Representation of three-dimensional mass distribution of the Earth's interior by biorthogonal series and its use for studying internal structure of the planet

This paper utilizes the mathematical concept of approximation within an ellipsoid from a single viewpoint to present the spatial mass distribution function of the Earth's interior and its internal potential.The primary focus lies in constructing the volume distribution of masses in the planet's interior, with the expansion coefficients being linear combinations of the Stokes constants. Several possible approaches are suggested for determining accurately these coefficients employing three-dimensional(biorthogonal)polynomials. By expressing the mass distribution function of the Earth's interior and its internal potential as a series, an algorithm is introduced for the calculation of gravitational energy. It allows us to estimate fluctuations in gravitational energy. The implementation of this algorithm offers the means of establishing the extent to which the Earth deviates from a state of hydrostatic equilibrium as a celestial body.Due to the aforementioned method, calculations have been conducted to validate its effectiveness and reliability. This example is given as an illustration of a given method for studying the internal structure of planets.

A quantitative analysis method for GPR signals based on optimal biorthogonal wavelet

Due to the disturbances arising from the coherence of reflected waves and from echo noise,problems such as limitations,instability and poor accuracy exist with the current quantitative analysis methods.According to the intrinsic features of GPR signals and wavelet time–frequency analysis,an optimal wavelet basis named GPR3.3 wavelet is constructed via an improved biorthogonal wavelet construction method to quantitatively analyse the GPR signal.A new quantitative analysis method based on the biorthogonal wavelet(the QAGBW method)is proposed and applied in the analysis of analogue and measured signals.The results show that compared with the Bayesian frequency-domain blind deconvolution and with existing wavelet bases,the QAGBW method based on optimal wavelet can limit the disturbance from factors such as the coherence of reflected waves and echo noise,improve the quantitative analytical precision of the GPR signal,and match the minimum thickness for quantitative analysis with the vertical resolution of GPR detection.

FREQUENCY OPTIMIZATION APPROACH FOR DESIGNING M-BAND BIORTHOGONAL SYMMETRIC WAVELETS

In this paper,a new method is presented for designing M-band biorthogonal symmetric wavelets.The design problem of biorthogonal linear-phase scaling filters and wavelet filters as a quadratic programming problem with the linear constraints is formulated.The closed-form solution is given and a design example is presented.

P wave onset time picking with the B-spline biorthogonal wavelet

The seismic wave consists of many seismic phases, which contain rich geophysical information from the hypocenter, medium of seismic wave passing through and so on. It is very important to detect and pick these seismic phases for understanding the mechanism of earthquake, the Earth structure and property of seismic waves. In order to reduce or avoid the loss resulted from the earthquake, one of the important goals of seismic event detecting is to obtain its related information before and after it occurs. Because of the particularity of P wave and S wave the seismic event detecting focuses on distinguishing P and S waves and picking their onset time, it has been becoming one of the research hotspots for many geophysicists to pick the P and S wave arrival accurately and effectively.

On the Zeros of Daubechies Orthogonal and Biorthogonal Wavelets

In the last decade, Daubechies’ wavelets have been successfully used in many signal processing paradigms. The construction of these wavelets via two channel perfect reconstruction filter bank requires the identification of necessary conditions that the coefficients of the filters and the roots of binomial polynomials associated with them should exhibit. In this paper, orthogonal and Biorthogonal Daubechies families of wavelets are considered and their filters are derived. In particular, the Biorthogonal wavelets Bior3.5, Bior3.9 and Bior6.8 are examined and the zeros distribution of their polynomials associated filters are located. We also examine the locations of these zeros of the filters associated with the two orthogonal wavelets db6 and db8.

Adaptive Biorthogonal Local Discrete Cosine Transform for Interference Excision in Direct Sequence Spread Spectrum Communications

A novel time-frequency domain interference excision technique is proposed. The technique is based on adaptive biorthogonal local discrete cosine trans form (BLDCT). It uses a redundant library of biorthogonal local discrete cosine bases and an efficient concave cost function to match the transform basis to the interfering signal. The main advantage of the algorithm over conventional trans form domain excision algorithms is that the basis functions are not fixed but ca n be adapted to the time-frequency structure of the interfering signal. It is w e ll suited to transform domain compression and suppression of various types of in terference. Compared to the discrete wavelet transform (DWT) that provides logar ithmic division of the frequency bands, the adaptive BLDCT can provide more flex ible frequency resolution. Thus it is more insensitive to variations of jamming frequency. Simulation results demonstrate the improved bit error rate (BER) perf ormance and the increased robustness of the receiver.

Modulated Lapped Biorthogonal Transform for Non-orthogonal Narrowband Interference Excision in Spread Spectrum Communications

Traditional lapped transform domain excision techniques obtain good performance at the expense of increased processing delay. Extension of transform domain filtering techniques to the lapped biorthogonal transform domain can help solve the problem. By incorporating biorthogonality into the lapped transforms, more flexibility is obtained in the design of windows. Thus transform bases with better stopband attenuation can be generated by designing windows, but not by increasing the overlapping factor. In this paper, a new modulated lapped biorthogonal transform (MLBT) with optimized windows is introduced for efficient compression of multi-tone interfering signal energy. The bit error rate (BER) performance of the receiver employing the proposed MLBT excision technique is analyzed and compared with that of the lapped transform domain excision-based receivers. Simulation results demonstrate the improved performance and increased robustness of the proposed technique.

The Biorthogonality of Multiple Vector-valued Bivariate Wavelet Packets

The notion of a sort of biorthogonal multiple vector-valued bivariate wavelet packets,which are associated with a quantity dilation matrix,is introduced.The biorthogonality property of the multiple vector-valued wavelet packets in higher dimensions is studied by means of Fourier transform and integral transform biorthogonality formulas concerning these wavelet packets are obtained.

Green Function and Perturbation Method for Dissipative Systems Based on Biorthogonal Basis

Based on the approach of biorthogonal basis, we carry out the quasinormal modes （QNMs） expansions for a class of open systems described by the wave equation with outgoing wave boundary conditions. For such a non-Hermitian system, the eigenfunction perturbation expansions and Green function method, which are based on the orthogonal eigenvectors of the Hermitian Hamiltonian for the dosed quantum system, can be generalized in terms of the biorthogonal basis, the two sets of eigenfunctions of H and its adjointness H . The time-independent perturbation theory for the complex frequencies can be also developed.

A General Algorithm for Biorthogonal Functions and Performance Analysis of Biorthogonal Scramble Modulation System

Applying the theorems of Mobius inverse and Dirichlet inverse, a general algorithm to obtain biorthogonal functions based on generalized Fourier series analysis is introduced. In the algorithm, the orthogonal function can be not only Fourier or Legendre series, but also can be any one of all orthogonal function systems. These kinds of biorthogonal function sets are used as scramble signals to construct biorthogonal scramble modulation (BOSM) wireless transmission systems. In a BOSM system, the transmitted signal has significant security performance. Several different BOSM and orthogonal systems are compared on aspects of BER performance and spectrum efficiency, simulation results show that the BOSM systems based on Chebyshev polynomial and Legendre polynomial are better than BOSM system based on Fourier series, also better than orthogonal MCM and OFDM systems.

A Wavelet Multigrid Method Using Symmetric Biorthogonal Wavelets

In [1], the author introduced a wavelet multigrid method that used the wavelet transform to define the coarse grid, interpolation, and restriction operators for the multigrid method. In this paper, we modify the method by using symmetric biorthogonal wavelet transforms to define the requisite operators. Numerical examples are presented to demonstrate the effectiveness of the modified wavelet multigrid method for diffusion problems with highly oscillatory coefficients, as well as for advection-diffusion equations in which the advection is moderately dominant.

A QUASI-REFINED ITERATIVE ALGORITHM BASED ON THE LANCZOS BIORTHOGONALIZATION PROCEDURE FOR LARGE EIGENPROBLEMS

The two-sided Lanczos method is popular for computing a few selected eigentriplets of large non-Hermitian matrices. However, it has been revealed that the Ritz vectors gained by this method may not converge even if the subspaces are good enough and the associated eigenvalues converge. In order to remedy this drawback, a novel method is proposed which is based on the refined strategy, the quasi-refined idea and the Lanczos biothogonalization procedure, the resulting algorithm is presented. The relationship between the new method and the classical oblique projection technique is also established. We report some numerical experiments and compare the new algorithm with the conventional one, the results show that the former is often more powerful than the latter.

便于快速计算的4带小波子带算子的范数

多带小波比2带小波具有更丰富的参数空间、更灵活的时频铺叠、提供更好的能量压缩,所以在信号处理、数值分析等领域有着广泛的应用。对子带算子理论进行了研究,建立了优化模型,能够在含有自由参数的紧支撑对称多带双正交小波中挑选子带算子范数最小的小波,为研究适合数字图像处理的小波理论及其快速算法提供参考。首先,给出了双无限维矩阵–子带算子的定义,发展了循环矩阵的理论,得到了便于快速计算的4带双正交小波子带算子的范数。通过构造一个特定的三角函数并计算其最大值,得到该子带算子的范数。然后,建立了范数最小化模型,构造并挑选一类具有快速计算结构的4带双正交小波滤波器。最后,通过实例验证了所得到的结论。

Super-Fast Approximation Algorithms Using Classical Fourier Tools

In the author’s recent publications, a parametric system biorthogonal to the corresponding segment of the exponential Fourier system was unusually effective. On its basis, it was discovered that knowledge of a finite number of Fourier coefficients of function f from an infinite-dimensional set of elementary functions allows f to be accurately restored (the phenomenon of over-convergence). Below, parametric biorthogonal systems are constructed for classical trigonometric Fourier series, and the corresponding phenomena of over-convergence are discovered. The decisive role here was played by representing the space L2 as an orthogonal sum of two corresponding subspaces. As a result, fast parallel algorithms for reconstructing a function from its truncated trigonometric Fourier series are proposed. The presented numerical experiments confirm the high efficiency of these convergence accelerations for smooth functions. In conclusion, the main results of the work are summarized, and some prospects for the development and generalization of the proposed approaches are discussed.

拟半双正交分数阶多小波框架的刻画

本文在半正交分数阶多小波框架的基础上,运用多分辨分析,泛函分析,矩阵理论研究L~2(R)上多元小波框架的特征,将半正交分数阶多小波框架推广到拟半双正交分数阶多小波框架,给出了拟半双正交分数阶多小波框架的概念,得到了严格拟半双正交分数阶多小波框架的等价条件,证明了广义多分辨分析分数阶规范对偶多小波框架与拟半双正交分数阶规范对偶多小波框架满足等价条件.

Dyadic Bivariate Fourier Multipliers for Multi-Wavelets in L^2(R^2)

The single 2 dilation orthogonal wavelet multipliers in one dimensional case and single A-dilation(where A is any expansive matrix with integer entries and|det A|=2) wavelet multipliers in high dimensional case were completely characterized by the Wutam Consortium(1998) and Z. Y. Li, et al.(2010). But there exist no more results on orthogonal multivariate wavelet matrix multipliers corresponding integer expansive dilation matrix with the absolute value of determinant not 2 in L~2(R~2). In this paper, we choose 2I2=(~2~0)as the dilation matrix and consider the 2 I2-dilation orthogonal multivariate waveletΨ = {ψ, ψ, ψ},(which is called a dyadic bivariate wavelet) multipliers. We call the3 × 3 matrix-valued function A(s) = [ f(s)], where fi, jare measurable functions, a dyadic bivariate matrix Fourier wavelet multiplier if the inverse Fourier transform of A(s)( ψ(s), ψ(s), ψ(s)) ~T=( g(s), g(s), g(s))~ T is a dyadic bivariate wavelet whenever(ψ, ψ, ψ) is any dyadic bivariate wavelet. We give some conditions for dyadic matrix bivariate wavelet multipliers. The results extended that of Z. Y. Li and X. L.Shi(2011). As an application, we construct some useful dyadic bivariate wavelets by using dyadic Fourier matrix wavelet multipliers and use them to image denoising.

Balance of multi-wavelets

The discrete scalar data need prefiltering when transformed by discrete multi-wavelet, but prefiltering will make some properties of multi-wavelets lost. Balanced multi-wavelets can avoid prefiltering. The sufficient and necessary condition of p-order balance for multi-wavelets in time domain, the interrelation between balance order and approximation order and the sampling property of balanced multi-wavelets are investigated. The algorithms of 1-order and 2-order balancing for multi-wavelets are obtained. The two algorithms both preserve the orthogonal relation between multi-scaling function and multi-wavelets. More importantly, balancing operation doesnt increase the length of filters, which suggests that a relatively short balanced multi-wavelet can be constructed from an existing unbalanced multi-wavelet as short as possible.

基于双正交样条小波的输油管道焊接缺陷漏磁信号识别技术

针对输油管道焊接缺陷漏磁信号量大,无法精准检测漏磁信号磁场强度,影响识别效果的问题,提出基于双正交样条小波的输油管道焊接缺陷漏磁信号识别技术。应用双正交样条小波技术分解信号,采用稀疏矩阵求解器分解磁感应强度。计算连续小波变换系数和缺陷内漏磁场场磁导率,明确不同介质磁感应强度和场磁导率关系,识别凹、凸缺陷漏磁信号。实验结果表明,该技术凹形缺陷磁场强度检测结果只在峰值处出现了与实际数据最大为50 A/m的偏差,凸形缺陷磁场强度检测结果与实验数据一致。

调频斜率捷变联合BFT抑制密集假目标干扰算法

为实现相参雷达有效抑制密集假目标干扰,提出一种调频斜率捷变联合双正交傅里叶变换的干扰抑制算法。在自卫式干扰条件下,雷达发射调频斜率捷变线性调频信号波形并接收1个相干处理间隔回波进行处理。首先,通过匹配相关运算法估计受干扰段回波的平均时延,并根据时延截取受干扰段回波;然后,利用双正交傅里叶变换将回波信号转换至调频斜率域,利用调频斜率差异分离真实目标回波与干扰信号,并设计遮盖窗对干扰峰值位置进行遮盖;其次,通过双正交傅里叶逆变换恢复真实目标回波;最后进行脉冲压缩、相参积累进一步抑制干扰。通过仿真实验验证了方法的可行性以及在不同场景下的有效性。效能分析结果表明经干扰抑制后真实目标检测概率大幅提高,且算法在应对假目标数量较多场景时具备优势。

若干Krylov子空间方法及应用比较

文章首先介绍Krylov子空间方法类算法中最具代表性的几种算法:CGNR算法、GMRES算法、BiCG算法、CGS算法、BiCGSTAB算法及QMR算法等;其次讨论这几种算法之间的关系及各自的优点与不足;最后针对某一类数值算例来验证所得结论的正确性。