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Highlights • A wavelet deconvolution technique for impact force reconstruction is developed by using the Haar wavelet. • The ill-posed nature of deconvolution is mitigated by controlling not only scale but also shift components. • The optimal scales and shifts for reconstruction are determined based on the compromise between the solution and residual norms. • The present technique can be deployed by considering the impulse response function explicitly or implicitly. • The validity of the present technique is verified by numerical experiment on the dynamic crushing of thin-walled columns. Abstract Deconvolution technique has become a useful solution for indirect measurement of impact force, however, a simple deconvolution technique often encounters difficulty and leads to an unsatisfactory reconstruction due to unavoidable error and the ill-posed nature of the inverse problem. This paper is concerned with a deconvolution technique using wavelet approach for reconstructing impact force. Based on the ability of wavelet transform, impact force reconstruction was formulated and implemented through its expansion coefficients of different scaled and shifted versions of wavelets. The reconstruction of impact force thus becomes the reconstruction of its expansion coefficients which are then used to reproduce the profile of impact force. Considerations of appropriate scales and shifts of wavelets in reconstruction can be regarded as a kind of regularization in order to improve the ill-posed problem. The numerical verification on reconstruction of impact force acting on a load-cell during the impact buckling of thin-walled column showed that the present technique is indeed capable to mitigate the ill-posed nature when the scale level of wavelets is controlled properly. The result also revealed that with an appropriate choice of the shifting parameter at a certain scale level, the noise in reconstructed force has been suppressed further. As a consequence, it has been verified that the present wavelet deconvolution technique can successfully reconstruct the impact force with less noise and higher accuracy in comparison with the conventional methods. [ABSTRACT FROM AUTHOR]

In this paper, we propose an adaptive approach to estimate the energies of the wavelet and scale coefficients of the Haar wavelet transform used in the multifractal modeling of network traffic traces. Simulation results confirm that the estimates obtained for the modeling parameters in the wavelet domain are precise. In addition, we propose an equation to calculate the autocorrelation function of the underlying multifractal model in terms of these wavelet domain parameters. In order to enhance the prediction performance of network traffic traces, the autocorrelation function is used to update orthonormal basis functions in a fuzzy system. To validate the adaptive fuzzy prediction approach, simulations with real network traffic traces are carried out, showing that the proposed algorithm provides lower mean square errors than other algorithms in the literature. [ABSTRACT FROM AUTHOR]

We study a nonparametric regression model for sample data which is defined on an N -dimensional lattice structure and which is assumed to be strong spatial mixing: we use design adapted multidimensional Haar wavelets which form an orthonormal system w.r.t. the empirical measure of the sample data. For such orthonormal systems, we consider a nonparametric hard thresholding estimator. We give sufficient criteria for the consistency of this estimator and we derive rates of convergence. The theorems reveal that our estimator is able to adapt to the local smoothness of the underlying regression function and the design distribution. We illustrate our results with simulated examples. [ABSTRACT FROM AUTHOR]

In this work, we focus on the buildup correction of dose calculation using Haar wavelets in the Tunisian gamma irradiation facility. The buildup effect of gamma rays was used to adjust absorbed dose calculation for different depth in the irradiated products. A buildup study with different product densities was carried out to generalize the dose adjustment approach to any product at any depth. [ABSTRACT FROM AUTHOR]

A new collocation method based on Haar wavelet is presented for numerical solution of three-dimensional elliptic partial differential equations with Dirichlet boundary conditions. An important advantage of the method is that it can be applied to both linear as well as nonlinear problems. The algorithm based on this new method is simple and can be easily implemented in any programming language. Experimental rates of convergence of the proposed method are calculated which are in agreement with theoretical results. The proposed method is applied to several benchmark problems from the literature including linear and nonlinear elliptic problems as well as systems of elliptic partial differential equations. The numerical experiments confirm the accuracy and diverse applicability of the method. [ABSTRACT FROM AUTHOR]

A fast preconditioned policy iteration method is proposed for the Hamilton–Jacobi–Bellman (HJB) equation involving tempered fractional order partial derivatives, governing the valuation of American options whose underlying asset follows exponential Lévy processes. An unconditionally stable upwind finite difference scheme with shifted Grünwald approximation is first developed to discretize the established HJB equation under the tempered fractional diffusion models. Next, the policy iteration method as an outer iterative method is utilized to solve the discretized HJB equation and proven to be convergent in finite steps to its numerical solution. Given the Toeplitz-like structure of the coefficient matrix in each policy iteration, the resulting linear system can be fast solved by the Krylov subspace method as an inner iterative method via fast Fourier transform (FFT). Furthermore, a novel preconditioner is proposed to speed up the convergence rate of the inner Krylov subspace iteration with theoretical analysis to ensure the linear system can be solved in O ( N log N ) operations under some mild conditions, where N is the number of spatial node points. Numerical examples are given to demonstrate the accuracy and efficiency of the proposed fast preconditioned policy method. [ABSTRACT FROM AUTHOR]

Current study contains adaption of Haar wavelet discretization method (HWDM) for FG beams and its accuracy estimates. The convergence analysis is performed for differential equations covering a wide class of composite and nanostructures. Corresponding error bound has been derived. It has been shown that the order of convergence of the HWDM can be increased from two to four by applying Richardson extrapolation method. The theoretical estimates are validated by numerical samples considering FGM beam as a model problem. The results obtained by applying HWDM are compared with the results of finite difference method (FDM). [ABSTRACT FROM AUTHOR]

The accuracy issues of Haar wavelet method are studied. The order of convergence as well as error bound of the Haar wavelet method is derived for general n th order ODE. The Richardson extrapolation method is utilized for improving the accuracy of the solution. A number of model problems are examined. The numerically estimated order of convergence has been found in agreement with convergence theorem results in the case of all model problems considered. [ABSTRACT FROM AUTHOR]

In this paper, we present a numerical scheme using uniform Haar wavelet approximation and quasilinearization process for solving some nonlinear oscillator equations. In our proposed work, quasilinearization technique is first applied through Haar wavelets to convert a nonlinear differential equation into a set of linear algebraic equations. Finally, to demonstrate the validity of the proposed method, it has been applied on three type of nonlinear oscillators namely Duffing, Van der Pol, and Duffing–van der Pol. The obtained responses are presented graphically and compared with available numerical and analytical solutions found in the literature. The main advantage of uniform Haar wavelet series with quasilinearization process is that it captures the behavior of the nonlinear oscillators without any iteration. The numerical problems are considered with force and without force to check the efficiency and simple applicability of method on nonlinear oscillator problems. [ABSTRACT FROM AUTHOR]

In this paper the numerical solution for the fractional order stationary neutron transport equation is presented using Haar wavelet Collocation Method (HWCM). Haar wavelet collocation method is efficient and powerful in solving wide class of linear and nonlinear differential equations. This paper intends to provide an application of Haar wavelets to nuclear science problems. This paper describes the application of Haar wavelets for the numerical solution of fractional order stationary neutron transport equation in homogeneous medium with isotropic scattering. The proposed method is mathematically very simple, easy and fast. To demonstrate about the efficiency and applicability of the method, two test problems are discussed. [ABSTRACT FROM AUTHOR]

Abstract: The boundary layer (BL) is a lower layer of the atmosphere that is directly influenced by the Earth''s surface and responds to surface forcing. In this study, a Mie lidar (light detection and ranging) system was utilized to measure the boundary layer over Wuhan, China. We employed the Haar wavelet covariance transform (WCT) method to find the boundary layer top to our lidar dataset. We thoroughly analyzed the well-mixed case and cases involving cloud droplets and pollution aerosols that were detected above, within and below the entrainment zone. In addition, the primary results indicate that the WCT method based on the parameters selected we can detect the BL top and the cloud base of a profile automatically, effectively and accurately, of which the entrainment zone is not concealed by an optical thick layer. Therefore, a general pattern can be concluded after a long-term monitoring is completed in the future. Furthermore, the BL top exhibits nearly a real-time feedback to cloud existence. [Copyright &y& Elsevier]

Abstract: In this paper we propose a procedure which allows the construction of a large family of FIR matrix wavelet filters by exploiting the one-to-one correspondence between QMF systems and orthogonal operators which commute with the shifts by two. A characterization of the class of filters of full rank type that can be obtained with such procedure is given. In particular, we restrict our attention to a special construction based on the representation of in terms of the elements of its Lie algebra. Explicit expressions for the filters in the case are given, as a result of a local analysis of the parameterization obtained from perturbing the Haar system. [Copyright &y& Elsevier]

Abstract: In this article, Haar wavelets have been employed to obtain solutions of boundary value problems for linear fractional partial differential equations. The differential equations are reduced to Sylvester matrix equations. The algorithm is novel in the sense that it effectively incorporates the aperiodic boundary conditions. Several examples with numerical simulations are provided to illustrate the simplicity and effectiveness of the method. [Copyright &y& Elsevier]

Abstract: This study proposes Haar wavelet (HW) approximation method for solving magnetohydrodynamic flow equations in a rectangular duct in presence of transverse external oblique magnetic field. The method is based on approximating the truncated double Haar wavelets series. Numerical solution of velocity and induced magnetic field is obtained for steady-state, fully developed, incompressible flow for a conducting fluid inside the duct. The calculations show that the accuracy of the Haar wavelet solutions is quite good even in the case of a small number of grid points. The HW approximation method may be used in a wide variety of high-order linear partial differential equations. Application of the HW approximation method showed that it is reliable, simple, fast, least computation at costs and flexible. [Copyright &y& Elsevier]

Abstract: In this work, we present a computational method for solving eigenvalue problems of high-order ordinary differential equations which based on the use of Haar wavelets. The variable and their derivatives in the governing equations are represented by Haar function and their integral. The first transform the spectral coefficients into the nodal variable values. The second, solve the obtained system of algebraic equation. The efficiency of the method is demonstrated by four numerical examples. [Copyright &y& Elsevier]

Abstract: Time series data mining is a rather complex branch, which is a technique that extracts the most valuable information from large amount of hisatory time series data. Haar wavelet technology will be used in time series similarity-pattern mining. This dissertation discusses the problem on time series similarity-pattern mining from similarity measurement, storage structure, searching integrality. Finally simulation results show that the Haar wavelet technology used in time-series similar pattern mining in a good performance. [Copyright &y& Elsevier]

Abstract: A discretization algorithm is proposed by Haar wavelet approximation theory for the fractional order integral. In this paper, the integration time is divided into two parts, one presents the effect of the past sampled data, calculated by the iterative method, and the other presents the effect of the recent sampled data at a fixed time interval, calculated by the Haar wavelet. This method can reduce the amount of the stored data effectively and be applied to the design of discrete-time fractional order PID controllers. Finally, several numerical examples and simulation results are given to illustrate the validity of this discretization algorithm. [Copyright &y& Elsevier]

Abstract: The Haar system is an alternative to the classical Fourier bases, being particularly useful for the approximation of discontinuities. The article tackles the construction of a set of fractal functions close to the Haar set. The new system holds the property of constitution of bases of the Lebesgue spaces of -integrable functions on compact intervals. Likewise, the associated fractal series of a continuous function is uniformly convergent. The case owns some peculiarities and is studied separately. [Copyright &y& Elsevier]

Abstract: Real-time monitoring of power quality requires extensive data-handling and data-processing capabilities. These requirements limit the scope of monitoring, despite the fact that microprocessor-based monitoring systems have seen significant increase in their storage and computational power. Development of compact algorithms will benefit power quality in two ways; they will allow monitoring of more points simultaneously for large systems, and they will help create powerful embeddable monitoring architectures within small power devices, such as a breaker, motors, or power drives. This paper proposes use of the distance L 1 norm as an indicator of power quality. We show how this approach will improve the computational and storage requirements. This work presents analyses of the proposed norm, how it compares with traditional approaches, and examples of its applications. [Copyright &y& Elsevier]

Between Haar and Walsh, there exist other Haar-type orthogonal matrixes (HTOMs), which are rarely utilized in practice. In this paper, we introduce HTOMs, which have fast algorithm, to the mechanic signal analysis. Concretely speaking, the mechanic signals are transformed by various HTOMs, which can be generated easily by varying any one of two parameters in the same program, then the performance of the transform results is compared by viewing FDC as the evaluation criterion, and the most optimal HTOM is achieved, which provides guidance and reference for the HTOMs applied in the signal analysis. [Copyright &y& Elsevier]

Abstract: Face-detection methods based on cascade architecture have demonstrated a fast and robust performance. In most of these methods, each node of the cascade employs the simple Haar-like features from the central eye–nose–mouth region using the boosting method. However, it can be empirically observed that, in the deeper nodes of the boosting process, the non-face examples collected by bootstrapping are in fact very similar to the face examples, and the error rate of those feature-based weak classifiers is very close to 50%. Consequently, the performance of the face detector is hardly further improved. In this paper, we propose a novel and simple solution to this problem by imitating the characteristics of the human visual system. The main idea of our solution is to boost the cascade based on a hierarchical strategy, which employs the information from the central and surrounding parts of the face regions step by step. We argue that the context information about a face can be advantageously used in the deeper nodes of the boosting process when the features derived from the central region of the face do not provide any further benefit. Furthermore, we also propose a simplified Gabor feature to extend the feature set for the training of deeper nodes. Experiments show that our proposed method can improve not only the detection performance, but also the detection speed, by about 10% when compared to the original AdaBoost face-detection method for our implementation. [Copyright &y& Elsevier]

Abstract: The Haar wavelet based discretization method for solving differential equations is developed. Nonlinear Burgers equation is considered as a test problem. Both, strong and weak formulations based approaches are discussed. The discretization scheme proposed is based on the weak formulation. An attempt is made to combine the advantages of the FEM and Haar wavelets. The obtained numerical results have been validated against a closed form analytical solution as well as FEM results. Good agreement with the exact solution has been observed. [Copyright &y& Elsevier]

Abstract: In this work, we present a computational method for solving nonlinear Fredholm integral equations of the second kind which is based on the use of Haar wavelets. Error analysis is worked out that shows efficiency of the method. Finally, we also give some numerical examples. [Copyright &y& Elsevier]

Abstract: In this paper, an algorithm for computing the Hilbert transform based on the Haar multiresolution approximation is proposed and the -error is estimated. Experimental results show that it outperforms the library function ‘hilbert’ in Matlab (The MathWorks, Inc. 1994–2007). Finally it is applied to compute the instantaneous phase of signals approximately and is compared with three existing methods. [Copyright &y& Elsevier]

A hardware-oriented image coding processing scheme based on the Haar wavelet transform is presented. The procedure computes a variant of the Haar wavelet transform that uses only addition and subtraction operations, after that, an optimized methodology performs the selection and coding of the coefficients, tailored for it with the main aim of attaining the lowest circuit complexity hardware implementation. A selection strategy, which does not require the previous ordering of coefficients, has been used. A non-conventional coding methodology, which uses an optimized combination of techniques adapted to the various groups of coefficients, has been devised for the coding of the selected coefficients leading to a compressed representation of the image and reducing the coding problems inherent in threshold selection. The compression level reached for images of 512×512 pixels with 256 grey levels is just over 22:1, (0.4bits/pixel) with a normalized mean square error, nrmse, of 2–3%, with subjective qualities which can be classified as good. The whole compression circuitry has been described and simulated at HDL level for up to 4 consecutive images, obtaining consistent results. The complete processor (excluding memory) for images of 256×256 pixels has been implemented using only one general-purpose low-cost FPGA chip, thus proving the design reliability and its relative simplicity. [Copyright &y& Elsevier]

Abstract: Recently, we have proposed a new multi-rate/multi-media system called wavelet based scale-code division multiple access (W/S-CDMA) that depends on the code, time and scale orthogonality introduced by pseudo-noise (PN) sequences, and wavelets. Wavelets are used as an orthogonal set of symbols for signaling, and their orthogonality is exploited over scale and time. In this system, the channel is divided into different scales, and each scale into time slots. In addition, the PN sequences are used in each scale to accommodate multiple users. In other words, each user encodes its successive information symbols with time-shifted replicas of the same basic wavelet in a specific scale and spreads its scaled and translated wavelets with its PN sequence. The finer the scale used, more symbols are transmitted. In this paper, we analyze the performance of Haar wavelet based S-CDMA (HW/S-CDMA) over a synchronous additive white Gaussian noise (AWGN) channel using a decorrelating multi-user detector. Results reveal that HW/S-CDMA holds promise since the users of HW/S-CDMA can achieve variable and higher data rates than those of direct sequence (DS)-CDMA for a similar bit error rate performance when real-valued PN sequences are used. However, HW/S-CDMA achieves better performance than DS-CDMA when complex-valued PN sequences are used. In addition, for all rates the same performance is achieved in HW/S-CDMA and the multi-user detector has no processing delay for different rates and has the same features as the standard decorrelating detector. Because of the reuse capability introduced by the scales, HW/S-CDMA is also capable of employing many available PN sequences from the optimal PN sequence families with limited number of sequences such as Kasami, Bent, etc. [Copyright &y& Elsevier]

Abstract: is defined to be and is called a cyclically ordered 3-subset. A compatibly ordered orthogonal group divisible design (briefly, compatibly ordered-OGDD or COGDD) is a set X and a partition of X into classes (usually called groups), and two sets and of cyclically ordered 3-subsets of X (usually called blocks), so that is an orthogonal group divisible design, and if appears in a block of then appears in a block of , where and . A compatibly ordered orthogonal Steiner triple system of order n (briefly, compatibly ordered- or ) can be viewed as a COGDD of type . In this article, we will show that for , [(i)] there exists a ; [(ii)] there exists a COGDD of type ; and [(iii)] there exists a COGDD of type for or 2 and . [Copyright &y& Elsevier]

Presents the survey on basic definitions of Haar wavelet transform. Illustration on the sign versions of transform; Discussion on the efficient symbolic calculation of Haar system; Frequency of ordering trigonometric functions.

Examines the concept of paired Haar transform (PHT) for representation and efficient optimization of systems for incomplete Boolean functions. Calculation on PHT for switching functions through binary decision programs; Advantages of computational and memory requirements of PHT; Emphasis on engineering practices for peculiar properties.

Presents the non-linear transform from ternary vectors into ternary spectral domain. Calculation on the signs of Haar transform; Details on the essential properties of logic functions and variables in spectral domain of quantized transform; Emphasis on the theory of error correcting codes for digital filtering.

For a given group Γ with a generating set A, a dipole with |A| parallel arcs (directed edges) labeled by elements of A gives rise to a voltage graph whose covering graph, denoted by H(Γ,A) is a bipartite, regular graph, called a bi-Cayley graph. In the case when Γ is abelian we refer to H(Γ,A) as the Haar graph of Γ with respect to the symbol A. In particular for Γ cyclic the above graph is referred to as a cyclic Haar graph. A basic theory of cyclic Haar graphs is presented. [Copyright &y& Elsevier]