A recursive formula for the correlations of Walsh functions

The correlation function of two Walsh functions appears in image processing, signal multiplexing, detection and spectral analysis using Walsh functions. Though Walsh functions are two-valued, their correlation functions are real-valued and rather difficult to evaluate. A recursive formula is developed in this paper to evaluate the correlation functions. Some other properties of the correlation function are also discussed.     

Chebyshev series approach to system identification,analysis and optimal control

The problems of identification, analysis and optimal control have been recently studied via orthogonal functions. The particular orthogonal functions used up to now are the Walsh, the block-pulse and the Laguerre functions. In this paper, the Chebyshev functions are introduced and solutions for the aforementioned problems are established. The algorithms proposed are analogous to those already derived for the Walsh, block-pulse and Laguerre functions. The Chebyshev series approach presented here appears to have certain advantages over other orthogonal series, and they may therefore be more suitable for the study of the problems of identification, analysis and optimal control.     

Walsh Operational Matrices for Fractional Calculus and Their Application to Distributed Systems

The Walsh operational matrix for performing integration and solving state equations is generalized to fractional calculus for investigating distributed systems. A new set of orthogonal functions is derived from Walsh functions. By using the new functions, the generalized Walsh operational matrices corresponding to √s, √(s²+ 1), e^-s and e^-√s etc. are established. Several distributed parameter problems are solved by the new approach.     

Improved Haar and Walsh functions over triangular domains

Piecewise constant orthogonal functions over triangular domains play an important role in many applications. In the present paper, some Haar and Walsh functions over triangular domains are constructed. Compared with the previously proposed Haar functions in [5], the new Haar functions take only integer. For any continuous function, the uniform convergence of the new Haar-Fourier series is proved. Moreover, based on the relation between the new Haar and Walsh functions, the uniform convergence of the Walsh-Fourier series is studied. Additionally, we obtain the relation between the Walsh functions in the two different orders, Paley order and Hadamard order, which have not been discussed previously.     

Bilinear system identification by block pulse functions

A new method is employed to identify the unknown parameters of a bilinear system. This method expands the system input and output by block pulse functions and reduces the original identification problem to an algebraic form. Furthermore, the dyad formed by block pulse functions and its integral are in diagonal forms, whereas the integration of the “triple-product” matrix can be reduced to the upper triangular form. Consequently, only very few calculations are required to find the solution for the algebraic equation. Two examples are given to show that the use of this method is considerably more economical in computation time than the use of Walsh function expansion.     

Identification of voltera kernels of a class of nonlinear systems by walsh function techniques

This paper discusses the identification problem for a class of nonlinear systems. A member of this class may be represented by a single-valued power-law type nonlinearity preceded and succeeded by linear dyadic invariant systems. Such an arrangement allows for a Voltera functional series representation. The identification problem is then concerned with the specification of the associated Voltera kernels.Two approaches are presented for dealing with this problem. Both approaches are, however, based on Walsh function techniques. The first approach relies on direct output measurements when the input is a Walsh function. This approach is suitable for a deterministic case. The second approach assumes ergodic processes for the input. Based on measurements drawn from an input-output dyadic correlation function, determination of the Voltera kernels is made.     

部分Bent函数的几种构造方法

给出了部分Bent函数几种一般性构造方法,同时得到了由上述方法所构造出的部分Bent函数的Walsh循环谱特征.特别给出了由变元个数少的Bent函数构造变元个数多的部分Bent函数的一般方法,由此方法可构造出大量的部分Bent函数.     

A Walsh function method for a non-linear Volterra integral equation

A new method for the numerical solution of linear and non-linear Volterra integral equations, using the discontinuous wavelet packets known as the Walsh functions, is proposed and investigated. Sufficient conditions for the method to converge are derived and a priori error estimates are obtained. Given sufficient regularity conditions on the integral equation, the method is shown to be locally of order two.     

Direct transformation of variational problems into Cauchy systems—II. scalar-semi-quadratic case

This series of papers addresses three interrelated problems: the solution of a variational problem, the solution of integral equations, and the solution of an initial valued system of integrodifferential equations. It will be shown that a large class of variational problems requires the solution of a nonlinear integral equation. It has also been shown that the solution of a nonlinear integral equation is identical to the solution of a Cauchy system. In this paper, we by-pass the nonlinear integral equations and show that the minimization problems directly imply a solution of the Cauchy system. This second paper in the series looks at semi-quadratic functional and scalar functions.     

Parameter Estimation of Bilinear Systems Via Walsh Functions

Walsh product matrix is formed by the multiplication of Walsh vector and its transpose. The operation of Walsh product matrix on a coefficient vector equals the product of a coefficient matrix and a Walsh vector. This unique property of Walsh function is used to determine the unknown parameters of a general bilinear system from the input-output data. An example with satisfactory result is given.     

Variational analysis of electrical networks

A unified variational approach to network analysis is presented. From an initial classification of variables, expressed in topological terms, Lagrangian energy functions are generated which may be used in Hamilton's principles. This topological formalism allows extension into nonconservative networks and enables the complete retention of classical formalism in all networks.By using a topological version of the Brayton-Moser mixed potential function, several power variational principles are developed. These principles lead to comparison of the roles played by classical energy and power state functions in analytical mechanics.     

Walsh series approach to lumped and distributed system identification

This paper considers the problem of identifying the parameters of dynamic systems from input-output records. Both lumped-parameter and distributed-parameter systems, deterministic and stochastic, are studied. The approach adopted is that of expanding the system variables in Walsh series. The key point is an operational matrix P which relates the coefficient matrix Г of the Walsh series of a given function with the coefficient matrix of its first derivative. Using this operational matrix P one overcomes the necessity to use differentiated data, a fact that usually is avoided either by integration of the data or by using discrete-time models. Actually, the original differential input-output model is converted to a linear algebraic (or regression) model convenient for a direct (or a least squares) solution. A feature of the method is that it permits the identification of unknown initial conditions simultaneously with the parameter identification. The results are first derived for single-input single-output systems and then are extended to multi-input multi-output systems. The case of non-constant parameters is treated by assuming polynomial forms. Some results are also included concerning the identification of state-space and integral equation models. The theory is supported by two examples, which give an idea of how effective the method is expected to be in the real practice.     

A variational method for the numerical simulation of a boundary controllability problem for the linear and semilinear 1D wave equations

We pursue the numerical implementation of a boundary controllability problem for the 1D wave equation based on a recent variational approach to deal with such situations that consists in analyzing an error functional defined for feasible functions complying with appropriate initial, boundary, and final constraints. The nature of such scheme, as a minimization process for a certain error functional, leads to a natural numerical implementation by using typical descent algorithms. Our aim here is to explore the basic ingredients to set up such practical numerical approximations which allow us to address linear and semilinear equations with the same numerical scheme.     

Optimal control of a distributed parameter system with applications to beam vibrations using piezoelectric actuators

This paper addresses the issue of the active vibration control of the transverse modes in a flexible elastic systems. The control is implemented by discrete sets of piezoelectric actuators that apply the optimal forces. The performance index is a time-dependent quadratic functional of state variables and their time derivatives, and control forces which are determined by minimizing the objective functional subject to a penalty term on the control functions. A combination of Galerkin and variational approaches are employed to determine the control forces in the time domain explicitly in terms of coupled amplitudes and velocities. The effectiveness of the proposed method is demonstrated by applying it to a physical problem controlled by piezoelectric patch actuators.     

Variational approach to some damped Dirichlet nonlinear impulsive differential equations

We use critical point theory and variational methods to investigate the solutions of a Dirichlet boundary value problem for damped nonlinear impulsive differential equations. The conditions for the existence of solution are established.     

基于游程统计和Walsh谱能量分布的调色板图像隐写分析

提出一种基于游程统计和Walsh谱能量分布的调色板图像隐写分析方法,利用调色板颜色的奇偶值(parity value,P值)构造P值图像,通过游程统计和Walsh变换提取出46维特征,并用校准技术削弱载体图像差异对分析统计量的干扰.实验表明,该方法适用于EzStego、分量和以及最佳奇偶隐写等调色板图像隐写算法,比奇异颜色分析准确.     

An efficient method for solving Emden–Fowler equations

In this paper, He's variational iteration method (VIM) is applied to solve the Emden–Fowler type equations in the second-order ordinary differential equations (ODEs). In this method, general Lagrange multipliers are introduced to construct correction functionals for the problems. The Lagrange multipliers in the functionals can be identified optimally via variational theory. This technique provides a sequence of functions which convergence to the exact solutions of the Emden–Fowler equations. Comparison with the exact solutions and the solutions by the Adomian decomposition method (ADM) show efficiency of VIM in solving equations with singularity.     

On the application of distribution theory to variational problems

The fixed end-point problem in variational calculus for an extremal containing corners is considered in a distributional context. The distributional Euler equation obtained is valid over the entire extremal and implicitly contains the Weierstrass-Erdmann corner conditions.     

一类特殊的Sturm—Liouville问题第一特征值的估计

为了更精确地研究Sturm—Liouville问题第一特征值，本文主要讨论一类特殊的Sturm—Liouville问题第一特征值关于势函数的依赖性。应用变分原理，估计了此类特殊的研Sturm-Liouville问题第一特征值在不同参数值下的界限，并证明出其中的一些界限是可以达到的，为Sturm-Liouville问题第一特征值的逼近提供了更好的理论基础。     

On optimization of circular fins with heat generation

The problem of minimizing the maximum temperature of a purely conducting circular fin with heat generation is studied via a variational technique. An exact solution is obtained for an arbitrary heat source distribution and a general non-linear heat conductivity. The problem of minimum weight design of convecting circular fin is then considered. An exact solution for temperature distribution and shape of the optimum fin with constant heat generation is obtained and discussed.