Table Of ContentMultiscale Signal Analysis and Modeling
Xiaoping Shen • Ahmed I. Zayed
Editors
Multiscale Signal Analysis
and Modeling
123
Editors
XiaopingShen AhmedI.Zayed
DepartmentofMathematics DepartmentofMathematicalSciences
OhioUniversity DePaulUniversity
OhioUniversity1 NorthKenmoreAvenue2320
Athens,Ohio Chicago,Illinois
MortonHall SchmittAcademicCenter(SAC)524
USA USA
ISBN978-1-4614-4144-1 ISBN978-1-4614-4145-8(eBook)
DOI10.1007/978-1-4614-4145-8
SpringerNewYorkHeidelbergDordrechtLondon
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In recognitionof Dr.GilbertG. Walterhavingoutlived
hisprimeof 79.
From XiaopingA. Shen andAhmed I. Zayed
Preface
Thismonographis a collection of chaptersauthoredor coauthoredby friendsand
colleagues of Professor Gilbert Walter in celebration of his 80th birthday. The
authors represent a spectrum of disciplines, mathematics, applied mathematics,
electrical engineering,and statistics; yet, the monographhas one common theme:
multiscaleanalysis.
Multiscale analysis has recently become a topic of increasing interest because
ofitsimportantapplications,inparticular,inanalyzingcomplexsystemsinwhich
thedatabehavedifferentlydependinguponwhichscalethedataarelookedat.The
adventofwaveletshasgivenanimpetustomultiscaleanalysis,butothertechniques
suchassamplingandsubsamplinghavebeenusedsuccessfullyasatoolinanalyzing
multiscalesignals.Forthisreasonwehavedecidedtoincludeavarietyofchapters
coveringdifferentaspectsandapplicationsofmultiscaleanalysis.
Themonographisdividedintothreemainparts:PartIisacollectionofchapters
on sampling theory while Parts II and III contain chapters on multiscale analysis
andstatisticalanalysis,respectively.Thelevelofpresentationvaries.Fewchapters
are very specialized, while others are self-contained or of expository nature, but
mostchaptersshouldbeaccessibletograduatestudentsinmathematicsorelectrical
engineering.
Part I, which consists of eight chapters, has chapters on sampling, interpo-
lation, and approximation in the space of bandlimited functions, shift-invariant
and reproducing-kernel Hilbert spaces, and the Hardy space H2(D) of analytic
functions in the open unit disk. Part II contains four chapters on a unified theory
for multiscale analysis, multiscale signal processing, developing algorithms for
signal and image classification problems in large data sets, and wavelet analysis
of ECG (electrocardiogram) signals. Part III is comprised of three chapters on
characterizationofcontinuousprobabilitydistributions,Bayesianwaveletshrinkage
methods, and multiparameter regularization for the construction of estimators in
statisticallearningtheory.
In Chapter1, W. Madych revisits the classical cardinal series and considers
its symmetric partial sums. He derives necessary and sufficient conditions for its
convergence under growth conditions on the coefficients that imply analogous
vii
viii Preface
asymptotic behavior of the function represented by the series. Severalcorollaries,
includingsamplingtypetheorems,areobtained.
Chapter 2, by F. Stenger, M. Youssef, and J. Niebsch, is also related to the
cardinal series and the Sinc function , but in the contextof function interpolation
and approximation. Function interpolation may be carried out using algebraic
polynomials, splines, Fourier polynomials, rational functions, wavelets, or Sinc
methods. Function interpolation by one of the above methods frequently has one
of the following features: (1) the modulusof the error of approximationnear one
endpointofanintervaldiffersconsiderablyfromthemodulusoftheerrornearthe
otherend-point;(2)themodulioftheerrorsnearthetwoendpointsoftheinterval
are roughly the same, but differ appreciably from the modulus of the error in the
mid-rangeoftheinterval.Theauthorscalltheformerthe(E-E)caseandthelatter
the(E-M)case.
In this chapter the authors describe methods for getting a more uniform ap-
proximation throughout the interval of approximation in the two aforementioned
cases(E-E)and(E-M).Theyalsodiscussimprovingapproximationofthederivative
obtainedbydifferentiatingtheconstructedinterpolationapproximations.
In Chapter3, H. R. Ferna´ndez-Morales, A. G. Garcia, and G. Pe´rez-Villalo´n
considersamplinginageneralshift-invariantspaceV (φ)ofL2(IRd)withasetΦof
2
rstablegeneratorsandinwhichthedataaresamplesofsomefilteredversionsofthe
signalitselftakenatasub-latticeofZZd.Theauthorscallthisproblemtheproblem
ofgeneralizedsamplinginshift-invariantspaces.Assumingthatthe(cid:2)2-normofthe
generalized samples of any f ∈V (φ) is stable with respect to the L2(IRd)-norm
2
ofthesignal f, theauthorsderiveframeexpansionsintheshift-invariantsubspace
V (φ)allowingtherecoveryofsignalsinthisspacefromtheavailabledata.
2
AsimilarsamplingproblemisconsideredbyM.NashedandQ.SuninChapter4,
where they consider a variety of Hilbert and Banach spaces of functions that
admit sampling expansionsof the form f(t)=∑∞ f(t )S (t), where {S (t)}∞
n=1 n n n n=1
is a family of functions that depend on the sampling points {t } but not on the
n
function f. Those function spaces, which arise in connection with sampling ex-
pansions,includereproducing-kernelspaces,Sobolevspaces,shift-invariantspaces,
translation-invariantspacesandspacesofsignalswithfiniterateofinnovation.The
authors first discuss the engineering approach to the Shannon sampling theorem
which is based on trains of delta functions and then try to provide rigorous
justification to the engineeringapproachusing distributiontheoryand generalized
functions.Theyalsodiscusssamplinginsomereproducing-kernelBanachspaces.
Chapter 5 by P. Vaidyanathan and P. Pal gives an overview of the concept of
coprimesamplinganditsapplications.Coprimesamplingwasrecentlyintroduced
by the authors first for the case of one-dimensional signals and then extended to
multidimensionalsignals.Thebasicideaisthatacontinuous-time(orspatial)signal
issampledsimultaneouslybytwosetsofsamplers,withsamplingrates1/NT and
1/MT whereMandNarecoprimeintegersandT>0.Oneofthemainresultsisthat
itispossibletoestimatetheautocorrelationofthesignalatamuchhigherrate1/T
thanthetotalsamplingrate.Thus,anyapplicationwhichisbasedonautocorrelation
will benefit from such sampling and reconstruction. An interesting mathematical
Preface ix
problem that comes up when coprime sampling is extended to higher dimensions
is how to generate a pair of integer matrices M and N which are commuting and
coprime.
Chromaticderivativesandseriesexpansionshaverecentlybeenintroducedasan
alternativerepresentationto Taylor series forbandlimitedfunctionsand they have
beenshowntobemoreusefulinpracticalapplicationsthanTaylorseries.Chromatic
series have similar properties to those of the Whittaker–Shannon–Kotel’nikov
samplingseries.InChapter6,A.Zayedgivesanoverviewofchromaticderivatives
and series in one and several variables and then use the Bargmann transform to
showthatfunctions,intheBargmann–Segal–FochspaceF,whichisareproducing-
kernel Hilbert Space of entire functions, can be represented by chromatic series
expansions.Asaresult,somepropertiesoftheBargmann–Segal–Fochspacecanbe
deducedfromthoseofchromaticseries.
In Chapter7, by D. Alpay, P. Jorgensen, I. Lewkowicz, and I. Marziano, the
authorsusefunctionalanalysistechniquestosolveinterpolationproblemsnotinthe
context of bandlimited functions or shift-invariantspaces but in the setting of the
HardyspaceH2(D)offunctionsanalyticintheopenunitdiskD.ThespaceH2(D)
playsanimportantroleincomplexanalysis,signalprocessing,andlineardynamical
∗
systems. Recently the Cuntz semigroups of C -algebras and the Cuntz relations
∗
for positive elements in a C -algebra have attracted some attention because of
theirnewlydiscoveredconnectionswithapplicationsinsignalprocessing,sub-band
filters,andwavelets,whichallfallwithinalargercontextofmultiscaleanalysis.
In this work the authors study the Cuntz relations in a different context. They
introduceconnectionsbetweentheCuntzrelationsandtheHardyspaceH2(D)and
thenusea decompositionofelementsin H2(D) associatedwithcertainisometries
which satisfy the Cuntz relations, to solve a new kind of multipoint interpolation
problem in H2(D) where for instance only a linear combination of the values
of a function at given points is preassigned, rather than the values at the points
themselves.
Chapter 8 by J. Benedetto and S. Datta deals with the autocorrelation of se-
quencesandtheconstructionofconstantamplitudezeroautocorrelation(CAZAC)
sequencesx on the integersZZ by meansof Hadamardmatrices. Recall that a real
Hadamardmatrixisasquarematrixwhoseentriesareeither+1or−1andwhose
rows are mutually orthogonal. First, the authors explain from a practical point of
view why constant amplitude and zero autocorrelation sequences are important.
The zero autocorrelation property ensures minimum interference between signals
sharingthesamechannel.
The authorsreview propertiesand problemsrelated to Hadamardmatrices and
then establish the relation between CAZAC sequences on ZZ/NZZ, Hadamard
matrices, and the discrete Fourier transform. They proceed to construct CAZAC
sequences on ZZ by means of Hadamard matrices and construct unimodular
functionsonZZ whoseautocorrelationsaretriangles.
PartII,MultiscaleAnalysis,consistsofChapters9–12.TheauthorsinChapter9
explain that chaos and random fractal theories, which have been used in the
analysis of complex data, are fundamentally two different theories. Chaos theory
