localization of the generalized sampling series and its numerical application we study the localization of the sampling series with general kernel and obtain its error estimates to achieve exponentially decaying accuracy for the series to approximate a band limited function and its derivatives a sufficient condition is given and a rigorous criterion is provided for the sampling to obtain any desired accuracy the result includes several known results as special cases and gives new results about wavelet sampling series

on simple oversampled a d conversion in shift invariant spaces it has been found that the quantization error e for a conventional oversampled analog to digital a d conversion behaves like spl par e spl par sup 2 o spl tau sup 2 with respect to the sampling rate spl tau recently conventional a d conversion has been extended to a d conversion based on shift invariant spaces as consequences of such extension it offers rich choices to build a nonideal a d conversion system of high accuracy and low computational complexity as well as reduces the noise sensitivity and computational complexity in digital to analog d a conversion therefore it is necessary to establish the estimate of quantization error for the extended a d conversion based on shift invariant spaces in this paper we introduce a constructive method to establish an estimate of the quantization error as e sup 2 o spl tau sup 2 for oversampled a d conversion in shift invariant spaces meanwhile we demonstrate that the bit rate required to encode the converted digital signal in such a d conversion scheme only increases as the logarithm of the sampling ratio therefore the quantization error is an exponentially decaying function of the bit rate in order to establish such an estimate we need the nonuniform sampling theorem for shift invariant spaces which as the necessary preparation is studied prior to introducing the constructive method

reproducing kernel structure and sampling on time warped spaces with application to warped wavelets time warped signal spaces have received attention in the literature among the topics of particular interest are sampling of time warped signals and signal analysis using warped analysis functions including wavelets this correspondence introduces a reproducing kernel rk structure for time warped signal spaces that unifies multiple perspectives on sampling in such spaces

nonuniform sampling and reconstruction in shift invariant spaces this article discusses modern techniques for nonuniform sampling and reconstruction of functions in shift invariant spaces it is a survey as well as a research paper and provides a unified framework for uniform and nonuniform sampling and reconstruction in shift invariant subspaces by bringing together wavelet theory frame theory reproducing kernel hilbert spaces approximation theory amalgam spaces and sampling inspired by applications taken from communication astronomy and medicine the following aspects will be emphasized a the sampling problem is well defined within the setting of shift invariant spaces b the general theory works in arbitrary dimension and for a broad class of generators c the reconstruction of a function from any sufficiently dense nonuniform sampling set is obtained by efficient iterative algorithms these algorithms converge geometrically and are robust in the presence of noise d to model the natural decay conditions of real signals and images the sampling theory is developed in weighted l p spaces

exponential weighting algorithms for reconstruction of bandlimited signals the problem of reconstruction of bandlimited signals from discrete noisy observations is considered whittaker shannon interpolation series based estimates involving the weighting factors are proposed it is shown that they converge as the sampling rate increases to infinity the rate of convergence of the double exponential weighting algorithm is established these results are corroborated in computer simulations

an overview of wavelet based multiresolution analyses in this paper an overview of wavelet based multiresolution analyses is presented first the continuous wavelet transform in its simplest form is discussed then the definition of a multiresolution analysis is given and how wavelets fit into it is shown the authors take a closer look at orthogonal biorthogonal and semiorthogonal wavelets the fast wavelet transform wavelets on an interval multidimensional wavelets and wavelet packets are discussed several examples of wavelet families are introduced and compared finally the essentials of two major applications are outlined data compression and compression of linear operators

generalized sampling a variational approach i theory we consider the problem of reconstructing a multidimensional vector function f sub in spl ropf sup m spl rarr spl ropf sup n from a finite set of linear measures these can be irregularly sampled responses of several linear filters traditional approaches reconstruct in an a priori given space e g the space of bandlimited functions instead we have chosen to specify a reconstruction that is optimal in the sense of a quadratic plausibility criterion j first we present the solution of the generalized interpolation problem later we also consider the approximation problem and we show that both lead to the same class of solutions imposing generally desirable properties on the reconstruction largely limits the choice of the criterion j linearity leads to a quadratic criterion based on bilinear forms specifically we show that the requirements of translation rotation and scale invariance restrict the form of the criterion to essentially a one parameter family we show that the solution can be obtained as a linear combination of generating functions we provide analytical techniques to find these functions and the solution itself

robust control with respect to coprime factors of infinite dimensional positive real systems it is shown that infinite dimensional positive real systems can be robustly stabilized with respect to coprime factor perturbations with a robustness margin of at least 1 square root 2 this result is applied to dissipative colocated systems which arise in pde models of flexible structures

free vibration analysis of orthotropic elliptical plates resting on arbitrarily distributed point supports the free vibration of point supported elliptical plates is studied analytically with consideration of rectangular orthotropy apparently this problem has been previously unsolved in the literature an analytical method used is an accurate variational approach the ritz lagrange multiplier method which constitutes an extension of the classical ritz method in numerical examples elliptical plates on some symmetric point supports are considered and it is shown that the method is straightforwardly applicable to determination of frequencies for elliptical plates resting on various support configurations

kernel induced sampling theorem a perfect reconstruction of functions in a reproducing kernel hilbert space from a given set of sampling points is discussed a necessary and sufficient condition for the corresponding reproducing kernel and the given set of sampling points to perfectly recover the functions is obtained in this paper the key idea of our work is adopting the reproducing kernel hilbert space corresponding to the gramian matrix of the kernel and the given set of sampling points as the range space of a sampling operator and considering the orthogonal projector defined via the range space onto the closed linear subspace spanned by the kernel functions corresponding to the given sampling points we also give an error analysis of a reconstructed function by incomplete sampling points

design of optimum nyquist signals based on generalized sampling theory for data communications a new method is given for the optimal design of bandlimited nyquist type signal shapes for data communications which maximizes its energy in a given time interval the method is based on the periodically nonuniform sampling pns theory making use of the linear splines the computation is straightforward and the constraint for intersymbol interference is shown to be easy to include in the problem a numerical example is given and performance of the optimal signal shapes is compared with that resulting from the use of previously obtained signal shapes in the literature it is also concluded that the optimal signal shapes thus obtained are almost immune to small offsets at the sampling instants

a sampling theorem for wavelet subspaces the classical shannon sampling theorem is extended to the subspaces used in the multiresolution analysis in wavelet theory under weak hypotheses these subspaces are first shown to have a riesz basis formed from the reproducing kernels these in turn are used to construct the sampling sequences examples are given

vector sampling expansions in shift invariant subspaces multi input multi output mimo sampling scheme which is motivated by applications in multi channel deconvolution and multi source separation has been investigated in many aspects common for most of results on mimo systems is that the input signals are supposed to be band limited in this paper we study vector sampling expansions on general finitely generated shift invariant subspaces necessary and sufficient conditions for a vector sampling theorem to hold are given we also give several examples to illustrate the main result

moving average restoration of bandlimited signals from noisy observations the purpose of this paper is to describe the extension of the whittaker shannon sampling theorem to reconstruction of bandlimited functions in the presence of zero mean uncorrelated noise it is shown that the classical whittaker shannon sampling scheme is not consistent in the case of noisy measurements and new reconstruction algorithms based on the moving average smoothing are proposed the weak and strong consistency of the algorithms is established and the rate of convergence is investigated the theory is verified in the computer simulations

m band compactly supported orthogonal symmetric interpolating scaling functions in many applications wavelets are usually expected to have the following properties compact support orthogonality linear phase regularity and interpolation to construct such wavelets it is crucial designing scaling functions with the above properties in two and three band cases except for the haar functions there exists no scaling function with the above five properties in m band case m spl ges 4 more free degrees available in design enable us to construct such scaling functions a novel approach to designing such scaling functions is proposed first we extend the two band dubuc 1986 filters to the m band case next the m band fir regular symmetric interpolating scaling filters are parameterized and then m band fir regular orthogonal symmetric interpolating scaling filters osisfs are designed via optimal selection of parameters finally two family of four band and five band osisfs and scaling functions are developed and their smoothness are estimated

error analysis of frame reconstruction from noisy samples this paper addresses the problem of reconstructing a continuous function defined on rd from a countable collection of samples corrupted by noise the additive noise is assumed to be i i d with mean zero and variance sigma2 we sample the continuous function on the uniform lattice 1 m zd and show for large enough m that the variance of the error between the frame reconstruction fepsiv m from noisy samples of f and the function f satisfy var fepsiv m x f x ap sigma2 md cx where cx is the best constant for every x isin rd we also prove a similar result in the case that our data are weighted average samples of corrupted by additive noise

sampling theorem for wavelet subspaces error estimate and irregular sampling the error estimate is useful in the application of the sampling theorem for the classical shannon sampling theorem various errors are widely studied but for the sampling theorem in general wavelet subspaces only the aliasing error is studied in this paper we study three other errors truncation error amplitude error and time jitter error with the same technique a result on irregular sampling is improved

sampling theorem and irregular sampling theorem for multiwavelet subspaces we generalize the multiwavelet sampling theorem by reproducing a kernel that is easy to use multiscaling functions that have orthogonality regularity short compact support symmetry and high approximation order satisfy the conditions in the sampling theorem which is not possible in the case of the scalar wavelet we consider the general cases of the uniform noninteger and irregular sampling a reconstruction from more general sets of points is necessary since the measurements may not be made at uniform points finally we establish a general irregular sampling theorem for multiwavelet subspaces and derive an estimate for the perturbations of uniform noninteger sampling in shift invariant spaces

signal extrapolation in wavelet subspaces the papoulis gerchberg pg algorithm is well known for band limited signal extrapolation the authors consider the generalization of the pg algorithm to signals in the wavelet subspaces in this research the uniqueness of the extrapolation for continuous time signals is examined and sufficient conditions on signals and wavelet bases for the generalized pg gpg algorithm to converge are given a discrete gpg algorithm is proposed for discrete time signal extrapolation and its convergence is investigated numerical examples are given to illustrate the performance of the discrete gpg algorithm

vector valued wavelets and vector filter banks in this paper we introduce vector valued multiresolution analysis and vector valued wavelets for vector valued signal spaces we construct vector valued wavelets by using paraunitary vector filter bank theory in particular we construct vector valued meyer wavelets that are band limited we classify and construct vector valued wavelets with sampling property as an application of vector valued wavelets multiwavelets can be constructed from vector valued wavelets we show that certain linear combinations of known scalar valued wavelets may yield multiwavelets we then present discrete vector wavelet transforms for discrete time vector valued or blocked signals which can be thought of as a family of unitary vector transforms

oversampling in shift invariant spaces with a rational sampling period it is well known that under appropriate hypotheses a sampling formula allows us to recover any function in a principal shift invariant space from its samples taken with sampling period one whenever the generator of the shift invariant space satisfies the strang fix conditions of order r this formula also provides an approximation scheme of order r valid for smooth functions in this paper we obtain sampling formulas sharing the same features by using a rational sampling period less than one with the use of this oversampling technique there is not one but an infinite number of sampling formulas whenever the generator has compact support among these formulas it is possible to find one whose associated reconstruction functions have also compact support

efficient wavelet prefilters with optimal time shifts a wavelet prefilter maps sample values of an analyzed signal to the scaling function coefficient input of standard discrete wavelet transform dwt algorithms the prefilter is the inverse of a certain postfilter convolution matrix consisting of integer sample values of a noninteger shifted wavelet scaling function for the prefilter and the dwt algorithms to have similar computational complexity it is often necessary to use a short enough approximation of the prefilter in addition to well known quadrature formula and identity matrix prefilter approximations we propose a neumann series approximation which is a band matrix truncation of the optimal prefilter and derive simple formulas for the operator norm approximation error this error shows a dramatic dependence on how the postfilter noninteger shift is chosen we explain the meaning of this shift in practical applications describe how to choose it and plot optimally shifted prefilter approximation errors for 95 different daubechies symlet and b spline wavelets whereas the truncated inverse is overall superior the neumann filters are by far the easiest ones to compute and for some short support wavelets they also give the smallest approximation error for example for daubechies 1 5 wavelets the simplest neumann prefilter provide an approximation error reduction corresponding to 100 10 000 times oversampling in a nonprefiltered system

a sampling theorem for shift invariant subspace a sampling theorem for regular sampling in shift invariant subspaces is established the sufficient necessary condition for which it holds is found then the theorem is modified to the shift sampling in shift invariant subspaces by using the zak transform finally some examples are presented to show the generality of the theorem

on the aliasing error in wavelet subspaces shannon s sampling formula has been extended to subspaces v j j z in multiresolution analysis when we apply the corresponding sampling formula for v 0 to a function which does not belong to this space the so called aliasing error arises this paper deals with the aliasing error ef whenever f is in v 1 we obtain an expression for its fourier transform ef which allows us to give estimations of the aliasing error both in l 2 r norm and in l r norm finally we carry out a study of the aliasing error in the case when f belongs to v 2

estimate of aliasing error for non smooth signals prefiltered by quasi projections into shift invariant spaces an ideal analog to digital a d converter prefilters a signal by an ideal lowpass filter recent research on a d conversion based on shift invariant spaces reveals that prefiltering signals by quasiprojections into shift invariant spaces provides more flexible choices in designing an a d conversion system of high accuracy this paper focuses on the accuracy of such prefiltering in which the aliasing error e sub f sup spl lambda is found to behave like e sub f sup spl lambda sub 2 o spl lambda sup spl alpha with respect to the dilation spl lambda of the underlying shift invariant space provided that the input signal f is lipschitz spl alpha continuous a formula to calculate the coefficient of the decay rate is also figured out in this paper

translation invariance and sampling theorem of wavelet the sampling theorem for wavelet spaces built by walter 1992 lacks the translation invariance except for walter s weak translation invariant wavelet i e meyer s wavelet indeed we must know a priori the shift offset a in the samples f n a n spl isin z otherwise the waveform cannot be recovered since the interpolation function is dependent on this offset in this correspondence we generalize our metric functional to metrize weak shiftability and find a somewhat surprising result that the b spline wavelets of order n spl ges 3 are degenerate shiftable thus we can recover approximately the waveform by double sampling without any information on shift offset a

m band scaling function with filter having vanishing moments two and minimal length abstract in this paper we consider the holder continuity local linearity linear independence and interpolation problem of the m band scaling function with a filter having vanishing moments two and minimal length in particular we find some new properties that are not true when m 2 such as local linearity local linear dependence differentiability at adjoint m adic points and interpolation problem at integer knots

computation of wavelet coefficients from average samples there exist efficient methods to compute the wavelet coefficients of a function f t from its point samples f t email protected email protected n however in many applications the available samples are average samples of the type f t email protected u t dt where the averaging function u t reflects the characteristic of the acquisition device in this work methods to compute the coefficients in a biorthogonal wavelet system from average samples are studied error estimations are obtained and using them the optimal values for the parameters in the proposed approximation rules are calculated the obtained error estimations can also be applied to the rules that compute the coefficients from point samples and thus these estimations can be used to compare and to choose between the different methods proposed in the literature the methods proposed here also allow us to compute the biorthogonal wavelet coefficients from the coefficients in another biorthogonal wavelet system

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

continuous and discrete wavelet transforms this paper is an expository survey of results on integral representations and discrete sum expansions of functions in l 2 bf r in terms of coherent states two types of coherent states are considered weyl heisenberg coherent states which arise from translations and modulations of a single function and affine coherent states called wavelets which arise as translations and dilations of a single function in each case it is shown how to represent any function in l 2 bf r as a sum or integral of these states most of the paper is a survey of literature most notably the work of i daubechies a grossmann and j morlet a few results of the authors are included

oversampling and reconstruction functions with compact support assume that a sequence of samples of a filtered version of a function in a shift invariant space is given this paper deals with the existence of a sampling formula involving these samples and having reconstruction functions with compact support this is done in the light of the generalized sampling theory by using the oversampling technique a necessary and sufficient condition is given in terms of the smith canonical form of a polynomial matrix finally we prove that the aforesaid oversampled formulas provide nice approximation schemes with respect to the uniform norm

biorthogonal partners and applications two digital filters h z and f z are said to be biorthogonal partners of each other if their cascade h z f z satisfies the nyquist or zero crossing property biorthogonal partners arise in many different contexts such as filterbank theory exact and least squares digital interpolation and multiresolution theory they also play a central role in the theory of equalization especially fractionally spaced equalizers in digital communications we first develop several theoretical properties of biorthogonal partners we also develop conditions for the existence of biorthogonal partners and fir biorthogonal pairs and establish the connections to the riesz basis property we then explain how these results play a role in many of the above mentioned applications

on sampling theorem wavelets and wavelet transforms the classical shannon sampling theorem has resulted in many applications and generalizations from a multiresolution point of view it provides the sine scaling function in this case for a band limited signal its wavelet series transform wst coefficients below a certain resolution level can be exactly obtained from the samples with a sampling rate higher than the nyquist rate the authors study the properties of cardinal orthogonal scaling functions cosf which provide the standard sampling theorem in multiresolution spaces with scaling functions as interpolants they show that cosf with compact support have and only have one possibility which is the haar pulse they present a family of cosf with exponential decay which are generalizations of the haar function with these cosf an application is the computation of wst coefficients of a signal by the mallat 1989 algorithm they present some numerical comparisons for different scaling functions to illustrate the advantage of cosf for signals which are not in multiresolution spaces they estimate the aliasing error in the sampling theorem by using uniform samples

the cardinal orthogonal scaling function and sampling theorem in the wavelet subspaces in this paper we derive that there is a relation between the lowpass filter coefficient and wavelet s samples in its integer points when a scaling function is a cardinal orthogonal scaling function and we give some examples in which the lowpass filter coefficients are constructed from the wavelets then we discuss the symmetry property of cardinal orthogonal scaling function and give some useful characterizations therefore we construct a family of cardinal orthogonal scaling functions with exponential decay and higher approximation order furthermore some examples with exponential decay and higher approximation order are given and they will testify our results in the end we deduce that there no exists a cardinal orthogonal scaling function with exponential decay high approximation order and symmetry property in this family of cardinal orthogonal scaling functions constructed in our paper

generalized sampling theorems in multiresolution subspaces it is well known that under very mild conditions on the scaling function multiresolution subspaces are reproducing kernel hilbert spaces rkhss this allows for the development of a sampling theory in this paper we extend the existing sampling theory for wavelet subspaces in several directions we consider periodically nonuniform sampling sampling of a function and its derivatives oversampling multiband sampling and reconstruction from local averages all these problems are treated in a unified way using the perfect reconstruction pr filter bank theory we give conditions for stable reconstructions in each of these cases sampling theorems developed in the past do not allow the scaling function and the synthesizing function to be both compactly supported except in trivial cases this restriction no longer applies for the generalizations we study here due to the existence of fir pr banks in fact with nonuniform sampling oversampling and reconstruction from local averages we can guarantee compactly supported synthesizing functions moreover local averaging schemes have additional nice properties robustness to the input noise and compression capabilities we also show that some of the proposed methods can be used for efficient computation of inner products in multiresolution analysis after this we extend the sampling theory to random processes we require autocorrelation functions to belong to some subspace related to wavelet subspaces it turns out that we cannot recover random processes themselves unless they are bandlimited but only their power spectral density functions we consider both uniform and nonuniform sampling

on sampling in shift invariant spaces a necessary and sufficient condition for sampling in the general framework of shift invariant spaces is derived then this result is applied respectively to the regular sampling and the perturbation of regular sampling in shift invariant spaces a simple necessary and sufficient condition for regular sampling in shift invariant spaces is attained furthermore an improved estimate for the perturbation is derived for the perturbation of regular sampling in shift invariant spaces the derived estimate is easy to calculate and shown to be optimal in some shift invariant spaces the algorithm to calculate the reconstruction frame is also presented

approximating functions from sampled fourier data using spline pseudofilters recently new polynomial approximation formulas were proposed for the reconstruction of compactly supported piecewise smooth functions from fourier data formulas for zero and first degree polynomials were presented for higher degree approximations polynomial formulas become extremely complicated to be handled in this paper we solve this problem by introducing spline approximations the new approach can be used in the same way as the polynomial one but producing computable formulas for any degree of approximation in fourier reconstruction we present general error estimates and numerical experiments

on restoring band limited signals the problem of reconstruction of band limited signals from discrete and noisy data is studied the reconstruction schemes employing cardinal expansions are proposed and their asymptotical properties are examined in particular the conditions for the convergence of the mean integrated squared error are found and the rate of convergence is evaluated the main difference between the proposed reconstruction scheme and the classical one is in treating the sampling rate and the reconstruction rate differently this distinction is necessary to ensure consistency of the reconstruction scheme in the presence of noise

sampling and reconstruction of signals in function spaces associated with the linear canonical transform the linear canonical transform lct has been shown to be useful and powerful in signal processing optics etc many results of this transform are already known including sampling theory most existing sampling theories of the lct consider the class of bandlimited signals however in the real world many analog signals arising in engineering applications are non bandlimited in this correspondence we propose a sampling and reconstruction strategy for a class of function spaces associated with the lct which can provide a suitable and realistic model for real applications first we introduce definitions of semi and fully discrete convolutions for the lct then we derive necessary and sufficient conditions pertaining to the lct under which integer shifts of a chirp modulated function generate a riesz basis for the function spaces by applying the results we present a more comprehensive sampling theory for the lct in the function spaces and further a sampling theorem which recovers a signal from its own samples in the function spaces is established moreover some sampling theorems for shift invariant spaces and some existing sampling theories for bandlimited signals associated with the fourier transform ft the fractional ft or the lct are noted as special cases of the derived results finally some potential applications of the derived theory are presented