System Of Walsh Functions Research Articles

On the distribution of pseudorandom vectors generated by elliptic curves

In the present paper, the distribution of pseudorandom vectors, derived from a specific set of rational points on elliptic curves over finite fields, is studied by estimating its discrepancy based on the system of Walsh functions and exponential sums over elliptic curves. It turns out these pseudorandom vectors have good behaviors, such as uniform distribution and strong pseudorandomness. Moreover, we apply the incomplete sum to the elliptic curve version of sum-product and Sárközy problems.

APPLICATION OF SPLINE FUNCTIONS AND WALSH FUNCTIONS IN PROBLEMS OF PARAMETRIC IDENTIFICATION OF LINEAR NONSTATIONARY SYSTEMS

Context. In this article, a generalized parametric identification procedure for linear nonstationary systems is proposed, which uses spline functions and orthogonal expansion in a series according to the Walsh function system, which makes it possible to find estimates of the desired parameters by minimizing the integral quadratic criterion of discrepancy based on solving a system of linear algebraic equations for a wide class of linear dynamical systems. The accuracy of parameter estimation is ensured by constructing a spline with a given accuracy and choosing the number of terms of the Walsh series expansion when solving systems of linear algebraic equations by the A. N. Tikhonov regularization method. To improve the accuracy of the assessment, an algorithm for adaptive partitioning of the observation interval is proposed. The partitioning criterion is the weighted square of the discrepancy between the state variables of the control object and the state variables of the model. The choice of the number of terms of the expansion into the Walsh series is carried out on the basis of adaptive approximation of non-stationary parameters in the observation interval, based on the specified accuracy of their estimates. The quality of the management of objects with variable parameters is largely determined by the accuracy of the evaluation of their parameters. Hence, obtaining reliable information about the actual nature of parameter changes is undoubtedly an urgent task.
 Objective. Improving the accuracy of parameter estimation of a wide class of linear dynamical systems through the joint use of spline functions and Walsh functions.
 Method. A generalized parametric identification procedure for a wide class of linear dynamical systems is proposed. The choice of the number of terms of the expansion into the Walsh series is made on the basis of the proposed algorithm for adaptive partitioning of the observation interval.
 Results. The results of modeling of specific linear non-stationary systems confirm the effectiveness of using the proposed approaches to estimating non-stationary parameters.
 Conclusions. The joint use of spline functions and Walsh functions makes it possible, based on the proposed generalized parametric identification procedure, to obtain analytically estimated parameters, which is very convenient for subsequent use in the synthesis of optimal controls of real technical objects. This procedure is applicable to a wide class of linear dynamical systems with concentrated and distributed parameters.

Chromatic Differentiation of Functional Mappings of the Composition of Nucleic Acids

Color visualization of the DNA of diverse living beings can help in the exploration of the issue of chromatic differentiation of functional mappings of the nucleotide composition of DNA molecules. By “chromatic differentiation”, we mean the coloring of these mappings. Algorithms for coloring genetic representations improve the perception of complex genetic information using color. Methodologically, to build the chromatic differentiation of functional mappings of the nucleotide composition of DNA, we employed the system of nucleotide Walsh functions and the Chaos Game Representation (CGR) algorithm. The authors compared these two approaches and proposed a modified CGR algorithm. The work presents various algorithms of chromatic differentiation based on the nucleotide Walsh functions at a specific location of the fragment in the nucleotide chain and on the frequencies of those fragments. The results of the analysis provide examples of chromatic differentiation in a variety of parametric spaces. The paper describes various approaches to coloring and video animation of DNA molecules in their chromatically differentiated spans of physicochemical parameters.

Improved bounds on the gain coefficients for digital nets in prime power base

We study randomized quasi-Monte Carlo integration by scrambled nets. The scrambled net quadrature has long gained its popularity because it is an unbiased estimator of the true integral, allows for a practical error estimation, achieves a high order decay of the variance for smooth functions, and works even for Lp-functions with any p≥1. The variance of the scrambled net quadrature for L2-functions can be evaluated through the set of the so-called gain coefficients.In this paper, based on the system of Walsh functions and the concept of dual nets, we provide improved upper bounds on the gain coefficients for digital nets in general prime power base. Our results explain the known bound by Owen (1997) for Faure sequences, the recently improved bound by Pan and Owen (2022) for digital nets in base 2 (including Sobol' sequences as a special case), and their finding that all the nonzero gain coefficients for digital nets in base 2 must be powers of two, all in a unified way.

Spectral Decomposition of Mappings of Molecular Genetic Information in the System Basis of Single Nucleotide Functions

This paper presents and visualizes examples of large amounts of genetic information using a new class of cognitive computer graphics algorithms. These algorithms are related to the semiotics of perception and allow the interpretation of those properties of nucleotide sequences that are difficult to perceive by simple reading or by standard means of statistical analysis. This article summarizes previously presented algorithms for visualizing long nucleic acids based on the primary Hadamard–Walsh function system. The described methods allow us to produce one-dimensional mappings of nucleic acids by levels corresponding to their scale-integral physicochemical parameters and construct a spectral decomposition of the nucleotide composition. An example of the spectral decomposition of parametric representations of molecular genetic structures is given. In addition, a multiscale composition of genetic functional mappings visualizing the structural features of nucleic acids is discussed.

Оптимизация двоичных групповых последовательностей, полученных в результате нелинейного кодового уплотнения

This paper presents the challenge of optimization of binary group sequences obtained from the nonlinear code multiplexing. It shows that at optimal element wise methodology for group sequences in a full system of orthogonal Walsh functions, a maximum criterion for the minimum distance of group sequences is equal to a maximum criterion for the minimum correlation response for the information orthogonal Walsh functions which contain group sequences. After a nonlinear code multiplexing group sequences are often contained errors, that makes using this method of multiplexing more difficult. To eliminate this source of errors the algorithm of optimization of group sequences was suggested. In this algorithm some elements of group sequences can be replaced with the opposite elements. Additionally, the algorithm of receiving of the entire group sequence among the nearest group sequences was developed. This algorithm provides considerable increasing of immunity while group sequence receiving unlike the other algorithm of element-to-element receiving in a full system of orthogonal Walsh functions.

Linear Functions Based on Walsh Functions

In this paper, the method of constructing an orthonormal system of linear functions for approximating the function of changing the wind speed by linear approximations is considered in order to optimize the operation of the wind generator and to maximize energy recovery from renewable sources. In the construction of this system, the system of orthonormal Walsh functions was taken as the basis. In the application of the Gram-Schmidt process to the system of the basic system of functions an analytic expression for linear functions based on Walsh functions was obtained. The developed system of functions possesses all properties, such as the system of continuous Walsh functions. The discrete version of the developed functions has all the parameters of the base system of functions, and also has the same graphs as the system of discrete Walsh functions. Most common ways of organizing functions within the system were considered. It is established that the error of the approximation of the function of changing the speed of the wind does not depend on the way of the ordering of functions. An example of an approximation of the function of change in wind speed during the day is given with the help of linear functions based on Walsh functions with a dimension of 8, as well as errors of the approximation of the wind speed change function are calculated. The mean square error of approximation of linear processes does not exceed 88%, and the average relative error of approximation in knots is 40%. This system is subjected to the Gibbs effect. That is, the reduced accuracy of approximation at the ends of the approximation interval. Given the Gibbs effect, the developed system allows for the approximation of linear processes with a mean square error that does not exceed 33%, the relative error of approximation in units of approximation decreases to 23%. The choice of the dimension of the system affects the frequency spectrum of the approximating signal, which in turn affects the accuracy of the approximation. Namely, with the increase in the number of functions in the spectrum of the approximating function, the content of high-frequency harmonics increases. Therefore, when increasing the signal frequency, which is subject to approximation, it is expedient to increase the dimension of the system. In turn, for the low-frequency signal approximation, it is expedient to use as little as possible the dimensions of the system. Thus, by choosing the appropriate dimension of the system, we can minimize the error of approximation. Ref. 11, fig. 2, add. 1.

The geometry of two-valued subsets of Lp -spaces

Abstract Let 𝓜(Ω, μ) denote the algebra of all scalar-valued measurable functions on a measure space (Ω, μ). Let B ⊂ 𝓜(Ω, μ) be a set of finitely supported measurable functions such that the essential range of each f ∈ B is a subset of {0,1}. The main result of this paper shows that for any p ∈ (0, ∞), B has strict p-negative type when viewed as a metric subspace of L p (Ω, μ) if and only if B is an affinely independent subset of 𝓜(Ω, μ) (when 𝓜(Ω, μ) is considered as a real vector space). It follows that every two-valued (Schauder) basis of L p (Ω, μ) has strict p-negative type. For instance, for each p ∈ (0, ∞), the system of Walsh functions in L p [0,1] is seen to have strict p-negative type. The techniques developed in this paper also provide a systematic way to construct, for any p ∈ (2, ∞), subsets of L p (Ω, μ) that have p-negative type but not q-negative type for any q > p. Such sets preclude the existence of certain types of isometry into L p -spaces.

ABOUT PHASE INTERDEPENDENCE AND POSSIBILITY OF WALSH FUNCTIONS SYSTEM REDUCTION

The system of Walsh functions is the multiplicative group of Rademacher- and Gray-function systems. The system contains discrete-harmonic sin-components of Rademacher functions, cos-components of Gray functions, and also discrete-nonharmonic components of Walsh functions. Pair phase interdependence of complete Walsh system functions is established. Subsystems of odd (sin-components) and even (cos-components) of Walsh functions as bases of theoretic-number transformations are constructed. The perspective of the future researches of transformations efficiency for digital signal processing in the proposed function systems is defined.

Walsh Matrix’s Copy Generation and Its Computer Images

Walsh functions are widely used in many areas such as signal and image processing, digital communication etc. Walsh function system is orthogonal and completed. Among many methods to generate Walsh functions, Swick’s copy theory is famous. This method takes Walsh function’s order numbers as the copy information and can generate an arbitrary function given certain order number, which in fact is the vector processing method and is not applicable for such 2D signal processing applications as fast transforms. Walsh function system can be expressed by Walsh matrix Wk. In this paper, row copy and block copy methods are put forward based on Wk. Copy operators are designed based on symmetries and a new ordering is discovered (named Walsh-Like ordering). After the iterative formulas of 6 ordering Walsh matrices are deduced by Kronecker product, the computer images of these matrices are illustrated. It is proved that the proposed method is more advanced than the former. The later can achieve higher performance and is applicable to the fast transforms designing. The fourth symmetric ordering of Walsh function system is discovered (Walsh-Like ordering), and the conjecture that the new ordering is likely to be the reverse form of Walsh ordering is made by analyzing and comparing with these images.

Motion compensated image reconstruction and holographic Doppler velocimetry using an encoded wavefront

In this paper, an image reconstruction method for relatively moving targets with acoustical holography or a synthetic aperture focusing technique is proposed. A high-speed imaging system using transmitting pulses modulated with a system of Walsh functions was proposed by the authors. The method, which utilizes multiplex transmission, is called encoded wavefront transmission. Moving targets are simultaneously observed with a multiple transmission of pulses modulated by the Walsh functions. For each repetition period, a 3-D image is reconstructed. From a sequence of the reconstructed images, the relative velocity is estimated. Then the images are combined to a single motion compensated image. In the paper, the influence of the quantization of delay on the quality of reconstructed images is discussed. Allowable errors of velocity estimation are evaluated using computer simulations. Also discussed are the velocity vector estimation algorithms and the feasibility of this method to a ‘‘Holographic Doppler Velocimetry,’’ which visualizes the spatial distribution of the velocity vector. The experimental results demonstrate the ability of the method for a simple moving object.

Development of transducers for a holographic 3-D imaging sonar

This report describes the processes of development and the performances of the transducers applied to a real-time 3-D acoustical holographic imaging system based on the encoded wavefront technique. In this system, the acoustical wave modulated its phases by a system of Walsh functions, emitted from the transducers. Thus the performance of the broad bandwidth was needed for transducers to make clear the points of the phase modulation on the waveforms. Piezorubber film was selected as a piezoelectric material and it was laminated to improve the sensitivity of the transmitters. The receivers have a copolymer film (PVDF) as the piezoelectric material. The thin film and a preamplifier mounted close to the film give superior performances to the receiver. It is made clear that the directivity of the transducers agrees with the results of theoretical analysis of the acoustic field generated by circular plane vibration; hemispherical shapes of the transducers distorted the waveform in some conditions; the broad bandwidth will be realized to set the resonance frequency lower than the operational frequency.

Analysis of the equilibrium states of the hopfield associative memory using the system of walsh functions

AbstractThis paper considers the situation where the multiple state‐patterns satisfying the orthogonality condition are memorized in the Hopfield associative memory model. Also discussed is the relation between the equilibrium state produced in the associative memory model and the memorized patterns. Numerous studies have been made of the associative memory model by the statistical approach. By contrast, this study considers the problem from the algebraic standpoint, using the Walsh functional system.As the first step, a sufficient condition is derived for the associative memory model to be in the equilibrium state. Using this condition, the equilibrium states are classified into three types and the relation between the memorized pattern and the generated equilibrium state is analyzed. As applications, a method of estimating the number of equilibrium states and their state pattern from the data concerning the memorized pattern, as well as a method of making the given pattern an equilibrium state, are shown through examples. the results are obtained by the component analysis of the state of the associative memory model using the Walsh functional system. In other words, the usefulness of applying the Walsh functional system is demonstrated.

Walsh power spectrum for wide-sense stationary stochastic processes (Corresp.)

Existing concepts of Walsh power spectra for wide-sense stationary stochastic processes are restricted to the case of autopower spectra because they are based on real Walsh functions. In this paper a Walsh power spectrum is developed which is based on a system of complex Walsh functions and thus applies to auto and cross power spectra as well. It is shown that this Walsh power spectrum is related to the Fourier power spectrum by a linear transformation. This fact makes it possible to calculate Fourier power spectrum estimates from corresponding Walsh power spectrum estimates. An estimation algorithm for the Walsh power spectrum is given in the paper.

Walsh-function analysis of a certain class of time series

This paper is concerned with investigating the use of the orthonormal system of Walsh functions in the analysis of a dyadic-stationary series. The main emphasis is on the finite Walsh transform of a sequence of values coming from such a series. Under a certain mixing condition, given in terms of the dyadic auto-covariance function of the series, we derive a central limit theorem for the finite Walsh transform. This, in turn, allows us to consider estimates of the Walsh spectrum, and we discuss briefly the Walsh periodogram.

B70-2 Transmission of Information by Orthogonal Functions

The book provides a new and interesting approach to the field of information and communication theory and is a welcome and thought- provoking exposition on a subject too often reviewed by conventional Fourier methods. Dr. Harmuth develops a generalized theory of signal transmission based on the concepts of orthogonal functions, the subject of his first chapter. His presentation is soon directed to the particular class of orthogonal functions known as the system of Walsh functions which have the convenient property of taking on only plus and minus unity values. Thus his subject becomes particularly relevant to the transistor and integrated circuit technology of today. The mathematical theories expounded in the text are unique in that experimental implementation is not far away. The first chapter develops the analytic tools necessary for applying the generalized orthogonal function theory to Walsh function applications in various phases of the communication process. The chapter is exemplary of Dr. Harmuth's historical accuracy and scholarly style in his exacting references of authors of both published and unpublished works contributing to this area. The fifteen pages of references at the end of the text attest to this fact.