bi hamiltonian structure of the kermack mckendrick model for epidemics the dynamical system proposed by kermack and mckendrick 1933 to model the spread of epidemics is shown to admit bi hamiltonian structure without any restrictions on the rate constants these two inequivalent hamiltonian structures are compatible

discrete prolate spheroidal wave functions and interpolation we describe an algorithm for the interpolation of burst errors in discrete time signals that can be modelled as being band limited the algorithm correctly restores a mutilated signal that is indeed band limited the behavior of the algorithm when applied to signals containing noise or out of band components can be analysed satisfactorily with the aid of asymptotic properties of the discrete prolate spheroidal sequences and wave functions the effect of windowing can also be described conveniently in terms of these sequences and functions

on the dynamics of lotka volterra equations having an invariant hyperplane the dynamics of n dimensional lotka volterra equations having an invariant hyperplane is investigated herein there exists a fundamental difference between the even and the odd dimensional case for even dimensions the dynamics are generated by a hamiltonian system with respect to an appropriately chosen poisson structure for odd dimensions the dynamics are hamiltonian if there is a continuum of interior fixed points and gradient like if the interior fixed point is unique in this case the invariant hyperplane is part of the omega limit set and the dynamics restricted to this set are those of a hamiltonian system furthermore an example of a completely integrable four dimensional hamiltonian lotka volterra system is presented

channel capacity under sub nyquist nonuniform sampling this paper investigates the effect of sub nyquist sampling upon the capacity of an analog channel the channel is assumed to be a linear time invariant gaussian channel where perfect channel knowledge is available at both the transmitter and the receiver we consider a general class of right invertible time preserving sampling methods which includes irregular nonuniform sampling and characterize in closed form the channel capacity achievable by this class of sampling methods under a sampling rate and power constraint our results indicate that the optimal sampling structures extract out the set of frequencies that exhibits the highest signal to noise ratio among all spectral sets of measure equal to the sampling rate this can be attained through filterbank sampling with uniform sampling grid employed at each branch with possibly different rates or through a single branch of modulation and filtering followed by uniform sampling these results reveal that for a large class of channels employing irregular nonuniform sampling sets while are typically complicated to realize in practice does not provide capacity gain over uniform sampling sets with appropriate preprocessing our findings demonstrate that aliasing or scrambling of spectral components does not provide capacity gain in this scenario which is in contrast to the benefits obtained from random mixing in spectrum blind compressive sampling schemes

on the hamiltonian structure of 2d ode possessing an invariant the authors discuss the hamiltonian structure with respect to the canonical bracket for a 2d dynamical system and prove that this structure can be obtained simply with a rescaling of the independent variable the rescaling will provide a hamiltonian when a time independent invariant is known the hamiltonian coinciding with the invariant applications to different systems of ode in two dimensions are presented

generalization of solutions of the jacobi pdes associated to time reparametrizations of poisson systems the determination of solutions of the jacobi partial differential equations pdes for finite dimensional poisson systems is considered in particular a novel procedure for the construction of solution families is developed such a procedure is based on the use of time reparametrizations preserving the existence of a poisson structure as a result a method which is valid for arbitrary values of the dimension and the rank of the poisson structure under consideration is obtained in this article two main families of time reparametrizations of this kind are characterized in addition these results lead to a novel application which is also developed namely the global and constructive determination of the darboux canonical form for poisson systems of arbitrary dimension and rank two thus improving the local result provided by darboux theorem for such a case

efficient reconstruction of band limited sequences from nonuniformly decimated versions by use of polyphase filter banks an efficient polyphase structure for the reconstruction of a band limited sequence from a nonuniformly decimated version is developed theoretically the reconstruction involves the implementation of a bank of multilevel filters and it is shown that how all these reconstruction filters can be obtained at the cost of one mth band low pass filter and a constant matrix multiplier the resulting structure is therefore more general than previous schemes in addition the method offers a direct means of controlling the overall reconstruction distortion t z by appropriate design of a low pass prototype filter p z extension of these results to multiband band limited signals and to the case of nonconsecutive nonuniform subsampling are also summarized along with generalizations to the multidimensional case design examples are included to demonstrate the theory and the complexity of the new method is seen to be much lower than earlier ones

new solution family of the jacobi equations characterization invariants and global darboux analysis a new family of skew symmetric solutions of the jacobi partial differential equations for finite dimensional poisson systems is characterized and analyzed such family has some remarkable properties firstly it is defined for arbitrary values of the dimension and the rank secondly it is described in terms of arbitrary differentiable functions namely it is not limited to a given degree of nonlinearity additionally it is possible to determine explicitly the fundamental properties of those solutions such as their casimir invariants and the algorithm for the reduction to the darboux canonical form which have been reported only for a very limited sample of finite dimensional poisson structures moreover such analysis is carried out globally in phase space thus improving the usual local scope of the darboux theorem

computationally attractive reconstruction of bandlimited images from irregular samples an efficient method for the reconstruction of bandlimited images and the approximation of arbitrary images from nonuniform sampling values is developed the novel method is based on the observation that the reconstruction problem can be formulated as linear system of equations using two dimensional 2 d trigonometric polynomials where the matrix is of block toeplitz type with toeplitz blocks this system is solved iteratively by the conjugate gradient cg method we show that the use of so called adaptive weights in the establishment of the block toeplitz matrix can be seen as efficient preconditioning the superiority of the new method over conventional approaches is demonstrated by numerical experiments

perfect reconstruction versus mmse filter banks in source coding classically the filter banks fbs used in source coding schemes have been chosen to possess the perfect reconstruction pr property or to be maximally selective quadrature mirror filters qmfs this paper puts this choice back into question and solves the problem of minimizing the reconstruction distortion which in the most general case is the sum of two terms a first one due to the non pr property of the fb and the other being due to signal quantization in the subbands the resulting filter banks are called minimum mean square error mmse filter banks several quantization noise models are considered first under the classical white noise assumption the optimal positive bit rate allocation in any filter bank possibly nonorthogonal is expressed analytically and an efficient optimization method of the mmse filter banks is derived then it is shown that while in a pr fb the improvement brought by an accurate noise model over the classical white noise one is noticeable it is not the case for the mmse fb the optimization of the synthesis filters is also performed for two measures of the bit rate the classical one which is defined for uniform scalar quantization and the order one entropy measure finally the comparison of rate distortion curves where the distortion is minimized for a given bit rate budget enables us to quantify the snr improvement brought by mmse solutions

efficient recovery of sub nyquist sampled sparse multi band signals using reconfigurable multi channel analysis and modulated synthesis filter banks sub nyquist cyclic nonuniform sampling cnus of a sparse multi band signal generates a nonuniformly sampled signal assuming that the corresponding uniformly sampled signal satisfies the nyquist sampling criterion the sequence obtained via cnus can be passed through a reconstructor to recover the missing uniform grid samples at present these reconstructors have very high design and implementation complexity that offsets the gains obtained due to sub nyquist sampling in this paper we propose a scheme that reduces the design and implementation complexity of the reconstructor in contrast to the existing reconstructors which use only a multi channel synthesis filter bank fb the proposed reconstructor utilizes both analysis and synthesis fbs which makes it feasitble to achieve an order of magnitude reduction of the complexity the analysis filters are implemented using polyphase networks whose branches are allpass filters with distinct fractional delays and phase shifts in order to reduce both the design and the implementation complexity of the synthesis fb the synthesis filters are implemented using a cosine modulated fb in addition to the reduced complexity of the reconstructor the proposed multi channel recovery scheme also supports online reconfigurability which is required in flexible multi mode systems where the user subband locations vary with time

local stability and lyapunov functionals for n dimensional quasipolynomial conservative systems abstract we present a method for determining the local stability of equilibrium points of conservative generalizations of the lotka volterra equations these generalizations incorporate both an arbitrary number of species including odd dimensional systems and nonlinearities of arbitrarily high order in the interspecific interaction terms the method combines a reformulation of the equations in terms of a poisson structure and the construction of their lyapunov functionals via the energy casimir method these lyapunov functionals are a generalization of those traditionally known for lotka volterra systems examples are given

filter design for mimo sampling and reconstruction we address the problem of finite impulse response fir filter design for uniform multiple input multiple output mimo sampling this scheme encompasses papoulis generalized sampling and several nonuniform sampling schemes as special cases the input signals are modeled as either continuous time or discrete time multiband input signals with different band structures we present conditions on the channel and the sampling rate that allow perfect inversion of the channel additionally we provide a stronger set of conditions under which the reconstruction filters can be chosen to have frequency responses that are continuous we also provide conditions for the existence of fir perfect reconstruction filters and when such do not exist we address the optimal approximation of the ideal filters using fir filters and a minmax l sub 2 end to end distortion criterion the design problem is then reduced to a standard semi infinite linear program an example design of fir reconstruction filters is given

multirate synchronous sampling of sparse multiband signals recent advances in electro optical systems make them ideal for undersampling multiband signals with very high carrier frequencies in this paper we propose a new scheme for sampling and reconstructing of a multiband sparse signals that occupy a small part of a given broad frequency range under the constraint of a small number of sampling channels the locations of the signal bands are not known a priori the scheme which we call synchronous multirate sampling smrs entails gathering samples synchronously at few different rates whose sum is significantly lower than the nyquist sampling rate the signals are reconstructed by finding a solution of an underdetermined system of linear equations by applying a pursuit algorithm and assuming that the solution is composed of a minimum number of bands the empirical reconstruction success rate is higher than obtained using previously published multicoset scheme when the number of sampling channels is small and the conditions for a perfect reconstruction in the multicoset scheme are not fulfilled the practical sampling system which is simulated in our work consists of three sampling channels our simulation results show that a very high empirical success rate is obtained when the total sampling rate is five times higher than the total signal support of a complex signal with four bands by comparison a multicoset sampling scheme obtains a very high empirical success rate with a total sampling rate which is three times higher than the total signal support however the multicoset scheme requires 14 channels

generalized hamiltonian structures for systems in three dimensions with a rescalable constant of motion the generalized hamiltonian structures of several three dimensional dynamical systems of interest in physical applications are considered in general hamiltonians exist only for systems that possess at least one time independent constant of motion systems with only time dependent constants of motion may sometimes be rescaled and their constant of motion made time independent when this is possible the transformed system may be cast in a generalized hamiltonian formalism with non canonical structure functions

determination of the number of errors in dft codes subject to low level quantization noise this paper analyzes the effects of quantization or other low level noise on the error correcting capability of a popular class of real number bose chaudhuri hocquenghem bch codes known as discrete fourier transform dft codes in the absence of low level noise a modified version of the peterson gorenstein zierler pgz algorithm allows the correction of up to corrupted entries in the real valued code vector of an dft code in this paper we analyze the performance of this modified pgz algorithm in the presence of low level quantization or other noise that might affect each entry of the code vector and not simply of them we focus on the part of the algorithm that determines the number of errors that have corrupted the real number codeword our approach for determining the number of errors is more effective than existing systematic approaches in the literature and results in an explicit lower bound on the precision needed to guarantee the correct determination of the number of errors our simulations suggest that this bound can be tight finally we prove that the optimal bit allocation for dft codes in terms of correctly determining the number of errors is the uniform one

new global solutions of the jacobi partial differential equations abstract a new family of solutions of the jacobi partial differential equations for finite dimensional poisson systems is investigated this family is mathematically remarkable as the functional dependences of the solutions appear to be associated to the distinguished invariants of the solutions themselves this kind of poisson structure termed distinguished solutions or d solutions is defined for every nontrivial combination of values of the dimension and the rank and is also determined in terms of functions of arbitrary nonlinearity properties usually not present simultaneously in the already known solution families in addition d solutions display several properties allowing the generation of an infinity of d solutions from a given one which is an uncommon feature in the framework of the jacobi equations furthermore a special family of d solutions complying with the previous requirements is constructively characterized and analyzed examples are discussed focusing on physical implications and including an application for the global construction of the darboux canonical form

regularized sampling of multiband signals this paper presents a regularized sampling method for multiband signals that makes it possible to approach the landau limit while keeping the sensitivity to noise and perturbations at a low level the method is based on band limited windowing followed by trigonometric approximation in consecutive time intervals the key point is that the trigonometric approximation inherits the multiband property that is its coefficients are formed by bursts of elements corresponding to the multiband components it is shown that this method can be well combined with the recently proposed synchronous multirate sampling smrs scheme given that the resulting linear system is sparse and formed by ones and zeroes the proposed method allows one to trade sampling efficiency for noise sensitivity and is specially well suited for bounded signals with unbounded energy like those in communications navigation audio systems etc besides it is also applicable to finite energy signals and periodic band limited signals trigonometric polynomials the paper includes a subspace method for blindly estimating the support of the multiband signal as well as its components and the results are validated through several numerical examples

generalized hamiltonian structures for ermakov systems we construct poisson structures for ermakov systems using the ermakov invariant as the hamiltonian two classes of poisson structures are obtained one of them is degenerate in which case we derive the casimir functions in some situations the existence of casimir functions can give rise to superintegrable ermakov systems finally we characterize the cases where linearization of the equations of motion is possible

poisson structure of dynamical systems with three degrees of freedom it is shown that the poisson structure of dynamical systems with three degrees of freedom can be defined in terms of an integrable one form in three dimensions advantage is taken of this fact and the theory of foliations is used in discussing the geometrical structure underlying complete and partial integrability techniques for finding poisson structures are presented and applied to various examples such as the halphen system which has been studied as the two monopole problem by atiyah and hitchin it is shown that the halphen system can be formulated in terms of a flat sl 2 r valued connection and belongs to a nontrivial godbillon vey class on the other hand for the euler top and a special case of three species lotka volterra equations which are contained in the halphen system as limiting cases this structure degenerates into the form of globally integrable bi hamiltonian structures the globally integrable bi hamiltonian case is a linear and the sl 2 r structure is a quadratic unfolding of an integrable one form in 3 1 dimensions it is shown that the existence of a vector field compatible with the flow is a powerful tool in the investigation of poisson structure and some new techniques for incorporating arbitrary constants into the poisson one form are presented herein this leads to some extensions analogous to q extensions of poisson structure the kermack mckendrick model and some of its generalizations describing the spread of epidemics as well as the integrable cases of the lorenz lotka volterra may leonard and maxwell bloch systems admit globally integrable bi hamiltonian structure

noniterative and fast iterative methods for interpolation and extrapolation in this correspondence we study the band limited interpolation and extrapolation problems for finite dimensional signals we show that these problems can be easily reduced to the solution of a set of linear equations with a real symmetric positive definite matrix s with spectral radius spl rho s

the eigenvalues of matrices that occur in certain interpolation problems the eigenvalues of the matrices that occur in certain finite dimensional interpolation problems are directly related to their well posedness and strongly depend on the distribution of the interpolation knots that is on the sampling set we study this dependency as a function of the sampling set itself and give accurate bounds for the eigenvalues of the interpolation matrices the bounds can be evaluated in as few as four arithmetic operations and therefore they greatly simplify the assessment of sampling sets regarding numerical stability the accuracy and usefulness of the bounds are illustrated with examples

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

estimation of time limited channel spectra from nonuniform samples this paper deals with the estimation of a time invariant channel spectrum from its own nonuniform samples assuming there is a bound on the channel s delay spread except for this last assumption this is the basic estimation problem in systems providing channel spectral samples however as shown in the paper the delay spread bound leads us to view the spectrum as a band limited signal rather than the fourier transform of a tapped delay line tdl using this alternative model a linear estimator is presented that approximately minimizes the expected root mean square rms error for a deterministic channel its main advantage over the tdl is that it takes into account the spectrum s smoothness time width thus providing a performance improvement the proposed estimator is compared numerically with the maximum likelihood ml estimator based on a tdl model in pilot assisted channel estimation pace for ofdm

sampling theorems for uniform and periodic nonuniform mimo sampling of multiband signals we examine a multiple input multiple output mimo sampling scheme for a linear time invariant continuous time mimo channel the input signals are modeled as multiband signals with different spectral supports and the channel outputs are sampled on either uniform or periodic nonuniform sampling sets with possibly different but commensurate intervals on the different outputs this scheme encompasses papoulis generalized sampling and several nonuniform sampling schemes as special cases we derive necessary and sufficient conditions on the channel and the sampling rate that allow stable perfect reconstruction of the inputs or equivalently perfect inversion of the channel from an implementation viewpoint we note that it is desirable that the reconstruction filters have continuous frequency responses we derive necessary and sufficient conditions that guarantee this continuity property the frequency responses of the reconstruction filters are specified as solutions to a system of linear equations finally we demonstrate that perfect reconstruction may be possible even when the channel outputs are sampled at an average rate that does not allow the reconstruction of any output from its samples alone in certain instances this average rate can achieve the recently presented fundamental bounds on mimo sampling density

incomplete sampling series and the recovery of missing samples from oversampled band limited signals it is well known that a band limited oversampled signal is completely determined even if an arbitrary finite number of samples is lost it is shown that an alternative simple proof of this fact carries over to generalized sampling expansions more precisely it is shown that any finite number of missing samples can be recovered from the remaining ones in the case of generalized kramer sampling expansions if an appropriate oversampling constraint is satisfied the recovery can be accomplished either iteratively or noniteratively

the feasible solution in signal restoration the feasible solution to the signal restoration problem is defined as the one which satisfies all constraints which can be imposed on the true solution a very important set of constraints can be obtained by examining the statistics of the noise these and other constraints can be described as closed convex sets thus projection onto closed convex sets is the numerical method used to obtain a feasible solution examples of this method demonstrate its usefulness in one and two dimensional signal restoration the limitations of the method are discussed

optimal estimation for band limited signals including time domain considerations the problem of estimation for band limited signals is investigated the desired estimate is based on a finite number of time samples taken from a measurement interval of length t m with no additional information the estimate must be based on these samples alone when the signal is band limited with normalized bandwidth this provides additional information which is used to improve the estimate by extrapolating the signal an extrapolation based on the measured time samples and the known bandwidth is concentrated in time about the measurement interval the length of this effective extrapolation interval is t l approx t m 2 beta several band limited estimation methods of this type have been described a new estimation method for band limited signals is described which is also based on a time concentrated extrapolation of the measurements this new method improves estimates by specifically incorporating the interval t l into the problem formulation and thereby improving the extrapolation on this relevant interval estimation error bounds and examples are presented in comparing these two types of band limited estimation procedures estimation with imperfect data is considered sensitivity of the band limited methods to errors in the data is discussed estimation with smoothing is utilized in reducing the effects of this noise

bogoyavlensky toda systems of type dn in this paper we confirm a conjecture of flaschka which relates the hierarchy of poisson brackets and master symmetrie sf or the bogoyavlesky toda systems of type dn with fundamental invariants of the corresponding lie group the areas investigated include master symmetries recursion operators higher poisson brackets and invariants the results are presented both in flaschka coordinates a b and i nt henatural q p coordinates

the local structure of poisson manifolds varietes de poisson et applications decomposition structures de poisson lineaires approximation lineaire systemes hamiltoniens le probleme de linearisation groupes de fonction realisations et applications impulsion paires duales et groupes de jauge existence des realisations unicite des realisations les problemes des 3 corps restreints et autres exemples

recovery of signals from nonuniform samples using iterative methods an iterative method to recover a bandlimited signal from its ideal nonuniform samples is proposed the convergence of iterations is proved and general regions for convergence are found it is shown that the iterative method is also applicable to other forms of nonuniform sampling i e natural sampling and interpolated sampling such as sample and hold signal simulation results show that this method works effectively and fairly fast and the errors after a few iterations are negligible if a particular sufficient condition is satisfied or the sampling rate is higher than the nyquist rate

on the minimax capacity loss under sub nyquist universal sampling this paper investigates the information rate loss in analog channels when the sampler is designed to operate independent of the instantaneous channel occupancy specifically a multiband linear time invariant gaussian channel under universal sub nyquist sampling is considered the entire channel bandwidth is divided into n subbands of equal bandwidth at each time only k constant gain subbands are active where the instantaneous subband occupancy is not known at the receiver and the sampler we study the information loss through an information rate lossmetric that is the gap of achievable rates caused by the lack of instantaneous subband occupancy information we characterize the minimax information rate loss for the sub nyquist regime provided that the number n of subbands and the snr are both large the minimax limits depend almost solely on the band sparsity factor and the undersampling factor modulo some residual terms that vanish as n and snr grow our results highlight the power of randomized sampling methods i e the samplers that consist of random periodic modulation and low pass filters which are able to approach the minimax information rate loss with exponentially high probability

congruence method for global darboux reduction of finite dimensional poisson systems a new procedure for the global construction of the casimir invariants and darboux canonical form for finite dimensional poisson systems is developed this approach is based on the concept of matrix congruence and can be applied without the previous determination of the casimir invariants recall that their prior knowledge is unavoidable for the standard reduction methods thus requiring either the integration of a system of pdes or solving some equivalent problem well the opposite in the new congruence method both the darboux coordinates and the casimir invariants arise simultaneously as the outcome of the reduction algorithm in fact the congruence algorithm proceeds only in terms of matrix algebraic transformations and direct quadratures thus avoiding the need of previously integrating a system of pdes and therefore improving previously known approaches physical examples illustrating different aspects of the theory are provided

efficient reconstruction algorithms using shifted lattices in this paper we consider sampling theory for irregular sampling sets of minimal density which are structured an efficient method for the reconstruction of band limited discrete signals from sampling sets which are unions of shifted lattices is developed these sets are not necessarily periodic a signal can be reconstructed from its samples provided the sampling set and the spectrum of the signal satisfy certain compatibility conditions while explicit reconstruction formulas for unions of sampling lattices are possible it is more convenient to use a recursive algorithm a numerical example utilizing nonperiodic sampling sets implemented in matlab is given the theory and the related algorithm has been applied to speech and image processing examples

an iterative method to compensate for the interpolation distortion the author shows that iterative techniques can be used to compensate for sample and hold or in general interpolation distortion it is proven that this iterative method converges to the true signal in the limit the simulation results of the iterative technique are compared to results of the modular method simulation results show that the iterative method is a powerful tool for reducing distortion compared to other well known techniques

minimum mean squared error reconstruction for generalized undersampling of cyclostationary processes the generalized sampling problem is considered using a filter bank model with a stochastic process framework a signal reconstruction solution minimizing the time averaged mean squared error is found using a filter bank optimization technique for cyclostationary signals the use of a stochastic model approach to find a minimized error solution rather than a perfect reconstruction solution allows consideration of sampling problems that deterministic approaches cannot solve for instance a particular sampling density is not required allowing reconstruction optimization in cases of undersampling an extension is also presented for optimal reconstruction in the presence of additive noise and interference the result is a generalized sampling model which can be used for an extensive variety of scenarios combining aspects of undersampling multiple input multiple output mimo sampling and periodic nonuniform sampling

robust patterns in recurrent sampling of multiband signals periodic nonuniform sampling can be used to achieve sub nyquist sampling of bandlimited multiband signals in this paper we examine the question of selecting the sampling pattern in such a scheme so that the reconstruction robustness measured by the condition number of the modulation matrix is maximized contrary to previous work where the sampling pattern was chosen from a discrete set we let the sampling patterns vary continuously but impose a structural constraint using this approach we derive necessary and sufficient conditions on the spectral support of the signal for which perfect conditioning exist namely for which a sampling pattern can be found so that the resulting modulation matrix has a condition number equal to 1 a simple test to check for these conditions is developed and the desired sampling patterns are found an algorithm for choosing the sampling pattern when the aforementioned conditions are not satisfied is also introduced finally we present some simulation results

fundamental distortion limits of analog to digital compression representing a continuous time signal by a set of samples is a classical problem in signal processing we study this problem under the additional constraint that the samples are quantized or compressed in a lossy manner under a limited bitrate budget to this end we consider a combined sampling and source coding problem in which an analog stationary gaussian signal is reconstructed from its encoded samples these samples are obtained by a set of bounded linear functionals of the continuous time path with a limitation on the average number of samples per unit time given in this setting we provide a full characterization of the minimal distortion in terms of the sampling frequency the bitrate and the signal s spectrum assuming that the signal s energy is not uniformly distributed over its spectral support we show that for each compression bitrate there exists a critical sampling frequency smaller than the nyquist rate such that the distortion in signal reconstruction when sampling at this frequency is minimal our results can be seen as an extension of the classical sampling theorem for bandlimited random processes in the sense that they describe the minimal amount of excess distortion in the reconstruction due to lossy compression of the samples and provide the minimal sampling frequency required in order to achieve this distortion finally we compare the fundamental limits in the combined source coding and sampling problem to the performance of pulse code modulation where each sample is quantized by a scalar quantizer using a fixed number of bits

an analogue to a result of takahashi ii abstract in this paper we address the question of the existence of an operator x such that a c x is of various appropriate types a closed range right invertible left invertible or a fredholm operator we point out to a connection between the existence of such x and the problem of completion of operator matrices of type a c b b h h and present necessary and sufficient conditions for the operator matrix a c b b h h to be completed to a fredholm operator also under certain conditions we solve the corresponding completion problem to a closed range operator

perfect reconstruction formulas and bounds on aliasing error in sub nyquist nonuniform sampling of multiband signals we examine the problem of periodic nonuniform sampling of a multiband signal and its reconstruction from the samples this sampling scheme which has been studied previously has an interesting optimality property that uniform sampling lacks one can sample and reconstruct the class spl bscr spl fscr of multiband signals with spectral support spl fscr at rates arbitrarily close to the landau 1969 minimum rate equal to the lebesgue measure of spl fscr even when spl fscr does not tile r under translation using the conditions for exact reconstruction we derive an explicit reconstruction formula we compute bounds on the peak value and the energy of the aliasing error in the event that the input signal is band limited to the span of spl fscr the smallest interval containing spl fscr which is a bigger class than the valid signals spl bscr spl fscr band limited to spl fscr we also examine the performance of the reconstruction system when the input contains additive sample noise

optimal sub nyquist nonuniform sampling and reconstruction for multiband signals we study the problem of optimal sub nyquist sampling for perfect reconstruction of multiband signals the signals are assumed to have a known spectral support spl fscr that does not tile under translation such signals admit perfect reconstruction from periodic nonuniform sampling at rates approaching landau s 1967 lower bound equal to the measure of spl fscr for signals with sparse spl fscr this rate can be much smaller than the nyquist rate unfortunately the reduced sampling rates afforded by this scheme can be accompanied by increased error sensitivity in a previous study we derived bounds on the error due to mismodeling and sample additive noise adopting these bounds as performance measures we consider the problems of optimizing the reconstruction sections of the system choosing the optimal base sampling rate and designing the nonuniform sampling pattern we find that optimizing these parameters can improve system performance significantly furthermore uniform sampling is optimal for signals with spl fscr that tiles under translation for signals with nontiling spl fscr which are not amenable to efficient uniform sampling the results reveal increased error sensitivities with sub nyquist sampling however these can be controlled by optimal design demonstrating the potential for practical multifold reductions in sampling rate

the intersection of the spectra of operator completions abstract let a b h b b k c b k h x b h k and m x a c x b be an operator completion of the partial operator matrix q a c b in this note we consider the intersection of the spectra of mx when x runs over b h k denote by a b c the set of scalar c such that either a c or b c is not right invertible we prove that x b h k m x a b c if dim r c a b c a b c if dim r c where a b c is the set of scalars c such that r a c h r b c k and ind a ind b 0 we also prove that the intersection is empty if and only if a c and b c are controllable

on the eigenvalues of matrices for the reconstruction of missing uniform samples in this correspondence we derive the relationship between the eigenvalues associated with the matrices of the minimum dimension time domain and frequency domain approaches used for reconstructing missing uniform samples the dependency of the eigenvalues of the weighted toeplitz matrix on positive weights are explored simple bounds for the maximum and minimum eigenvalues of the weighted toeplitz matrix are also presented alternative matrices possessing the same nonzero eigenvalues as that of the weighted toeplitz matrix are provided we verify the theory by the examples presented

hamiltonian equations in r3 the hamiltonian formulation of n 3 systems is considered in general the most general solution of the jacobi equation in r3 is proposed the form of the solution is shown to be valid also in the neighborhood of some irregular points compatible poisson structures and corresponding bi hamiltonian systems are also discussed hamiltonian structures the classification of irregular points and the corresponding reduced first order differential equations of several examples are given

multidimensional manhattan sampling and reconstruction this paper introduces manhattan sampling in two and higher dimensions and proves sampling theorems for them in 2 d manhattan sampling which takes samples densely along a manhattan grid of lines can be viewed as sampling on the union of two rectangular lattices one dense horizontally and the other vertically with the coarse spacing of each being a multiple of the fine spacing of the other the sampling theorem shows that the images bandlimited to the union of the nyquist regions of the two rectangular lattices can be recovered from their manhattan samples and an efficient procedure for doing so is given such recovery is possible even though there is an overlap among the spectral replicas induced by manhattan sampling in three and higher dimensions there are many possible configurations for manhattan sampling each consisting of the union of special rectangular lattices called bi step lattices this paper identifies them proves a sampling theorem showing that the images bandlimited to the union of the nyquist regions of the bi step rectangular lattices are recoverable from manhattan samples presents an efficient onion peeling procedure for doing so and shows that the union of nyquist regions is as large as any bandlimited region such that all images supported by such can be stably reconstructed from samples taken at the rate of the manhattan sampling it also develops a special representation for the bi step lattices with a number of useful properties while most of this paper deals with continuous space images manhattan sampling of discrete space images is also considered for infinite as well as finite support images

filterbank reconstruction of bandlimited signals from nonuniform and generalized samples this paper introduces a filterbank interpretation of various sampling strategies which leads to efficient interpolation and reconstruction methods an identity which is referred to as the interpolation identity is developed and is used to obtain particularly efficient discrete time systems for interpolation of generalized samples as well as a class of nonuniform samples to uniform nyquist samples either for further processing in that form or for conversion to continuous time the interpolation identity also leads to new sampling strategies including an extension of papoulis 1977 generalized sampling expansion

irregular sampling theorems and series expansions of band limited functions abstract we present a new approach to the problem of irregular sampling of band limited functions that is based on the approximation and factorization of convolution operators a special case of our main result is the following theorem if r n is compact g l 1 rn a band limited function with g t 0 on and xi i i and yj j j are two discrete sets in rn which are dense enough then every band limited function f lwp 1 p our methods work a for a very general class of function spaces not only weighted lwp spaces b in all dimensions even in general locally compact abelian groups c to provide the correct behavior of the coefficients in contrast to the traditional estimates currently used by engineers furthermore the method is constructive and stable with respect to input errors such as round off errors truncation effors jitter errors or aliasing errors

unified approach to hamiltonian systems poisson systems gradient systems and systems with lyapunov functions or first integrals systems with a first integral i e constant of motion or a lyapunov function can be written as linear gradient systems dot x l x nabla v x for an appropriate matrix function l with a generalization to several integrals or lyapunov functions the discrete time analogue delta x delta t l bar nabla v where bar nabla is a discrete gradient preserves v as an integral or lyapunov function respectively

interpolation and the discrete papoulis gerchberg algorithm analyze the performance of an iterative algorithm similar to the discrete papoulis gerchberg algorithm and which can be used to recover missing samples in finite length records of band limited data no assumptions are made regarding the distribution of the missing samples in contrast with the often studied extrapolation problem in which the known samples are grouped together indeed it is possible to regard the observed signal as a sampled version of the original one and to interpret the reconstruction result studied as a sampling result the authors show that the iterative algorithm converges if the density of the sampling set exceeds a certain minimum value which naturally increases with the bandwidth of the data they give upper and lower bounds for the error as a function of the number of iterations together with the signals for which the bounds are attained also they analyze the effect of a relaxation constant present in the algorithm on the spectral radius of the iteration matrix from this analysis they infer the optimum value of the relaxation constant they also point out among all sampling sets with the same density those for which the convergence rate of the recovery algorithm is maximum or minimum for low pass signals it turns out that the best convergence rates result when the distances among the missing samples are a multiple of a certain integer the worst convergence rates generally occur when the missing samples are contiguous

reconstruction algorithms in irregular sampling a constructive solution of the irregular sampling problem for band limited functions is given we show how a band limited function can be com pletely reconstructed from any random sampling set whose density is higher than the nyquist rate and give precise estimates for the speed of convergence of this iteration method variations of this algorithm allow for irregular sampling with derivatives reconstruction of band limited functions from local averages and irregular sampling of multivariate band limited functions in the irregular sampling problem one is asked whether and how a band limited function can be completely reconstructed from its irregularly sam pled values f xi this has many applications in signal and image processing seismology meteorology medical imaging etc finding constructive solutions of this problem has received considerable attention among mathematicians and engineers the mathematical literature provides several uniqueness results 1 2 17 18 19 it is now part of the folklore that for stable sampling the sampling rate must be at least the nyquist rate 18 these results as deep as they are have had little impact for the applied sciences because they were not constructive if the sampling set is just a perturbation of the regular oversampling then a reconstruction method has been obtained in a seminal paper by duffin and schaeffer 6 see also 29 if for some l 0 a 0 and o 0 the sampling points xk k e z satisfy a xk ok a k i then the norm equivalence a ir f x 2dx with w n o this norm equivalence implies that it is possible to reconstruct through an iterative procedure the so called frame method most of the later work on constructive methods consists of variations of this method 3 21 22 26 the above conditions on the sampling set exclude random irregular sampling sets e g sets with regions of higher sampling density a partial but undesirable remedy to handle highly irregular sampling sets would be to force the above conditions by throwing away information on part of the points and accept a very slow convergence of the iteration

a discrete theory of irregular sampling abstract we derive algorithms for the iterative reconstruction of discrete band limited signals from their irregular samples they converge at a geometric rate and all constants are computable explicitly we also treat the related problem of interpolation of trigonometric polynomials and give estimates for the condition number of certain positive definite toeplitz matrices

efficient numerical methods in non uniform sampling theory summary we present a new second generation reconstruction algorithm for irregular sampling i e for the problem of recovering a band limited function from its non uniformly sampled values the efficient new method is a combination of the adaptive weights method which was developed by the two first named authors and the method of conjugate gradients for the solution of positive definite linear systems the choice of adaptive weights can be seen as a simple but very efficient method of preconditioning further substantial acceleration is achieved by utilizing the toeplitztype structure of the system matrix this new algorithm can handle problems of much larger dimension and condition number than have been accessible so far furthermore if some gaps between samples are large then the algorithm can still be used as a very efficient extrapolation method across the gaps

efficient algorithms for burst error recovery using fft and other transform kernels we show that the problem of signal reconstruction from missing samples can be handled by using reconstruction algorithms similar to the reed solomon rs decoding techniques usually the rs algorithm is used for error detection and correction of samples in finite fields for the case of missing samples of a speech signal we work with samples in the field of real or complex numbers and we can use fft or some new transforms in the reconstruction algorithm dsp implementation and simulation results show that the proposed methods are better than the ones previously published in terms of the quality of recovered speech signal for a given complexity the burst error recovery method using the fft kernel is sensitive to quantization and additive noise like the other techniques however other proposed transform kernels are very robust in correcting bursts of errors with the presence of quantization and additive noise

acceleration of the frame algorithm shows how polynomial acceleration techniques which have been developed for the solution of large linear systems can be employed to improve and accelerate the frame algorithm these methods permit a reduction in the number of necessary iterations by an order of magnitude when the frame algorithm is slow the author gives several examples from the theory of irregular sampling from wavelet theory and from gabor theory where these methods are probably mandatory for efficient reconstruction

image restoration by the method of convex projections part 1ߞtheory a projection operator onto a closed convex set in hilbert space is one of the few examples of a nonlinear map that can be defined in simple abstract terms moreover it minimizes distance and is nonexpansive and therefore shares two of the more important properties of ordinary linear orthogonal projections onto closed linear manifolds in this paper we exploit the properties of these operators to develop several iterative algorithms for image restoration from partial data which permit any number of nonlinear constraints of a certain type to be subsumed automatically their common conceptual basis is as follows every known property of an original image f is envisaged as restricting it to lie in a well defined closed convex set thus m such properties place f in the intersection e0 ei of the corresponding closed convex sets e1 e2 em given only the projection operators pi onto the individual ei s i 1 m we restore f by recursive means clearly in this approach the realization of the pi s in a hilbert space setting is one of the major synthesis problems section i describes the geometrical significance of the three main theorems in considerable detail and most of the underlying ideas are illustrated with the aid of simple diagrams section ii presents rules for the numerical implementation of 11 specific projection operators which are found to occur frequently in many signal processing applications and the appendix contains proofs of all the major results

analysis of error in reconstruction of two dimensional signals from irregularly spaced samples we consider the problem of estimating a band limited two dimensional 2 d signal based on a finite set of irregularly spaced samples we derive the minimum mean squared error estimator of the form of a sum of weighted interpolating functions that are identical in shape but are centered at the irregularly spaced sample points and show that this estimator is identical to the minimum energy band limited interpolator that has been previously obtained by others the relation between the mean squared error and the set of irregularly spaced sampling points is studied using the minimax principle the maximum of the mean squared error over a class of signals is minimized by proper choice of the set of sampling points three criteria for evaluating the performance of the sampling point set are derived from a candidate group of sets of irregularly spaced sampling points these criteria may be used to select that point set which is optimal for recovery over a class of signals

two families of nonstandard poisson structures for newton equations two families of nonstandard two dimensional poisson structures for systems of newton equations are studied they are closely related either with separable systems or with the so called quasi lagrangian systems a theorem characterizing the general form of bi hamiltonian formulation for separable systems in two and in n dimensions is formulated and proved

sufficient conditions for dynamical systems to have pre symplectic or pre implectic structures abstract we present a number of alternative sufficient conditions for the existence of pre symplectic or pre implectic poisson structures for both ordinary differential ode and ordinary difference o e equations four alternative sets of conditions are obtained for odes and o es in n dependent variables 1 a vector field in involution with the ode and an integral or two symmetries for an o e imply a pre implectic structure 2 volume preservation and n 2 symmetries imply a pre symplectic structure 3 volume preservation and n 2 integrals imply a pre implectic structure 4 complex implectic structure implies infinitely many real implectic structures in all but the first case the methods can give a set of distinct structures

minimum rate sampling and reconstruction of signals with arbitrary frequency support we examine the question of reconstruction of signals from periodic nonuniform samples this involves discarding samples from a uniformly sampled signal in some periodic fashion we give a characterization of the signals that can be reconstructed at exactly the minimum rate once a nonuniform sampling pattern has been fixed we give an implicit characterization of the reconstruction system and a design method by which the ideal reconstruction filters may be approximated we demonstrate that for certain spectral supports the minimum rate can be approached or achieved using reconstruction schemes of much lower complexity than those arrived at by using spectral slicing as in earlier work previous work on multiband signals have typically been those for which restrictive assumptions on the sizes and positions of the bands have been made or where the minimum rate was approached asymptotically we show that the class of multiband signals which can be reconstructed exactly is shown to be far larger than previously considered when approaching the minimum rate this freedom allows us in certain cases to have a far less complex reconstruction system

a stable nonuniform sampling expansion involving derivatives the work of d a linden and n m abramson inf control vol 3 p 26 31 1960 and vol 4 p 95 6 1961 is completed in the sense that a full investigation of the convergence properties of the expansion is made by adapting the work of n levinson am math soc colloq pub vol 26 1940 on nonharmonic fourier series and general results from the theory of b splines the author extends the result to the case of sampling with uniformly spaced samples a stability property is proved

the hamiltonization problem from a global viewpoint owing to the so called lie koenigs theorem any finite dimensional dynamical system can be put into hamiltonian form locally to that end the introduction of dummy variables is not necessary provided that the original system is even dimensional and that we allow for an explicitly time dependent symplectic structure in this paper inquiry is made as to which conditions are needed for a universality statement similar to the lie koenigs theorem to hold true globally in addition the passage to the lagrangian formalism is discussed and the results are illustrated by three examples

bi hamiltonian structure in frenet serret frame abstract we reduce the problem of constructing bi hamiltonian structure in three dimensions to the solutions of a riccati equation in moving coordinates of frenet serret frame all explicitly constructed examples in the literature are exhausted by constant solutions we conclude that in three dimensions vector fields which are not eigenvectors of the curl operator are locally bi hamiltonian

image restoration by the method of convex projections part 2 applications and numerical results the image restoration theory discussed in a previous paper by youla and webb 1 is applied to a simulated image and the results compared with the well known method known as the gerchberg papoulis algorithm the results show that the method of image restoration by projection onto convex sets by providing a convenient technique for utilizing a priori information performs significantly better than the gerchberg papoulis method

an average sampling theorem for bandlimited stochastic processes in this correspondence we give an average sampling theorem for bandlimited stochastic processes which shows that many average sampling theorems have their counterparts for stochastic signals

on the zeros of the generalized euler frobenius laurent polynomial and reconstruction of cardinal splines of polynomial growth from local average samples abstract let h t be a nonnegative measurable function supported in 1 2 1 2 and m d t 1 2 1 2 1 2 1 2 1 2 1 2 t d 1 times be the central b spline of degree d we show that the roots of the generalized euler frobenius laurent polynomial defined by e h d z n z h m d n z n are simple negative and all are different from 1 as a consequence of this result we show that for every sequence y n n z r z of samples having polynomial growth and nonnegative measurable function h supported in 1 2 1 2 there is a unique spline f of degree d with polynomial growth satisfying f h n y n n z the presented work answers affirmatively the open problem posed in perez villalon and portal 2012 9