Walsh Summing and Differencing Transforms

Analogous to Fourier frequency transforms of the integration and differentiation of a continuous-time function, Walsh sequency transforms of the summing and differencing of an arbitrary discrete-time function have been derived. These transforms can be represented numerically in the form of matrices of simple recursive structure. The matrices are not orthogonal, but they are the inverse of each other, and the value of their determinants is one.     

Transforms of generalized Walsh functions

The Laplace transforms of the generalized Walsh functions are presented, and from these the Fourier transforms and z transforms are easily deduced. The Fourier series representations of those generalized functions that are periodic are also given, and the bandwidth of finite-termed approximations is determined.     

Fourier Transforms of Walsh Functio ns

The Z-transform of a discrete square-wave function simplifies the derivation of an expression for the Fourier transform of a continuous square-wave function defined over a finite interval. In particular, since the Z-transform of any Walsh function can be written in closed form by inspection of the G ray code representation of its index, the Fourier transform can be readily written down as a trigonometric product function.     

Comments on "Walsh Summing and Differencing Transforms"

To obtain the Walsh transform (WT) of the summing or the differencing of a discrete-time function f from the WT of f, transform matrices may be used, for which recursive relationships have been found. The present letter is intended to state them rigorously.     

Spectral Decomposition of EEG Intervals Using Walsh and Fourier Transforms

This paper presents a comparison of the use of features derived from Walsh and Fourier transforms for classification of short segments of EEG data. The result was that features obtained from Fourier transforms produced consistently better classification results than Walsh function-derived features.     

Classification of Vectorcardiograms Using Walsh And Cosine Orthogonal Transforms

In this study, the usefulness of two-dimensional transforms in classifying human vectorcardiograms is investigated. The transforms used are the Walsh-Hadamard (WHT) and the discrete cosine (DCT). Experimental results included in the paper demonstrate that about 80-85 percent correct classification may be achieved.     

整数剩余类环上的Walsh变换

本文提出了整数剩余类环上的Walsh变换,它们与一般的Walsh变换相比用于CDMA通信中有着更加优越的 性能。它们具有数据压缩的功能,提高了信息传输的可靠性。整数剩余类环上的运算具有速度快、无舍入误差的优点。     

A Note of the Relation between Reed-Muller Expansions and Walsh Transforms

In this note we will show that there is a direct relation between the Reed-Muller expansions and orthogonal expansions for Boolean functions which, from the mathematical point of view, rest upon completely different bases.     

Noise Analysis of Soft Errors in Combinational Digital Circuits Via Walsh Transforms

The noise produced at the output of combinational logic circuits by individual gate failures is analyzed through the use of Walsh functions. Soft errors are modeled by allowing the output of each gate in a particular realization to fail temporarily, possibly introducing an error in the single binary output. The input variables also are allowed to be stochastically driven. The output probability of error contains the Walsh transform of an extended logic function and the Walsh characteristic functions of the input variables as well as the individual gate failure variables. These results are specialized to the case where the inputs are statistically independent of the soft errors. A discussion of the transform of the extended logic function is included.     

The Constructive Solution of Linear Systems of Partial Difference and Differential Equations with Constant Coefficients

This paper gives a survey of past work in the treated subject and also contains several new results. We solve the Cauchy problem for linear systems of partial difference equations on general integral lattices by means of suitable transfer operators and show that these can be easily computed with the help of standard implementations of Gröbner basis algorithms. The Borel isomorphism permits to transfer these results to systems of partial differential equations. We also solve the Cauchy problem for the function spaces of convergent power series and for entire functions of exponential type. The unique solvability of the Cauchy problem implies that the considered function spaces are large injective cogenerators for which the duality between finitely generated modules and behaviours holds. Already in the beginning of the last century C. Riquier considered and solved problems of the type discussed here.     

A Frequency-Domain Method for Time-Shift Estimation and Alignment of Seismic Signals

Alignment of seismic signals using time shifts obtained from first breaks or through cross-correlation techniques in the time domain without interpolation can only be done to within an integer multiple of the sampling period. In most seismic applications, the accuracy of such techniques is not sufficient. On the other hand, alignment between only two signals can be achieved to within a fraction of the sampling period by fitting a parabola to the cross-correlation peak. In this paper, we present a new alignment procedure in which the time shifts necessary to align simultaneously an ensemble of signals are determined through a parameter-optimization problem in the discretefrequency omain. A comparison of this procedure with the parabolic fit procedure is performed on synthetic data with various levels of signal-to-noise ratios. The effectiveness of this technique is also demonstrated on a real vertical seismic data set.     

Sub-exponential Solutions of Multidimensional Difference Equations

We shall prove that practically any recursive solutionof a linear multidimensional difference equation increases sub-exponentially.If all the steps of the recursive procedure are of the samedirection, useful growth estimates can be given. The resultsare also generalized to some cases of non-linear difference equationsand sequences in the linear normed space.     

Variational Solution of Integral Equations

A variational solution of the Fredholm integral equation of the first kind resulting from Laplace's equation with Dirichlet boundary conditions is discussed. Positive-definiteness of the integral operator is used to guarantee convergence. The square parallel plate capacitor is given as an example with several different types of trial functions. Special singular functions to handle known field behavior are shown to result in improved accuracy with reduced computing cost. The air-dielectric interface condition is related to a general Neumann-mixed boundary condition for which a variational method with a positive-definite integral operator is presented. Multiple boundary conditions are handled by mutually constraining separate variational expressions for each boundary condition. A T-shaped conductor on a dielectric slab, representative of quasi-static solutions of microstrip discontinuities, is presented as a three-dimensional example with multiple boundary conditions. Generally, it is shown how the finite-element method for the solution of partial differential equations may be extended to handle integral equation formulations.     

Multiplexing system for Walsh waves

A sequency-division multiplexing system is described that operates on multilevel data signals, where the output carrier does not contain more levels than the input. The system also possesses an efficient tradeoff between error correction and the number of channels in use. All operations are done in the residue number system, where the ideal p is any odd number. The system is particularly suited for data coded in is-complement fashion, and results for a 16-channel system are shown.     

Rigorous Numerical Solution to Complex Transcendental Equations

The Kuhn algorithm is introduced firstly and the numerical methods for resolving the complex transcendental equations in the electromagnetic theory are researched. The numerical examples are given to confirm the validity and efficiency of the numerical methods.     

Computation of Upper Bounds for the Solution of Continuous Algebraic Riccati Equations

This paper is considered with the computation of upper bounds for the solution of continuous algebraic Riccati equations (CARE). A parameterized upper bound for the solution of CARE is proposed by utilizing some linear algebraic techniques. Based on this bound, more precise estimation can be achieved by means of carefully choosing the bound’s parameters. Iterative algorithm is also developed to obtain more sharper solution bounds. Comparing with some existing results in the literature, the proposed bounds are less restrictive and more effective. The effectiveness and advantages of the proposed approach are illustrated via a numerical example.     

The Finite Difference Solution of Microwave Circuit Problems

Using finite difference methods this paper shows how solutions may be obtained with the aid of a digital machine to a wide range of microwave circuit problems. These problems include the parameters of TEM-mode transrnission lines, the equivalent circuits of obstacles in these lines, the cutoff frequencies of the fundamental mode in a waveguide of very general cross section, and the equivalent circuits of obstacles in rectangular waveguide. Methods for deriving the appropriate finite difference equations are presented and optimum methods for their solution set out; singularities are also included in the treatment. The paper ends with a resume of some typical results to problems of practical interest which have been obtained by these methods.