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walsh functions, scrambled (0, m, s)-nets, and negative covariance: Applying symbolic computation to quasi-Monte Carlo integration

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MATHEMATICS AND COMPUTERS IN SIMULATION 2021年 182卷 277-295页

作者： Wiart, Jaspar Wong, Elaine Austrian Acad Sci Johann Radon Inst RICAM Altenberger Str 69 A-4040 Linz Austria Johannes Kepler Univ Linz Altenberger Str 69 A-4040 Linz Austria

We investigate base b walsh functions for which the variance of the integral estimator based on a scrambled (0, m, s)-net in base b is less than or equal to that of the Monte-Carlo estimator based on the same number of points. First we compute the walsh decomposition for the joint probability density function of two distinct points randomly chosen from a scrambled (t, m, s)-net in base b in terms of certain counting numbers and simplify it in the special case t is zero. Using this, we obtain an expression for the covariance of the integral estimator in terms of the walsh coefficients of the function. Finally, we prove that the covariance of the integral estimator is negative when the walsh coefficients of the function satisfy a certain decay condition. To do this, we use creative telescoping and recurrence solving algorithms from symbolic computation to find a sign equivalent closed form expression for the covariance term. (C) 2020 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.

关键词： Quasi-Monte Carlo integration Scrambled digital nets walsh functions Symbolic computation Creative telescoping Symbolic summation

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CORRESPONDENCE BETWEEN walsh-functions AND BINARY RATE MULTIPLIER PULSE TRAINS

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INTERNATIONAL JOURNAL OF ELECTRONICS 1993年第1期74卷 47-50页

作者： NIEZNANSKI, J Electrical Engineering Department Technical University of Gdańsk Gdańsk ul. Majakowskiego 11 Poland

A direct correspondence is shown to exist between even walsh functions and binary rate multiplier (BRM) output trains. It can be concluded from this correspondence that walsh functions can be conceived of as derived f... 详细信息

关键词： walsh functions MULTIPLIERS (Mathematical analysis)

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Optimum pulse-width modulated patterns in induction motors using walsh functions

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ELECTRIC POWER SYSTEMS RESEARCH 1995年第2期35卷 87-91页

作者： Mohsen, R Nazarzadeh, J Department of Mathematics and Statistics Mississippi State University Mississippi State MS 39762 USA Department of Mathematics Amirkabir University Tehran Iran Department of Electrical Engineering Amirkabir University Tehran Iran

A new method is described for optimizing pulse-width modulated (PWM) inverter waveforms by the use of walsh functions. This method would minimize the torque ripple in inverter fed induction motors. The set of systems of linear equations obtained replaces the system of nonlinear transcendental equations used in the Fourier analysis approach. The present algorithms are computationally more efficient and faster than those based on Fourier analysis.

关键词： induction machines pulse-width modulation techniques inverters torque distortion walsh functions

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ROBUST COEFFICIENT ESTIMATION OF walsh-functions

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IEE PROCEEDINGS-D CONTROL THEORY AND APPLICATIONS 1990年第6期137卷 357-363页

作者： DAI, H SINHA, NK Department of Electrical and Computer Enginering McMaster University Hamilton Canada

An iterative least squares method with modified residuals is presented which is dedicated to the robust coefficient estimation of walsh functions for time series contaminated with noise, some of which may even be outliers. Instead of the mean-square approximation error (MSE), a robust criterion is proposed for estimating the coefficients of the time series. It is minimised by applying the ordinary iterative Gauss-Newton approach so that an arbitrary function, which is absolutely integrable in the interval [0, T), can be properly approximated by the first M walsh functions. A proof of convergence of the proposed method is provided. Results of simulation confirming robustness and convergence of the robust estimates are included. This method should be of great value in real-life situations.

关键词： iterative least squares method convergence least squares approximations noise Simulation, modelling and identification coefficient estimation iterative methods walsh functions time series robustness Information theory Interpolation and function approximation (numerical analysis) proof of convergence iterative Gauss-Newton approach

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TIME-DOMAIN SYNTHESIS VIA walsh-functions

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PROCEEDINGS OF THE INSTITUTION OF ELECTRICAL ENGINEERS-LONDON 1975年第5期122卷 565-570页

作者： CHEN, CF HSIAO, CH UNIV HOUSTON DEPT ELECT ENGNHOUSTONTX 77004

The paper deals with the application of walsh functions to the time-domain-synthesis problem, i.e., the determination of a suitable internal structure for a system from its prescribed external (input/output) behaviour. The method is based on repeated integration, and a new operational matrix, which relates walsh functions and their integrations, is defined. Examples of transfer-function and state-equation synthesis are used to illustrate the technique.

关键词： operational matrix transfer function state equation time domain synthesis repeated integration walsh functions General circuit analysis and synthesis methods time-domain analysis

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SINE-WAVE SYNTHESIS USING walsh-functions

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IEE PROCEEDINGS-G CIRCUITS DEVICES AND SYSTEMS 1987年第1期134卷 1-8页

作者： EVANS, WA Department of Electrical and Electronic Engineering University College of Swansea Swansea UK

An investigation of the synthesis of sine waves from a truncated series of walsh waves of ascending sequency is described. Computer simulation of the model is carried out in establishing potential circuit performance.... 详细信息

关键词： sine wave synthesis digital implementation truncated series walsh functions Function generators Other digital circuits digital circuits Signal processing and detection function generators signal processing function generator

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IMPROVED ALGORITHMS FOR PARAMETER-IDENTIFICATION IN CONTINUOUS SYSTEMS VIA walsh-functions

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IEE PROCEEDINGS-D CONTROL THEORY AND APPLICATIONS 1983年第1期130卷 9-16页

作者： RAO, GP PALANISAMY, KR INDIAN INST TECHNOL DEPT ELECT ENGN KHARAGPUR 721302 W BENGAL INDIA GOVT COLL TECHNOL DEPT ELECTR & COMMUN ENGN COIMBATORE 641013 INDIA

The paper first reviews the status of the walsh-functions (WF) approach to the problem of parameter identification in continuous dynamical systems. The WF method of analysis and parameter estimation relies on the use of the so-called operational matrix for integration. For higher-order systems, the existing operational matrix has so far been applied after being raised to the required power. This procedure results in an accumulation of error at each stage of integration. To improve this situation, the paper introduces one-shot operational matrices for repeated integration (OSOMRI) via walsh functions. The superiority of OSOMRI is demonstrated in problems of analysis and identification of higher-order systems. The paper then presents a direct noniterative method of parameter estimation for lumped linear systems in the presence of small unknown delays employing walsh delay matrices developed recently by the authors. When the delay terms are not small, the results of this noniterative method will be considerably in error. However, these results can be taken as good initial values for the inevitable refining iterative shift algorithm of Prasada Rao and Sivakumar. The paper also presents an efficient multistep algorithm for parameter estimation in systems with large unknown time delays.

关键词： Simulation, modelling and identification direct noniterative method continuous systems walsh functions one-shot operational matrices OSOMRI lumped linear systems Distributed parameter control systems dynamical systems higher-order systems control system analysis parameter identification parameter estimation delays Control system analysis and synthesis methods

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Algebraic modelling of linear systems by means of walsh functions

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MATHEMATICAL AND COMPUTER MODELLING OF DYNAMICAL SYSTEMS 2006年第6期12卷 589-605页

作者： Konigorski, Ulrich Tech Univ Darmstadt Inst Automat Control D-64283 Darmstadt Germany

In this paper a new algebraic representation of linear time-variant dynamic systems is developed. It is shown that walsh functions can be used to provide such a representation up to any desired precision. Due to the orthogonality of the walsh functions, the required precision only depends on the number of walsh functions used in the underlying walsh-Fourier analysis. The resulting linear algebraic model bears some resemblance to the well-known Laplace transform and especially in the case of linear time-invariant systems there is even a direct link between the two descriptions. Based upon this result new procedures for simulation, system identification and controller design can be obtained. This is illustrated by calculating stairstep approximations of the inverse Laplace transform of rational and irrational systems as well as the design of a time-variant multivariable PI controller for a sixth-order linear time-variant system.

关键词： linear time-variant systems walsh functions Laplace transform system identification and approximation

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TIME-DOMAIN SYNTHESIS VIA walsh-functions

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PROCEEDINGS OF THE INSTITUTION OF ELECTRICAL ENGINEERS-LONDON 1976年第5期123卷 461-462页

作者： GOPALSAMI, N DEEKSHATULU, BL INDIAN INST SCI DEPT ELECT ENGNBANGALORE 560012INDIA

Reference is made to a paper by C. F. Chen nd C. H. Hsiao, published in Proc Inst Electr Eng (Lond) v 122 n 5 May 1975 p 565-570. In that paper, the authors proposed a method of identification using walsh functions. In this correspondence, it is shown that their method is essentially the same as the repeated trapezoidal type of numerical integration procedure. A reply by the original authors is included.

关键词： operational matrix time domain synthesis Signal processing and detection sampling walsh functions Simulation, modelling and identification identification time-domain analysis Codes Information theory signal processing

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ANALYSIS OF BILINEAR-SYSTEMS USING walsh-functions

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL 1990年第1期35卷 119-123页

作者： LEWIS, FL MERTZIOS, VG VACHTSEVANOS, G CHRISTODOULOU, MA SYRACUSE UNIV DEPT ELECT & COMP ENGNSYRACUSENY 13210 DEMOCRITUS UNIV DEPT ELECT ENGNGR-67100 XANTHEGREECE

By using walsh functions to analyze bilinear systems, it is shown that the nonlinear differential system equation can be converted to a linear algebraic generalized Lyapunov equation that can be solved for the coefficients of the state x(t) in terms of the walsh basis functions. This Lyapunov equation provides an approximate closed-form solution for a bilinear system. Some guidelines are given for selecting the number of terms in the walsh approximating series.

关键词： Closed-form solutions Lyapunov equation System of equations walsh functions bilinear systems

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