generalized sampling expansions with multiple sampling rates for lowpass and bandpass signals in the fractional fourier transform domain the objective of generalized sampling expansion gse is the reconstruction of an unknown continuously defined function f left t right from samples of the responses from m linear time invariant lti systems that are each sampled using the 1 m th nyquist rate in this paper we investigate the gse for lowpass and bandpass signals with multiple sampling rates in the fractional fourier transform frft domain first we propose an improvement of papoulis gse which has multiple sampling rates in the frft domain based on the proposed gse we derive the periodic nonuniform sampling scheme and the derivative interpolation method by designing different fractional filters and selecting specific sampling rates in addition the papoulis gse and the previous gse associated with frft are shown to be special instances of our results second we address the problem of the gse of fractional bandpass signals a new gse for fractional bandpass signals with equal sampling rates is derived we show that the restriction of an even number of channels in the gse for fractional bandpass signals is unnecessary and perfect signal reconstruction is possible for any arbitrary number of channels further we develop the gse for a fractional bandpass signal with multiple sampling rates lastly we discuss the application of the proposed method in the context of single image super resolution reconstruction based on gse illustrations and simulations are presented to verify the validity and effectiveness of the proposed results

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

nonuniform sampling of periodic bandlimited signals digital processing techniques are based on representing a continuous time signal by a discrete set of samples this paper treats the problem of reconstructing a periodic bandlimited signal from a finite number of its nonuniform samples in practical applications only a finite number of values are given extending the samples periodically across the boundaries and assuming that the underlying continuous time signal is bandlimited provides a simple way to deal with reconstruction from finitely many samples two algorithms for reconstructing a periodic bandlimited signal from an even and an odd number of nonuniform samples are developed in the first the reconstruction functions constitute a basis while in the second they form a frame so that there are more samples than needed for perfect reconstruction the advantages and disadvantages of each method are analyzed specifically it is shown that the first algorithm provides consistent reconstruction of the signal while the second is shown to be more stable in noisy environments next we use the theory of finite dimensional frames to characterize the stability of our algorithms we then consider two special distributions of sampling points uniform and recurrent nonuniform and show that for these cases the reconstruction formulas as well as the stability analysis are simplified significantly the advantage of our methods over conventional approaches is demonstrated by numerical experiments

sinc interpolation of nonuniform samples it is well known that a bandlimited signal can be uniquely recovered from nonuniformly spaced samples under certain conditions on the nonuniform grid and provided that the average sampling rate meets or exceeds the nyquist rate however reconstruction of the continuous time signal from nonuniform samples is typically more difficult to implement than from uniform samples motivated by the fact that sinc interpolation results in perfect reconstruction for uniform sampling we develop a class of approximate reconstruction methods from nonuniform samples based on the use of time invariant lowpass filtering i e sinc interpolation the methods discussed consist of four cases incorporated in a single framework the case of sub nyquist sampling is also discussed and nonuniform sampling is shown as a possible approach to mitigating the impact of aliasing

spectral analysis and reconstruction for periodic nonuniformly sampled signals in fractional fourier domain nonuniform sampling occurs in many applications due to imperfect timebase or random events periodic nonuniform sampling is a special type of nonuniform sampling the properties and applications of periodic nonuniform sampling signals in traditional fourier domain have been extensively studied but so far none of the research papers covering the spectral analysis and reconstruction of periodic nonuniformly sampled signals in fractional fourier domain have been published this correspondence is to explore the spectral properties of periodic nonuniformly sampled signals associated with the fractional fourier transform first based on the uniform fractional fourier transform kernel the general spectral representation of periodic nonuniformly sampled signals has been derived second detailed analysis of periodic nonuniformly sampled chirp signals in the fractional fourier domain have been performed the results can be used to estimate the chirp rate and the sampling offsets finally a more simple relationship between the discrete fractional spectrum of periodic nonuniformly sampled signals and the continuous spectrum has been obtained based on this relationship the original continuous spectrum can be reconstructed from periodic nonuniformly sampled signals in fractional fourier domain in addition the simulations are carried out to verify the correctness of the results

nonuniform interpolation of noisy signals using support vector machines the problem of signal interpolation has been intensively studied in the information theory literature in conditions such as unlimited band nonuniform sampling and presence of noise during the last decade support vector machines svm have been widely used for approximation problems including function and signal interpolation however the signal structure has not always been taken into account in svm interpolation we propose the statement of two novel svm algorithms for signal interpolation specifically the primal and the dual signal model based algorithms shift invariant mercer s kernels are used as building blocks according to the requirement of bandlimited signal the sine kernel which has received little attention in the svm literature is used for bandlimited reconstruction well known properties of general svm algorithms sparseness of the solution robustness and regularization are explored with simulation examples yielding improved results with respect to standard algorithms and revealing good characteristics in nonuniform interpolation of noisy signals

functionally weighted lagrange interpolation of band limited signals from nonuniform samples a modification of the conventional lagrange interpolator is proposed in this paper that allows one to approximate a band limited signal from its own nonuniform samples with high accuracy the modification consists in applying the lagrange method to the signal but pre multiplied by a fixed function and then solving for the desired signal value its efficiency lies in the fact that the fixed function is independent of the sampling instants it is shown in this paper that the function can be selected so that the interpolation error decreases exponentially with the number of samples for the case in which the sampling instants have a maximum deviation from a uniform grid this paper includes a low complexity recursive implementation of the method its accuracy is validated in the numerical examples by comparison with several interpolators in the literature and by deriving upper and lower bounds for its maximum error

variational image reconstruction from arbitrarily spaced samples a fast multiresolution spline solution we propose a novel method for image reconstruction from nonuniform samples with no constraints on their locations we adopt a variational approach where the reconstruction is formulated as the minimizer of a cost that is a weighted sum of two terms 1 the sum of squared errors at the specified points and 2 a quadratic functional that penalizes the lack of smoothness we search for a solution that is a uniform spline and show how it can be determined by solving a large sparse system of linear equations we interpret the solution of our approach as an approximation of the analytical solution that involves radial basis functions and demonstrate the computational advantages of our approach using the two scale relation for b splines we derive an algebraic relation that links together the linear systems of equations specifying reconstructions at different levels of resolution we use this relation to develop a fast multigrid algorithm we demonstrate the effectiveness of our approach on some image reconstruction examples

reconstruction of multidimensional bandlimited signals from nonuniform and generalized samples this paper addresses the problem of multidimensional signal reconstruction from nonuniform or generalized samples typical solutions in the literature for this problem utilize continuous filtering the key result of the current paper is a multidimensional interpolation identity which establishes the equivalence of two multidimensional processing operations one of these uses continuous domain filters whereas the other uses discrete processing this result has obvious benefits in the context of the afore mentioned problem the results here expand and generalize earlier work by other authors on the one dimensional 1 d case potential applications include two dimensional 2 d images and video signals

multidimensional filter bank signal reconstruction from multichannel acquisition we study the theory and algorithms of an optimal use of multidimensional signal reconstruction from multichannel acquisition by using a filter bank setup suppose that we have an channel convolution system referred to as analysis filters in dimensions instead of taking all the data and applying multichannel deconvolution we first reduce the collected data set by an integer uniform sampling matrix and then search for a synthesis polyphase matrix which could perfectly reconstruct any input discrete signal first we determine the existence of perfect reconstruction pr systems for a given set of finite impulse response fir analysis filters second we present an efficient algorithm to find a sampling matrix with maximum sampling rate and to find a fir pr synthesis polyphase matrix for a given set of fir analysis filters finally once a particular fir pr synthesis polyphase matrix is found we can characterize all fir pr synthesis matrices and then find an optimal one according to design criteria including robust reconstruction in the presence of noise

an exactly soluble schrodinger equation with smooth position dependent mass the one dimensional generalized schrodinger equation for a system with smooth potential and mass step is resolved exactly the wave function depends on the heun s function which is a solution of a second order fuchsian equation with four singularities the behavior of the transmission coefficient as a function of energy is compared to that of the case of an abrupt potential and mass step two limiting cases are also studied when the width of the mass step is vanishing and when the smooth potential and mass step tend to an abrupt potential and mass step

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

asymmetric multi channel sampling in shift invariant spaces abstract we develop an asymmetric multi channel sampling on a shift invariant space v with a riesz generator t in l 2 r where each channeled signal is assigned a uniform but distinct sampling rate we use fourier duality between v and l 2 0 2 to find conditions under which there is a stable asymmetric multi channel sampling formula on v

algebraic solution for the natanzon confluent potentials an algebraic method is used to solve the bound state problem for the most general natanzon potential for which the schrodinger equation can be reduced to the confluent hypergeometric form hence confluent potentials the solution is obtained straightforwardly a simple argument is given to sustain that the natanzon potential is the most general confluent potential

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

block based methods for the reconstruction of finite length signals from nonuniform samples two novel block based algorithms are presented for the reconstruction of uniform samples given the nonuniform samples the first algorithm uses a sinc interpolator whereas the second one uses a dft based interpolator it is shown that the proposed algorithms are stable and the error due to noise and sampling jitter is bounded by the corresponding error norms of noise and jitter respectively we show that both of the block based algorithms provide nearly perfect reconstruction for a class of practically time and bandlimited signals boundary effects are considered and single and multiblock processing is discussed a modified block based algorithm is developed by using the windowing technique in order to improve the mean squared error mse performance for nonbandlimited signals it is shown that this algorithm performs better than a group of alternative algorithms including yen s third algorithm for a variety of signal noise and sampling grids

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

reconstruction of nonuniformly sampled bandlimited signals by means of digital fractional delay filters this paper considers the problem of reconstructing a class of nonuniformly sampled bandlimited signals of which a special case occurs in e g time interleaved analog to digital converter adc systems due to time skew errors to this end we propose a synthesis system composed or digital fractional delay filters the overall system i e nonuniform sampling and the proposed synthesis system can be viewed as a generalization of time interleaved adc systems to which the former reduces as a special case compared with existing reconstruction techniques our method has major advantages from an implementation point of view to be precise 1 we can perform the reconstruction as well as desired in a certain sense by properly designing the digital fractional delay filters and 2 if properly implemented the fractional delay filters need not be redesigned in case the time skews are changed the price to pay for these attractive features is that we need to use a slight oversampling it should be stressed however that the oversampling factor is less than two as compared with the nyquist rate the paper includes error and quantization noise analysis the former is useful in the analysis of the quantization noise and when designing practical fractional delay filters approximating the ideal filters

fractional spectrum analysis for nonuniform sampling in the presence of clock jitter and timing offset this paper focuses on analyzing the fractional spectrum of nonuniform sampling which is affected by clock jitter and timing offset both cases of single channel and recurrent samplings with limited bandwidth in the fractional fourier domain are considered for which we study two types of discrete time fractional fourier transforms dtfrfts for the first type dtfrfts we perform their statistical analyses and meanwhile establish relationships between the fractional spectra of the input and output signals of non ideal single multi channel analog to digital converter such relationships indicate that the statistical mean of each fractional spectrum contains periodic replicas of the spectrum for the input signal and these replicas are particularly modulated by the characteristic function of the perturbations introduced by the clock jitter moreover we interpret the first type dtfrfts from the perspective of linear frequency modulation signal decomposition for nonuniform sampling the second type dtfrfts serve as approximate dtfrfts and they especially deal with the general situation when the nonuniform sampling instants are unknown to this end we study modified forms of the first type dtfrfts with known inputs in kernels based on which the statistical mean of the approximate fractional spectra for both cases are derived then we compensate the fractional spectrum bias with high accuracy by developing optimal filters that minimize the mean square error between the original and the compensated fractional spectra for both cases different from single channel sampling the compensation for recurrent sampling involves an additional step before optimal filtering which gives better accuracy simulation results show that the proposed methods outperform existing methods

solvable potentials generated by sl 2 r canonical transformations are used to build realisations of sl 2 r in terms of the basic quantum mechanical operators q and p the results are used to construct solvable potentials in the framework of one dimensional quantum mechanics

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

on the relationship between uniform and recurrent nonuniform discrete time sampling schemes recurrent nonuniform discrete time signal samples can be regarded as a combination of k mutual delayed sequences of uniform discrete time signal samples taken at one kth of the nyquist sampling rate this paper introduces a new alternative discrete time analysis model of the recurrent nonuniform sampling scenario this model can be described by the analysis part of a uniform discrete fourier transform dft modulated filterbank from which the k uniformly distributed and down sampled frequency bands are mixed in a very specific way this description gives a clear relationship between uniform and recurrent nonuniform discrete time sampling schemes a side benefit of this model is an efficient structure with which one can reconstruct uniform discrete time nyquist signal samples from recurrent nonuniform samples with known mutual delays between the nonuniform distributed samples this reconstruction structure can be viewed as a natural extension of the synthesis part of an uniform dft modulated filterbank

fft interpolation from nonuniform samples lying in a regular grid this paper presents a method to interpolate a periodic band limited signal from its samples lying at nonuniform positions in a regular grid which is based on the fft and has the same complexity order as this last algorithm this kind of interpolation is usually termed the missing samples problem in the literature and there exists a wide variety of iterative and direct methods for its solution the one presented in this paper is a direct method that exploits the properties of the so called erasure polynomial and provides a significant improvement on the most efficient method in the literature which seems to be the burst error recovery ber technique of marvasti s the paper includes numerical assessments of the method s stability and complexity

algebraic approach to the scattering matrix a purely algebraic procedure is presented for calculating recursion relations for s matrices belonging to problems associated with the group su 1 1 the procedure involves supplementing a recently introduced group theoretic approach to scattering by an algebraic framework that characterizes asymptotic behavior our formulation makes use of the euclidean group which contains the symmetry transformations for a free particle the coulomb and poeschl teller potentials are discussed to illustrate our methods

single variable realisation of the su 1 1 spectrum generating algebra and discrete eigenvalue spectra of a class of potentials a wide range of operators can be expressed in terms of the generators of the su 1 1 algebra the spectral properties of which are well known we attempt to find a realisation of the algebra in its most general form and thus evolve a unified approach to the problem of finding the spectra of hamiltonians amenable to the technique rather than taking each case separately we extend the analysis to obtain the spectra of potentials whose coordinate dependence is implicit

lp exponential stability for the equilibrium solutions of the navier stokes equations the exponential stability of the solution of the navier stokes equation in lp p 2 p 3 in bounded domain is considered in this paper under some assumptions on the external force it can be shown that the bounded solution of the navier stokes equation with initial and boundary conditions approaches the stationary solution of the system exponentially when time t goes to infinite

a lie algebraic approach to effective mass schrodinger equations we use lie algebraic techniques to obtain exact solutions of the effective mass schrodinger equation in particular we use the su 1 1 algebra both as a spectrum generating algebra and as a potential algebra to obtain exact solutions of effective mass schrodinger equations corresponding to a number of potentials we also discuss the construction of isospectral hamiltonians for which both the mass and the potential are different

on sampling of band limited signals associated with the linear canonical transform sampling is one of the fundamental topics in the signal processing community theorems proposed under this topic form the bridge between the continuous time signals and discrete time signals several sampling theorems which aid in the reconstruction of signals in the linear canonical transform lct domain have been proposed in the literature however two main practical issues associated with the sampling of the lct still remain unresolved the first one relates to the reconstruction of the original signal from nonuniform samples and the other issue relates to the fact that only a finite number of samples are available practically focusing on these issues this paper seeks to address the above from the lct point of view first we extend several previously developed theorems for signals band limited in the fourier domain to signals band limited in the lct domain followed by the derivation of the reconstruction formulas for finite uniform or recurrent nonuniform sampling points associated with the lct simulation results and the potential applications of the theorem are also proposed