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We use the eigenfunction expansion of Green's function of Dirichlet problems to obtain sampling theorems. The analytic properties of the sampled integral transforms as well as the uniform convergence of the sampling series are proved without any restrictions on the integral transforms. We obtain a one- and multi-dimensional versions of sampling theorems. In both cases the sampling series are written in terms of Lagrange-type interpolation expansions. Some examples and the truncation error as well as the stability of the obtained sampling expansions are discussed at the end of the paper. © 1998 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.

This paper deals with the identification of the inverse dynamics of nonlinear plants using a neural network (NN) modelling approximation. The problem of neural identification is tackled using static feedforward networks in a NARX configuration. The NN is periodically replaced by a new version trained outside the control loop using input-output patterns gathered on-line during operation. The selection of patterns for the training of the NN is discussed in the framework of multidimensional sampling theory. A modulation signal prevents the use of the NN when the expected approximation is poor. Some examples of application are given.

For pt.I see ibid., vol.43, no.1, p.190-207 (1997). We consider the sampling of bandlimited spatiotemporal signals subject to the time-sequential (TS) constraint that only one spatial position can be sampled at any given time. Part I of this paper developed a new unifying theory linking TS sampling with generalized multidimensional sampling. It provided a complete characterization of time-sequential lattice patterns, including tight bounds on the temporal parameters of those time-sequential sampling patterns that produce zero aliasing error. In this paper we present large space-spatial-bandwidth product asymptotics for these bounds. One of the surprising results is that in many cases, there exist optimal patterns, for which, asymptotically, there is no extra penalty for lattice sampling subject to the time-sequential constraint, as compared to unconstrained multidimensional sampling. The implication to source coding is that an optimum encoder for spatiotemporal signals can be implemented with no buffering or other processing using a time-sequential sampler. The results apply to very general multidimensional spatial and spectral supports (star shaped, or at most convex).

Universal (pointwise uniform and time shifted) truncation error upper bounds are presented for the Whittaker--Kotel'nikov--Shannon (WKS) sampling restoration sum for Bernstein function classes $B_{��,d}^q,\, q>1,\, d\in \mathbb N$, when the decay rate of the sampled functions is unknown. The case of regular sampling is discussed. Extremal properties of related series of sinc functions are investigated., 18 pages, 2 figures. This is an Author's Accepted Manuscript of an article published in the Georgian Mathematical Journal. Vol.17, No. 4. (2010), 765-786. The final publication is available at De Gruyter. DOI: 10.1515/gmj.2010.033

It is shown that half of the points on the diamond grid are redundant for sampling and reconstructing a bandlimited 3-D signal. This redundancy is then exploited to show that the tetrahedral digital waveguide mesh requires four times more computational density and twice the memory storage for the same approximation as a finite difference scheme on the face-centered cubic lattice.

Universal (pointwise uniform and time shifted) truncation error upper bounds are presented in Whittaker--Kotel'nikov--Shannon (WKS) sampling restoration sum for Bernstein function class $B_{\pi,d}^q\,,\ q \ge 1,$ $d\in \mathbb N\,,$ when the sampled functions decay rate is unknown. The case of multidimensional irregular sampling is discussed., Comment: 13 pages. This is an Author's Accepted Manuscript of an article published in the Applicable Analysis, Vol.90, No. 3-4. (2011), 595--608. [copyright Taylor & Francis], available online at: http://www.tandfonline.com/ [DOI:10.1080/00036810903437754]

Today we collect large amounts of data and we receive more than we can handle, the accumulated data are often raw and far from being of good quality they contain Missing Values and noise.

Papoulis' generalized sampling expansion (GSE) can be carried out if a set of linear equations has a solution. This enables the reconstruction of a signal from its filtered and downsampled versions. We show that the existence of this solution is also a necessary condition for the reconstruction of the signal from the available data. Special attention is given to the case of recurrent multidimensional sampling.

Consider two uniform samplers operating simultaneously on a signal, with sample spacings MT and NT where M and N are coprime integers, and T has time or space dimension. It can be shown that the difference coarray of this pair of sampling arrays has elements at all integer multiples of T, regardless of how large M and N are. This implies that any application which depends only on second order statstics, such as angle of arrival estimation, beamforming, and multiple frequency detection, can be carried out at high resolution with the help of sparse sampling arrays. One manifestation is that two sensor arrays withM and N sensors can actually identify O(MN) independent sources. This paper extends these results to the case of multidimensional signals. The multidimensional sampling arrays operate on a lattice geometry. The coarray of such a system is studied. Even though the two lattice arrays are sparse (with respect to the integer grid), the coarray contains all integer vectors. It is also shown how to achieve the effect of a high resolution multidimensional DFT filter bank by combining coprime low resolution filter banks.

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 N-channel convolution system, referred to as N analysis filters, in M dimensions. Instead of taking all the data and applying multichannel deconvolution, we first reduce the collected data set by an integer M×M uniform sampling matrix [Formula: see text], 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.

Universal (pointwise uniform and time shifted) truncation error upper boundsare presented for the Whittaker-Kotel'nikov-Shannon (WKS) sampling restoration sum for Bernstein function classes $B_{; ; \pi, d}; ; ^q, q>1, d\in \mathbb N$, when the decay rate of the sampled functions is unknown. The case of regular sampling is discussed. Extremal properties of related series of sinc functions are investigated.

In this paper, a novel multiresolution algorithm for registering multimodal images, using an adaptive Monte Carlo scheme is presented. At each iteration, random solution candidates are generated from a multidimensional solution space of possible geometric transformations, using an adaptive sampling approach. The generated solution candidates are evaluated based on the Pearson type-VII error between the phase moments of the images to determine the solution candidate with the lowest error residual. The multidimensional sampling distribution is refined with each iteration to produce increasingly more plausible solution candidates for the optimal alignment between the images. The proposed algorithm is efficient, robust to local optima, and does not require manual initialization or prior information about the images. Experimental results based on various real-world medical images show that the proposed method is capable of achieving higher registration accuracy than existing multimodal registration algorithms for situations, where little to no overlapping regions exist.

This paper describes an automatic methodology for optimizing sample point selection for using in the framework of model order reduction (MOR). The procedure, based on the maximization of the dimension of the subspace spanned by the samples, iteratively selects new samples in an efficient and automatic fashion, without computing the new vectors and with no prior assumptions on the system behavior. The scheme is general, and valid for single and multiple dimensions, with applicability on rational nominal MOR approaches, and on multi-dimensional sampling based parametric MOR methodologies. The paper also presents an integrated algorithm for multi-point MOR, with automatic sample and order selection based on the transfer function error estimation. Results on a variety of industrial examples demonstrate the accuracy and robustness of the technique.

New magnitude of truncation error upper bound and convergence rate are obtained for Whittaker - Kotel'nikov - Shannon (WKS) sampling restoration sum for Bernstein function class $B_{; ; ; \pi, d}; ; ; ^q, q \ge 1, d\in \mathbb N$, when the sampled functions decay rate is unknown. The case of multidimensional irregular sampling is discussed.

Wireless sensor networks have emerged as versatile tools to monitor the evolution of physical fields over large areas by means of self-powered and low-cost sensing devices. Although their development was originally motivated by military applications, they are nowadays used in numerous civilian application areas. In many sensor network scenarios, including the ones studied in this thesis, the goal of data acquisition by means of sensor nodes is to reproduce at a base station the observed field under a fidelity constraint, using limited communication resources. Due to the scarcity of the energy resources, the physical nature of the observed fields, the constraints on spatial sampling, the high cost of inter-node communication and the reliance on a shared communication medium, wireless sensor networks raise challenging questions in the areas of multidimensional sampling, multiterminal source coding and network communications. In this thesis we address the issue of how to efficiently represent the data collected by the sensor nodes. This data has some inherent structure that is determined by the laws of physics. Since the design of efficient sampling geometries and source coding schemes requires a thorough understanding of the physical properties of the observed phenomenon, the first part of our work is devoted to setting up a mathematical model for physical fields that is amenable to the tools of information theory while applying to most physical phenomena of interest. The question regarding the sufficiency of a discrete representation of an analog field is addressed in two steps. First, we prove a multidimensional sampling theorem for homogeneous random fields with compactly supported spectral measures. While, under appropriate conditions, a discrete-space and -time representation of the analog field is sufficient, discretizing the field's amplitude, also referred to as source coding, implies an unavoidable loss of information, which may be expressed in terms of a rate distortion function. In the second step we study various source coding schemes, differing by the amount of required inter-node communication and the complexity of the involved maps. We derive lower and upper bounds for the rate distortion function of the multiterminal source coding scheme, which is of particular interest in sensor network engineering as it makes use of the spatio-temporal structure of the collected data without requiring inter-node communication. In the second part of the thesis we study some applications of sensor networks. In particular, we consider the acquisition of acoustic waves, the monitoring of temperature distributions and the compression of data sequences generated by random walks. For the setup of sound field acquisition, we show that, under the assumption of spectral whiteness of the sound field, the afore-mentioned bounds coincide, thus establishing the multiterminal rate distortion function for this setup.

The authors state and prove a theorem solving the problem of commutativity in two dimensions. It is shown under which conditions upsampling and downsampling can be interchanged in two dimensions. This is the generalization to arbitrary two-dimensional lattices of the result that one-dimensional upsampling and downsampling commute if and only if their sampling rates are coprime. Some illustrative examples are given. The result holds for arbitrary sampling lattices. >

Digital waveguide mesh (DWM) models are numerical solvers for the wave equation in -dimensions. They are used for obtaining the traveling-wave solution in practical acoustical modeling applications. Although unstructured meshes can be used with DWMs, regular mesh topologies are traditionally used due to their implementation simplicity. This letter discusses the accuracy of first-order approximations to numerical derivatives on more general unstructured mesh topologies. The results are applied to structured, regular mesh topologies as used in DWM modeling. A comparison of 2-D and 3-D DWM topologies with respect to the accuracy of first-order approximations to numerical derivatives is presented.

This paper reviews the fundamentals of multidimensional multirate signal processing. Central to these discussions is the idea of generalized sampling-lattice. Topics discussed include nonrectangular decimators, interpolators, generalized DFT, and filter banks. The multidimensional polyphase decomposition is developed, with applications in decimation filtering and perfect reconstruction filter banks.

Some of the key mathematical limitations encountered in multidimensional digital signal processing research, the useful general settings in which various topical investigations are facilitated, and directions for future research are summarized. The discussion covers multidimensional sampling, multidimensional signal processing over polydisc algebra, nonlinear multidimensional filtering and prediction, and nonlinear nonstationary multidimensional modelling approaches. >

A general mathematical model for digital signal processing, which is based onLp function spaces, is introduced. This model is used to derive a new class of reconstruction filters for the reconstruction ofN-dimensional images which are not necessarily band-limited. A basic proposition in this work is that the best reconstruction algorithm will depend on the pre-sample filter (or point spread) function. The reconstruction filters described here are optimal in the sense that they result in a reconstructed image which is as close as possible, with respect to a given measure of fidelity, to the unsampled image. The filter is similar in form to the optimal filter derived by Peterson and Middleton (Inform. and Control5, 1962 , 279–323) in their comprehensive paper on multidimensional sampling. However the reconstruction filter of Peterson and Middleton is optimal for random fields, whereas the filter described here is optimal for individual images. The optimal reconstruction filter has certain practical advantages over empirically derived reconstruction methods such as cubic interpolation. One of these advantages is that the method produces positive-valued images without loss of image resolution. The performance of the optimal reconstruction filter can be understood without reference to aliasing and truncation errors and is quantified in terms of a simple error metric.

A new type of three-dimensional (3-D) display recently introduced on the market holds great promise for the future of 3-D visualization, communication, and entertainment. This so-called automultiscopic display can deliver multiple views without glasses, thus allowing a limited "look-around" (correct motion-parallax). Central to this technology is the process of multiplexing several views into a single viewable image. This multiplexing is a complex process involving irregular subsampling of the original views. If not preceded by low-pass filtering, it results in aliasing that leads to texture as well as depth distortions. In order to eliminate this aliasing, we propose to model the multiplexing process with lattices, find their parameters and then design optimal anti-alias filters. To this effect, we use multidimensional sampling theory and basic optimization tools. We derive optimal anti-alias filters for a specific automultiscopic monitor using three models: the orthogonal lattice, the nonorthogonal lattice, and the union of shifted lattices. In the first case, the resulting separable low-pass filter offers significant aliasing reduction that is further improved by hexagonal-passband low-pass filter for the nonorthogonal lattice model. A more accurate model is obtained using union of shifted lattices, but due to the complex nature of repeated spectra, practical filters designed in this case offer no additional improvement. We also describe a practical method to design finite-precision, low-complexity filters that can be implemented using modern graphics cards.

Time shifted alisaing error upper bound extremals for the sampling reconstruction procedure are fully characterized. Sharp upper bounds are found on the alisaing error of truncated cardinal series and the corresponding extremals are described for the entire functions from certain specific $L^p, p \ge 1$ classes. Analogous results are obtained in multidimensional regular sampling. Truncation error analysis is provided in all cases considered. Sharpness of bounding inequalities. convergence rates and various sufficient conditions are discussed.

Two types of 3D MAS NMR experiments are introduced, which combine standard (NC,CC) transfer schemes with (1H,1H) mixing to simultaneously detect connectivities and structural constraints of uniformly 15N,13C-labeled proteins with high spectral resolution. The homonuclear CCHHC and CCC experiments are recorded with one double-quantum evolution dimension in order to avoid a cubic diagonal in the spectrum. Depending on the second transfer step, spin systems or proton–proton contacts can be determined with reduced spectral overlap. The heteronuclear NHHCC experiment encodes NH–HC proton–proton interactions, which are indicative for the backbone conformation of the protein. The third dimension facilitates the identification of the amino acid spin system. Experimental results on U-[15N,13C]valine and U-[15N,13C]ubiquitin demonstrate their usefulness for resonance assignments and for the determination of structural constraints. Furthermore, we give a detailed analysis of alternative multidimensional sampling schemes and their effect on sensitivity and resolution.

In this paper, the sampling and aliasing consequences of employing a quarter-detector-offset (QDO) in helical computed tomography (CT) are analyzed. QDO is often used in conventional CT to reduce in-plane aliasing by eliminating data redundancies to improve radial sampling. In helical CT, these same redundancies are exploited to improve longitudinal sampling and so it might seem ill-advised to employ QDO. The relative merit of the two geometries for helical CT is studied by conducting a multidimensional sampling analysis of projection-space sampling as well as a Fourier crosstalk analysis of crosstalk among the object's Fourier basis components. Both a standard fanbeam helical CT geometry and a hypothetical parallel-beam CT geometry, which helps illuminate the more complicated fanbeam results, are analyzed. Using the sampling analysis, it was found that the use of QDO leads to very different spectral tiling than arise when not using QDO. However, due to the shape of the essential support of the projection data spectra that arises in practice, both configurations lead to very similar or identical amounts of spectral overlap. This perspective also predicts the spatially variant longitudinal aliasing that has been observed in helical CT. The crosstalk results were consistent with those of the multidimensional sampling analysis. Thus, from the standpoint of aliasing and crosstalk, no compelling difference is found between the two geometries.

Multiview passive 3-D displays, such as those based on lenticular or parallax-barrier technologies, require multiplexing of views into a single same-size RGB image. Thus, multiplexing of N views necessitates N :1 sub-sampling of each view and must be preceded by suitable lowpass filtering to prevent, or at least reduce, aliasing. Without such filtering, objectionable "jagged" edges, distorted textures, or Moire patterns are perceived although, admittedly, these effects are not as disturbing as in the case of single-view sub-sampling without multiplexing with other views. In this paper, unlike in our previous work, we consider anti-alias filtering derived from a non-orthogonal lattice. First, we approximate pixel layout for each view (sampling pattern) by a two-dimensional lattice; we find parameters of the lattice by minimizing a mismatch error between lattice and single-view points. Then, based on lattice parameters, we find frequency-domain specifications of the anti-alias filter. The filter has hexagonal passband and thus is non-separable. Although previously we designed such filters for floating-point implementations, here we opt for the more practical fixed-point arithmetic; the resulting filters can be easily implemented on ubiquitous fixed-point DSP chipsets. The fixed-point filters slightly depart from the desired magnitude specifications, but when applied to actual multiview images they produce almost indistinguishable results from those obtained by floating-point counterparts.© (2004) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Multi-slice helical CT scanners allow for much faster scanning and better x-ray utilization than do their single-slice predecessors, but they engender considerably more complicated data sampling patterns due to the interlacing of the samples from different rows as the patient is translated. Characterizing and optimizing this sampling is challenging because the conebeam geometry of such scanners means that the projections measured by each detector row are at least slightly oblique, making it difficult to apply standard multidimensional sampling analyses. In this study, we seek to apply a more general framework for analyzing sampled imaging systems known as Fourier crosstalk analysis. Our purpose in this preliminary work is to compare the information content of the data acquired in three different scanner geometries and operating conditions with ostensibly equivalent volume coverage and average longitudinal sampling interval: a single-slice scanner operating at pitch 1, a four-slice scanner operating at pitch 3 and a 15-slice scanner operating at pitch 15. We find that moving from a single-slice to a multi-slice geometry introduces longitudinal crosstalk characteristic of the longitudinal sampling interval between periods of individual each detector row, and not of the overall interlaced sampling pattern. This is attributed to data inconsistencies caused by the obliqueness of the projections in a multi-slice/conebeam configuration. However, these preliminary results suggest that the significance of this additional crosstalk actually decreases as the number of detector rows increases.

One- or multidimensional sampling of sensor information is the origin of natural- source digital multimedia signals. The methodologies of sampling have an eminent impact on any subsequent processing steps, including encoding and content analysis of signals. In particular for video signals, full understanding of sampling phenomena is one key to further improve compression technology. To introduce the conditions behind multi-dimensional sampling, this chapter starts by a general introduction to multi-dimensional coordinate systems and related dependencies in the Fourier spectral domain. This also includes aspects of geometric modifications and their impact on the properties of the spectrum.

This paper addresses the issue of artifact visibility in automultiscopic 3-D lenticular displays. A straightforward extension of the two-view lenticular autostereoscopic principle to M views results in an M-fold loss of horizontal resolution due to the subsampling needed to properly multiplex the views. In order to circumvent the imbalance between the horizontal and vertical resolution, a tilt can be applied to the lenticules to orient them at a small angle to the vertical direction, as is done in the SynthaGram display from Stereographics Corp. In either case, to avoid aliasing the subsampling should be preceded by suitable lowpass pre-filtering. Although for purely vertical lenticules a sufficiently narrowband lowpass horizontal filtering suffices, the situation is more complicated for diagonal lenticules; the subsampling of each view is no more orthogonal, and more complex sampling models need to be considered. Based on multidimensional sampling theory, we have studied multiview sampling models based on lattices. These models approximate pixel positions on a lenticular automultiscopic display and lead to optimal anti-alias filters. In this paper, we report results for a separable approximation to non-separable 2-D anti-alias filters based on the assumption that the lenticule slant is small. We have carried out experiments on a variety of images, and different filter bandwidths. We have observed that the theoretically-optimal bandwidth is too restrictive; aliasing artifacts disappear, but some image details are lost as well. Somewhat wider bandwidths result in images with almost no aliasing and largely preserved detail. For subjectively-optimized filters, the improvements, although localized, are clear and enhance the 3-D viewing experience.

Direct finite interpolation formulas are developed for the Paley-Wiener function spaces $L^2_\Diamond$ and $L^2_{; ; [-\pi, \pi]^d}; ; $, where $L^2_\Diamond$ contains all bivariate functions whose Fourier spectrum is supported by the set $\Diamond = {; ; \rm Cl}; ; \{; ; (u, v)| |u|+|v|< \pi\}; ; $, while in $L^2_{; ; [-\pi, \pi]^d}; ; $ the Fourier spectrum support set of its $d$-variate entire elements is $[-\pi, \pi]^d$. The multidimensional Kotel'nikov - Shannon sampling formula reamins valid when only finitely many sampling knots are deviated from the uniform sampling. By using this interpolation procedure, we truncate a sampling sum to its irregularly sampled part. Upper bounds of the truncation error are obtained in both cases. According to the Sun-Zhou extension of the Kadets 1/4 - theorem, the magnitude of deviations are limited coordinatewise to 1/4. To avoid this inconvenience, we introduce the weighted Kotel/nikov - Shannon sampling sums. For $L^2_{; ; [-\pi, \pi]^d}; ; $ Lagrange-type direct finite interpolation fromulas are given. Finally, convergence - rate questions are discussed.

This paper presents a vector quantization scheme, based on the current understanding of early vision. This allows one to capture more details around the region of interest, and less details farther from that region, which can be achieved by a variable resolution in the feature domain. A cone beam form, nonuniform, multidimensional sampling strategy has been introduced, where the size of each cell increases proportionally to the inverse of the cell's distance from the region of interest. To simplify the actual realization, a discrete-ray-distance has been introduced in the LP/sub /spl infin// space, generating hyper-cubic cells. This approach can be used in MRI image representation, the 3-D Radon transformation, and volumetric image reconstruction.

This paper deals with multidimensional sampling and optimal reconstruction of the input random field or image class in the mean squared sense. It considers the interpolation and approximation sampling systems and presents closed form expressions for the filters in these systems as well as the mean squared errors between the input and the reconstructed fields. For the approximation sampling systems the optimal filters turn out to be spectral factors of an ideal brickwall filter whose support is determined by the sampling lattice. Finally, it presents examples of filters matched to an input image class derived from a multispectral LANDSAT image. In an image coding context, these filters can be considered as providing the best tradeoff between coding rate reduction by undersampling and distortion in reconstruction. The performance of these filters is tested in the presence of a quantizer for different bit rates and is compared with that of some standard filters to demonstrate this.

We present sampling results for certain classes of 2-D signals that are not bandlimited, but have a parametric representation with a finite number of degrees of freedom, such as 2-D Diracs, polygons and bilevel signals with piecewise polynomial boundaries. As opposed to standard multidimensional sampling schemes, the proposed methods exploit the properties of the Radon transform of such signals. In particular, we demonstrate that by using an appropriate sampling kernel, one can perfectly reconstruct the signal from a finite set of samples of its Radon transform, and thus significantly reduce a computational load. The novel approach we present in the paper, offers practical algorithmic implementation and is potentially applicable to a large class of two-dimensional signals.

Waveguide Meshes are efficient and versatile models of wave propagation along a multidimensional ideal medium. The choice of the mesh geometry affects both the computational cost and the accuracy of simulations. In this paper, we focus on 2D geometries and use multidimensional sampling theory to compare the square, triangular, and hexagonal meshes in terms of sampling efficiency and dispersion error under conditions of critical sampling. The analysis shows that the triangular geometry exhibits the most desirable tradeoff between accuracy and computational cost., 9 pages, 6 figures, 1 table, to appear on IEEE Transactions on Speech and Audio Processing, vol. 9, no. 2, february 2001

Bidemsional langrange - Yen type interpolation formula is given base on the Whittaker - Kolel'nikov Shannon sampling reconstruction sum for bivariate deterministic signals living in the Paley - Wiener functions space L2◊. The interpolation parametar size is computed when the approximation relative error level and coordinate wise sample derivations are prescribed.

In this chapter we extend the efficient multidimensional sampling results well known in tomography to the rotation invariant Radon transform with a polynomial weight. Numerical experiments are given. Application of this result to Doppler Imaging, for the particular stellar inclination angle 7r/2 is presented.

Image based rendering using the plenoptic function is an efficient technique for re-rendering at different viewpoints. We study the sampling and reconstruction problem of plenoptic function as a multidimensional sampling problem. The spectral support of plenoptic function is found to be an important quantity in the efficient sampling and reconstruction of such a function. A spectral analysis for the light field, a 4D plenoptic function, is performed. Its spectrum, as a function of the depth function of the scene, is then derived. This result enables us to estimate the spectral support of the light field given some prior estimate of the depth function. Results using a piecewise constant depth model show significant improvement in rendering of the light field images. The design of the reconstruction filter is also discussed.

This is basically a review of the field of Quasi-Monte Carlo intended for computational physicists and other potential users of quasi-random numbers. As such, much of the material is not new, but is presented here in a style hopefully more accessible to physicists than the specialized mathematical literature. There are also some new results: On the practical side we give important empirical properties of large quasi-random point sets, especially the exact quadratic discrepancies; on the theoretical side, there is the exact distribution of quadratic discrepancy for random point sets., Comment: 51 pages. Full paper, including all figures also available at: ftp://ftp.nikhef.nl/pub/preprints/96-017.ps.gz Accepted for publication in Comp.Phys.Comm. Fixed some typos, corrected formula 108,figure 11 and table 2

This chapter discusses theoretical foundations of modelling of nonlinear control systems with neural networks. Both feedforward and recurrent networks are described with emphasis on the practical implications of the mathematical results. The major approaches based on approximation and interpolation theories are presented: Stone-Weierstrass’ theorem, Kolmogorov’s theorem and multidimensional sampling. These are compared within a unified framework and the relevance for neural modelling of nonlinear control systems is stressed. Also, approximation of functionals with feedforward networks is briefly explained. Finally, approximation of dynamical systems with recurrent networks is described with emphasis on the concept of differential approximation.

We consider the sampling of band-limited spatio-temporal signals subject to the time- sequential (TS) constraint that only one spatial sample can be taken at a given time. Using the powerful techniques of lattice theory, we develop a new unifying theory linking TS sampling with generalized multidimensional sampling. The theory, which applies to arbitrary spatial and spectral supports in any dimension, includes tight fundamental performance bounds, which are asymptotically achievable. Simple graphical design algorithms are provided for lattice sampling patterns minimizing the sampling rates, and illustrated by examples.

It has been almost thirty years since Shannon introduced the sampling theorem to communications theory. In this review paper we will attempt to present the various contributions made for the sampling theorems with the necessary mathematical details to make it self-contained. We will begin by a clear statement of Shannon's sampling theorem followed by its applied interpretation for time-invariant systems. Then we will review its origin as Whittaker's interpolation series. The extensions will include sampling for functions of more than one variable, random processes, nonuniform sampling, nonband-limited functions, implicit sampling, generalized functions (distributions), sampling with the function and its derivatives as suggested by Shannon in his original paper, and sampling for general integral transforms. Also the conditions on the functions to be sampled will be summarized. The error analysis of the various sampling expansions, including specific error bounds for the truncation, aliasing, jitter and parts of various other errors will be discussed and summarized. This paper will be concluded by searching the different recent applications of the sampling theorems in other fields, besides communications theory. These include optics, crystallography, time-varying systems, boundary value problems, spline approximation, special functions, and the Fourier and other discrete transforms.

Multidimensional sampling of real data, e.g., in space and time, often requires observations at non-uniformly spaced intervals. A discrete transform of a multidimensional stationary stochastic process transforms a multivariate problem into an asymptotically univariate one if the spacing is uniform in at least one dimension. For both uniform and non-uniform sampling and a model of ‘signal’ imbedded in a ‘noise’ process, asymptotic normality and independence justifies statistical testing in each cell of the transformed domain of the hypothesis ‘noise alone’ versus the alternate ‘signal plus noise’.

The well-known Whittaker-Kotel'nikov-Shannon sampling theorem for frequency-bandlimited functions of time is extended to functions of multidimensional arguments. It is shown that a function whose spectrum is restricted to a finite region of wave-number space may be reconstructed from its samples taken over a periodic lattice having suitably small repetition vectors. The most efficient lattice (i.e., requiring minimum sampling points per unit hypervolume) is not in general rectangular, nor is a unique reconstruction function associated with a given sampling lattice. The above results also apply to homogeneous wave-number-limited stochastic processes in the sense of a vanishing mean-square error. It is also found that, given a particular sampling lattice, the optimum (mean-square) presampling filter for nonwave-number-limited processes effects an ideal wave-number cutoff appropriate to the specified sampling lattice. Particular attention is paid to isotropic processes: minimum sampling lattices are specified up to eight-dimensional spaces, and a number of typical reconstruction functions are calculated.

All sampling representations of band-limited signals involve infinite sums. The truncation error associated with a given representation is defined as the difference between the signal and an approximating sum utilizing a finite number of terms. In this paper truncation error is expressed as a contour integral for Lagrange interpolation, general Hermite interpolation, the Shannon series (cardinal series), the Fogel derivative series, and multidimensional sampling expansions. Truncation error bounds are obtained under various constraints on the signal magnitude, spectral smoothness, and energy content.

We consider the problem of designing sparse sampling strategies for multidomain signals, which can be represented using tensors that admit a known multilinear decomposition. We leverage the multidomain structure of tensor signals and propose to acquire samples using a Kronecker-structured sensing function, thereby circumventing the curse of dimensionality. For designing such sensing functions, we develop low-complexity greedy algorithms based on submodular optimization methods to compute near-optimal sampling sets. We present several numerical examples, ranging from multi-antenna communications to graph signal processing, to validate the developed theory., 13 pages, 7 figures

The authors state and prove a theorem solving the problem of commutativity in two dimensions. It is shown under which conditions upsampling and downsampling can be interchanged in two dimensions. This is the generalization to arbitrary two-dimensional lattices of the result that one-dimensional upsampling and downsampling commute if and only if their sampling rates are coprime. Some illustrative examples are given. The result holds for arbitrary sampling lattices.

This paper deals with results from the n-D (n-dimensional) sampling theory relevant to neurocontrol. These form a proper setting for modelling with feedforward neural networks for the purposes of nonlinear control. The focus is on RBF networks in the context of n-D sampling. RBF neural structure may be applied as an interpolating filter, according to the n-D sampling theory.

Two nonlinear control systems modelling methodologies are presented and compared. The nonlinear system is given in the form of a NARX model. One modelling method is based on decomposition of the nonlinear function (right-hand side of the NARX model) into multiple local linear mappings, while the second takes the global approach by using a modified Fourier series technique. The local mappings are represented by a self-organising neural network and a local linear mapping is associated with each neuron. The network is trained by a combination of vector quantisation and winner takes most procedure. The global approach is based on a nonuniform multidimensional sampling theory. It replaces the original nonuniform sampling problem in the space domain by the uniform sampling problem in the frequency domain. The application of the proposed methods to modelling of nonlinear control systems is also given.