wavelets and filter banks theory and design the wavelet transform is compared with the more classical short time fourier transform approach to signal analysis then the relations between wavelets filter banks and multiresolution signal processing are explored a brief review is given of perfect reconstruction filter banks which can be used both for computing the discrete wavelet transform and for deriving continuous wavelet bases provided that the filters meet a constraint known as regularity given a low pass filter necessary and sufficient conditions for the existence of a complementary high pass filter that will permit perfect reconstruction are derived the perfect reconstruction condition is posed as a bezout identity and it is shown how it is possible to find all higher degree complementary filters based on an analogy with the theory of diophantine equations an alternative approach based on the theory of continued fractions is also given these results are used to design highly regular filter banks which generate biorthogonal continuous wavelet bases with symmetries

dynamic shakedown analysis and bounds for elastoplastic structures with nonassociative internal variable constitutive laws abstract reference is made to discrete finite element structural models formulated in terms of generalized variables the constitutive laws adopted are elastic plastic nonlinear hardening with internal variables and generally nonassociative the notion of reduced domain is employed as a generalization of a similar concept introduced in earlier developments of shakedown theory on this basis a unified theory is presented which encompasses necessary and separately sufficient conditions for shakedown by a static approach as a further generalization of classical melan s theorem bounds on various post shakedown quantities sufficient and necessary shakedown criteria by a kinematic approach as a further extension of neal symonds koiter theorem by suitable specializations and relaxations of the achievable results the criteria and bounding inequalities established here are formulated as mathematical programming problems in view of numerical applications

complementarity framework for non linear dynamic analysis of skeletal structures with softening plastic hinges in this paper we describe an algorithm for the incremental state update of elasto plastic systems with softening the algorithm uses a complementary pivoting technique and is based on casting the incremental state update as a complementarity problem in developing the algorithm we take advantage of the special features of solid and structural mechanics problems to achieve good computational performance and hence the ability to compute numerical solutions to practical size problems for example the notion of a tangent stiffness matrix arises numerical examples using models of skeletal structures are presented to demonstrate the practicability of the algorithm the numerical examples also raise some interesting questions about multiplicity of the solutions copyright 2010 john wiley sons ltd

harmonic wavelet analysis a new harmonic wavelet is suggested unlike wavelets generated by discrete dilation equations whose shape cannot be expressed in functional form harmonic wavelets have the simple structure w x exp i4 pi x exp i2 pi x i2 pi x this function w x is concentrated locally around x 0 and is orthogonal to its own unit translations and octave dilations its frequency spectrum is confined exactly to an octave band so that it is compact in the frequency domain rather than in the x domain an efficient implementation of a discrete transform using this wavelet is based on the fast fourier transform fft fourier coefficients are processed in octave bands to generate wavelet coefficients by an orthogonal transformation which is implemented by the fft the same process works backwards for the inverse transform

on local smoothing of nonparametric curve estimators abstract we develop new local versions of familiar smoothing methods such as cross validation and smoothed cross validation in the contexts of density estimation and regression these new methods are locally adaptive in the sense that they capture smooth local fluctuations in the curve by using smoothly varying bandwidths that change as the character of the curve changes moreover the new methods are accurate easy to apply and computationally expedient

the scattering of sound by two semi infinite parallel staggered plates ii evaluation of the velocity potential for an incident plane wave and an incident duct mode in a recent paper abrahams wickham showed how to exactly solve the diffraction problem for two parallel semi infinite plates that are arranged to form a duct the edges of the plates are misaligned so that the geometry is asymmetric with respect to the central line of the duct in this paper we examine in detail the effect of this asymmetry on the field radiated from the duct when a fundamental travelling mode is excited within it and the scattered field when a plane wave is excited at infinity to evaluate the field in either case it is first necessary to solve the integral equations derived in our paper here we show how this is effected using the expansion theorem also enunciated in that paper this solution is then used to calculate a number of physically interesting quantities such as the variation of the energy capture by the duct when irradiated by an incident plane wave as the stagger is increased and the asymmetric radiation pattern from an incident duct mode the numerical computations are checked using an energy balance argument and finally a simple heuristic approximation is given for the scattering problem that in many situations accurately reproduces the essential physical characteristics of the solution

stochastic joint time frequency response analysis of nonlinear structural systems abstract a novel approximate analytical approach for determining the response evolutionary power spectrum eps of nonlinear hysteretic structural systems subject to stochastic excitation is developed specifically relying on the theory of locally stationary processes and utilizing a recently proposed representation of non stationary stochastic processes via wavelets a versatile formula for determining the nonlinear system response eps is derived this is done in conjunction with a stochastic averaging treatment of the problem and by resorting to the orthogonality properties of harmonic wavelets further the nonlinear system non stationary response amplitude probability density function pdf which is required as input for the developed approach is determined either by utilizing a numerical path integral scheme or by employing a time dependent rayleigh pdf approximation technique a significant advantage of the approach relates to the fact that it is readily applicable for treating not only separable but non separable in time and frequency eps as well the hardening duffing and the versatile preisach hysteretic oscillators are considered in the numerical examples section comparisons with pertinent monte carlo simulations demonstrate the reliability of the approach

wavelet expansion based stochastic response of chain like mdof structures abstract this paper presents a wavelet expansion based approach for response determination of a chain like multi degree of freedom mdof structure subject to full non stationary stochastic excitations specifically the generalized harmonic wavelet ghw is first utilized as the expansion basis to solve the dynamic equation of structures via the galerkin treatment in this way a linear matrix relationship between the deterministic response and excitation can be derived further considering the ghw based representation of the stochastic processes a time varying power spectrum density psd relationship on a certain wavelet scale or frequency band between the excitation and response is derived finally pertinent numerical simulations including deterministic dynamic analysis and monte carlo simulations of both the response psd and the story drift based reliability are utilized to validate the proposed approach

on the controllability properties of circulant networks this paper examines the controllability of a group of first order agents adopting a weighted consensus type coordination protocol over a circulant network specifically it is shown that a circulant network with laplacian eigenvalues of maximum algebraic multiplicity q is controllable from q nodes our approach leverages on the cauchy binet formula which in conjunction with the popov belevitch hautus test leads to new insights on structural aspects of network controllability

an introduction to wavelets with applications to andrews plots a brief introduction to wavelets targeted to the statistician is given several types of wavelets are described and compared with the fourier functions it is shown how wavelets might extend the multivariate clustering technique of andrews plots

travelling waves on a membrane reflection and transmission at a corner of arbitrary angle ii this is the second part of an investigation into the reflection and transmission of fluid coupled membrane waves at a corner of arbitrary angle in part one of this work abrahams lawrie 1995 an exact solution was obtained for a model problem comprising of a fluid wedge of arbitrary angle 4 bounded by two identical semi infinite plane membranes and forced by a surface wave incident along one face of the wedge the problem was decomposed into a symmetric and an antisymmetric sub problem and closed form expressions for the reflection coefficients r s and r a respectively were derived the solution method incorporates several fundamental advancements on the work of maliuzhinets 1958 and offers a constructive approach by which wedge problems with higher order boundary conditions can be solved easily in this part of the investigation it is demonstrated how for rational wedge angles the formulae of part i can be exploited to yield simple exact or asymptotic expressions for r s r a and therefore the reflection and transmission coefficients for the full problem further a numerical implementation of the analytic solution enables these coefficients to be determined for all wedge angles 0 both for heavy and moderate fluid loading the results confirm the reflection coefficients known previously for a few special wedge angles and highlight several interesting trends in particular it is found that in the heavy fluid loading limit there is a remarkably simple relationship between the phases of r s r a and

wavelet processes and adaptive estimation of the evolutionary wavelet spectrum this article defines and studies a new class of non stationary random processes constructed from discrete non decimated wavelets which generalizes the cramer fourier representation of stationary time series we define an evolutionary wavelet spectrum ews which quantifies how process power varies locally over time and scale we show how the ews may be rigorously estimated by a smoothed wavelet periodogram and how both these quantities may be inverted to provide an estimable time localized autocovariance we illustrate our theory with a pedagogical example based on discrete non decimated haar wavelets and also a real medical time series example

harmonic wavelet based data filtering for enhanced machine defect identification a filter construction technique is presented for enhanced defect identification in rotary machine systems based on the generalized harmonic wavelet transform a series of sub frequency band wavelet coefficients are constructed by choosing different harmonic wavelet parameter pairs the energy and entropy associated with each sub frequency band are then calculated the filtered signal is obtained by choosing the wavelet coefficients whose corresponding sub frequency band has the maximum energy to entropy ratio experimental studies using rolling bearings that contain different types of structural defects have confirmed that the developed new technique enables high signal to noise ratio for effective machine defect identification

wavelet shrinkage using cross validation summary wavelets are orthonormal basis functions with special properties that show potential in many areas of mathematics and statistics this paper concentrates on the estimation of functions and images from noisy data by using wavelet shrinkage a modified form of twofold cross validation is introduced to choose a threshold for wavelet shrinkage estimators operating on data sets of length a power of 2 the cross validation algorithm is then extended to data sets of any length and to multidimensional data sets the algorithms are compared with established threshold choosers by using simulation an application to a real data set arising from anaesthesia is presented

general solution for a class of diffraction problems using a special mathematical technique previously developed by malyuzhinets the general solution of two dimensional plane wave diffraction problems in an angular region of any angle with boundary conditions containing spatial derivatives of arbitrary orders is constructed

effects of softening in elastic plastic structural dynamics softening understood as unstable behaviour of material or structural components is considered herein for its possible consequences on the overall behaviour of discrete dynamic models of elastic plastic beam structures in the absence of geometric effects it is shown that multiplicity of incremental solutions response bifurcations and manifestations of overall instability may occur the bifurcated responses may exhibit different scenarios for the same excitation e g shakedown or local damage up to failure an insight into and criteria for such occurrences are achieved by formulating the rate and the finite increment problem as linear complementarity problems period doubling and deterministic chaos under harmonic excitation are observed when reversible holonomic softening behaviour is assumed the strong sensitivity of the phenomena investigated with respect to the choice of the structural model is pointed out the findings are elucidated by illustrative examples

sets of matrices all infinite products of which converge an infinite product iit lmi of matrices converges on the right if limi m mi exists a set z ai i l of n x n matrices is called an rcp set rightconvergent product set if all infinite products with each element drawn from z converge such sets of matrices arise in constructing self similar objects like von koch s snowflake curve in various interpolation schemes in constructing wavelets of compact support and in studying nonhomogeneous markov chains this paper gives necessary conditions and also some sufficient conditions for a set x to be an rcp set these are conditions on the eigenvalues and left eigenspaces of matrices in 2 and finite products of these matrices necessary and sufficient conditions are given for a finite set z to be an rcp set having a limit function m d rit lad where d d d which is a continuous function on the space of all sequences d with the sequence topology finite rcp sets of column stochastic matrices are completely characterized some results are given on the problem of algorithmically deciding if a given set x is an rcp set

quasi linear wavelet estimation abstract the main paradigm of the modern wavelet theory of spatial adaptation formulated by donoho and johnstone is that there is a divergence between the linear minimax adaptation theory and the heuristic guiding algorithm development that leads to the necessity of using strongly nonlinear adaptive thresholded methods on the other hand it is well known that linear adaptive estimates are the best whenever an estimated function is smooth is it possible to suggest a quasi linear wavelet estimate by adding to a linear adaptive estimate a minimal number of nonlinear terms on finest scales that offers advantages of linear adaptive estimates and at the same time matches asymptotic properties of strongly nonlinear procedures like the benchmark sureshrink the answer is yes and we discuss quasi linear estimation both theoretically and via a monte carlo study in particular i show that asymptotically a quasi linear procedure not only matches properties of sureshrink over the besov scale but also allows

diffraction by a thin dielectric half plane a uniform diffraction coefficient is derived for a thin dielectric half plane illuminated by an h polarized incident plane wave this is accomplished by modeling the half plane using a pair of resistive and modified conductive sheets and then solving the resulting integral equations via the wiener hopf technique in contrast to a recently published solution the one presented here satisfies reciprocity and the numerical results which are included confirm its accuracy

adaptive harmonic wavelet transform with applications in vibration analysis an adaptive harmonic wavelet transform is developed by taking advantage of the flexibility of the generalized harmonic wavelets it first constructs a partition tree which contains a great number of disjoint partitions of the frequency axis of a signal with each corresponding to an orthogonal harmonic wavelet basis then it searches the tree for the partition to represent the signal most sparsely since the corresponding basis is adapted to the composition of the signal the transform can well reveal its characteristics this is demonstrated with analysis examples of some simulated and vibration signals as well as comparisons with the conventional orthogonal harmonic wavelet transforms and wavelet packet transform

wavelet transforms versus fourier transforms this note is a very basic introduction to wavelets it starts with an orthogonal basis of piecewise constant functions constructed by dilation and translation the wavelet transform maps each f x to its coefficients with respect to this basis the mathematics is simple and the transform is fast faster than the fast fourier transform which we briefly explain but approximation by piecewise constants is poor to improve this first wavelet we are led to dilation equations and their unusual solutions higher order wavelets are constructed and it is surprisingly quick to compute with them always indirectly and recursively

fast wavelet transforms and numerical algorithms i a class of algorithms is introduced for the rapid numerical application of a class of linear operators to arbitrary vectors previously published schemes of this type utilize detailed analytical information about the operators being applied and are specific to extremely narrow classes of matrices in contrast the methods presented here are based on the recently developed theory of wavelets and are applicable to all calderon zygmund and pseudo differential operators the algorithms of this paper require order o n or o n log n operations to apply an n n matrix to a vector depending on the particular operator and the version of the algorithm being used and our numerical experiments indicate that many previously intractable problems become manageable with the techniques presented here

on uniqueness of the dynamic finite step problem in gradient dependent softening plasticity the dynamic evolution of an elastoplastic softening solid is considered a material model including in the yield function the laplacian of the plastic multiplier is used to regularize the problem the dynamic finite step problem is formulated according to a generalized mid point integration scheme space discretization is carried out by a mixed finite element technique based on generalized variables a sufficient uniqueness condition of the finite step solution is proved for a one dimensional problem also a necessary and sufficient condition is presented a simple numerical test shows the regularizing properties mesh independence of the proposed model and the positive influence of the gradient term also on the time step amplitude ensuring uniqueness of solution

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

a multiscale wavelet solver with o n complexity in this paper we use the biorthogonal wavelets recently constructed by dahlke and weinreich to implement a highly efficient procedure for solving a certain class of one dimensional problems 21 x21 u f i z i 0 for these problems the discrete biorthogonal wavelet transform allows us to set up a system of wavelet galerkin equations in which the scales are uncoupled so that a true multiscale solution procedure may be formulated we prove that the resulting stiffness matrix is in fact an almost perfectly diagonal matrix the original aim of the construction was to achieve a block diagonal structure and we show that this leads to an algorithm whose cost is o n we also present numerical results which demonstrate that the multiscale biorthogonal wavelet algorithm is superior to the more conventional single scale orthogonal wavelet approach both in terms of speed and in terms of convergence

acoustic diffraction by a fluid loaded membrane corner the complete solution is derived for an acoustic plane wave incident on a membrane which is stretched over a line constraint forming a wedge shaped fluid region of interior angle 2 the exact two dimensional solution is developed using malyuzhinets functions and integral transforms explicit formulae are presented for the acoustic and structural diffraction coefficients for 2 and numerical results are given for the frequency and angular dependence of the membrane wave excitation coefficients for identical membranes on either side of the vertex

wavelets and dilation equations a brief introduction wavelets are new families of basis functions that yield the representation f x sum b jk w 2 j x k their construction begins with the solution phi x to a dilation equation with coefficients c k then w comes from phi and the basis comes by translation and dilation of w it is shown in part 1 how conditions on the c k lead to approximation properties and orthogonality properties of the wavelets part 2 describes the recursive algorithms also based on the c k that decompose and reconstruct f the object of wavelets is to localize as far as possible in both time and frequency with efficient algorithms

diffraction by a symmetric material junction simulated with generalized sheet transition conditions when diffraction by a material discontinuity in a thick dielectric ferrite slab is treated by the dual integral equation approach a variation of the weiner hopf method the resulting solution displays unknown constants characteristic of the incompleteness associated with the generalized transition conditions employed in modeling a thick slab the constants can be shown to depend on the geometry and properties of the discontinuity their explicit determination by enforcing field continuity across the slab junction is treated the field internal to the slab is determined by first expressing the field as an expansion of discrete and continuous eigenmodes valid in the exterior and interior slab region the expansion or mode coefficients are then identified by recasting the weiner hopf solution in a suitable form enabling the solution constants to be numerically computer by matching the field expansions to the left and right of the slab discontinuity at a discrete set of points

harmonic and musical wavelets the concept of a harmonic wavelet is generalized to describe a family of mixed wavelets with the structure w m n x exp i n 2 x exp i m 2 x i n m 2 x it is shown that this family provides a complete set of orthogonal basis functions for signal analysis by choosing the real numbers m and n not necessarily integers appropriately wavelets whose frequency content ascends according to the musical scale can be generated these musical wavelets provide greater frequency discrimination than is possible with harmonic wavelets whose frequency interval is always an octave an example of the wavelet analysis of music illustrates possible applications

short wavelets and matrix dilation equations scaling functions and orthogonal wavelets are created from the coefficients of a lowpass and highpass filter in a two band orthogonal filter bank for multifilters those coefficients are matrices this gives a new block structure for the filter bank and leads to multiple scaling functions and wavelets geronimo hardin and massopust see j approx theory vol 78 p 373 401 1994 constructed two scaling functions that have extra properties not previously achieved the functions spl phi sub 1 and spl phi sub 2 are symmetric linear phase and they have short support two intervals or less while their translates form an orthogonal family for any single function spl phi apart from haar s piecewise constants those extra properties are known to be impossible the novelty is to introduce 2 spl times 2 matrix coefficients while retaining orthogonality of the multiwavelets this note derives the properties of spl phi sub 1 and spl phi sub 2 from the matrix dilation equation that they satisfy then our main step is to construct associated wavelets two wavelets for two scaling functions the properties were derived by geronimo hardin and massopust from the iterated interpolation that led to spl phi 1 and spl phi sub 2 one pair of wavelets was found earlier by direct solution of the orthogonality conditions using mathematica our construction is in parallel with recent progress by hardin and geronimo to develop the underlying algebra from the matrix coefficients in the dilation equation in another language to build the 4 spl times 4 paraunitary polyphase matrix in the filter bank the short support opens new possibilities for applications of filters and wavelets near boundaries

bounded semigroups of matrices in this note are proved two conjectures of daubechies and lagarias the first asserts that if z is a bounded set of matrices such that all left infinite products converge then 8 generates a bounded semigroup the second asserts the equality of two differently defined joint spectral radii for a bounded set of matrices one definition involves the conventional spectral radius and one involves the operator norm

computational algorithms for multiscale identification of nonlinearities in hammerstein systems with random inputs simple and efficient computational algorithms for nonparametric wavelet based identification of nonlinearities in hammerstein systems driven by random signals are proposed they exploit binary grid interpolations of compactly supported wavelet functions the main contribution consists in showing how to use the wavelet values from the binary grid together with the fast wavelet algorithms to obtain the practical counterparts of the wavelet based estimates for irregularly and randomly spaced data without any loss of the asymptotic accuracy the convergence and the rates of convergence are examined for the new algorithms and in particular conditions for the optimal convergence speed are presented efficiency of the algorithms for a finite number of data is also illustrated by means of the computer simulations

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed