Showing papers on "Trigonometric interpolation published in 2018"

Lagrange-Polynomial-Interpolation-Based Keystone Transform for a Passive Radar

Abstract: In this paper, we address the problem of target's range migration in passive bistatic radar exploiting long coherent integration times with fairly wideband signals of opportunity. We resort to the well-known keystone transform (KT) to compensate for the range walk effect and to take advantage of a higher coherent integration gain against targets with nonnegligible radial velocity. Specifically, an efficient implementation of the KT is proposed, based on the Lagrange polynomial interpolation, in order to reduce the computational load of the method that mostly depends on the required slow-time interpolation stage. The analysis conducted against simulated data shows that the conceived approach allows us to achieve theoretical performance, while further reducing the KT complexity with respect to alternative solutions based on cardinal sine functions or chirp-Z transforms. Moreover, the application against experimental datasets collected by a DVB-T-based passive radar proves the practical effectiveness of the proposed algorithm and highlights its suitability for real-time air traffic surveillance applications.

Multiple Rank-1 Lattices as Sampling Schemes for Multivariate Trigonometric Polynomials

Abstract: We present a new sampling method that allows for the unique reconstruction of (sparse) multivariate trigonometric polynomials. The crucial idea is to use several rank-1 lattices as spatial discretization in order to overcome limitations of a single rank-1 lattice sampling method. The structure of the corresponding sampling scheme allows for the fast computation of the evaluation and the reconstruction of multivariate trigonometric polynomials, i.e., a fast Fourier transform. Moreover, we present a first algorithm that constructs a reconstructing sampling scheme consisting of several $$\text {rank}\text{- }1$$ lattices for arbitrary, given frequency index sets. Various numerical tests indicate the advantages of the constructed sampling schemes.

Asymmetric Pulse Modeling for FRI Sampling

Abstract: We consider sampling and reconstruction of finite-rate-of-innovation (FRI) signals such as a train of pulses, where the pulses have varying degrees of asymmetry. We address the problem of asymmetry modeling starting from a given symmetric prototype. We show that among the class of unitary operators that are linear and invariant to translation and scale, the fractional Hilbert (FrH) operator is unique for parametrically modeling pulse asymmetry. The FrH operator is obtained by a trigonometric interpolation between the standard Hilbert and identity operators, where the interpolation weights are determined by the degree of asymmetry. The FrH operators are also steerable , which allows for estimation of the asymmetry factors, in addition to the delays and amplitudes, using the high-resolution spectral estimation techniques that are used for solving standard FRI problems. We also develop the discrete counterpart using discrete FrH operators and show that all the desirable properties carry over smoothly to the discrete setting as well. We derive closed-form expressions for the Cramer–Rao bounds and Hammersley–Chapman–Robbins bound, on the variances of the estimators for continuous and discrete parameters, respectively. Experimental results show that the proposed estimators have variances that meet the lower bounds. We demonstrate an application of the proposed discrete FrH methodology on real electrocardiogram (ECG) signals in the presence of noise. Specifically, we show how the asymmetry of QRS complexes in various channels of an ECG signal could be modeled accurately.

Hybrid Interpolation Approach to Link Currents and Flux Linkages of Induction Machines

Abstract: This paper proposes a method to link currents and flux linkages of induction machines by combining trigonometric and scattered data interpolation. The currents and flux linkages of the stator and rotor are gained from transient electromagnetic finite-element-simulations. The relation between currents and flux linkages is a superposition of rotating angle dependency and nonlinear material properties. Both dependencies can be described with the developped hybrid interpolation approach. The validation results show a good agreement between the interpolated signals and comparative signals from finite-element-analysis. The benefit of the proposed method is that it can be coupled with the differential equations to obtain an induction machine model with high accuracy.

One-dimensional, non-local, first-order, stationary mean-field games with congestion: a Fourier approach

Abstract: Here, we study a one-dimensional, non-local mean-field game model with congestion. When the kernel in the non-local coupling is a trigonometric polynomial we reduce the problem to a finite dimensional system. Furthermore, we treat the general case by approximating the kernel with trigonometric polynomials. Our technique is based on Fourier expansion methods.

The cubic trigonometric automatic interpolation spline

Abstract: A class of cubic trigonometric interpolation spline curves with two parameters is presented in this paper. The spline curves can automatically interpolate the given data points and become C ² 2 interpolation curves without solving equations system even if the interpolation conditions are fixed. Moreover, shape of the interpolation spline curves can be globally adjusted by the two parameters. By selecting proper values of the two parameters, the optimal interpolation spline curves can be obtained.

An application of limiting interpolation to Fourier series theory

Abstract: The limiting real interpolation method is applied to describe the behavior of the Fourier coefficients of functions that belong to spaces which are “very close” to L2. The Fourier coefficients are taken with respect to bounded orthonormal systems.

Domain decomposition for quasi-periodic scattering by layered media via robust boundary-integral equations at all frequencies

Abstract: We develop a non-overlapping domain decomposition method (DDM) for the solution of quasi-periodic scalar transmission problems in layered media. Our approach relies on robust boundary-integral equation formulations of Robin-to-Robin (RtR) maps throughout the frequency spectrum, including at Wood, or cutoff, frequencies. We overcome the obstacle of non-convergent quasi-periodic Green functions at these frequencies by incorporating newly introduced shifted quasi-periodic Green functions. Using the latter in the definition of our quasi-periodic boundary-integral operators leads to rigorously stable computations of RtR operators. We develop Nystr\"om discretizations of the RtR maps that rely on trigonometric interpolation, singularity resolution, and fast convergent windowed quasi-periodic Green functions. We solve the tridiagonal DDM system via recursive Schur complements and we establish rigorously that this procedure is always completed successfully. We present a variety of numerical results concerning Wood frequencies in two and three dimensions as well as large numbers of layers.

Multidimensional Time Model for Probability Cumulative Function Applied to Geometrical Predictions

Abstract: Multidimensional Time Model for Probability Cumulative Function can be reduced to finite-dimensional time model, which can be characterized by Boolean algebra for operations over events and their probabilities and index set for reduction of infinite dimensional time model to finite number of dimensions of time model through application of Boolean prime ideal theorem and Stone duality and can be indexed by an index set considering also the fractal-dimensional time arising from alike supersymmetrical properties of probability through consideration of extension of the classical Stone duality to the category of Boolean spaces, locally compact Hausdorff spaces The introduction of probabilistical prediction philosophically based on Erdős–Renyi LLN for the prediction through Descartes’ cycles, Gauss methods of trigonometric interpolation and least squares to reduce error in determination of the orbits of planetary bodies, and Farey series continued by sampling on the Sierpinski gasket

Some Notes on Neville’s Algorithm of Interpolation with Applications to Trigonometric Interpolation

Abstract: In this paper is given a description of Neville’s algorithm which is generated from Lagrange interpolation polynomials. Given a summary of the properties of these polynomials with some applications. Then, using the Lagrange polynomials of lower degrees, Neville algorithm allows recursive computation of those of the larger degrees, including the adaption of Neville’s method to trigonometric interpolation. Furthermore, using a software application, such as in our case, Matlab, we will show the numerical experiments comparisons between the Lagrange interpolation and Neville`s interpolation methods and conclude for their advantages or disadvantages.

Observations on Interpolation by Total Degree Polynomials in Two Variables

Abstract: In contrast to the univariate case, interpolation with polynomials of a given maximal total degree is not always possible even if the number of interpolation points and the space dimension coincide. Due to that, numerous constructions for interpolation sets have been devised, the most popular ones being based on intersections of lines. In this paper, we study algebraic properties of some such interpolation configurations, namely the approaches by Radon–Berzolari and Chung–Yao. By means of proper H-bases for the vanishing ideal of the configuration, we derive properties of the matrix of first syzygies of this ideal that allow us to draw conclusions on the geometry of the point configuration.

Discrete Fourier Transforms

Abstract: This chapter deals with the discrete Fourier transform (DFT). In Sect. 3.1, we show that numerical realizations of Fourier methods, such as the computation of Fourier coefficients, Fourier transforms or trigonometric interpolation, lead to the DFT. We also present barycentric formulas for interpolating trigonometric polynomials. In Sect. 3.2, we study the basic properties of the Fourier matrix and of the DFT. In particular, we consider the eigenvalues of the Fourier matrix with their multiplicities. Further, we present the intimate relations between cyclic convolutions and the DFT. In Sect. 3.3, we show that cyclic convolutions and circulant matrices are closely related and that circulant matrices can be diagonalized by the Fourier matrix. Section 3.4 presents the properties of Kronecker products and stride permutations, which we will need later in Chap. 5 for the factorization of a Fourier matrix. We show that block circulant matrices can be diagonalized by Kronecker products of Fourier matrices. Finally, Sect. 3.5 addresses real versions of the DFT, such as the discrete cosine transform (DCT) and the discrete sine transform (DST). These linear transforms are generated by orthogonal matrices.

An Approximate Representation of the Fourier Spectra of Irregularly Sampled Multidimensional Functions: A Cost-Effective, Memory-Saving Algorithm

Abstract: This article discusses how to estimate the Fourier spectra of irregularly sampled multidimensional functions in an approximate way, using current fast Fourier transform (FFT) algorithms. This estimate may be an alternative for more rigorous approaches when the inversion of huge matrices is prohibitively expensive. The approximation results from a Taylor expansion of the Fourier transform kernel, which is a series on the product of wavenumbers and displacements from centers of a grid where a regular, discrete Fourier transform (DFT) is defined. Convergence and efficiency, as well as some shortcuts for implementation, is indicated. The problem of finding a Fourier spectrum of a function given a finite set of irregular measurements is usually associated with the idea of regularization/interpolation. This, in turn, raises the question of representativeness of continuous functions via its discrete measurements. Although this question is central for the very Fourier spectrum estimation, the scope of this article will be restricted to the trigonometric interpolation, postponing representativeness issues. For the sake of simplicity, the proposed approximation is first derived for the one-dimensional (problem where most related aspects are more clearly stated. Then, extensions to higher dimensions are straightforwardly presented.

Discrete Fourier Analysis on Lattice Grids

Abstract: Using group theory we describe the relation between lattice sampling grids and the corresponding non-aliasing Fourier basis sets, valid for all 1-periodic lattices. This technique enable us to extend the results established in [16]. We also provide explicit formula for the Lagrange functions and show how the FFT algorithm may be used to compute the expansion coefficients.

New Frequency-Dependent Trigonometric Interpolation Functions for the Dynamic Finite Element Analysis of Thin Rectangular Plates

Abstract: The Dynamic Finite Element (DFE) formulation is a superconvergent, semianalytical method used to perform vibration analysis of structural components during the early stages of design. It was presented as an alternative to analytical and numerical methods that exhibit various drawbacks, which limit their applicability during the preliminary design stages. The DFE method, originally developed by the second author, has been exploited heavily to study the modal behaviour of beams in the past. Results from these studies have shown that the DFE method is capable of arriving at highly accurate results with a coarse mesh, thus, making it an ideal choice for preliminary stage modal analysis and design of structural components. However, the DFE method has not yet been extended to study the vibration behaviour of plates. Thus, the aim of this study is to develop a set of frequency-dependent, trigonometric shape functions for a 4-noded, 4-DOF per node element as a basis for developing a DFE method for thin rectangular plates. To this end, the authors exploit a distinct quasi-exact solution to the plate governing equation and this solution is then used to derive the new, trigonometric basis and shape functions, based on which the DFE method would be developed.

Trefftz Approximations in Complex Media: Accuracy and Applications

Abstract: Approximations by Trefftz functions are rapidly gaining popularity in the numerical solution of boundary value problems of mathematical physics By definition, these functions satisfy locally, in weak form, the underlying differential equations of the problem, which often results in high-order or even exponential convergence with respect to the size of the basis set We highlight two separate examples of that in applied electromagnetics and photonics: (i) homogenization of periodic structures, and (ii) numerical simulation of electromagnetic waves in slab geometries Extensive numerical evidence and theoretical considerations show that Trefftz approximations can be applied much more broadly than is traditionally done: they are effective not only in physically homogeneous regions but also in complex inhomogeneous ones Two mechanisms underlying the high accuracy of Trefftz approximations in such complex cases are pointed out The first one is related to trigonometric interpolation and the second one -- somewhat surprisingly -- to well-posedness of random matrices

The Trigonometric Interpolation Method Based on Bordering Functions in Monitoring of Wave and Resonance Changes of Mechanical and Physical Fields

Abstract: The analysis of applied approaches of fast interpolation, recovery of discrete signals in multi-speed microprocessor systems for modeling and recognition of wave and resonance changes of mechanical and physical fields are given. A method of fast trigonometric interpolation based on bordering functions, which simplifies and accelerates the process of hierarchical recovery of signals is proposed. Expressions for simultaneous fast parallel filtering and interpolation of signals based on the application of new border-filtering functions are given. A set of interconnected sampling theorems for cases of equidistant and not equidistant samples of signal in time on a finite and infinite interval is formulated. The hierarchy of these theorems is determined and their interrelation with the well-known Kotelnikov's theorem is established.

Modelling of Moving Interfaces for Reduced-Order Finite Element Models using Trigonometric Interpolation

Abstract: Flexible multi-body simulation by means of reduced-order finite element models is gathering in importance for the simulation of mechanical and mechatronic systems like, e.g. machine tools or handling systems. Moving of the system’s axes involves changing of the coupling position between flexible bodies, and thus, changing of the finite element nodes involved in an interface. Because modern model reduction techniques are based on matching properties of the system’s input-output behaviour, the order of the reduced model strongly depends on the number of interfaces used. Therefore, it is not appropriate to use all finite element nodes of a potential interface area as independent inputs. The subject of this paper is a method for modelling of moving interfaces to flexible bodies which is compatible with model order reduction. First, a formalism is presented which allows the application of distributed loads onto the nodal degrees of freedom of finite element nodes. Next, elementary load distributions which allow the application of forces and torques to a surface are introduced and, in a further step, orthonormalised in order to achieve stationary interfaces which provide the desired six degrees of freedom with an arbitrary center of action. Subsequently, a method using trigonometric interpolation of a weighting function is presented which enables modelling of interfaces moving along a predefined path and acting on a known set of surfaces. As weighting function for geometrical restriction of the action on a surface, a trapezoidal function is suggested. As with the stationary interfaces, elementary load distributions for moving interfaces are presented and orthonormalised, what allows modelling of moving interfaces with six degrees of freedom. The resulting nodal degrees of freedom are visualised by means of examples and analysed for different meshes. For appropriate finite element meshes, the maximum relative error of action lies below 10−5 and the maximum crosscoupling between the interface’s degrees of freedom lies below 10−3 for the majority of the moving path length. Due to the trigonometric interpolation approach, only a low number of harmonics are to be used as interface matrices for the finite element system what qualifies the method for the use in combination with model order reduction.