM and
FrFT,includesimulationoffractionalprocesses,fractional-ordersystemmodeling,
fractional-order filtering, realization of fractional systems, etc. So, random signals
which exhibit evident ‘fractional’ properties should be investigated using FOSP
techniquestoobtainbetteranalysisresults.
This monograph includes four parts. The first part is the overview of fractional
processes and FOSP techniques. The second part presents fractional processes,
whicharestudiedastheoutputofthefractionalorderdifferentialsystems,includ-
ingconstant-orderfractionalprocessesandvariable-orderfractionalprocesses.The
thirdpartintroducestheFOSPtechniquesfromthe‘fractionalsignalsandfractional
systems’pointofview.Inthelastpartofthemonograph,someapplicationexamples
ofFOSPtechniquesarepresentedtohelpreadersunderstandandappreciatethefrac-
tional processes and fractional techniques. We sincerely wish that this monograph
willgiveourreadersanovelinsightintothecomplexrandomsignalscharacterized
by‘fractional’properties,andsomepowerfultoolstocharacterizethosesignals.
Dalian,Liaoning,China HuSheng
Logan,UT,USA YangQuanChen
Dalian,Liaoning,China TianShuangQiu