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 pr...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.展开更多
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 ...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.展开更多
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...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.展开更多
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 identificat...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.展开更多
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 d...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.展开更多
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 dom...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.展开更多
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 wavel...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.展开更多
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...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.展开更多
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 metho...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.展开更多
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 th...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.展开更多
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...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.展开更多
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 c...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.展开更多
文摘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.
文摘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.
文摘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.
文摘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.
基金Project supported by the National Natural Science Foundation of China(Grant No.6017201860372007)
文摘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.
文摘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.
基金Supported by Natural Science Foundation of Henan Province(0511013500)
文摘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.
文摘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.
文摘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.
基金Supported by National Natural Science Foundation of China under Grant Nos. 10647108, 10547101, and 10604002the National Fundamental Research Program of China under Grant No. 2006CB921200
文摘把直角的基础基于双性人的途径,我们执行为开的系统的一个类的扩大与出去的波浪边界条件由波浪方程描述了的伪正常模式(QNM ) 。为如此的一个 non-Hermitian 系统,特徵函数不安扩大和格林功能方法,为关上的量系统基于 Hermitian Hamiltonian 的直角的特徵向量,能以双性人被概括直角的基础, H 和它的伴随海角 H <SUP>†</SUP>的特徵函数的二个集合。为复杂频率的时间无关的不安理论能也被开发。
基金Supported by the National Natural Science Foundation of China (Project 10171021)
文摘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.
文摘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.
基金partially supported by the National Natural Science Foundation of China (Grant No. 11101142 and No. 11571107)
文摘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.