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Compactly supported orthogonal wavelet filters are extensively applied to the analysis and description of abrupt signals in fields such as multimedia. Based on the application of an elementary method for compactly supported orthogonal wavelet filters and the construction of a system of nonlinear equations for filter coefficients, we design compactly supported orthogonal wavelet filters, in which both the scaling and wavelet functions have many vanishing moments, by approximately solving the system of nonlinear equations. However, when solving such a system about filter coefficients of compactly supported wavelets, the most widely used method, the Newton Iteration method, cannot converge to the solution if the selected initial value is not near the exact solution. For such, we propose optimization algorithms for the Gauss-Newton type method that expand the selection range of initial values. The proposed method is optimal and promising when compared to other works, by analyzing the experimental results obtained in terms of accuracy, iteration times, solution speed, and complexity

The properties of low-frequency global seismic noise, represented by continuous records for 24 years, 1997–2020, are investigated at 229 broadband stations located around the world. The property of waveforms, known as the Donoho–Johnston threshold, which separates the absolute values of the orthogonal wavelet coefficients into "small" and "large" is analyzed. The ratio of the number of "large" coefficients to their total number is determined by the dimensionless DJ index, which takes values from 0 to 1. The DJ index is considered as a measure of the nonstationarity of noise: the larger is the DJ index, the more is nonstationary the waveform. For each station, daily DJ index values are calculated. An auxiliary network of 50 reference points is introduced, the positions of which are determined by clustering the positions of seismic stations. For each reference point, a time series is constructed with a time step of 1 day, which is calculated as the median of daily DJ index values from the 5 nearest operable stations. For all pairs of reference points, the coherence between the DJ index values is estimated in a sliding time window of 365 days with an offset of 3 days, and the maximum values of the coherence function and the frequency at which the maximum coherence is reached are determined. The average values of the maximum coherences show strong growth after 2003, and the maximum distances between the reference points, for which the maximum coherence exceeded the threshold of 0.9, undergo an explosive increase in values after 2012. By extrapolating and averaging the DJ index values at the reference points, the region of concentration of maximum DJ index values was determined at the North-East Siberia. The bursts in the mean value of the maximum coherences between the day length and the DJ index values at the control points precede the release of seismic energy with a delay of about 530 days.

To reveal the particle dynamic characteristic in the bend, high-speed particle image velocimetry (PIV) and wavelet transform were used to measure and analyze the particle velocity in a horizontal-vertical pneumatic conveying system. The pressure drop and particle velocity are analyzed to elucidate the macroscopic motion properties of particles in the different radius ratio bend firstly. Then the methods of continuous wavelet transform and one-dimensional discrete orthogonal wavelet transform are used to analyze the particle dynamic characteristic in the different regions of the bend pipe in terms of time–frequency characteristics of particle fluctuation velocity, fluctuation energy distributions of wavelet components, and auto-correlation of various frequencies. The results show that the particles are mainly small-scale motion in the rapidly decreasing region, while the large-scale motion increases in the accelerating region near the inlet and the stable region near the outlet. And the results of the wavelet component show that the acceleration and deceleration of particles in the bend will decrease the proportion of high-frequency fluctuation energy. The auto-correlation coefficient of the high-frequency component decays slower and has a longer period at the critical position of the three regions.

The irrational coefficients of orthogonal wavelet filters require a large amount of hardware for their implementation. This results in huge memory requirement, increased power dissipation and reduction in speed of operation. This paper presents a new class of multiplier free db3 orthogonal wavelet filter bank with significantly reduced power dissipation, adders and shifters. In this paper, dyadic filter coefficients of db3 have been obtained by slightly altering double-shifting orthogonality property for perfect reconstruction. The proposed db3 wavelet filter improved time–frequency product, Sobolev regularity and frequency selectivity. Also, the VLSI architecture is suggested for the proposed db3 wavelet filter banks to analyse the hardware computational complexity. The performance of the proposed db3 wavelets achieved satisfactory performance for feature extraction in medical image retrieval and fingerprint recognition of infants and toddlers.

Seismic signal can be expressed by nonstationary convolution model (NCM) which integrates acoustic impedance (AI), attenuation factor (AF) and source wavelet (SW) into a single formula. Although it provides attractive potential to invert AI, AF and SW, simultaneously, effective joint inversion algorithm has not been developed because of the extreme instability of this nonlinear inverse problem. In this paper, we propose an alternating optimization scheme to achieve this nonlinear joint inversion. Our algorithm repeatedly alternates among three subproblems corresponding to AI, AF and SW recovery until changes in inverted models become smaller than the user-defined tolerances. Also, when we optimize one parameter, other two parameters are fixed. NCM is an explicit linear formula for AI; therefore, AI recovery is accomplished by linear inversion which is regularized by low-frequency model and isotropy total variation domain sparse constraints. However, NCM is a complicated nonlinear formula for AF. To facilitate the AF inversion, we propose a centroid frequency-based attenuation tomography method whose forward operator and observations are acquired from the time-varying wavelet amplitude spectra which is estimated according to Gabor domain factorization of NCM. SW is decoupled from NCM based on Toeplitz structure constraint, and we obtain an orthogonal wavelet transform domain sparse regularized SW inverse subproblem. Split Bregman technique is adopted to optimize AI and SW inverse subproblems. Numerical test and field data application confirm that the proposed nonstationary seismic inversion algorithm can stably generate accurate estimates of AI, AF and SW, simultaneously.

Purpose In today’s digital world, real-time health monitoring is becoming a most important challenge in the field of medical research. Body signals such as electrocardiogram (ECG), electromyogram and electroencephalogram (EEG) are produced in human body. This continuous monitoring generates huge count of data and thus an efficient method is required to shrink the size of the obtained large data. Compressed sensing (CS) is one of the techniques used to compress the data size. This technique is most used in certain applications, where the size of data is huge or the data acquisition process is too expensive to gather data from vast count of samples at Nyquist rate. This paper aims to propose Lion Mutated Crow search Algorithm (LM-CSA), to improve the performance of the LMCSA model. Design/methodology/approach A new CS algorithm is exploited in this paper, where the compression process undergoes three stages: designing of stable measurement matrix, signal compression and signal reconstruction. Here, the compression process falls under certain working principle, and is as follows: signal transformation, computation of Θ and normalization. As the main contribution, the theta value evaluation is proceeded by a new “Enhanced bi-orthogonal wavelet filter.” The enhancement is given under the scaling coefficients, where they are optimally tuned for processing the compression. However, the way of tuning seems to be the great crisis, and hence this work seeks the strategy of meta-heuristic algorithms. Moreover, a new hybrid algorithm is introduced that solves the above mentioned optimization inconsistency. The proposed algorithm is named as “Lion Mutated Crow search Algorithm (LM-CSA),” which is the hybridization of crow search algorithm (CSA) and lion algorithm (LA) to enhance the performance of the LM-CSA model. Findings Finally, the proposed LM-CSA model is compared over the traditional models in terms of certain error measures such as mean error percentage (MEP), symmetric mean absolute percentage error (SMAPE), mean absolute scaled error, mean absolute error (MAE), root mean square error, L1-norm and L2-normand infinity-norm. For ECG analysis, under bior 3.1, LM-CSA is 56.6, 62.5 and 81.5% better than bi-orthogonal wavelet in terms of MEP, SMAPE and MAE, respectively. Under bior 3.7 for ECG analysis, LM-CSA is 0.15% better than genetic algorithm (GA), 0.10% superior to particle search optimization (PSO), 0.22% superior to firefly (FF), 0.22% superior to CSA and 0.14% superior to LA, respectively, in terms of L1-norm. Further, for EEG analysis, LM-CSA is 86.9 and 91.2% better than the traditional bi-orthogonal wavelet under bior 3.1. Under bior 3.3, LM-CSA is 91.7 and 73.12% better than the bi-orthogonal wavelet in terms of MAE and MEP, respectively. Under bior 3.5 for EEG, L1-norm of LM-CSA is 0.64% superior to GA, 0.43% superior to PSO, 0.62% superior to FF, 0.84% superior to CSA and 0.60% better than LA, respectively. Originality/value This paper presents a novel CS framework using LM-CSA algorithm for EEG and ECG signal compression. To the best of the authors’ knowledge, this is the first work to use LM-CSA with enhanced bi-orthogonal wavelet filter for enhancing the CS capability as well reducing the errors.

This paper develops a high performance scalable computational model for simulating coupled transient electric and finite deformation dynamic fields with widely discrepant frequencies, governing the behavior of piezoelectric materials in the finite element framework. The nonlinear piezoelectric constitutive laws are formulated in the context of continuum damage models (CDM) at finite deformation. A fully coupled, total Lagrangian finite element formulation is developed for modeling the electric and mechanical fields in a staggered way. To address the computational challenge in modeling piezoelectric devices with large discrepancy in the frequenlightsghtscies of the electrical signal and mechanical vibrations, a wavelet transformation induced multi-time scaling (WATMUS) algorithm is developed. The WATMUS method projects the field with higher frequency onto the lower frequency problem through the use of orthogonal wavelet transformation. No assumption of periodic response is needed in the WATMUS scheme. While being accurate, the WATMUS algorithm significantly enhances the computational efficiency in comparison with single time scale integration methods. The accuracy and efficiency of the WATMUS algorithm are validated through piezoelectric applications.

Seismic noise properties in Southern California are considered. The initial data are continuous records of vertical oscillations with a sampling frequency of 1 Hz at 81 broadband stations for 12 years, 2008–2019. These data were converted to a time step of 1 min by calculating the average values in successive time intervals of 60 samples in length. The time series of low-frequency seismic noise with a time step of 1 min from each station was further converted to a sequence of daily values of the minimum normalized entropy of the distribution of the squared orthogonal wavelet coefficients. Since entropy estimates can be obtained from each station with a time step of 1 day, this makes it possible to construct a daily map of its changes in space. A general map of the distribution of entropy values is obtained by averaging daily maps. The main attention is paid to areas in which maximum entropy values are most often realized. These areas are identified by estimating the spatial probability density of the distribution of points at which a given number of maximum values are realized in each daily map. It has been shown that since 2012, the region of the most frequent realizations of entropy maxima is in direct contact with the epicenter of a strong earthquake on July 6, 2019, with magnitude 7.1. We consider the “secondary” entropy calculated for the probability densities of the distribution of the maximum values of the “primary” entropy of waveforms of seismic noise in a semi-annual moving time window. It is shown that the time intervals of increasing seismic activity correspond to a decrease in secondary entropy, which is interpreted as the concentration in the space of the distribution of the maxima of the primary noise entropy. Estimates of the change in the correlation coefficients between the daily values of the noise entropy in a semi-annual time window at network nodes of 12 reference points covering the region under study made it possible to study the spatiotemporal dynamics of strong correlations. The area of the future strong earthquake is characterized by high correlations of entropy values at the nearest reference points with other points.

We present an effective numerical method for the recovery of biharmonic functions in a disk from continuous boundary values of these functions and of their normal derivatives using wavelets that are harmonic in the disk and interpolating on its boundary on dyadic rational grids. The expansions of solutions of boundary value problems into cumbersome interpolation series in the wavelet basis are replaced by sequences of their partial sums that are compactly presentable in the subspace bases of the corresponding multiresolution analysis (MRA) of Hardy spaces $$h_{\infty}(K)$$ of functions harmonic in the disk. Effective estimates are obtained for the approximation of solutions by partial sums of any order in terms of the best approximation of the boundary functions by trigonometric polynomials of a slightly smaller order. As a result, to provide the required accuracy of the representation of the unknown biharmonic functions, one can choose in advance the scaling parameter of the corresponding MRA subspace such that the interpolation projection to this space defines a simple analytic representation of the corresponding partial sums of interpolation series in terms of appropriate compressions and shifts of the scaling functions, skipping complicated iterative procedures for the numerical construction of the coefficients of expansion of the boundary functions into series in interpolating wavelets. We write solutions using interpolating and interpolating-orthogonal wavelets based on modified Meyer wavelets; the latter are convenient to apply if the boundary values of the boundary value problem are given approximately, for example, are found experimentally. In this case, one can employ the usual, well-known procedures of discrete orthogonal wavelet transformations for the analysis and refinement (correction) of the boundary values.

Epilepsy is a neural disorder that is associated with the central nervous system (CNS) in which the brain activity sometimes becomes abnormal, which may lead to seizures, loss of awareness, unusual sensations, and behavior. Electroencephalograms (EEG) are widely used to detect epilepsy accurately. However, the interpretation of a particular type of abnormality using the EEG signal is a subjective affair and may vary from clinician-to-clinician. Visual inspection of the EEG signal by observing a change in frequency or amplitude in long-duration signals is an arduous task for the clinicians. It may lead to an erroneous classification of EEGs. The proposed methodology focuses on automated detection of epilepsy using a novel stop-band energy (SBE) minimized orthogonal wavelet filter bank. Using the wavelet decomposition, we obtain subbands (SBs) of EEG signals. Subsequently, fuzzy entropy, logarithmic of the squared norm, and fractal dimension are computed for each SB. The different combinations of the extracted features were supplied to various classifiers for the classification of normal and abnormal EEG signals. In the proposed method, we have used a single-channel EEG dataset of Temple University Hospital. The dataset is the most substantial EEG data publicly available, which contains an EEG recording of 2130 distinct subjects. Our proposed system obtained the highest classification accuracy (CACC) of 78.4% and 79.34% during training and evaluation using the SVM classifier. We achieved the highest F1-score of 0.88.

—A new biorthogonal system of wavelet bases has been constructed, which is oriented toward reconstructing the useful signal of a measuring system if the measurement process is represented as a convolution model. New biorthogonal wavelet bases are obtained by using a instrumental function to modify a Kravchenko orthogonal wavelet system with a finite spectrum. The properties of new biorthogonal frequency-modified wavelets are studied, and digital filters that realize fast computational algorithms are constructed. Schemes for multiresolution analysis are proposed, which, during discrete wavelet transform, immediately solve the problem of reconstructing the useful signal, as well as effective noise suppression, which can significantly speed up computations.

One of the major concerns in 5G IoT networks is that most of the sensor nodes are powered through limited lifetime, which seriously affects the performance of the networks. In this article, Compressive sensing (CS) technique is used to decrease transmission cost in 5G IoT networks. Sparse basis is one of the important steps in the CS. However, most of the existing sparse basis-based method such as DCT (Discrete cosine transform) and DFT (Discrete Fourier Transform) basis do not capture data structure characteristics in the networks. They also do not take into consideration multi-resolution representations. In addition, some of sparse basis-driven methods exploit either spatial or temporal features, resulting in performance degradation of CS-based strategies. To address these challenging problems, we propose a novel spatial–temporal correlation basis algorithm (SCBA). Subsequently, an optimal basis algorithm (OBA) is provided considering greedy scoring criteria. To evaluate the efficiency of OBA, orthogonal wavelet basis algorithm (OWBA) by employing NS (Numerical Sparsity) and GI (Gini Index) sparse metrics is also presented. In addition, we discuss the complexity of the above three algorithms, and prove that OBA has low numerical rank. After experimental evaluation, we found that OBA is capable of the sparsest representing original signal compared to spatial, DCT, haar-1, haar-2, and rbio5.5. Furthermore, OBA has the low recovery error and the highest efficiency.

This study aims to address the problem of load identification using single-point sensing. The problem of load identification forms the basis for nonintrusive load monitoring, which is an important application in future smart metering infrastructure. This work introduces a novel approach that uses orthogonal wavelet design and the unsupervised clustering of wavelet basis functions for single-point sensing. The coefficients of the discrete wavelet filters of length-6 and length-8 are computed using the minimum properties of the orthogonal wavelet filters. Covariance analysis is then applied to identify the set of orthogonal wavelets that best fits the load signals after clustering the newly generated wavelets using the unsupervised machine learning. To automate the classification process, a decision tree classifier is then applied to induce the classification models. The results have shown that the use of length-8 filters may improve the classification accuracies when compared to length-6 filters, which is mainly due to the large number of wavelets functions.