Showing papers on "Bilinear interpolation published in 1973"
01 Jun 1973
TL;DR: In this article, the authors examined the relative merits of finite-duration impulse response (FIR) and infinite duration impulse response(IIR) digital filters as interpolation filters and showed that FIR filters are generally to be preferred for interpolation.
Abstract: In many digital signal precessing systems, e.g., vacoders, modulation systems, and digital waveform coding systems, it is necessary to alter the sampling rate of a digital signal Thus it is of considerable interest to examine the problem of interpolation of bandlimited signals from the viewpoint of digital signal processing. A frequency dmnain interpretation of the interpolation process, through which it is clear that interpolation is fundamentally a linear filtering process, is presented, An examination of the relative merits of finite duration impulse response (FIR) and infinite duration impulse response (IIR) digital filters as interpolation filters indicates that FIR filters are generally to be preferred for interpolation. It is shown that linear interpolation and classical polynomial interpolation correspond to the use of the FIR interpolation filter. The use of classical interpolation methods in signal processing applications is illustrated by a discussion of FIR interpolation filters derived from the Lagrange interpolation formula. The limitations of these filters lead us to a consideration of optimum FIR filters for interpolation that can be designed using linear programming techniques. Examples are presented to illustrate the significant improvements that are obtained using the optimum filters.
643 citations
TL;DR: The paper is concerned with the construction of bilinear state space descriptions from a prescribed nonlinear input-output map and shows the problem to be equivalent to that of matching an infinite sequence of constant parameters which uniquely identifies the given map.
Abstract: The paper is concerned with the construction of bilinear state space descriptions from a prescribed nonlinear input-output map. The problem is shown to be equivalent to that of matching an infinite sequence of constant parameters which uniquely identifies the given map. Both the problems of requiring the matching over a finite number of terms of the sequence (partial realization problem) and over the whole sequence (complete realization problem) are treated. In both cases explicit existence criteria and an algorithm for finding minimal realizations are given. The approach is based on the introduction of a suitable infinite matrix formed with the input-output parameters, which can be considered as a generalization of the Hankel matrix usually considered in the realization theory of linear systems.
146 citations
IBM1
90 citations
Patent•
04 May 1973
TL;DR: In this paper, a method for defining complex two-and threedimensional objects with a minimal amount of data is described, which employs a preprocessor which receives the coordinate values of an ordered set of points on the object and calculates one or more state vectors which concisely define the object.
Abstract: A method is described for defining complex two- and threedimensional objects with a minimal amount of data. The method employs a ''''preprocessor'''' which receives the coordinate values of an ordered set of points on the object and calculates one or more state vectors which concisely define the object. Two numerical control systems are described in which these state vectors are applied to a ''''mapping'''' interpolator that controls the servomechanisms of a machine tool. The mapping interpolator controls machine tool motion to generate a three-dimensional surface which not only passes through each of the data points used to describe the object, but which also assumes a smooth shape of minimum strain energy.
52 citations
TL;DR: In this paper, the authors gave explicit error bounds for bicubic spline interpolation and gave similar bounds for the cubic spline-blended interpolation scheme of Gordon.
37 citations
TL;DR: In this article, a trilinear and tricubic interpolation scheme for triangle boundary data is proposed, and its results are applied to the construction and analysis of potentially useful 15parameter and 12-parameter families of trianguZar$nite elements.
21 citations
17 citations
TL;DR: In this paper, error bounds for lacunary interpolation of certain functions by deficient quintic splines are extended to a wider class of functions and a stability result for such interpolation is also presented.
Abstract: In the previous paper by A Meir and A Sharma, error bounds for lacunary interpolation of certain functions by deficient quintic splines are developed In this note, we extend their results to a wider class of functions and indicate that the extended results are best possible In addition, a stability result for such interpolation is also presented
16 citations
TL;DR: An integer matrix method for implementation of the bilinear transformation is discussed and extended to a more general case that is useful in the design of digital filters in this paper, where an integer matrix is used to represent the input matrix.
Abstract: An integer matrix method for implementation of the bilinear transformation is discussed and extended to a more general case that is useful in the design of digital filters.
16 citations
TL;DR: In this paper, a generalization of B-spline is used to improve the numerical behavior of the interpolation process. But this generalization is not suitable for a large class of spline interpolation problems.
Abstract: This paper generates interpolatingM-splines in the sense of Lucas [J. of Approx. Th. 5, 1---14 (1972)] by a simple algebraic construction. The method yieldsM-spline interpolants for every finite family of functionals commuting with the remainder term of a generalized Taylor formula. These assumptions are fulfilled for a large class of spline interpolation problems (e.g. splines generated by certain singular differential operators and splines of several variables) without any further requirements about the geometrical distribution or denseness of the interpolation points. A generalization ofB-splines is used to improve the numerical behaviour of the interpolation process.
12 citations
TL;DR: In this article, the structural distribution for the parameters conditional on known linear relationship parameters α and β is derived for a general error distribution of the linear model, then have been applied to the normal error model.
Abstract: For a bilinear model the two sets of responses may be linearly related and these linearly related responses may follow linear regression models with known independent variables, usually known as design matrices. Assuming the linear relations expressed by the parameters α und β are known, the responses in the linear form have been described as a structural model. Then inference about the regression parameters and the scale parameters of the transformed responses have been made in the framework of a structural model, which provided the structural distribution for the parameters conditional on known linear relationship parameters α and β. Inference about α and β is based on the marginal probability element of the inverse image of the orbit of the transformed response. The results have been derived for a general error distribution of the linear model, then have been applied to the normal error model.
01 Jan 1973
TL;DR: In this paper, the authors deal with some observations arising in the realization of bilinear input-output maps, which are consequences of the definition of the state by the most natural way i.e. Nerode equivalence classes.
Abstract: This paper deals with some observations arising in the realization of bilinear input-output maps. It concerns essentially with some internal structural properties which are consequences of the definition of the state by the most natural way i.e. Nerode equivalence classes.
TL;DR: The currently available bicubic spline fit interpolation scheme for the rectangular coordinate system is not suitable for use with a polar grid pattern and must therefore be modified and a feature of physical significance has been added.
Abstract: Modification of the rectangular bicubic spline fit interpolation scheme so as to make it suitable for use with a polar grid pattern. In the proposed modified scheme the interpolation function is expressed in terms of the radial length and the arc length, and the shape of the patch, which is a wedge or a truncated wedge, is taken into account implicitly. Examples are presented in which the proposed interpolation scheme was used to reproduce the equations of a hemisphere.
TL;DR: Following four important papers on Birkhoff interpolation by Turan and his associates as discussed by the authors, the following theorems were proved by the following authors: [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15] and [16].
Abstract: Following four important papers on Birkhoff interpolation by Turan and his associates ([2], [3], [4], [14]), Kis ([8], [19]) proved the following theorems.
TL;DR: Among global interpolation techniques, bicubic splines and spline-blended are reviewed; among local, Hermite's and'serendipity' polynomials, the conclusion is that spline interpolation is most convenient for regular hyper-elements, while high precision finite elements become convenient for very fine or irregular partition as discussed by the authors.
Abstract: Interpolation techniques are reviewed in the context of the approximation of the solution of boundary value problems. From the variational formulation, the approximation error norm is related to the interpolation error norm.
Among global interpolation techniques, bicubic splines and spline-blended are reviewed; among local, Hermite's and ‘serendipity’ polynomials. The corresponding interpolation error norms are computed numerically on two test functions. The methods are compared for accuracy and for number of operations required in the solution of boundary value problems. The conclusion is that spline interpolation is most convenient for regular hyper-elements, while high precision finite elements become convenient for very fine or irregular partition.
01 Jan 1973
TL;DR: In this article, two approximate methods of finding the period of oscillations of non-linear conservative systems excited by step functions are presented, one based on an extension of the analysis presented by Jonckheere [4] and the other based on a weighted bilinear approximation of the nonlinear characteristic.
01 Mar 1973
TL;DR: In this paper, the union of two interpolation sets for a regular commutative convolution measure algebra is not necessarily an interpolation set, while every singleton is a Ditkin set.
Abstract: These are proved : (l)The union of two interpolation sets for a regular commutative convolution measure algebra is not necessarily an interpolation set. (2) There exists a regular com- mutative convolution measure algebra for which interpolation sets are not necessarily of spectral synthesis, while every singleton is a Ditkin set. (3) For every nondiscrete LCA group G, there exist compact interpolation sets for M(G) whose union is not an inter- polation set. A tensor algebra method is used.
TL;DR: In this paper, the authors established the minimum norm property, the existence and uniqueness of a solution of the interpolation problem, the property of best approximation, and the convergence of interpolation processes.
Abstract: The choice of function space allows us to make conclusions in the multidimensional case that are analogous to results in the theory of spline functions of one variable. We establish the minimum norm property, the existence and uniqueness of a solution of the interpolation problem, the property of best approximation, and the convergence of interpolation processes.
TL;DR: This paper shows Hermite-Fejér interpolation based on the so-called “practical” Chebyshev points, , 0(1)n, is uniformly convergent for all continuous functions on the interval [−1, 1].
Abstract: Berman has raised the question in his work of whether Hermite-Fejér interpolation based on the so-called “practical” Chebyshev points, , 0(1)n, is uniformly convergent for all continuous functions on the interval [−1, 1]. In spite of similar negative results by Berman and Szegö, this paper shows this result is true, which is in accord with the great similarities of Lagrangian interpolation based on these points versus the points , 1(1)n.
TL;DR: This paper has presented in some detail the digital filtering with sampled spectrum frequency shift technique and its application to spectral estimation and the design procedure of a bank of filters using the methods along with the performance results.
Abstract: In this paper along with a brief theoretical analysis of the design aspects of recursive and non-recursive digital filters, derivation of design algorithms and their performance characteristics are studied with the help of general-purpose digital computer, Honeywell-400. The basic mathematical tools used are the well-known z-transformation calculus and the bilinear z-transformation methods. We have presented in this paper in some detail the digital filtering with sampled spectrum frequency shift technique and its application to spectral estimation and the design procedure of a bank of filters using the methods along with the performance results.