Wavelets with convolution-type orthogonality conditions

Wavelets with free parameters are constructed using a convolution-type orthogonality condition. First, finer and coarser scaling function spaces are introduced with the help of a two-scale relation for scaling functions. An inner product and a norm having convolution parameters are defined in the finer scaling function space, which becomes a Hilbert space as a result. The finer scaling function space can be decomposed into the coarser one and its orthogonal complement. A wavelet function is constructed as a mother function whose shifted functions form an orthonormal basis in the complement space. Such wavelet functions contain the Daubechies' compactly supported wavelets as a special case. In some restricted cases, several symmetric and almost compactly supported wavelets are constructed analytically by tuning free convolution parameters contained in the wavelet functions     

Lyapunov matrix equations with positive semidefinite input matrices

An algebraic criterion is derived, which establishes whether or not the negative-semidefinite first difference of a quadratic-form Lyapunov function for a linear constant-coefficient difference equation, or the derivative of such a function for a linear constant-coefficient differential equation, can vanish identically along any trajectory of the free motion.     

Efficient numerical techniques for solving Pocklington's equation and their relationships to other methods

It is shown that testing Pocklington's equation with piecewise sinusoidal functions yields an integro-difference equation whose numerical solution is identical to that of the point-matched Hallen's equation when a common set of basis functions is used with each. For any choice of basis functions, the integro-difference equation has the simple kernel, the fast convergence, the simplicity of point-matching, and the adequate treatment of rapidly varying incident fields, but none of the additional unknowns normally associated with Hallen's equation. Furthermore, for the special choice of piecewise sinusoids as the basis functions, the method reduces to Richmond's piecewise sinusoidal reaction matching technique, or Galerkin's method. It is also shown that testing with piecewise linear (triangle) functions yields an integro-difference equation whose solution converges asymptotically at the same rate as that of Hallen's equation. The resulting equation is essentially that obtained by approximating the second derivative in Pocklington's equation by its finite difference equivalent. The authors suggest a simple and highly efficient method for solving Pocklington's equation. This approach is contrasted to the point-matched solution of Pocklington's equation and the reasons for the poor convergence of the latter are examined.     

Nonlinear system identification and prediction using orthogonalfunctions

We describe a systematic scheme for the nonlinear adaptive filtering of signals that are generated by nonlinear dynamical systems. The complete filter consists of three sections: a signal-independent standard orthonormal expansion, a scaling derived from an estimate of the vector probability density function (PDF), and an adaptive linear combiner. The orthonormal property of the expansions has two significant implications for adaptive filtering: first, model order reduction is trivial since the contribution of each term to the mean squared error is directly related to the coefficient in the final linear combiner; and second, consistent and rapid convergence of stochastic gradient algorithms is assured. A technique based on the inverse Fourier transform for obtaining a PDF estimate from the characteristic function is also presented. The prediction and identification performance of this nonlinear structure is examined for a number of signals, and it is contrasted with common radial basis function and linear networks     

On the asymptotic tightness of the Shannon lower bound

New results are proved on the convergence of the Shannon (1959) lower bound to the rate distortion function as the distortion decreases to zero. The key convergence result is proved using a fundamental property of informational divergence. As a corollary, it is shown that the Shannon lower bound is asymptotically tight for norm-based distortions, when the source vector has a finite differential entropy and a finite α th moment for some α>0, with respect to the given norm. Moreover, we derive a theorem of Linkov (1965) on the asymptotic tightness of the Shannon lower bound for general difference distortion measures with more relaxed conditions on the source density. We also show that the Shannon lower bound relative to a stationary source and single-letter difference distortion is asymptotically tight under very weak assumptions on the source distribution     

An integral transform technique for analysis of planar dielectricstructures

This paper presents a novel efficient technique for the study of planar dielectric waveguides for submillimeter-wave and optical applications. In an appropriate integral transform domain, which is determined by the Green's function of the substrate structure, higher-order boundary conditions are enforced in conjunction with Taylor expansions of the fields to derive an equivalent one-dimensional integral equation for the corresponding two-dimensional waveguide geometry. This reduction in the dimensionality of the boundary-value problem can easily be extended to three-dimensional planar structures, with equivalent two-dimensional integral equations being formulated. The reduced integral equations are solved numerically by invoking the method of moments, in which the transform-domain unknowns are expanded in a smooth localized entire-domain basis. It is demonstrated that using orthogonal Hermite-Gauss functions as an expansion basis provides very satisfactory results with only a few expansion terms. For the validation of the technique, single and coupled dielectric slab waveguides are treated     

截断差分概率的上界估计与应用

截断差分分析是差分分析的一个变形。为说明一个密码算法能够抵抗截断差分分析,需要给出截断差分概率的上界。Masayuki Kanda等人就密码算法中S盒为GF(256)上的乘法逆变换和仿射双射变换复合而成时,提出了截断差分概率的上界一个猜想。该文就一般双射S盒给出了该概率上界问题的一个估计,Masayuki Kanda的猜想是该估计所考虑问题的一个特例,在一些情况下,该估计给出的上界与Masayuki Kanda的猜想接近。利用该结论可以衡量密码算法截断差分传递链概率的上界。该结论为分组密码抗截断差分分析的可证明安全性提供了理论依据。     

Use of discrete Laguerre sequences to extrapolate wide-band response from early-time and low-frequency data

Extrapolation of wide-band response using early-time and low-frequency data has been accomplished by the use of the orthogonal polynomials, such as Laguerre polynomials, Hermite polynomials, and Bessel-Chebyshev functions. It is a good approach to reduce the computational loads and obtain stable results for computation intensive electromagnetic analysis. However, all the orthonormal basis functions that have been used are all continuous or analog functions, which means we have to sample the polynomials both in time and frequency domains before we can use them to carry out the extrapolation. The process of sampling will introduce some errors, especially for high degrees or small scaling factors and, hence, may destroy the orthogonality between the polynomials of various degrees in a discrete sense. In this paper, we introduce the discrete Laguerre functions, which are directly derived using the Z transform and, thus, are exactly orthonormal in a discrete sense. The discrete Laguerre polynomials are fundamentally different from its continuous counterparts, except asymptotically when the sampling interval approaches zero. The other advantage of using these discrete orthonormal functions is that they do not give rise to the Gibbs phenomenon unlike its continuous counterpart. Using it in the extrapolation, the range or convergence can be extended both for the scaling factor and order of expansion, and at the same time, the quality of performance can be improved. Since the error of extrapolation is sensitive to the scaling factor, an efficient way to estimate the error as a function of the scaling factor is explained and its feasibility for any problem is validated by numerical examples of antennas.     

Level set-based bimodal segmentation with stationary global minimum.

In this paper, we propose a new level set-based partial differential equation (PDE) for the purpose of bimodal segmentation. The PDE is derived from an energy functional which is a modified version of the fitting term of the Chan-Vese model. The energy functional is designed to obtain a stationary global minimum, i.e., the level set function which evolves by the Euler-Lagrange equation of the energy functional has a unique convergence state. The existence of a global minimum makes the algorithm invariant to the initialization of the level set function, whereas the existence of a convergence state makes it possible to set a termination criterion on the algorithm. Furthermore, since the level set function converges to one of the two fixed values which are determined by the amount of the shifting of the Heaviside functions, an initialization of the level set function close to those values can result in a fast convergence.     

On the upper bound of the probability of error of a linear pattern classifier for probabilistic pattern classes

Given a set of sample patterns for two pattern classes, some simple expressions for the upper bound of the probability of error for a linear pattern classifier and the optimal linear discriminant function minimizing the upper bound are obtained. Using these results, if the tolerable probability of error of classifying patterns in the two pattern classes is not smaller than this upper bound, not only a linear pattern classifier is known to be feasible, but also a satisfactory linear discriminant function is given. The results presented here are independent of the probability distribution of the patterns in the pattern classes. For some special cases, a smaller upper bound is found.     

Full wave analysis of microstrip floating line structures bywavelet expansion method

A full wave analysis of microstrip floating line structures by wavelet expansion method is presented. The surface integral equation developed from a dyadic Green's function is solved by Galerkin's method, with the integral kernel and the unknown current expanded in terms of orthogonal wavelets. Using the orthonormal wavelets (and scaling functions) with compact support as basis functions and weighting functions, the integral equation is converted into a set of linear algebraic equations, with the matrices nearly diagonal or block-diagonal due to the localization, orthogonality, and cancellation properties of the orthogonal wavelets. Limitations inherited in the traditional orthogonal basis systems are released: The problem-dependent normal modes have been replaced by the problem-independent wavelets, preserving the orthogonality; the trade-off between orthogonality and continuity (e.g. subsectional basis functions including pulse functions, roof-top functions, piecewise sinusoidal functions, etc.) is well balanced by the orthogonal wavelets. Numerical results are compared with measurements and previous published data with good agreement     

A bound on the multiple scattered power from a plasma medium

A measure of the validity of the first Born approximation for scattering from an inhomogeneous plasma is obtained. This measure is derived using the inequalities appropriate to quadratically summable functions and bounded operators. A simple upper bound for the ratio of multiple scattered power to first Born power is derived as a function ofB, the sufficiency condition for the convergence of the series.     

Online optimization of the time scale in adaptive Laguerre-basedfilters

A new online method to optimize the free parameter in adaptive Laguerre-based filters is presented, it is based on the minimization of a criterion that is equivalent to an upper bound for the quadratic approximation error. The proposed technique presents a fast convergence and a good robustness     

SAR图像提高分辨率的主要方法间的联系

SAR图像提高分辨率的几种典型方法有谱估计、正则化、基追踪及偏微分方程方法.讨论这几种方法之间的联系.为其综合应用提供参考.首先,揭示不同方法在先验信息利用、建模及参数估计等方面的异同.其次,给出不同方法使用的边界条件和工程应用过程中的优选方法.最后,给出了几种综合模型的构造思路,计算结果验证了综合方法的优势.     

Direct extrapolation of a causal signal using low-frequency and early-time data

In this paper, we provide three direct procedures to extrapolate the early-time and the low-frequency response of a causal signal simultaneously in the time-and frequency domain. Compared with the extrapolation by orthonormal basis functions, direct extrapolation is straightforward and we do not need to evaluate the basis functions and search for the optimal scaling factor and the optimal number of basis functions. We show that the extrapolation introduced by Adve and Sarkar is equivalent to a Neumann-series solution of an integral equation of the second kind. It is further shown that this iterative Neumann expansion is an error-reducing method. We propose to solve this integral equation efficiently by employing a conjugate gradient iterative scheme. The convergence of this scheme is also demonstrated. We provide the matrix equations and show the equivalence to the integral equations, and demonstrate that the method of singular value decomposition (SVD) of solving the matrix equation provides accurate and stable results. Finally, a number of illustrative numerical examples are presented and the performances of the three direct methods are compared.     

线性码广义汉明重量的上限函数

在广义汉明重量的研究过程中，限函数有着重要的研究价值。本文提出了广义汉明重量上限函数的Ｕ 条件，给出了在Ｕ 条件下的有限和表达式。并指出这样的条件是容易满足的。从而给出了广义汉明重量上限函数的一种简单的计算方法     

An error bound for Lagrange interpolation of low-pass functions (Corresp.)

The well-known error formula for Lagrange interpolation is used to derive an expression for a truncation error bound in terms of the sampling rate and Nyquist frequency for regular samples and central interpolation. The proof is restricted to pulse-type functions possessing a Fourier transform. The formula finds application to the estimation of convergence rate in iterative interpolation, thus providing a criterion for the choice of sampling rate to achieve a specified truncation error level in a given number of steps. The formula can also be used as a guide when the samples are not regular but fairly evenly distributed.     

Deconvolution by thresholding in mirror wavelet bases

The deconvolution of signals is studied with thresholding estimators that decompose signals in an orthonormal basis and threshold the resulting coefficients. A general criterion is established to choose the orthonormal basis in order to minimize the estimation risk. Wavelet bases are highly sub-optimal to restore signals and images blurred by a low-pass filter whose transfer function vanishes at high frequencies. A new orthonormal basis called mirror wavelet basis is constructed to minimize the risk for such deconvolutions. An application to the restoration of satellite images is shown.     

Second-order time-frequency synthesis of nonstationary randomprocesses

We present time-frequency methods for the synthesis of finite-energy, nonstationary random processes. The energetic characteristics of the process to be synthesized are specified in a joint time-frequency domain via a time-frequency model function. The synthesis methods optimize the autocorrelation function of the process such that the process' Wigner-Ville spectrum is closest to the given model function. An optional signal subspace constraint allows the incorporation of additional properties such as bandlimitation and also permits the reformulation of the synthesis methods in a discrete-time setting. The synthesized process is expressed either in terms of an orthonormal basis of the constraint subspace or via its Karhunen-Loeve expansion. An example involving the prolate spheroidal functions is given, and computer simulation results are provided