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In this article, the applications of the new Haar wavelet collocation methods called as Haar wavelet collocation method (HWCM) and higher-order Haar wavelet collocation method (H-HWCM) are developed for the solution of linear and nonlinear systems of ordinary differential equations. The proposed H-HWCM is compared with a variety of other methods including the well-known HWCM. The quasi-linearization technique is introduced in the nonlinear cases. The stability and convergence of both techniques is studied in detail, which are the important parts to analyze the proposed methods. The efficiency of the methods is illustrated with certain numerical examples, but the H-HWCM is more accurate with faster convergence than the HWCM and other methods reported in the literature. [ABSTRACT FROM AUTHOR]

In this study, for the first time, the approximate solution of Black–Scholes option pricing distributed order time-fractional partial differential equation by means of Legendre and Chebyshev wavelets is considered. The operational matrices of Legendre and Chebyshev wavelets for integer order derivative and distributed order fractional derivative are derived. Furthermore, the combination of Gauss–Legendre quadrature formula and standard Tau method along with the obtained operational matrices reduces the distributed order time-fractional Black–Scholes model (DOTFBSM) into the system of linear algebraic equations. Convergence analysis, error bounds and numerical stability of the proposed approach are discussed in detail. The presented scheme is applied on three test examples and numerical experiments confirm the theoretical results and illustrate robustness of the presented method. The results produced by current approach are found to be more accurate than some available results. [ABSTRACT FROM AUTHOR]

We study the singular values of certain triangular random matrices. When their elements are i.i.d. standard complex Gaussian random variables, the squares of the singular values form a biorthogonal ensemble, and with an appropriate change in the distribution of the diagonal elements, they give the biorthogonal Laguerre ensemble. [ABSTRACT FROM AUTHOR]

Design of time–frequency localized filters and functions is a classical subject in the field of signal processing. Gabor's uncertainty principle states that a function cannot be localized in time and frequency domain simultaneously and there exists a nonzero lower bound of 0.25 on the product of its time variance and frequency variance called time–frequency product (TFP). Using arithmetic mean (AM)– geometric mean (GM) inequality, product of variances and sum of variances can be related and it can be shown that sum of variances has lower bound of one. In this paper, we compute the frequency variance of the filter from its discrete Fourier transform (DFT) and propose an equivalent summation based discrete-time uncertainty principle which has the lower bound of one. We evaluate the performance of the proposed discrete-time time–frequency uncertainty measure in multiresolution setting and show that the proposed DFT based concentration measure generate sequences which are even more localized in time and frequency domain than that obtained from the Slepian, Ishii and Furukawa's concentration measures. The proposed design approach provides the flexibility in which the TFP can be made arbitrarily close to the lowest possible lower bound of 0.25 by increasing the length of the filter. In the other proposed approach, the sum of the time variance and frequency variance is used to formulate a positive definite matrix to measure the time–frequency joint localization of a bandlimited function from its samples. We design the time–frequency localized bandlimited low pass scaling and band pass wavelet functions using the eigenvectors of the formulated positive definite matrix. The samples of the time–frequency localized bandlimited function are obtained from the eigenvector of the positive definite matrix corresponding to its minimum eigenvalue. The TFP of the designed bandlimited scaling and wavelet functions are close to the lowest possible lower bound of 0.25 and 2.25 respectively. We propose a design method for time–frequency localized three-band biorthogonal linear phase (BOLP) wavelet perfect reconstruction filter bank (PRFB) wherein the free parameters can be optimized for time–frequency localization of the synthesis basis functions for the specified frequency variance of the analysis scaling function. The performance of the designed filter bank is evaluated in classification of seizure and seizure-free electroencephalogram (EEG) signals. It is found that the proposed filter bank outperforms other existing methods for the classification of seizure and seizure-free EEG signals. [ABSTRACT FROM AUTHOR]

The lifting style biorthogonal wavelet implementation has a nice property of enabling flexible design; it is immediately reversible and has a simple relation to subband filters. In this work, we present a method to design prediction (P) and update (U) filters of two-channel lifting structures by minimising the difference between Block Wavelet Transform (BWT) matrix of the wavelet and the Karhunen–Loéve Transform (KLT) of a stochastic process with certain autocorrelation. Here, BWTs are transform matrices that are generated by constructing columns through balanced wavelet trees fed by shifted impulse trains. Although wavelets already have fast implementations through subband filtering or lifting, parametric optimisation of the filter coefficients is still possible through indirectly mimicking the corresponding wavelet and a class of signals with certain KLT. This paper describes the above optimisation by putting constraints on the P and U filters for regularity and constructing the filter coefficients in a least-squares sense. In this part of the paper, the iterated orthogonality constraint over the BWT is resolved with a generalised 12 k + 1 -tap/ 6 k + 1 -tap P / U construction with numerical results for the 4 × 4 and 8 × 8 BWT cases. Experimental approximation to several KLT cases with numerical results in terms of filter coefficients, their spectral behaviour, compaction gain, etc. are provided. [ABSTRACT FROM AUTHOR]

A conventional quantum phase transition (QPT) not only occurs at zero temperature, but also exhibits finite-temperature quantum criticality. Motivated by the discovery of the pseudo-Hermiticity of non-Hermitian systems, we explore the finite-temperature quantum criticality in a non-Hermitian ƤƮ -symmetric Ising model. We present the complete set of exact eigenstates of the non-Hermitian Hamiltonian, based on which the mixed-state fidelity in the context of biorthogonal bases is calculated. Analytical and numerical results show that the fidelity approach to finite-temperature QPT can be extended to the non-Hermitian Ising model. This paves the way for experimental detection of quantum criticality in a complex-parameter plane. [ABSTRACT FROM AUTHOR]

Using a relation between a biorthogonal set of equiseparable bases and the weak values of the density matrix we derive an explicit formula for its tomographic reconstruction completely analogous to the standard mutually unbiased bases expansion. [ABSTRACT FROM AUTHOR]

Multiple approaches can be used to meet the n th order wavelet analysis consistency requirement between balanced multiwavelets and armlets. Due to the complexity of the n -balancing conditions, it is difficult to construct a family of n -balanced orthogonal multiwavelets for a relatively large n . In this paper, we introduce a convenient way to construct symmetric/antisymmetric biorthogonal multiwavelets provided that both the balanced order n ≥ 2 and the high armlets order conditions are met. First, we introduce the concept of biorthogonal armlets and illustrate a relationship among the four filter banks of a biorthogonal system. Then we present an algorithm for constructing biorthogonal balanced multiwavelets with armlets order (BBMA) and compute a novel class of such multiwavelets. Finally, we investigate the performances of BBMA and existing wavelets on the application of image denoising. Simulation results show that BBMA typically result in improved denoising. [ABSTRACT FROM AUTHOR]

Using as a prototype example the harmonic oscillator we show how losing self-adjointness of the Hamiltonian H changes drastically the related functional structure. In particular, we show that even a small deviation from strict self-adjointness of H produces two deep consequences, not well understood in the literature: First of all, the original orthonormal basis of H splits into two families of biorthogonal vectors. These two families are complete but, contrarily to what often claimed for similar systems, none of them is a basis for the Hilbert space H. Second, the so-called metric operator is unbounded, as well as its inverse. In the second part of the paper, after an extension of some previous results on the so-called V pseudobosons, we discuss some aspects of our extended harmonic oscillator from this different point of view. [ABSTRACT FROM AUTHOR]

Abstract: A new approach to estimate wind velocity and spectral width for weather radar is presented. In this paper, we deal with spectrum estimation and moments estimation of the received signal with a phased array Doppler weather radar. The lifted wavelet transform is applied on the estimated power spectral density prior to estimate the wind features (velocity, direction, spectral width). We apply the soft thresholding strategy on the lifted wavelet transform coefficients using the well-known minimax threshold as first guess. We estimate the velocity and the corresponding spectral width. We, iteratively, threshold the detail signal, in the wavelet domain, using an incremented threshold value until we obtain (using a stopping test) the optimal value of the initial threshold (minimax). In the first stage of the work, we have chosen a radar model used for meteorology to generate synthetic data on which we applied our algorithm. In the second stage, we used actual data got from the real radar that we used as model. The probative results we got with synthetic data are strengthened by the results obtained by the actual data. [Copyright &y& Elsevier]

Abstract: We discuss multiscale representations of discrete manifold-valued data. As it turns out that we cannot expect general manifold analogs of biorthogonal wavelets to possess perfect reconstruction, we focus our attention on those constructions which are based on upscaling operators which are either interpolating or midpoint-interpolating. For definable multiscale decompositions we obtain a stability result. [Copyright &y& Elsevier]

Most wavelet filters reported in the literature have irrational coefficients. Hardware implementation can be simpler if the filter coefficients are rational valued. In this paper, we present novel methods to rationalize both orthogonal and biorthogonal filter coefficients with perfect reconstruction and vanishing moments preservation. Rational orthogonal filter coefficients are obtained using lattice structures. Rational biorthogonal filter coefficients are obtained using the complementary filter technique in which one set of filter coefficients is expressed in terms of the other. The rationalized filters have characteristics that are very close to the original irrational filters. The techniques are simple yet general enough to be used for almost any filter bank unlike the techniques in previously reported works. [ABSTRACT FROM PUBLISHER]

Abstract: Instantaneous derivative estimates of a signal are obtained using the wighted least square (WLS) approximation of a Taylor (WLST) signal model, using classical windows as weighting factors. The WLST approximation in time corresponds to a Taylor approximation at the origin of the windowed signal spectrum. And the successive application of the WLST approximation leads to a filter bank whose frequency responses approach the set of ideal differentiator gains on the baseband, providing maximally flat differentiators on that band. Examples of these differentiator banks are designed with the Rectangular, Kaiser and Hamming windows, and their frequency and impulse responses are illustrated. Due to the strong symmetry of the signal model, this method achieves linear phase filter banks with equal delay for all the derivative estimates, which are very useful in applications where synchronized derivative of a bandlimited signal are desired. [Copyright &y& Elsevier]

Abstract: In this article, we introduce vector-valued multiresolution analysis and the biorthogonal vector-valued wavelets with four-scale. The existence of a class of biorthogonal vector-valued wavelets with compact support associated with a pair of biorthogonal vector-valued scaling functions with compact support is discussed. A method for designing a class of biorthogonal compactly supported vector-valued wavelets with four-scale is proposed by virtue of multiresolution analysis and matrix theory. The biorthogonality properties concerning vector-valued wavelet packets are characterized with the aid of time–frequency analysis method and operator theory. Three biorthogonality formulas regarding them are presented. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce a class of vector-valued wavelet packets of space , which are generalizations of multivariate wavelet packets. A procedure for constructing a class of biorthogonal vector-valued wavelet packets in higher dimensions is presented and their biorthogonality properties are characterized by virtue of matrix theory, time–frequency analysis method, and operator theory. Three biorthogonality formulas regarding these wavelet packets are derived. Moreover, it is shown how to gain new Riesz bases of space from these wavelet packets. Relation to some physical theories such as the Higgs field is also discussed. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce the biorthogonal vector-valued wavelets. We prove that, like in the scalar wavelet case, the existence of a pair of biorthogonal compactly supported vector-valued scaling functions guarantees the existence of a pair of biorthogonal compactly supported vector-valued wavelet functions. An algorithm for constructing a pair of biorthogonal compactly supported vector-valued wavelet functions is presented by means of vector-valued multiresolution analysis and matrix theory. The notion of biorthogonal vector-valued wavelet packets is introduced, and their properties are investigated by virtue of time–frequency analysis and algebra theory. Three biorthogonality formulas concerning the wavelet packets are established. Relation to some physical theories such as E-infinity Cantorian space–time theory is also discussed. [Copyright &y& Elsevier]

Abstract: Biorthogonal wavelets can be constructed from the biorthogonal scaling functions. In this note, we consider a inverse proposition in high dimension wavelet with arbitrary matrix dilation factor and prove that if the wavelets are biorthogonal, then the associated scaling functions should be biorthogonal. [Copyright &y& Elsevier]

Abstract: The notion of vector-valued multiresolution analysis is introduced and the concept of biorthogonal multiple vector-valued wavelets which are wavelets for vector fields, is introduced. It is proved that, like in the scalar and multiwavelet case, the existence of a pair of biorthogonal multiple vector-valued scaling functions guarantees the existence of a pair of biorthogonal multiple vector-valued wavelet functions. An algorithm for constructing a class of compactly supported biorthogonal multiple vector-valued wavelets is presented. Their properties are investigated by means of operator theory and algebra theory and time-frequency analysis method. Several biorthogonality formulas regarding these wavelet packets are obtained. [Copyright &y& Elsevier]

We describe how a 2-D rectangular lattice of inductors and capacitors can serve as an analog Fourier transform de- vice, generating an approximate discrete Fourier transform (DFT) of an arbitrary input vector of fixed length. The lattice displays diffractive and refractive effects and mimics the combined optical effects of a thin-slit aperture and lens. Diffraction theories in optics are usually derived for 3-D media, whereas our derivations proceed in 2-D. Analytical and numerical results show agreement between lattice output and the true DFT. Potentially, this lattice can be used for an extremely low latency and high throughput analog signal processing device. The lattice can be fabricated on-chip with frequency of operation of more than 10 GHz. [ABSTRACT FROM AUTHOR]

A new class of biorthogonal filter banks, called the Even-Triplet-Halfband-Filter-Bank (ETHFB), is introduced here. The filters are of even length and have linear phase response. There are two versions of the ETHFB, and they are modifications of the (odd-length) Triplet-Halfband-Filter-Bank. The Parametric Bernstein Polynomial is utilized in the construction of the three kernels that define the ETHFB. These filters will be used to match a given odd-length filter bank such that the equivalent wavelet functions of both filter banks are approximate Hilbert transform of each other, i.e., a Hilbert pair. The determination of the design parameters of the filter bank is achieved through an efficient least-squares method. [ABSTRACT FROM AUTHOR]

Abstract: The paper deals with a method of constructing orthonormal bases of coordinates which maximize, through redundant dictionaries (frames) of biorthogonal bases, a class separability index or distances among classes. The method proposes an algorithm which consists of biorthogonal expansions over two redundant dictionaries. Embedded classes are often present in multiclassification problems. It is shown how the biorthogonality of the expansion can really help to construct a coordinate system which characterizes the classes. The algorithm is created for training wavelet networks in order to provide an efficient coordinate system maximizing the Cross Entropy function between two complementary classes. Sine and cosine wavelet packets are basis functions of the network. Thanks to their packet structure, once selected the depth of the tree, an adaptive number of basis functions is automatically chosen. The algorithm is also able to carry out centering and dilation of the basis functions in an adaptive way. The algorithm works with a preliminary extracted feature through shrinkage technique in order to reduce the dimensionality of the problem. In particular, our attention is pointed out for time-frequency monitoring, detection and classification of transients in rail vehicle systems and the outlier problem. In the former case the goal is to distinguish transients as inrush current and no inrush current and a further distinction between the two complementary classes: dangerous inrush current and no dangerous inrush current. The proposed algorithm is used on line in order to recognize the dangerous transients in real time and thus shut-down the vehicle. The algorithm can also be used in a general application of the outlier detection. A similar structure is used in developed algorithms which are currently integrated in the inferential modeling platform of the unit responsible for Advanced Control and Simulation Solutions within ABB''s (Asea Brown Boveri) industry division. It is shown how impressive and rapid performances are achieved with a limited number of wavelets and few iterations. Real applications using real measured data are included to illustrate and analyze the effectiveness of the proposed method. [Copyright &y& Elsevier]

An MCHF atomic-structure package is presented based on dynamic memory allocation, sparse matrix methods, and a recently developed angular library. It is meant for large-scale calculations in a basis of orthogonal orbitals for groups of LS terms of arbitrary parity. For Breit–Pauli calculations, all operators—spin–orbit, spin–other orbit, spin–spin, and orbit–orbit—may be included. For transition probabilities the orbitals of the initial and final state need not be orthogonal. A bi-orthogonal transformation is used for the evaluation of matrix elements in such cases. In addition to transition rates of all types, isotope shifts and hyperfine constants can be computed as well as factors. New version summary: Title of program: atsp2K Version number: 1. 00 Catalogue identifier: ADLY_v2_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADLY_v2_0 Program obtainable from: CPC Program Library, Queen''s University of Belfast, N. Ireland Computer: Pentium III 500 MHz Installations: Vanderbilt University, Nashville, TN 37235, USA Operating systems under which the present version has been tested: Red Hat 8 Programming language used in the present version: FORTRAN 90 Memory required to execute with typical data: 256 Mbytes words No. Of bits in a word: 32 Supplementary material: User manuals for the program atsp2k and for the Spin-Angular library are available No. Of lines in distributed program, including test data, etc. :209 992 No. Of bytes in distributed package, including test data, etc. : 1 740 883 Distribution format: tar. Gz CPC Program Library subprograms used: none Does the new version supersede the previous version?: Yes Nature of physical problem: This program determines energy levels and associated wave functions for states of atoms and ions in the MCHF (LS) or Breit–Pauli (LSJ) approximation. [Copyright &y& Elsevier]

Quantization noise amplification (QNA) restricts the coding gain (CG) in biorthogonal subband coders. Here, a coloring filter is Introduced to color the quantization noise. The optimal coloring filter eliminating/minimizing QNA, with or without order re- striction, is found for a given finite-impulse response filter bank (FB). An efficient implementation of the coloring filter is proposed. With the coloring filter, the optimal biorthogonal ideal YB becomes the full whitening coder achieving maximum possible CG. Results verify, the CG improvement due to coloring for existing FEs. [ABSTRACT FROM AUTHOR]

Architectures and methods for the rapid design of silicon cores for implementing discrete wavelet transforms over a wide range of Specifications are described. These architectures are efficient, modular, scalable, and cover orthonormal and biorthogonal wavelet transform families. They offer efficient hardware utilization by exploiting a number of core wavelet filter properties and allow the creation of silicon designs that are highly parameterized, including in terms of wavelet type and wordlengths. Control circuitry is embedded within these systems allowing them to be cascaded for any desired level of decomposition without any interface glue logic. The time to produce chip designs for a specific wavelet application is typically less than a day and these are comparable in area and performance to handcrafted designs. They are also portable across a wide range of silicon foundries and suitable for field programmable gate array and programmable logic data implementation. The approach described has also been extended to wavelet packet transforms. [ABSTRACT FROM AUTHOR]

We present a fractional Gabor expansion on a non-rectangular time–frequency lattice. Sinusoidal analysis used in the traditional Gabor expansion is not appropriate for a compact representation for chirp-type signals. Basis functions of the proposed expansion are obtained via fractional Fourier basis. Completeness and biorthogonality conditions of the new expansion are derived. [Copyright &y& Elsevier]

Presents an approach for designing filter banks for image compression. Biorthogonal filter banks for image compression; Discussion on the optimization and generalization of based design; Analysis on the agent-based system development.

According to the Heisenberg uncertainty principle, an analysis window with a high time resolution in time domain will result in a low frequency resolution in frequency domain, and vice versa. To obtain the Gabor spectrum with high time-frequency resolution and concentration, weighted multiwindow discrete Gabor transform (M-DGT) using weights and the biorthogonal analysis method for analyzing long (or infinite) sequences is proposed in this paper, in which the combined Gabor coefficients constructed by a combination of M-DGT coefficients can be adaptively changed according to the time-frequency distributions of an analyzed signal containing multiple time-varying frequencies. To obtain the weights of the weighted M-DGT, the M-DGT is converted into a sparse problem with ℓ 1 - ℓ 2 regularization, then an efficient iterative algorithm for solving the weights in terms of real-valued matrix and real-valued vector is derived. The convergence of the iterative algorithm is proved by optimization theory. The experimental results demonstrate that the proposed method is an effective and efficient tool for nonstationary time-frequency analysis of signals. [ABSTRACT FROM AUTHOR]