Walsh Functions Research Articles

Orthogonal Chaotic Binary Sequences Based on Tent Map and Walsh Functions

Orthogonal Chaotic Binary Sequences Based on Tent Map and Walsh Functions

Boolean Spectral Analysis in Categorical Reservoir Modeling

This work introduces a new method for simulating facies distribution with two categories based on Fourier analysis of Boolean functions. According to this method, two categories of facies distributed along vertical wells are encoded as Boolean functions taking two values. The subsequent simulation process is divided into three consecutive steps. First, Boolean functions of the well data are decomposed into a binary version of a Fourier series. Decomposition coefficients are then simulated over the two-dimensional area as stationary random fields. Finally, synthetic data in the interwell space are reconstructed from simulated coefficients. The described method was implemented experimentally in software and tested on a case of a real oil field and on a case of a synthetic oil field model. Simulations on the synthetic model were used to test the performance of the method for two different bases in the Fourier expansion (Walsh functions and Haar wavelets). The simulation results were compared to those obtained on the same synthetic model via the classical sequential indicator simulation. It was shown that, for both bases, the new method reproduces statistical parameters of the well data better than sequential indicator simulation.

Method of High-Noise-Resistant Qudrature-Pulse Phase Modulation

The paper refers to the method of quadrature intra-pulse phase modulation (QIPPM). This method focuses on solving the relevant problem of improving the noise resistance of message transmission over communication channels with variable parameters. The problem is resolved by intra-pulse phase manipulation of signal quadratures by mutually orthogonal binary sequences of the Walsh function type. The advantage of the QIPPM modem over other modems is shown, and recommendations for its practical use in communication channels with pseudo-random tuning of operating frequency are given.

A Generalized Walsh System and Its Fast Algorithm

In this paper, we propose a new class of discontinuous orthogonal system called Walsh-U system, which is composed of piecewise polynomials of degree <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k$</tex-math></inline-formula> in Hilbert space <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L^{2}[0, 1]$</tex-math></inline-formula> . The Walsh-U system generalizes the Walsh system from piecewise constant to piecewise polynomials, which is capable of representing discontinuous signals. Besides, the basis functions of the Walsh-U system not only possess a more concise construction, but also have stronger sparse representation capabilities than the U-system. In the construction of the Walsh-U system, the Legendre polynomials are adopted as the generators at first, and then a series of basis functions are produced through the generators and basic operations of scaling and replication. These basis functions are formed into different groups based on the generation order. Furthermore, a fast algorithm for the Walsh-U system is designed in this paper. The experimental results demonstrate that when compared with Fourier series, Walsh functions, wavelet functions, and U-system, the Walsh-U system has fast convergence and high approximation precision in analog signal reconstruction and denoising. It is verified that the fast algorithm for the Walsh-U system can effectively improve the computation speed for discontinuous signals.

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.

Walsh functions, scrambled [formula omitted]-nets, and negative covariance: Applying symbolic computation to quasi-Monte Carlo integration

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.

Subdivision schemes on the dyadic half-line

We consider subdivision schemes, which are used for the approximation of functions and generation of curves on the dyadic half-line. In the classical case of functions on the real line, the theory of subdivision schemes is widely known because of its applications in constructive approximation theory and signal processing as well as for generating fractal curves and surfaces. We define and study subdivision schemes on the dyadic half-line (the positive half-line endowed with the standard Lebesgue measure and the digit-wise binary addition operation), where the role of exponentials is played by Walsh functions. We obtain necessary and sufficient conditions for the convergence of subdivision schemes in terms of the spectral properties of matrices and in terms of the smoothness of solutions of the corresponding refinement equation. We also investigate the problem of convergence of subdivision schemes with non-negative coefficients. We obtain an explicit criterion for the convergence of algorithms with four coefficients. As an auxiliary result, we define fractal curves on the dyadic half-line and prove a formula for their smoothness. The paper contains various illustrative examples and numerical results.

Generation of ultra-long multiple optical tubes using annular Walsh function filters

The tight focusing properties of an azimuthally polarized Bessel Gaussian beam phase modulated by annular Walsh function filter is studied numerically by vector diffraction theory. It is observed that upon suitable optimization of order and annular obstruction ratio of an annular Walsh function filter, one can generate multiple sub wavelength scale optical tubes (optical holes) with super long focal depth. Such a focal system is usable for Nano-lithography, particle trapping and transportation, as well as confocal and STED microscopy, microstructure fabrication etc.

Extraction of Walsh Harmonics by Linear Combinations of Dyadic Shifts

We solve two problems of extracting any term from a signal in the form of a finite sum of the Fourier series with respect to the discrete Walsh functions (in the first case) and the Walsh functions (in the second case) by a linear combination of group shifts of the original signal. We propose a vector version of the discrete time- and frequencythinning wavelet Haar bases, which widely used in encoding and decoding algorithms for the discrete Haar transform. Bibliography: 10 titles.

Single-frame far-field wavefront retrieval method based on Walsh function modulation

The reconstruction of wavefront from single far-field image data has unique advantages in simplicity of structure. However, the traditional wavefront reconstruction algorithm has multiple solutions based on single far-field image, its iterative process easily falls into stagnation. In this paper, based on the analysis of the multi-solution problem of single-frame phase retrieval method, a wavefront reconstruction method based on Walsh function two-dimensional discrete phase modulation is proposed. This method can effectively break the symmetry of near-field wavefront and overcome problem of multiple solutions. The simulation results show that the method can accurately reconstruct wavefront aberration with only one far-field image.

Multiband SHEPWM Control Technology Based on Walsh Functions

The drawback of modulation ratio limitation in selective harmonic elimination pulse width modulation (SHEPWM) technology based on Walsh function transformation. In order to solve this problem, a multiband SHEPWM control technology method for a multilevel inverter based on Walsh functions is proposed. By analyzing multiband SHEPWM for multilevel inverter voltage waveforms under a Fourier series transform, the unified nonlinear multiband SHEPWM equations for a multilevel inverter with a modulation index varying from 0 to 1 can be generalized. The equations can be solved by Walsh function transform. In this way, the difficulties faced in expanding the modulation index, online calculation, and real-time control are resolved simultaneously. A seven-level inverter is taken as the example, for which the piecewise linear equations of Walsh functions under different bands and the trajectories of switching angles are given. Meanwhile, the conditions for multiple sets of solutions at such a point where the modulation index is switched over are also taken into account. The feasibility of the proposed method is verified by simulation based on MATLAB and SIMULINK. Finally, the feasibility of the practical application is proved by the experiment based on Digital Signal Processor (DSP).

Vector-valued nonuniform multiresolution analysis related to Walsh function

In this paper, we introduce vector-valued nonuniform multiresolution analysis on positive half-line related to Walsh function. We obtain the necessary and sufficient condition for the existence of associated wavelets.

Heartbeat classification by using a convolutional neural network trained with Walsh functions

From recent studies, it is observed that convolutional neural networks are proved to be extremely successful in classification problems. Accurate and fast classification of electrocardiogram (ECG) beats is a crucial step in the implementation of real-time arrhythmia diagnosis systems. In this study, convolutional neural networks are employed to classify eleven different ECG beat types in the MIT-BIH arrhythmia database. We aimed to implement a computer-aided mobile diagnosis system equipped with artificial intelligence that detects the classes of heartbeats by visual inspection of the ECG records in the manner as the cardiologists do. Since doctors make their decisions basing heavily on the 2D visual appearances of the ECG signals without doing numerical calculations on 1D time samples, 2D images of 1D ECG records were given to the classifier as the input data. It would not be surprising that the structure of the network classifying 2D ECG data has to be larger than the one used to classify 1D ECG signals. The small size of a neural network is an important property for real-time use of the system. In this study, smaller network structures that provide high performances using the Walsh functions (WF), and drawbacks of converting 1D signals to 2D images have been investigated. The network structures using the WF during the training stage have been applied to different databases and successful results have been obtained. Classification results and sizes of the network structures are compared for the ECG beats in one dimension and in the form of 2D visuals. The training of ECG signals in the form of 2D visuals takes much longer time than that of the 1D signals. However, it is observed that training and testing times of both networks were quite fast. Moreover, average success rates of 99% were achieved for all beat types by using small-size networks.

The development of modern automated image processing and transfer systems for agriculture unmanned aerial vehicles

The paper contains results of analytic research of unmanned aerial vehicles using in agriculture. The main problems arising in the creation and subsequent large volumes of high-resolution images real time transfer in unmanned aerial vehicles are highlighted. The Automated image processing and transfer system using new methods of information compression on unmanned aerial vehicles board is proposed. The paper considers the issues of consider the problems of constructing new orderings of Walsh functions and constructing fast compression algorithms in synthesized systems of discrete Walsh functions. For processing and subsequent transmission of information from UAVs recommended to use the fast DWT procedure, it allows for a hardware implementation capable of the real-time conversion performing due to its simplicity. The introduction of the proposed solutions for UAVs in agriculture allows to increase accurasy of electronic cartographic material, to keep electronic records of agricultural operations, to carry out operational monitoring of the crops state and to respond quickly for violations and deviations, to predict crop yields and plan their activities for short-term and long-term prospects.

Расчет спектральных характеристик оператора интегрирования дробного порядка относительно функций Уолша и Хаара

Расчет спектральных характеристик оператора интегрирования дробного порядка относительно функций Уолша и Хаара

Завадостійка система імпульсної лазерної дальнометрії

Pulsed laser rangefinders prove to be cost-effective and practical devices when used at distances of several tens of kilometers due to their compactness, portability and energy efficiency. However, the measurement accuracy is significantly reduced by the presence of pulsed interference affecting the input of the optical receiver both during the sensing period and when the reflected signal is being received. Using the algorithms with the accumulation and subsequent processing of the results of several successive measurements reduces the speed of decision-making and does not guarantee the convergence of the results to the real value of the distance. The paper proposes a structural diagram of a laser rangefinder with the ability to detect pulsed interference in the range interval and correct errors that occur in the structure of the signal reflected from the target. The basis of the rangefinder circuit is a logical consistent filter, the structure of which contains multipliers (multiplication operations). The following requirements were formulated for the structure of the probe signal: — the first element should always be set to +1 to synchronize the receiver decider; — the weight of the coding sequence is equal to half its length; — the length of the coding sequence is even. Based on the requirements for coding sequences, the optimal structures of binary probing signals of length 8 were found, providing the best corrective ability. Comparison of the correlation properties of the found sequences and the sequences that are constructed using the Walsh functions showed the advantage of the optimal sequences by the criterion of the minimum level of the ACF side lobes. The simulation of the rangefinder under pulsed noise conditions has shown that the logical filter is advisable to use for those cases when the duration of the obstacle does not exceed 1/3 of the duration of the probing signal.

Prediction of the wind speed change function by linear regression method

In the article the approximation of the function of wind speed changes by linear functions based on Walsh functions and the prediction of function values by linear regression method is made. It is shown that under the condition of a linear change of the internal resistance of the wind generator over time, it is advisable to introduce the wind speed change function with linear approximation. The system of orthonormal linear functions based on Walsh functions is given. As an example, the approximation of the linear-increasing function with a system of 4, 8 and 16 linear functions based on the Walsh functions is given. The result of the approximation of the wind speed change function with a system of 8 linear functions based on Walsh functions is shown. Decomposition coefficients, mean-square and average relative approximation errors for such approximation are calculated. In order to find the parameters of multiple linear regression the method of least squares is applied. The regression equation in matrix form is given. The example of application of the prediction method of linear regression to simple functions is shown. The restoration result for wind speed change function is shown. Decomposition coefficients, mean-square and average relative approximation errors for restoration of wind speed change function with linear regression method are calculated.

Orthogonal Chaotic Binary Sequences Based on Bernoulli Map and Walsh Functions

The statistical properties of chaotic binary sequences generated by the Bernoulli map and Walsh functions are discussed. The Walsh functions are based on a Hadamard matrix. For general k (=), we will prove that Walsh functions can generate essentially different balanced and i.i.d. binary sequences that are orthogonal to each other.

Bed boundary identification from well log data using Walsh transform technique: A case study from NGHP Expedition-02 in the Krishna–Godavari basin, India

Identification of thin stratigraphic beds is often dealt with using the scale of core data analysis. The downhole wireline log data also provide accurate information on the characteristics of lithological boundaries. In the present study, we have used a Walsh low pass filter and bed boundary detection algorithm to develop an automated bed boundary identification approach. Initially, we ran a Walsh low pass filter against the wireline log responses. Afterwards, a bed boundary detection algorithm has been applied to the Walsh low pass filtered version of the wireline log data. The Walsh domain filter creates a stepped version of the well log data having a constant step width over the entire low pass version of the log signal. As the Walsh function basically represents the binary waveforms, the Walsh low pass filtering can appropriately identify the thin lithological units within a complex succession of sedimentary strata. Further, the proposed approach is an efficient way of resolving and understanding subsurface inhomogeneity from wireline log data. The feasibility of the technique is successfully tested on the downhole wireline log data acquired from the Krishna–Godavari basin during the Expedition 02 of Indian National Gas Hydrate Program (MGHP).

Visual Feedback Without Geometric Features Against Occlusion: A Walsh Basis

For a visual feedback without geometric features, this brief suggests to apply a basis made by the Walsh functions in order to reduce the off-line experimental cost. Depending on the resolution, the feedback is implementable and achieves the closed-loop stability of dynamical systems as long as the input-output linearity on matrix space exists. Remarkably, a part of the whole occlusion effects is rejected, and the remaining part is attenuated. The validity is confirmed by the experimental feedback for nonplanar sloshing.