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

an adaptive finite element method with perfectly matched absorbing layers for the wave scattering by periodic structures we develop a finite element adaptive strategy with error control for the wave scattering by periodic structures the unbounded computational domain is truncated to a bounded one by an extension of the perfectly matched layer pml technique which attenuates both the outgoing and evanescent waves in the pml region pml parameters such as the thickness of the layer and the medium property are determined through sharp a posteriori error estimates numerical experiments are included to illustrate the competitive behavior of the proposed adaptive method

peak value estimation of bandlimited signals from their samples noise enhancement and a local characterization in the neighborhood of an extremum the paper addresses the problem of estimating the peak value of bandlimited signals from their samples with and without oversampling this problem has significant relevance to orthogonal frequency division multiplexing ofdm signal processing and system design in particular an upper bound on the peak value is established given the peak value of the samples and the oversampling rate moreover it is shown that the bounds are sharp for all practical rates by constructing bandlimited signals taking on this bound the proof also provides a local characterization of bandlimited signals in the neighborhood of an extremum a different analysis examines the effect of small errors in the samples it is shown that oversampling can provide robust recovery in the sense that small errors in the samples lead to small errors in the reconstructed signal again an upper bound is derived relating the peak error in the samples and the peak error in the signals furthermore both problems are shown to be coupled and put in a unifying context the bounds are compared and applied to problems concerning ofdm

digital representations of operators on band limited random signals sampling representations of bounded linear operators blos acting on extended classes of band limited signals are derived these are classes of deterministic signals that are not necessarily square integrable and random signals with covariance functions that are not necessarily square integrable on the plane these classes consist of signals whose fourier transforms defined as generalized functions or distributions have supports that are subsets of compact sets the merit of these representations lies in the fact that the image of a signal under an operator may be reconstructed from the samples of the original signal rather than the samples of the image for example the derivative of a signal of any order may be expressed in terms of the samples of the signal itself instead of the samples of the derivative

sampling theory for not necessarily band limited functions a historical overview shannon s sampling theorem is one of the most powerful results in signal analysis the aim of this overview is to show that one of its roots is a basic paper of de la vallee poussin of 1908 the historical development of sampling theory from 1908 to the present especially the matter dealing with not necessarily band limited functions which includes the duration limited case actually studied in 1908 is sketched emphasis is put on the study of error estimates as well as on the delicate point wise behavior of sampling sums at discontinuity points of the signal to be reconstructed

boundedness behavior of the spectral factorization for polynomial data in the wiener algebra spectral factorization is of fundamental importance in many areas of signal processing this paper investigates the boundedness behavior of the spectral factorization mapping in the wiener algebra thereby the focus lies on the factorization of polynomial spectral densities with a finite degree n since such spectra are especially important for practical applications the paper presents a lower and an upper bound on the boundedness behavior which will show that the boundedness constant of the spectral factorization mapping gets worse as the degree n of the spectra increases therewith one obtains independently the known result that the spectral factorization mapping is unbounded on the wiener algebra

a galerkin least squares finite element method for the two dimensional helmholtz equation in this paper a galerkin least squares gls finite element method in which residuals in least squares form are added to the standard galerkin variational equation is developed to solve the helmholtz equation in two dimensions an important feature of gls methods is the introduction of a local mesh parameter that may be designed to provide accurate solutions with relatively coarse meshes previous work has accomplished this for the one dimensional helmholtz equation using dispersion analysis in this paper the selection of the gls mesh parameter for two dimensions is considered and leads to elements that exhibit improved phase accuracy for any given direction of wave propagation an optimal gls mesh parameter is determined using two dimensional fourier analysis in general problems the direction of wave propagation will not be known a priori in this case an optimal gls parameter is found which reduces phase error for all possible wave vector orientations over elements the optimal gls parameters are derived for both consistent and lumped mass approximations several numerical examples are given and the results compared with those obtained from the galerkin method the extension of gls to higher order quadratic interpolations is also presented

numerical solution of optimal design problems for binary gratings in this paper we describe recent developments in the application of mathematical and computational techniques to the problem of designing binary gratings on top of a multilayer stack in such a way that the propagating modes have a specified intensity or phase pattern for a chosen range of wavelengths or incidence angles the diffraction problems are transformed to strongly elliptic variational formulations of quasi periodic transmission problems for the helmholtz equation in a bounded domain coupled with boundary integral representations in the exterior we obtain analytic formulae for the gradients of cost functionals with respect to the parameters of the grating profile and the thickness of the layers so that the optimal design problems can be solved by minimization algorithms based on gradient descent for the computation of diffraction efficiencies and gradients the variational problems are solved by using a generalized finite element method with minimal pollution we provide some numerical examples to demonstrate the convergence properties for evaluating diffraction efficiencies and gradients the method is applied to optimal design problems for polarisation gratings and beam splitters

dispersion induced by the pollution for the wave equation in this paper we give a computational method to measure the dispersion which is directly related to the pollution effect this pollution effect is observed by various researchers when the helmholtz equation is solved for high wave numbers the main effect of the pollution is that the wave number of the solution obtained via a finite element method is different from the wave number of the exact solution and this is what is called dispersion using fourier analysis we present a numerical method to measure the dispersion corresponding to a finite element method

on simple oversampled a d conversion in l sup 2 r the accuracy of oversampled analog to digital a d conversion the dependence of accuracy on the sampling interval spl tau and on the bit rate r are characteristics fundamental to a d conversion but not completely understood these characteristics are studied for oversampled a d conversion of band limited signals in l sup 2 r we show that the digital sequence obtained in the process of oversampled a d conversion describes the corresponding analog signal with an error which tends to zero as spl tau sup 2 in energy provided that the quantization threshold crossings of the signal constitute a sequence of stable sampling in the respective space of band limited functions further we show that the sequence of quantized samples can be represented in a manner which requires only a logarithmic increase in the bit rate with the sampling frequency r o log spl tau and hence that the error of oversampled a d conversion actually exhibits an exponential decay in the bit rate as the sampling interval tends to zero

a generalized finite element method for solving the helmholtz equation in two dimensions with minimal pollution when using the galerkin fem for solving the helmholtz equation in two dimensions the error of the corresponding solution differs substantially from the error of the best approximation and this effect increases with higher wave number k in this paper we will design a generalized finite element method gfem for the helmholtz equation such that the pollution effect is minimal

second harmonic generation in nonlinear optical films second harmonic generation an important phenomenon in nonlinear optics is modeled in this work the model is derived from a nonlinear system of maxwell s equations which overcomes the known shortcomings of some commonly used models in the literature existence and uniqueness of solutions are established by a combination of a variational approach and the contraction mapping principle some numerical results are also presented

single bit oversampled a d conversion with exponential accuracy in the bit rate a scheme for simple oversampled analog to digital a d conversion using single bit quantization is presented the scheme is based on recording positions of zero crossings of the input signal added to a deterministic dither function this information can be represented in a manner such that the bit rate increases only logarithmically with the oversampling factor r the input band limited signal can be reconstructed from this information locally with o 1 r pointwise error resulting in an exponentially decaying distortion rate characteristic in the course of studying the accuracy of the proposed a d conversion scheme some new results are established about reconstruction of band limited signals from irregular samples using linear combination of functions with fast decay schemes for local interpolation of band limited signals from quantized irregular samples are also proposed

deterministic analysis of oversampled a d conversion and decoding improvement based on consistent estimates discusses the deterministic analysis of oversampled a d conversion adc the properties derivable from such an analysis and the consequences on reconstruction using nonlinear decoding given a band limited input x producing a quantized version c the authors consider the set of all input signals that are band limited and produce c they call any element of this set a consistent estimate of x regardless of the type of encoder simple predictive or noise shaping they show that this set is convex and as a consequence any nonconsistent estimate can be improved they also show that the classical linear decoding estimates are not necessarily consistent numerical tests performed on simple adc single loop and multiloop spl sigma spl delta modulation show that consistent estimates yield a mean square error mse that decreases asymptotically with the oversampling ratio faster than the linear decoding mse by approximately 3 db octave this implies an asymptotic mse of the order of spl oscr r sup spl minus 2n 2 instead of spl oscr r sup spl minus 2n 1 in linear decoding where r is the oversampling ratio and n the order of the modulator methods of improvement of nonconsistent estimates based on the deterministic knowledge of the quantized signal are proposed for simple adc predictive adc single loop and multiloop spl sigma spl delta modulation

oversampled a d conversion and error rate dependence of nonbandlimited signals with finite rate of innovation we study the problem of a d conversion and error rate dependence of a class of nonbandlimited signals with finite rate of innovation in particular we analyze a continuous periodic stream of diracs characterized by a finite set of time positions and weights previous research has only considered sampling of this type of signals ignoring the presence of quantization necessary for any practical implementation to this end we first define the concept of consistent reconstruction and introduce corresponding oversampling in both time and frequency high accuracy in a consistent reconstruction is achieved by enforcing the reconstructed signal to satisfy three sets of constraints related to low pass filtering quantization and the space of continuous periodic streams of diracs we provide two schemes to reconstruct the signal for the first one we prove that the estimation mean squared error of the time positions is o 1 r sub t sup 2 r sub f sup 3 where r sub t and r sub f are the oversampling ratios in time and frequency respectively for the second scheme it is experimentally observed that at the cost of higher complexity the estimation accuracy lowers to o 1 r sub t sup 2 r sub f sup 5 our experimental results show a clear advantage of consistent over nonconsistent reconstruction regarding the rate we consider a threshold crossing based scheme where as opposed to previous research both oversampling in time and in frequency influence the coding rate we compare the error rate behavior resulting on the one hand from increasing the oversampling in time and or frequency and on the other hand from decreasing the quantization stepsize

finite element approximation of time harmonic waves in periodic structures consider the diffraction of a time harmonic wave incident on a periodic surface of some inhomogeneous material it is shown that the scattering diffraction problem may be modeled by a helmholtz equation with transparent boundary conditions the diffraction problem may be solved by a finite element method in this paper optimal error estimates for the finite element method are established the error estimates are also established when the truncation of the nonlocal transparent boundary operators takes place

entropy of highly correlated quantized data this paper considers the entropy of highly correlated quantized samples two results are shown the first concerns sampling and identically scalar quantizing a stationary continuous time random process over a finite interval it is shown that if the process crosses a quantization threshold with positive probability then the joint entropy of the quantized samples tends to infinity as the sampling rate goes to infinity the second result provides an upper bound to the rate at which the joint entropy tends to infinity in the case of an infinite level uniform threshold scalar quantizer and a stationary gaussian random process specifically an asymptotic formula for the conditional entropy of one quantized sample conditioned on the previous quantized sample is derived at high sampling rates these results indicate a sharp contrast between the large encoding rate in bits sec required by a lossy source code consisting of a fixed scalar quantizer and an ideal sampling rate adapted lossless code and the bounded encoding rate required by an ideal lossy source code operating at the same distortion

maxwell s equations in a periodic structure consider a diffraction of a beam of particles in r3 when the dielectric coefficient is a constant cl above a surface s and a constant e2 below a surface s and the magnetic permeability is constant throughout rii s is assumed to be periodic in the xl direction and of the form x1 f1 s x3 f3 s x2 arbitrary we prove that there exists a unique solution to the time harmonic maxwell equations in r 3 having the form of refracted waves for x3 1 and of transmitted waves for x3 1 if and only if there exists a unique solution to a certain system of two coupled fredholm equations thus in particular for all the e s except for a discrete number there exists a unique solution to the maxwell equations introduction in this paper we consider the maxwell equations for time harmonic solutions in the entire space r3 with piecewise constant dielectric coefficient having jump across a periodic surface the magnetic permeability u is assumed to be constant whereas the dielectric coefficient e is given by e 8 above a surface s x3 f xl and e e2 below the surface s e and 82 are different constants if s is a half space x3 0 then the solution eo ho can be computed explicitly we assume in this paper that s is periodic i e f xl l f xl vx1 e r l 0 we wish to find a solution e h such that e eo and h ho are superpositions of transmitted waves 0 1 in x3 a where a maxlfl in 1 7 we assume that f e c2 and we reduce the solution of the maxwell equations to a fredholm system of four integral equations in 8 we reduce it further to a fredholm system of two integral equations thus for all but a discrete sequence of values of the physical parameters there exists a unique solution to the integral equations yielding a solution of the maxwell equations received by the editors january 25 1989 1980 mathematics subject classification 1985 revision primary 78a10 78a45 secondary 35p25 47a40

positive solutions of semilinear elliptic problems in unbounded domains di a ax i l n each a e c yq b e cg q h x 3 b 0 0 ci 1 and f x u satisfies assumptions f fs below in particular it is assumed that f x 0 0 for all x e 52 implying that the boundary value problem 1 1 always has the trivial solution our main purpose is to establish the existence of a positive solution of 1 1 throughout 1 2 in cases for which the nonlinearity in 1 1 is unbounded above i e f x r t co as t co locally uniformly in q we treat the case of bounded nonlinearities elsewhere 20 the main theorem 4 5

numerical solution of nonlinear diffraction problems an adaptive finite element method is developed for solving maxwell s equations in a nonlinear periodic structure the medium or computational domain is truncated by a perfect matched layer pml technique error estimates are established numerical examples are provided which illustrate the efficiency of the method

optimal design of nonlinear diffraction gratings the goal of the paper is to study the optimal design problem for nonlinear diffraction gratings the problem arises in the study of surface enhanced nonlinear optical effects of second harmonic generation in order to apply certain gradient based optimization methods an explicit formula for the partial derivatives of the rayleigh coefficients with respect to the parameters of the grating profile is derived using the formula numerical results are presented on an optimal design problem of nonlinear binary gratings

a d conversion with imperfect quantizers this paper analyzes mathematically the effect of quantizer threshold imperfection commonly encountered in the circuit implementation of analog to digital a d converters such as pulse code modulation pcm and sigma delta sigmadelta modulation sigmadelta modulation which is based on coarse quantization of oversampled redundant samples of a signal enjoys a type of self correction property for quantizer threshold errors bias that is not shared by pcm although classical sigmadelta modulation is inferior to pcm in the rate distortion sense this robustness feature is believed to be one of the reasons why sigmadelta modulation is preferred over pcm in a d converters with imperfect quantizers motivated by these facts other encoders are constructed in this paper that use redundancy to obtain a similar self correction property but that achieve higher order accuracy relative to bit rate compared to classical sigmadelta more precisely two different types of encoders are introduced that exhibit exponential accuracy in the bit rate in contrast to the polynomial type accuracy of classical sigmadelta while possessing the self correction property

high resolution distributed sampling of bandlimited fields with low precision sensors the problem of sampling a discrete time sequence of spatially bandlimited fields with a bounded dynamic range in a distributed communication constrained processing environment is studied a central unit having access to the data gathered by a dense network of low precision sensors is required to reconstruct the field snapshots to maximum accuracy both deterministic and stochastic field models are considered for stochastic fields results are established in the almost sure sense the feasibility of having a flexible tradeoff between the oversampling rate sensor density and the analog to digital converter adc precision while achieving an exponential accuracy in the number of bits per nyquist interval per snapshot is demonstrated this exposes an underlying conservation of bits principle the bit budget per nyquist interval per snapshot the rate can be distributed along the amplitude axis sensor precision and space sensor density in an almost arbitrary discrete valued manner while retaining the same exponential distortion rate characteristics

on accuracy conditions for the numerical computation of waves abstract the helmholtz equation k 2 n 2 u f with a variable index of refraction n and a suitable radiation condition at infinity serves as a model for a wide variety of wave propagation problems such problems can be solved numerically by first truncating the given unbounded domain imposing a suitable outgoing radiation condition on an artificial boundary and then solving the resulting problem on the bounded domain by direct discretization for example using a finite element method in practical applications the mesh size h and the wave number k are not independent but are constrained by the accuracy of the desired computation it will be shown that the number of points per wavelength measured by kh 1 is not sufficient to determine the accuracy of a given discretization for example the quantity k 3 h 2 is shown to determine the accuracy in the l 2 norm for a second order discretization method applied to several propagation models

the time harmonic maxwell equations in a doubly periodic structure abstract consider the diffraction of a beam of particles in r 3 when the dielectric coefficient is a constant e1 above a surface s and a different constant e2 below s and the magnetic permeability is constant throughout r 3 s is assumed to be a doubly periodic surface say z f x y with f x ml1 y nl2 f x y for integers m n the existence and uniqueness of a solution satisfying a radiation condition at infinity is reduced to a system of fredholm equations thus for all but a discrete set of e s there exists a unique solution

the domain derivative in rough surface scattering and rigorous estimates for first order perturbation theory we investigate frechet differentiability of the scattered field with respect to variation in the boundary in the case of time harmonic acoustic scattering by an unbounded sound soft one dimensional rough surface we rigorously prove the differentiability of the scattered field and derive a characterization of the frechet derivative as the solution to a dirichlet boundary value problem as an application of these results we give rigorous error estimates for first order perturbation theory justifying small perturbation methods that have a long history in the engineering literature as an application of our rigorous estimates we show that a plane acoustic wave incident on a sound soft rough surface can produce an unbounded scattered field

an lq rn theory of subcritical semilinear elliptic problems abstract existence and asymptotic decay estimates are obtained for solutions u l 2n n 2 r n n 3 of a class of semilinear elliptic problems including the subcritical emden fowler type the problems are not assumed to be radially symmetric the method is based on the convergence in cloc2 rn of a sequence of approximate solutions uk w01 2 rn for which uk is bounded in lq rn for all q 2n n 2 in one case the solution is shown to have a removable singularity at

complete characterization of stable bandlimited systems under quantization and thresholding in this paper we analyze the approximation behavior of sampling series where the sample values taken equidistantly at nyquist rate are disturbed either by the nonlinear threshold operator or the nonlinear quantization operator we perform the analysis for several spaces of bandlimited signals and completely characterize the spaces for which an approximation is possible additionally we study the approximation of outputs of stable linear time invariant systems by sampling series with disturbed samples for signals in pw pi 1 we show that there exist stable systems that become unstable under thresholding and quantization and that the approximation error is unbounded irrespective of how small the quantization step size is chosen further we give a necessary and sufficient condition for the pointwise and the uniform convergence of the series surprisingly this condition is the well known condition for bounded input bounded output bibo stability finally we discuss the special case of finite impulse response fir filters and give an upper bound for the approximation error

behavior of the quantization operator for bandlimited nonoversampled signals the process of quantization generates a loss of information and thus the original signal cannot be reconstructed exactly from the quantized samples in general however it is desirable to keep the error as small as possible in this paper the quantization error is quantified in terms of several distortion measures all these measures employ the difference between the original signal and the reconstructed signal which is obtained by bandlimited interpolation of the quantized samples we assume that the signals are bandlimited and that the samples are taken at nyquist rate it is shown that for signals in the paley wiener space pw 1 the supremum of the reconstructed signal and hence the quantization error cannot be bounded in the sense that there exists a bounded subset of pw 1 on which both quantities can increase unboundedly this unexpected behavior is due to the nonlinearity of the quantization operator and the slow decay of the sinc function the nonlinearity is essential for this behavior because every linear operator that fulfills a certain property of the quantization operator would otherwise have to be bounded furthermore it is proven that for a fixed signal the possible quantization error increases as the quantization step size tends to zero the treatment of the quantization error in this paper is completely deterministic

the shannon sampling series and the reconstruction of signals in terms of linear quadratic and cubic splines the sine kernel function of the sampling series is replaced by spline functions having compact support all built up from the b splines m n the resulting generalized sampling series reduces to finite sums so that no truncation error occurs moreover the approximation error generally decreases more rapidly than for the classical series when w tends to infinity 1 w being the distance between the sampling points for the kernel function varphi t 5m 4 t 4m 5 t with support tfrac 5 2 tfrac 5 2 e g it decreases with order o w 4 provided the signal has a fourth order derivative for the classical series the order is o w 4 log w seven characteristic examples are treated in detail

efficient quantization for overcomplete expansions in spl ropf sup n we study construction of structured regular quantizers for overcomplete expansions in spl ropf sup n our goal is to design structured quantizers which allow simple reconstruction algorithms with low complexity and which have good performance in terms of accuracy most related work to date in quantized redundant expansions has assumed that the same uniform scalar quantizer was used on all the expansion coefficients several approaches have been proposed to improve the reconstruction accuracy with some of these methods having significant complexity instead we consider the joint design of the overcomplete expansion and the scalar quantizers allowing different step sizes in such a way as to produce an equivalent vector quantizer evq with periodic structure the construction of a periodic quantizer is based on lattices in spl ropf sup n and the concept of geometrically scaled similar sublattices the periodicity makes it possible to achieve good accuracy using simple reconstruction algorithms e g linear reconstruction or a small lookup table

reconstruction of oversampled band limited signals from spl sigma spl delta encoded binary sequences considers the application of spl sigma spl delta modulators to analog to digital conversion the authors have previously shown that for constant input signals optimal nonlinear decoding can achieve large gains in signal to noise ratio snr over linear decoding the present paper shows a similar result for band limited input signals the new nonlinear decoding algorithm is based on projections onto convex sets pocs and alternates between a time domain operation and a band limitation to find a signal invariant under both the time domain operation results in a quadratic programming problem the band limitation can be based on singular value decomposition of a certain matrix the authors show simulation results for the snr performance of a pocs based decoder and a linear decoder for the single loop double loop and two stage spl sigma spl delta modulators and for a specific fourth order interpolative modulator depending on the modulator and the oversampling ratio improvements in snr of up to 10 20 db can be achieved

reduction of the mse in r times oversampled a d conversion o 1 r to o 1 r sup 2 in oversampled analog to digital conversion the usual reconstruction method using lowpass filtering leads to a mean squared error mse inversely proportional to the oversampling ratio r the authors prove under certain assumptions and with periodic analog input signals that optimal reconstruction achieves an mse with an oversampling ratio dependence order of at least o 1 r sup 2 that is an mse slope of spl minus 6 db per octave of oversampling is obtained rather than the conventional spl minus 3 db octave slope of classical schemes