A newly revised inverse scattering transform（IST） for the derivative nonlinear Schrdinger（DNLS＋） equation with non-vanishing boundary condition（NVBC） and normal group velocity dispersion is proposed by introduc...A newly revised inverse scattering transform（IST） for the derivative nonlinear Schrdinger（DNLS＋） equation with non-vanishing boundary condition（NVBC） and normal group velocity dispersion is proposed by introducing a suitable affine parameter in Zakharov-Shabat integral kern.The explicit breather-type one-soliton solution,which can reproduce one pure soliton at the de-generate case and one bright soliton solution at the limit of van-ishing boundary,has been derived to verify the validity of the revised IST.展开更多
Gradient nanostructured(GNS)metallic materials are commonly achieved by gradient severe plastic de-formation with a gradient of nano-to micro-sized structural units from the surface/boundaries to the center.Certainly,...Gradient nanostructured(GNS)metallic materials are commonly achieved by gradient severe plastic de-formation with a gradient of nano-to micro-sized structural units from the surface/boundaries to the center.Certainly,such GNS can be inversely positioned,which however has not yet been reported.The present work reports a facile method in deformation gradient control to attain inverse gradient nanostructured(iGNS),i.e.,tailoring the cross-section shape,successfully demonstrated in Ti-50.3Ni shape memory alloy(SMA)wire through cold rolling.The microstructure of the rolled wire is characterized by a macroscopic inverse gradient from boundaries to the center—the average sizes of grain and martensite domain evolve from micrometer to nanometer scale.The iGNS leads to a gradient martensitic transforma-tion upon stress,which has been proved to be effectively reversible via in-situ bending scanning electron microscopy(SEM)observations.The iGNS Ti-50.3Ni SMA exhibits quasi-linear superelasticity(SE)in a wide temperature range from 173 to 423 K.Compared to uniform cold rolling,the gradient cold rolling with less overall plasticity further improves SE strain(up to 4.8%)and SE efficiency.In-situ tensiling synchrotron X-ray diffraction(SXRD)analysis reveals the underlying mechanisms of the unique SE in the iGNS SMAs.It provides a new design strategy to realize excellent SE in SMAs and sheds light on the advanced GNS metallic materials.展开更多
The inverse scattering transform of a coupled Sasa–Satsuma equation is studied via Riemann–Hilbert approach. Firstly, the spectral analysis is performed for the coupled Sasa–Satsuma equation, from which a Riemann–...The inverse scattering transform of a coupled Sasa–Satsuma equation is studied via Riemann–Hilbert approach. Firstly, the spectral analysis is performed for the coupled Sasa–Satsuma equation, from which a Riemann–Hilbert problem is formulated. Then the Riemann–Hilbert problem corresponding to the reflection-less case is solved.As applications, multi-soliton solutions are obtained for the coupled Sasa–Satsuma equation. Moreover, some figures are given to describe the soliton behaviors, including breather types, single-hump solitons, double-hump solitons, and two-bell solitons.展开更多
One of the basic problems about the inverse scattering transform for solving a completely integrable nonlinear evolutions equation is to demonstrate that the Jost solutions obtained from the inverse scattering equatio...One of the basic problems about the inverse scattering transform for solving a completely integrable nonlinear evolutions equation is to demonstrate that the Jost solutions obtained from the inverse scattering equations of Cauchy integral satisfy the Lax equations. Such a basic problem still exists in the procedure of deriving the dark soliton solutions of the NLS equation in normal dispersion with non-vanishing boundary conditions through the inverse scattering transform. In this paper, a pair of Jost solutions with same analytic properties are composed to be a 2 × 2 matrix and then another pair are introduced to be its right inverse confirmed by the Liouville theorem. As they are both 2 × 2 matrices, the right inverse should be the left inverse too, based upon which it is not difficult to show that these Jost solutions satisfy both the first and second Lax equations. As a result of compatibility condition, the dark soliton solutions definitely satisfy the NLS equation in normal dispersion with non-vanishing boundary conditions.展开更多
Since the Jost solutions of the DNLS equation does not tend to the free Jost solutioins as ｜λ｜→∞, the usual inverse scattering transform （IST） must be revised. Beside the Kaup and Newell＇s approach, we propose...Since the Jost solutions of the DNLS equation does not tend to the free Jost solutioins as ｜λ｜→∞, the usual inverse scattering transform （IST） must be revised. Beside the Kaup and Newell＇s approach, we propose a simple revision in constructing the equations of IST, where the usual Zakharov-Shabat kern is revised by multiplying λ^-2 or λ^-1. To justify the revision we show that the Jost solutions obtained do satisfy the pair of compatibility equations.展开更多
N-soliton solutions of the hierarchy of non-isospectral mKdV equation with self-consistent sources andthe hierarchy of non-isospectral sine-Gordon equation with self-consistent sources are obtained via the inverse sca...N-soliton solutions of the hierarchy of non-isospectral mKdV equation with self-consistent sources andthe hierarchy of non-isospectral sine-Gordon equation with self-consistent sources are obtained via the inverse scatteringtransform.展开更多
A new set of relative orbit elements (ROEs) is used to derive a new elliptical formation flying model in previous work. In-plane and out-of-plane relative motions can be completely decoupled, which benefits elliptical...A new set of relative orbit elements (ROEs) is used to derive a new elliptical formation flying model in previous work. In-plane and out-of-plane relative motions can be completely decoupled, which benefits elliptical formation design. In order to study the elliptical control strategy and perturbation effects, it is necessary to derive the inverse transformation of the relative state transition matrix based on relative orbit elements. Poisson bracket theory is used to obtain the linear transformations between the two representations: the relative orbit elements and the geocentric orbital frame. In this paper, the details of these transformations are presented.展开更多
This study presents a novel two-step approach to assess plate-like structural laminar damages,particularly for delamination damage detection of composite structures.Firstly,a 2-D continuous wavelet transform is employ...This study presents a novel two-step approach to assess plate-like structural laminar damages,particularly for delamination damage detection of composite structures.Firstly,a 2-D continuous wavelet transform is employed to identify the damage location and sizes from vibration curvature data.An inverse method is subsequently then used to determine the bending stiffness reduction ratio along a specified direction,enabling the quantification of the delamination severity.The method employed in this study is an extension of the one-dimensional inverse method developed in a previous work of the authors.The applicability of the two-step inverse approach is demonstrated in a simulation analysis and by an experimental study on a cantilever composite plate containing a single delamination.The inverse method is shown to have the capacity to reveal the detailed damage information of delamination within a constrained searching space and can be used to determine the effective flexural stiffness of composite plate structures,even in cases of complex delamination damage.展开更多
A new reflection approach for field distribution measurement of ultrasonic transducers was investigated. Instead of a point-like reflection target （rigid sphere） or thin wires （line-like targets）, a line response ...A new reflection approach for field distribution measurement of ultrasonic transducers was investigated. Instead of a point-like reflection target （rigid sphere） or thin wires （line-like targets）, a line response function of experimental knife-edge distribution combined with the inverse Abel transforms was used to estimate the lateral beam distributions of ultrasonic transducers. The measurement steps were as follows：① A knife-edge was scanned perpendicularly to acoustic beam axis of the transducer using an ultrasonic C-scan system to obtain its ultrasonic image line response function, ② the transverse beam distribution was solved by the inverse Abel transforms, and ③ experiments were performed to obtain the lateral beam profiles of two transducers, with and without focus, and the results were compared with those from a hydrophone. The results showed that this method was effective for ultrasonic field measurement and could be as a substitute for hydrophone in most cases.展开更多
In order to find stable, accurate, and computationally efficient methods for performing the inverse Laplace transform, a new double transformation approach is proposed. To validate and improve the inversion solution o...In order to find stable, accurate, and computationally efficient methods for performing the inverse Laplace transform, a new double transformation approach is proposed. To validate and improve the inversion solution obtained using the Gaver-Stehfest algorithm, direct Laplace transforms are taken of the numerically inverted transforms to compare with the original function. The numerical direct Laplace transform is implemented with a composite Simpson’s rule. Challenging numerical examples involving periodic and oscillatory functions, are investigated. The numerical examples illustrate the computational accuracy and efficiency of the direct Laplace transform and its inverse due to increasing the precision level and the number of terms included in the expansion. It is found that the number of expansion terms and the precision level selected must be in a harmonious balance in order for correct and stable results to be obtained.展开更多
In this paper, we estimate the Fekete-Szego functional with k-th root transform for the inverse of certain classes of analytic univalent functions using quasi-subordination.
By combined power evolution laws of the spectral parameter and the initial constants of integration,a new differential-difference hierarchy is presented from the Toda spectral problem.The hierarchy contains the classi...By combined power evolution laws of the spectral parameter and the initial constants of integration,a new differential-difference hierarchy is presented from the Toda spectral problem.The hierarchy contains the classic Toda lattice equation,the nonisospectral Toda lattice equation and the mixed Toda lattice equation as reduced cases.The evolution of the scattering data in the inverse scattering transform is analyzed in detail and exact soliton solutions are computed through the corresponding inverse scattering transform.展开更多
A high-performance, low cost inverse integer transform architecture for advanced video standard （AVS） video coding standard was presented. An 8 × 8 inverse integer transform is required in AVS video system whic...A high-performance, low cost inverse integer transform architecture for advanced video standard （AVS） video coding standard was presented. An 8 × 8 inverse integer transform is required in AVS video system which is compute-intensive. A hardware transform is inevitable to compute the transform for the real-time application. Compared with the 4 × 4 transform for H.264/AVC, the 8 × 8 integer transform is much more complex and the coefficient in the inverse transform matrix Ts is not inerratic as that in H.264/AVC. Dividing the Ts into matrix Ss and Rs, the proposed architecture is implemented with the adders and the specific CSA-trees instead of multipliers, which are area and time consuming. The architecture obtains the data processing rate up to 8 pixels per-cycle at a low cost of area. Synthesized to TSMC 0.18 μm COMS process, the architecture attains the operating frequency of 300 MHz at cost of 34 252 gates with a 2-stage pipeline scheme. A reusable scheme is also introduced for the area optimization, which results in the operating frequency of 143 MHz at cost of only 19 758 gates.展开更多
A new method for approximating the inerse Laplace transform is presented. We first change our Laplace transform equation into a convolution type integral equation, where Tikhonov regularization techniques and the Four...A new method for approximating the inerse Laplace transform is presented. We first change our Laplace transform equation into a convolution type integral equation, where Tikhonov regularization techniques and the Fourier transformation are easily applied. We finally obtain a regularized approximation to the inverse Laplace transform as finite sum展开更多
In this paper, we built upon the estimating primaries by sparse inversion （EPSI） method. We use the 3D curvelet transform and modify the EPSI method to the sparse inversion of the biconvex optimization and Ll-norm r...In this paper, we built upon the estimating primaries by sparse inversion （EPSI） method. We use the 3D curvelet transform and modify the EPSI method to the sparse inversion of the biconvex optimization and Ll-norm regularization, and use alternating optimization to directly estimate the primary reflection coefficients and source wavelet. The 3D curvelet transform is used as a sparseness constraint when inverting the primary reflection coefficients, which results in avoiding the prediction subtraction process in the surface-related multiples elimination （SRME） method. The proposed method not only reduces the damage to the effective waves but also improves the elimination of multiples. It is also a wave equation- based method for elimination of surface multiple reflections, which effectively removes surface multiples under complex submarine conditions.展开更多
The main problems in three-dimensional gravity inversion are the non-uniqueness of the solutions and the high computational cost of large data sets. To minimize the high computational cost, we propose a new sorting me...The main problems in three-dimensional gravity inversion are the non-uniqueness of the solutions and the high computational cost of large data sets. To minimize the high computational cost, we propose a new sorting method to reduce fluctuations and the high frequency of the sensitivity matrix prior to applying the wavelet transform. Consequently, the sparsity and compression ratio of the sensitivity matrix are improved as well as the accuracy of the forward modeling. Furthermore, memory storage requirements are reduced and the forward modeling is accelerated compared with uncompressed forward modeling. The forward modeling results suggest that the compression ratio of the sensitivity matrix can be more than 300. Furthermore, multiscale inversion based on the wavelet transform is applied to gravity inversion. By decomposing the gravity inversion into subproblems of different scales, the non-uniqueness and stability of the gravity inversion are improved as multiscale data are considered. Finally, we applied conventional focusing inversion and multiscale inversion on simulated and measured data to demonstrate the effectiveness of the proposed gravity inversion method.展开更多
This study deduces a general inversion of continuous wavelet transform (CWT) with timescale being real rather than positive. In conventional CWT inversion, wavelet’s dual is assumed to be a reconstruction wavelet or ...This study deduces a general inversion of continuous wavelet transform (CWT) with timescale being real rather than positive. In conventional CWT inversion, wavelet’s dual is assumed to be a reconstruction wavelet or a localized function. This study finds that wavelet’s dual can be a harmonic which is not local. This finding leads to new CWT inversion formulas. It also justifies the concept of normal wavelet transform which is useful in time-frequency analysis and time-frequency filtering. This study also proves a law for CWT inversion: either wavelet or its dual must integrate to zero.展开更多
基金Supported by the National Natural Science Foundation of China(10775105)
文摘A newly revised inverse scattering transform（IST） for the derivative nonlinear Schrdinger（DNLS＋） equation with non-vanishing boundary condition（NVBC） and normal group velocity dispersion is proposed by introducing a suitable affine parameter in Zakharov-Shabat integral kern.The explicit breather-type one-soliton solution,which can reproduce one pure soliton at the de-generate case and one bright soliton solution at the limit of van-ishing boundary,has been derived to verify the validity of the revised IST.
基金supported by the National Natural Science Foundation of China(Grant Nos.52171007,52101166,51931004)the 111 Projects 2.0(Grant No.BP0618008).
文摘Gradient nanostructured(GNS)metallic materials are commonly achieved by gradient severe plastic de-formation with a gradient of nano-to micro-sized structural units from the surface/boundaries to the center.Certainly,such GNS can be inversely positioned,which however has not yet been reported.The present work reports a facile method in deformation gradient control to attain inverse gradient nanostructured(iGNS),i.e.,tailoring the cross-section shape,successfully demonstrated in Ti-50.3Ni shape memory alloy(SMA)wire through cold rolling.The microstructure of the rolled wire is characterized by a macroscopic inverse gradient from boundaries to the center—the average sizes of grain and martensite domain evolve from micrometer to nanometer scale.The iGNS leads to a gradient martensitic transforma-tion upon stress,which has been proved to be effectively reversible via in-situ bending scanning electron microscopy(SEM)observations.The iGNS Ti-50.3Ni SMA exhibits quasi-linear superelasticity(SE)in a wide temperature range from 173 to 423 K.Compared to uniform cold rolling,the gradient cold rolling with less overall plasticity further improves SE strain(up to 4.8%)and SE efficiency.In-situ tensiling synchrotron X-ray diffraction(SXRD)analysis reveals the underlying mechanisms of the unique SE in the iGNS SMAs.It provides a new design strategy to realize excellent SE in SMAs and sheds light on the advanced GNS metallic materials.
基金Supported by the National Natural Science Foundation of China under Project Nos.11331008 and 11171312the Collaborative Innovation Center for Aviation Economy Development of Henan Province
文摘The inverse scattering transform of a coupled Sasa–Satsuma equation is studied via Riemann–Hilbert approach. Firstly, the spectral analysis is performed for the coupled Sasa–Satsuma equation, from which a Riemann–Hilbert problem is formulated. Then the Riemann–Hilbert problem corresponding to the reflection-less case is solved.As applications, multi-soliton solutions are obtained for the coupled Sasa–Satsuma equation. Moreover, some figures are given to describe the soliton behaviors, including breather types, single-hump solitons, double-hump solitons, and two-bell solitons.
基金The project supported by National Natural Science Foundation of China under Grant Nos. 10474076 and 10375041
文摘One of the basic problems about the inverse scattering transform for solving a completely integrable nonlinear evolutions equation is to demonstrate that the Jost solutions obtained from the inverse scattering equations of Cauchy integral satisfy the Lax equations. Such a basic problem still exists in the procedure of deriving the dark soliton solutions of the NLS equation in normal dispersion with non-vanishing boundary conditions through the inverse scattering transform. In this paper, a pair of Jost solutions with same analytic properties are composed to be a 2 × 2 matrix and then another pair are introduced to be its right inverse confirmed by the Liouville theorem. As they are both 2 × 2 matrices, the right inverse should be the left inverse too, based upon which it is not difficult to show that these Jost solutions satisfy both the first and second Lax equations. As a result of compatibility condition, the dark soliton solutions definitely satisfy the NLS equation in normal dispersion with non-vanishing boundary conditions.
基金The project supported by National Natural Science Foundation of China under Grant Nos. 10534030 and 10375041
文摘Since the Jost solutions of the DNLS equation does not tend to the free Jost solutioins as ｜λ｜→∞, the usual inverse scattering transform （IST） must be revised. Beside the Kaup and Newell＇s approach, we propose a simple revision in constructing the equations of IST, where the usual Zakharov-Shabat kern is revised by multiplying λ^-2 or λ^-1. To justify the revision we show that the Jost solutions obtained do satisfy the pair of compatibility equations.
基金Supported by the National Natural Science Foundation of China under Grant Nos.10371070,10671121the Foundation of Shanghai Education Committee for Shanghai Prospective Excellent Young Teachers+1 种基金Shanghai Leading Academic Discipline Project under Grant No.J50101 the President Foundation of East China Institute of Technology under Grant No.DHXK0810
文摘N-soliton solutions of the hierarchy of non-isospectral mKdV equation with self-consistent sources andthe hierarchy of non-isospectral sine-Gordon equation with self-consistent sources are obtained via the inverse scatteringtransform.
文摘A new set of relative orbit elements (ROEs) is used to derive a new elliptical formation flying model in previous work. In-plane and out-of-plane relative motions can be completely decoupled, which benefits elliptical formation design. In order to study the elliptical control strategy and perturbation effects, it is necessary to derive the inverse transformation of the relative state transition matrix based on relative orbit elements. Poisson bracket theory is used to obtain the linear transformations between the two representations: the relative orbit elements and the geocentric orbital frame. In this paper, the details of these transformations are presented.
文摘This study presents a novel two-step approach to assess plate-like structural laminar damages,particularly for delamination damage detection of composite structures.Firstly,a 2-D continuous wavelet transform is employed to identify the damage location and sizes from vibration curvature data.An inverse method is subsequently then used to determine the bending stiffness reduction ratio along a specified direction,enabling the quantification of the delamination severity.The method employed in this study is an extension of the one-dimensional inverse method developed in a previous work of the authors.The applicability of the two-step inverse approach is demonstrated in a simulation analysis and by an experimental study on a cantilever composite plate containing a single delamination.The inverse method is shown to have the capacity to reveal the detailed damage information of delamination within a constrained searching space and can be used to determine the effective flexural stiffness of composite plate structures,even in cases of complex delamination damage.
文摘A new reflection approach for field distribution measurement of ultrasonic transducers was investigated. Instead of a point-like reflection target （rigid sphere） or thin wires （line-like targets）, a line response function of experimental knife-edge distribution combined with the inverse Abel transforms was used to estimate the lateral beam distributions of ultrasonic transducers. The measurement steps were as follows：① A knife-edge was scanned perpendicularly to acoustic beam axis of the transducer using an ultrasonic C-scan system to obtain its ultrasonic image line response function, ② the transverse beam distribution was solved by the inverse Abel transforms, and ③ experiments were performed to obtain the lateral beam profiles of two transducers, with and without focus, and the results were compared with those from a hydrophone. The results showed that this method was effective for ultrasonic field measurement and could be as a substitute for hydrophone in most cases.
文摘In order to find stable, accurate, and computationally efficient methods for performing the inverse Laplace transform, a new double transformation approach is proposed. To validate and improve the inversion solution obtained using the Gaver-Stehfest algorithm, direct Laplace transforms are taken of the numerically inverted transforms to compare with the original function. The numerical direct Laplace transform is implemented with a composite Simpson’s rule. Challenging numerical examples involving periodic and oscillatory functions, are investigated. The numerical examples illustrate the computational accuracy and efficiency of the direct Laplace transform and its inverse due to increasing the precision level and the number of terms included in the expansion. It is found that the number of expansion terms and the precision level selected must be in a harmonious balance in order for correct and stable results to be obtained.
基金Supported by the National Natural Science Foundation of China (Grant No. 11561001)the Natural Science Foundation of Inner Mongolia of China (Grant No. 2018MS01026)+4 种基金the Natural Science Foundation of Anhui Provincial Department of Education (Grant Nos. KJ2018A0833KJ2020A0993KJ2020ZD74)Provincial Quality Engineering Project of Anhui Colleges and Universities (Grant No. 2018mooc608)the Key Cultivated Project at School Level of the National Science Fund of Guangzhou Civil Aviation College (Grant No. 18X0428)。
文摘In this paper, we estimate the Fekete-Szego functional with k-th root transform for the inverse of certain classes of analytic univalent functions using quasi-subordination.
基金Supported by the National Natural Science Foundation of China(Grant Nos.11671177,11771186)Postgraduate Research&Practice Innovation Program of Jiangsu Province(Grant No.KYCX212566)。
文摘By combined power evolution laws of the spectral parameter and the initial constants of integration,a new differential-difference hierarchy is presented from the Toda spectral problem.The hierarchy contains the classic Toda lattice equation,the nonisospectral Toda lattice equation and the mixed Toda lattice equation as reduced cases.The evolution of the scattering data in the inverse scattering transform is analyzed in detail and exact soliton solutions are computed through the corresponding inverse scattering transform.
文摘A high-performance, low cost inverse integer transform architecture for advanced video standard （AVS） video coding standard was presented. An 8 × 8 inverse integer transform is required in AVS video system which is compute-intensive. A hardware transform is inevitable to compute the transform for the real-time application. Compared with the 4 × 4 transform for H.264/AVC, the 8 × 8 integer transform is much more complex and the coefficient in the inverse transform matrix Ts is not inerratic as that in H.264/AVC. Dividing the Ts into matrix Ss and Rs, the proposed architecture is implemented with the adders and the specific CSA-trees instead of multipliers, which are area and time consuming. The architecture obtains the data processing rate up to 8 pixels per-cycle at a low cost of area. Synthesized to TSMC 0.18 μm COMS process, the architecture attains the operating frequency of 300 MHz at cost of 34 252 gates with a 2-stage pipeline scheme. A reusable scheme is also introduced for the area optimization, which results in the operating frequency of 143 MHz at cost of only 19 758 gates.
文摘A new method for approximating the inerse Laplace transform is presented. We first change our Laplace transform equation into a convolution type integral equation, where Tikhonov regularization techniques and the Fourier transformation are easily applied. We finally obtain a regularized approximation to the inverse Laplace transform as finite sum
基金supported by the National Science and Technology Major Project (No.2011ZX05023-005-008)
文摘In this paper, we built upon the estimating primaries by sparse inversion （EPSI） method. We use the 3D curvelet transform and modify the EPSI method to the sparse inversion of the biconvex optimization and Ll-norm regularization, and use alternating optimization to directly estimate the primary reflection coefficients and source wavelet. The 3D curvelet transform is used as a sparseness constraint when inverting the primary reflection coefficients, which results in avoiding the prediction subtraction process in the surface-related multiples elimination （SRME） method. The proposed method not only reduces the damage to the effective waves but also improves the elimination of multiples. It is also a wave equation- based method for elimination of surface multiple reflections, which effectively removes surface multiples under complex submarine conditions.
基金This work was supported by the Key National Research Project of China （Nos. 2017YFC0601900 and 2016YFC0303100） and the Key Program of National Natural Science Foundation of China （Nos. 41530320 and 41774125）.
文摘The main problems in three-dimensional gravity inversion are the non-uniqueness of the solutions and the high computational cost of large data sets. To minimize the high computational cost, we propose a new sorting method to reduce fluctuations and the high frequency of the sensitivity matrix prior to applying the wavelet transform. Consequently, the sparsity and compression ratio of the sensitivity matrix are improved as well as the accuracy of the forward modeling. Furthermore, memory storage requirements are reduced and the forward modeling is accelerated compared with uncompressed forward modeling. The forward modeling results suggest that the compression ratio of the sensitivity matrix can be more than 300. Furthermore, multiscale inversion based on the wavelet transform is applied to gravity inversion. By decomposing the gravity inversion into subproblems of different scales, the non-uniqueness and stability of the gravity inversion are improved as multiscale data are considered. Finally, we applied conventional focusing inversion and multiscale inversion on simulated and measured data to demonstrate the effectiveness of the proposed gravity inversion method.
文摘This study deduces a general inversion of continuous wavelet transform (CWT) with timescale being real rather than positive. In conventional CWT inversion, wavelet’s dual is assumed to be a reconstruction wavelet or a localized function. This study finds that wavelet’s dual can be a harmonic which is not local. This finding leads to new CWT inversion formulas. It also justifies the concept of normal wavelet transform which is useful in time-frequency analysis and time-frequency filtering. This study also proves a law for CWT inversion: either wavelet or its dual must integrate to zero.
