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Chapter1
SchurParametrizationandOrthogonal
Modelingofp-StationarySecond-Order
StochasticProcesses
AgnieszkaWielgus,JanZarzycki,FelicjaLwowandBartłomiejGolenko
AbstractTheSchuralgorithmisthewell-knowntechniquetoemcientlyperform
parametrizationofstationarysecond-orderstochasticsignals.Thelinearmodeling
filter,drivenbywhitenoise,producesaprocesswhichisstochasticallyequivalentto
theoriginalprocessinaweaksecond-ordersense.Iftheoriginalprocessisnonsta-
tionary,itsparametrizationrequiresthegeneralizedSchuralgorithmofmuchhigher
complexity.WeproposeemcientSchur-typeparametrizationalgorithms,associated
withaclassofstochasticprocesess,calledhereP-stationary,whosecovariance
matricesareblock-Toeplitz.Next,wederivetheresultingSchurparametrization,in-
novationsfilteringandorthogonalstochasticmodelingalgorithmsfornear-stationary
processes,andtheirrealizations.
101Introduction
Thelinearparametrizationandorthogonalmodelingproblemssolutionsforsecond-
orderwide-sensestationarystochasticprocessesarebasedonthecelebratedSchur
algorithm[7,8,9],employingsecond-orderstatistics(i.e.,thecovariancematrix)of
theunderlyingprocess.Inthestationarycase,thiscovariancematrixisapositive-
AgnieszkaWielgus,JanZarzycki,BartłomiejGolenko
DepartmentofAcoustics,MultimediaandSignalProcessing
FacultyofElectronics,PhotonicsandMicrosystems
WrocławUniversityofScienceandTechnology
W.Wyspianskiego27,50-370Wroclaw,Poland
e-mail:agnieszka.wielgus@pwr.edu.pl,jan.zarzycki@pwr.edu.pl,bartlomiej.golenko@pwr.edu.pl
FelicjaLwow
FacultyofPhysiotherapy
UniversitySchoolofPhysicalEducationinWroclaw
Al.I.J.Paderewskiego35,51-617Wroclaw,Poland
e-mail:felicja.lwow@awf.wroc.pl