Walsh Functions Research Articles

Universal quantum processors in spin systems via robust local pulse sequences

We propose a protocol to realize quantum simulation and computation in spin systems with long-range interactions. Our approach relies on the local addressing of single spins with external fields parametrized by Walsh functions. This enables a mapping from a class of target Hamiltonians, defined by the graph structure of their interactions, to pulse sequences. We then obtain a recipe to implement arbitrary two-body Hamiltonians and universal quantum circuits. Performance guarantees are provided in terms of bounds on Trotter errors and total number of pulses. Additionally, Walsh pulse sequences are shown to be robust against various types of pulse errors, in contrast to previous hybrid digital-analog schemes of quantum computation. We demonstrate and numerically benchmark our protocol with examples from the dynamics of spin models, quantum error correction and quantum optimization algorithms.

Synchronization of M-sequences based on fast Нadamard transform

Options have been developed for constructing circulant matrices of any M-sequence (MS) based on automorphic multiplicative groups of the extended Galois field, constructed using an irreducible primitive polynomial, on the basis of which the original MS is formed. The result of this approach is the identification of new methods for transforming MS circulant matrices to a matrix of Walsh functions, ordered by the powers of the antiderivative element of the field. It is shown for the first time that, depending on the initial conditions of the transformation, a set of any number of any cyclic shifts of the MP, shifted relative to each other by one symbol, can be transformed to any rows of the ordered matrix of Walsh functions, following one another. This circumstance makes it possible to simplify the MS synchronization algorithm for a known range of its cyclic shifts, especially in the case of large periods of its repetition, and also to reduce the computational complexity of the processing algorithm when working in a truncated basis of Walsh–Hadamard functions.

Synthesis of Singular Systems Walsh and Walsh-like Functions of Arbitrary Order

Functionally complete systems of Walsh functions (bases), a particular case of alternating piecewise constant sequential functions, are widely used in various scientific and technological fields. As applied to the tasks of spectral analysis of discrete signals, the most interesting are those Walsh bases that deliver linear coherence of the frequency scales of fast Fourier transform (FFT) processors. By the frequency scales of an FFT processor, we mean the scale on which the normalized frequencies of the input signal are arranged (input scale) and the scale on which the signal's spectral components are arranged (output scale). The frequency scales of the FFT processor are considered linearly coherent if the processor responses with maximum amplitudes and phases of the same sign are located on the bisector of the Cartesian coordinate system formed by the frequency scales of the processor. None of the known Walsh bases ordered by Hadamard, Kaczmage, or Paley provide linear coherence of the frequency scales of the FFT processor. In this study, we develop algorithms to synthesize two systems, called Walsh-Cooley and Walsh-Tukey systems, which turn out to be the only ones in the set of classical Walsh systems and sequents Walsh-like systems, respectively, that deliver linear coherence to the frequency scales of FFT processors.

POLYNOMIAL SOLUTIONS FOR THE KOLMOGOROV–WIENER PREDICTION OF MODELED SMOOTHED HEAVY-TAIL PROCESS

Nowadays telecommunication traffic in systems with data packet transfer is considered as a heavy-tail random process. In a couple of rather simple models traffic is considered to be stationary one. In our recent papers we generated modeled heavy-tail data, which is based on the smoothing of the fractional Gaussian noise. In particular, the applicability if the continuous Kolmogorov–Wiener filter to the prediction of such data was investigated, the corresponding Wiener–Hopf integral equation was solved on the basis of the truncated Walsh function expansion. However, a question occurs – may another truncated orthogonal function expansion be applied to the problem under consideration? So, the corresponding investigation may be an actual question. In our recent papers we investigated theoretical fundamentals of the Kolmogorov–Wiener filter construction for different models, in particular, on the basis of the truncated polynomial expansion method and on the basis of the truncated trigonometric Fourier series expansion method. In this paper we restrict ourselves to the investigation of the applicability of the truncated polynomial expansion method to the problem under consideration, the corresponding method is based on the Сhebyshev polynomials of the first kind. The applicability of another polynomial or trigonometric expansions to the problem under consideration may be discussed in other papers. The aim of the work is to investigate the applicability of the Galerkin method based on the Chebyshev polynomials of the first kind to the Kolmogorov–Wiener prediction of smoothed heavy-tail data. The methodology consists in the solving of the Wiener–Hopf integral equation on the basis of the truncated polynomial expansion method which is based on the Chebyshev polynomials of the first kind. The scientific novelty consists in the proof of the fact that the Galerkin method based on the Chebyshev polynomials of the first kind may be applied to the Kolmogorov–Wiener prediction of smoothed heavy-tail data. The conclusions are as follows. The truncated polynomial expansion method based on the Chebyshev polynomials of the first kind may give reliable results in the framework of the Kolmogorov–Wiener prediction of smoothed heavy-tail data.

A sequency match filtering algorithm based on the generalized Walsh transform for processing rectangular wave signals.

The conventional Walsh function just takes values of +1 and -1 and can only track limited signal states. However, based on the characteristics of Walsh functions that can capture the frequency of square wave signals, this article proposes a generalized Walsh transform algorithm to process multi-valued rectangular wave signals. As an extension of the traditional Walsh functions, generalized Walsh functions have advantages in signal frequency matching and sequency spectrum amplitude extraction, which makes them well suited to express the valuable signal. First, we infer the invariance displacement theory in the time and sequency domains, limiting the value of a circular time shift to ensure the concentrated distribution of the signal sequency energy. This limitation lays a good foundation for constructing generalized Walsh functions. Then, two types of generalized Walsh functions are built by combining the characteristics of different periodic rectangular waves. We deduce two properties of orthogonality and completeness to prove the ability of the constructed functions to match the frequency and the extracted energy. Finally, we compare multiple filtering methods to verify the reliability of the proposed method.

2023—A twofold commemoration: the 100th birthday of Walsh functions and the 50th anniversary of Professor Joseph Leonard Walsh’s death

In 2023, it is 100 years after the introduction of Walsh functions and 50 years after the death of their introducer Joseph Leonard Walsh. This paper is an homage to J. L. Walsh and presents some insights into the development of Walsh and dyadic analysis with applications of Walsh functions in both mathematics and various areas of computing and engineering. The presentation is based on reminiscences of the authors acting in these areas.

Widths and rigidity

We consider the Kolmogorov widths of finite sets of functions. Any orthonormal system of $N$ functions in $L_2$ is rigid, that is, it cannot be well approximated by linear subspaces of dimension essentially smaller than $N$. This is not true for weaker metrics: it is known that in every $L_p$ for $p<2$ the first $N$ Walsh functions can be $o(1)$-approximated by a linear space of dimension $o(N)$. We present some sufficient conditions for rigidity. We prove that the independence of functions (in the probabilistic meaning) implies rigidity in $L_1$ and even in $L_0$, the metric that corresponds to convergence in measure. In the case of $L_p$ for $1<p<2$ the condition is weaker: any $S_{p'}$-system is $L_p$-rigid. Also we obtain some positive results, for example, that the first $N$ trigonometric functions can be approximated by very low-dimensional spaces in $L_0$, and by subspaces generated by $o(N)$ harmonics in $L_p$ for ${p<1}$. Bibliography: 34 titles.

Walsh Functions with Barycentric Symmetry over Triangular Domain

Walsh Functions with Barycentric Symmetry over Triangular Domain

Dynamic mode decomposition for Koopman spectral analysis of elementary cellular automata.

We apply dynamic mode decomposition (DMD) to elementary cellular automata (ECA). Three types of DMD methods are considered, and the reproducibility of the system dynamics and Koopman eigenvalues from observed time series is investigated. While standard DMD fails to reproduce the system dynamics and Koopman eigenvalues associated with a given periodic orbit in some cases, Hankel DMD with delay-embedded time series improves reproducibility. However, Hankel DMD can still fail to reproduce all the Koopman eigenvalues in specific cases. We propose an extended DMD method for ECA that uses nonlinearly transformed time series with discretized Walsh functions and show that it can completely reproduce the dynamics and Koopman eigenvalues. Linear-algebraic backgrounds for the reproducibility of the system dynamics and Koopman eigenvalues are also discussed.

Numerical approximation of nonlinear stochastic Volterra integral equation based on Walsh function

Numerical approximation of nonlinear stochastic Volterra integral equation based on Walsh function

Generation of sub-wavelength longitudinal magnetization needle and multiple longitudinal spots using circularly polarized beam through an annular walsh function filter

Generation of sub-wavelength longitudinal magnetization needle and multiple longitudinal spots using circularly polarized beam through an annular walsh function filter

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.

GENERALIZED FRACTIONAL GAUSSIAN NOISE PREDICTION BASED ON THE WALSH FUNCTIONS

Context. Some of the authors’ recent papers were devoted to the Kolmogorov-Wiener filter for telecommunication traffic prediction in some stationary models, such as the fractional Gaussian noise model, the power-law structure function model, and the GFSD (Gaussian fractional sum-difference) model. Recently, the so-called generalized fractional Gaussian noise model was proposed for stationary telecommunication traffic description in some cases. So, in this paper the theoretical fundamentals of the continuous Kolmogorov-Wiener filter used for the prediction of the generalized fractional Gaussian noise are investigated.
 Objective. The aim of the work is to obtain the filter weight function as an approximate solution of the corresponding Wiener– Hopf integral equation with the kernel equal to the generalized fractional Gaussian noise correlation function.
 Method. A truncated Walsh function expansion is proposed in order to obtain the corresponding solution. This expansion is a special case of the Galerkin method, in the framework of which the unknown function is sought as a truncated series in orthogonal functions. The integral brackets and the results for the mean absolute percentage errors, which are a measure of discrepancy between the left-hand side and the right-hand side of the Wiener-Hopf integral equation, are calculated numerically on the basis of the Wolfram Mathematica package.
 Results. The investigation is made for approximations up to sixty four Walsh functions. Different model parameters are investigated. It is shown that for different model parameters the proposed method is convergent and leads to small mean absolute percentage errors for approximations of rather large numbers of Walsh functions.
 Conclusions. The paper is devoted to a theoretical construction of the continuous Kolmogorov-Wiener filter weight function for the prediction of a stationary random process described by the generalized fractional Gaussian noise model. As is known, this model may give a good description of some actual telecommunication traffic data in systems with packet data transfer. The corresponding weight function is sought on the basis of the truncated Walsh function expansion method. The corresponding discrepancy errors are small and the method is convergent.

WALSH Functions and WALSH Transform. Mathematical Foundations and Some Applications

The purpose of this article is to popularize the Walsh functions. A methodology for teaching of Walsh functions is presented. The Descrete Walsh-Hadamard transform is described as well as their applications.

MAGIC-DHT: Fast in-memory computing for Discrete Hadamard Transform

Discrete Hadamard transform (DHT) is a signal processing tool that decomposes an arbitrary input vector into a superposition of Walsh functions. Due to its wide range of applications in processing big data, a fast and energy-efficient hardware design for DHT with high throughput capability is essential. Processing in memory (PIM) allows the in-place computation to reduce the data traffic, which is a major speed bottleneck in the existing computing. In this work, we propose an efficient hybrid parallel PIM-based computation for DHT. Our proposed method explores the recursive computation of DHT and is based on the memristor-aided logic (MAGIC) gates in which the arithmetic operations are carried out via simple logic NOR operation. We propose two in-memory computing methods for the DHT encoding process. At the arithmetic level, to improve efficiency, we propose to share the intermediate results between addition and subtraction in DHT in the first method called MAGIC-DHT-1D which provides an average speedup of 1.12× over the recently proposed DigitalPIM for 1D DHT. Furthermore,MAGIC-DHT-1D also outperforms SIMPLER in terms of energy and energy density in average. We also propose a second method, called MAGIC-DHT-2D, to share the carrier independent computation cycles among multi-bit parallel addition and subtraction. At the algorithm level, we also explore both row and column-based PIM NOR computing in the same crossbar to avoid the transposition operation required in the 2D DHT process. MAGIC-DHT-2D provides an average speedup of 4.84× and 7.25× over two state-of-the-art methods DigitalPIM and SIMPLER, respectively for each complete set of 2D DHT computing cycles. Our numerical results further show that our proposed optimized methods can lead up to 56.19× and 6.90× speed-up, as well as 57.84× and 5.96× higher throughput over NVIDIA RTX Titan GPU to compute 1D DHT and 2D DHT, respectively.

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.

Time domain model order reduction of discrete-time bilinear systems by discrete Walsh functions

In this paper, the discrete Walsh functions are first used for time domain model order reduction (MOR) of discrete-time bilinear systems. Firstly, in terms of the discrete Walsh functions, the reduced order system is generated by means of the explicit solution formula for the discrete-time bilinear system. Meanwhile, we expand the discrete-time bilinear system under space spanned by the discrete Walsh functions. Then the reduced order system is produced by the orthogonal projection matrix that is defined by discrete Walsh coefficient matrix. Moreover, MOR method via discrete Walsh functions is investigated for discrete-time bilinear systems with nonzero initial conditions, where nonzero initial condition is also utilized to create the orthogonal projection matrix. Theoretical analyses show that the outputs of the resulting reduced order systems can match the first several outputs of the original system in finite intervals. Finally, an example is given to illustrate the effectiveness of the proposed methods.

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.

Conceptual Bases of Gray Transformations and Discrete Vilenkin-Crestenson Functions Systems

The article is devoted to the theoretical foundations of the construction of generalized Gray codes. These include the classical "left-side" and the proposed "right-side" Gray's codes. Left-side codes develop from left to right and right-side codes from right to left. Left- and right-handed Gray's codes augmented with reverse permutation operators form a subset of composite Gray's codes. Such codes have found constructive application in solving synthesis and analysis problems of discrete systems of Vilenkin-Crestenson functions (VCF). Particular cases of VCF systems are systems of discrete exponential functions and Walsh functions. The so-called indicator matrices (IM), bijective connected to the VCF systems, form the basis for VCF systems synthesis. The order of IMs is determined by the logarithmic dependence on the order of the VCF system. Unique Walsh-Cooley function systems, the only ones in the set of Walsh function systems that provide linear connectivity of the frequency scales of DFT processors, are developed. Examples of trees of VCF systems give. The orders of IMs are related by logarithmic dependence with the orders of VCF systems. Using IM: (1) estimates of the number of symmetric VCF systems in an unbounded range of changes of their parameters are obtained, (2) structures are determined and (3) rules of the interconnection of the systems established. The fundamental axioms and lemmas corresponding to the VCF systems formulating and directions for further research are outlined.

Features of the Expansion of Multiple Stochastic Stratonovich Integrals Using Walsh and Haar Functions

The problem of the root-mean-square convergence for approximations of multiple stochastic Stratonovich integrals based on the generalized multiple Fourier series method using Walsh and Haar functions is considered. It is shown that when they are chosen to expand multiple stochastic integrals, the proof of the root-mean-square convergence of a subsequence of series partial sums, which is formed in a way that is quite natural for these functions, does not require the explicit fulfillment of any additional conditions, except for the condition of the existence of the multiple stochastic Stratonovich integral.