Orthogonal functionals of independent-increment processes

In analogy with the Wiener-Itô theory of multiple integrals and orthogonal polynominals, a set of functionals of general square-integrable martingales is presented which, in the case of independent-increments processes, is orthogonal and complete in the sense that everyL^{2}-functional of the independent-increment process can be represented as an infinite sum of these elementary functionals. The functionals are iterated integrals of the basic martingales, similar to the multiple iterated integrals of Itô and can be also thought of as being the analogs of the powers1,x,x^{2}, cdotsof the usual calculus. The analogy is made even clearer by observing that expanding the Doleans-Dade formula for the exponential of the process in a Taylor-like series leads again to the above elementary functionals. A recursive formula for these functionals in terms of the basic martingale and of lower order functionals is given, and several connections with the theory of reproducing kernel Hilbert spaces associated with independent-increment processes are obtained.     

Lower bounds on estimator performance for energy-invariantparameters of multidimensional Poisson processes

Using rate distortion theory, lower bounds are developed for the mean-square error of estimates of a random parameter of an M-dimensional inhomogeneous Poisson process with respect to which the energy, i.e. the average number of points, is invariant. The bounds are derived without stringent assumptions on either the form of the intensity or the prior distribution of the parameter, and they can handle random nuisance parameters. The derivation makes use of a side-information averaging principle applied to the distortion-rate function and a maximum-entropy property of energy-constrained Poisson processes. Under the additional assumption of conditional entropy invariance of the point process with respect to the parameter of interest, an explicit bound is given which depends on the information discrimination between the inhomogeneous conditionally Poisson process and a nearly homogeneous Poisson process. The application of the explicit bound is illustrated through a treatment of the problems of time-shift estimation and relative time-shift estimation for Poisson streams     

Optimal functional expansion for estimation from counting observations

A systematic method to develop approximate nonlinear estimators is presented, in the form of a functional series, for the signal that modulates the rate of a counting process. The estimators are optimal for the given structure and approach the minimum variance (MV) estimator when the approximation order increases. Two kinds of functional series, the iterated integral (II) series and the Fourier-Charlier (FC) series, are used. Product-to-sum formulas for the II and FC functionals are derived. By using the formulas, the MV estimate is projected onto the Hilbert subspaces of the II and the FC series driven by the counting observations with the given index set. The projection results in a Wiener-Hopf type equation for the II kernels and a system of linear algebraic equations for the FC coefficients. The FC series estimator consists of finitely many single Wiener integrals of the counting observations and a nonlinear postprocessor. The nonlinear postprocessor, however, is not memoryless.     

Multiparameter discrete transforms based on discrete orthogonal polynomials and their application to image watermarking

Applications of discrete orthogonal polynomials (DOPs) in image processing have been recently emerging. In particular, Krawtchouk, Chebyshev, and Charlier DOPs have been applied as bases for image analysis in the frequency domain. However, fast realizations and fractional-type generalizations of DOP-based discrete transforms have been rarely addressed. In this paper, we introduce families of multiparameter discrete fractional transforms via orthogonal spectral decomposition based on Krawtchouk, Chebyshev, and Charlier DOPs. The eigenvalues are chosen arbitrarily in both unitary and non-unitary settings. All families of transforms, for varieties of eigenvalues, are applied in image watermarking. We also exploit recently introduced fast techniques to reduce complexity for the Krawtchouk case. Experimental results show the robustness of the proposed transforms against watermarking attacks.     

A computationally efficient multivariate maximum-entropy density estimation (MEDE) technique

Density estimation is the process of taking a set of multivariate data and finding an estimate for the probability density function (pdf) that produced it. One approach for obtaining an accurate estimate of the true density f(x) is to use the polynomial-moment method with Boltzmann-Shannon entropy. Although rigorous mathematically, the method is difficult to implement in practice because the solution involves a large set of simultaneous nonlinear integral equations, one for each moment or joint moment constraint. Solutions available in the literature are generally not easily applicable to multivariate data, nor computationally efficient. In this paper, we take the functional form that was developed in this problem and apply pointwise estimates of the pdf as constraints. These pointwise estimates are transformed into basis coefficients for a set of Legendre polynomials. The procedure is mathematically similar to the multidimensional Fourier transform, although with different basis functions. We apply this technique, called the maximum-entropy density estimation (MEDE) technique, to a series of multivariate datasets.     

I.I.R. Volterra Filtering with Application to Bilinear Systems

A new nonlinear filtering technique by means of infinite impulse response (IIR) Volterra functionals is developed. It yields the projection onto the closed class of finite Volterra series with separable kernels of arbitrary degree k. Such an optimal estimator is finitely realizable as a bilinear system with parameters that are computable off line. Moreover, if the original system model is itself bilinear, this computation is finitely recursive through higher moments of degree 2 k. Two simple illustrating examples are developed: (i) estimation of the covariance of the internal white noise driving a linear system and (ii) filtering of a non-Gaussian linear system (driven by a Poisson process). The robustness with respect to the observation noise distribution is finally examined     

A New Stability Criterion for a Partial Element Equivalent Circuit Model of Neutral Type

This brief is concerned with stability for a partial element equivalent circuit model of neutral type. First, the relationship between two recently established integral inequalities is presented. Second, a new Lyapunov-Krasovskii functional is introduced based on the fact that the delay interval is nonuniformly divided into multiple subintervals, and different functionals are chosen on different subintervals. Then, some new delay-dependent criteria are derived. Finally, a numerical example is given to show that the results obtained by the new stability criteria can significantly improve some existing results.     

Series expansion of wide-sense stationary random processes

This paper presents a general approach to the derivation of series expansions of second-order wide-sense stationary mean-square continuous random process valid over an infinite-time interval. The coefficients of the expansion are orthogonal and convergence is in the mean-square sense. The method of derivation is based on the integral representation of such processes. It covers both the periodic and the aperiodic cases. A constructive procedure is presented to obtain an explicit expansion for a given spectral distribution.     

On digital smoothing filters: A brief review of closed form solutions and two new filter approaches

The concept of smoothing noisy data using appropriate polynomials turns out to be equivalent to the application of suitable nonrecursive digital filters having the following properties: They process the data in such a way that the moments are conserved up to a desired order while the energy of their impulse response is minimum. Flatness constraints of their frequency response at =0 are equivalent to the moment condition. By using orthogonal polynomials, an explicit solution is known from the literature. A second approach which uses a special decomposition also yields closed form solutions. The realization is simplified, especially in the case where a large number of moments is supposed to be conserved.     

Multiple availability on stochastic demand

Stochastic models for multiple availability are analyzed for a system with periods of operation and repair that form an alternating process. The system is defined as available in time interval (0, T] if it is available at each moment of demand. System unavailability at the moment of demand is called a breakdown. The approximate probability of functioning without breakdowns is derived and analyzed for the nonhomogeneous Poisson point process of demand. Specific cases, which can be of interest in practical applications, are investigated. The integral equation for the multiple availability for arbitrary Cdf's of periods of operation and repair is developed     

A solution of nonuniformly linear antenna array with the help of Chebyshev polynomials

An analysis of the problem of synthesizing nonuniform linear arrays is carried out; in this, the orthogonal method is combined with the analysis of the functions by Chebyshev's polynomials. The method development can be applied to any array and for a variety of patterns, including those given by Dolph's method. Finally, some applications are presented.     

The robustness properties of univariate and multivariate reciprocalpolynomials

The author investigates the robustness properties of univariate and multivariate reciprocal polynomials that are nonzero on the unit-circle and the unit-polycircle, respectively. He shows that any polytope of univariate reciprocal polynomials are nonzero on the unit-circle, if and only if a set of real-valued rationals corresponding to its vertices are entirely either positive or negative on the unit-circle. Ensuring that these vertex rationals are entirely either positive or negative on the unit-circle can be carried out by the tests described by Lakshmanan (1992). When these existing tests are combined with the results contained in this paper, it provides a complete procedure for testing the nonzeroness of polytopes of univariate reciprocal polynomials over the unit-circle. He shows that this result generalizes to the case of multivariate polynomials. For any polytope of multivariate polynomials to be nonzero on the unit-polycircle, it is necessary and-sufficient that a set of real-valued multivariate rationals corresponding to its vertices are entirely either positive or negative on the unit-polycircle. Again, by using the test, the positivity or the negativity of the vertex rationals can be ensured as well, thereby resulting in a complete procedure for testing the nonzeroness of an entire polytope of multivariate reciprocal polynomials over the unit-polycircle. Although he develops the results for polytopic families, he then extends those results to the case of non-polytopic reciprocal polynomial families     

基于近区后向散射场幅度的二维导体目标的逆散射

本文给出了一种改进的基于多方向多入射频率的平均波照射下的近区散射场度测量值反演导体目标轮廓的逆散射场的幅度值反演导体目标轮廓的逆散射方法。     

Modeling and analysis of stochastic differential equations driven by point processes

The modeling and analysis of nonlinear systems described by differential equations driven by point process noise are considered. The stochastic calculus of McShane is generalized to include such differential equations, and a more general canonical extension is defined. It is proved that this canonical extension possesses the same desirable properties for point process noise that it does for the noise processes, such as Brownian motion, considered by McShane. In addition, a new stochastic integral with respect to a point process is defined; this alternative integral obeys the rules of ordinary calculus. As a special case of the analysis of such systems, linear systems with multiplicative point process noise are investigated. The consistency of the canonical extension is studied by means of the product integral. Finally, moment equations and criteria for the stochastic stability of linear systems with multiplicative Poisson noise are derived.     

1D and 2D economical FIR filters generated by Chebyshev polynomials of the first kind

Christoffel–Darboux formula for Chebyshev continual orthogonal polynomials of the first kind is proposed to find a mathematical solution of approximation problem of a one-dimensional (1D) filter function in the z domain. Such an approach allows for the generation of a linear phase selective 1D low-pass digital finite impulse response (FIR) filter function in compact explicit form by using an analytical method. A new difference equation and structure of corresponding linear phase 1D low-pass digital FIR filter are given here. As an example, one extremely economic 1D FIR filter (with four adders and without multipliers) is designed by the proposed technique and its characteristics are presented. Global Christoffel–Darboux formula for orthonormal Chebyshev polynomials of the first kind and for two independent variables for generating linear phase symmetric two-dimensional (2D) FIR digital filter functions in a compact explicit representative form, by using an analytical method, is proposed in this paper. The formula can be most directly applied for mathematically solving the approximation problem of a filter function of even and odd order. Examples of a new class of extremely economic linear phase symmetric selective 2D FIR digital filters obtained by the proposed approximation technique are presented.     

A Hybrid of Cooperative Particle Swarm Optimization and Cultural Algorithm for Neural Fuzzy Networks and Its Prediction Applications

This study presents an evolutionary neural fuzzy network, designed using the functional-link-based neural fuzzy network (FLNFN) and a new evolutionary learning algorithm. This new evolutionary learning algorithm is based on a hybrid of cooperative particle swarm optimization and cultural algorithm. It is thus called cultural cooperative particle swarm optimization (CCPSO). The proposed CCPSO method, which uses cooperative behavior among multiple swarms, can increase the global search capacity using the belief space. Cooperative behavior involves a collection of multiple swarms that interact by exchanging information to solve a problem. The belief space is the information repository in which the individuals can store their experiences such that other individuals can learn from them indirectly. The proposed FLNFN model uses functional link neural networks as the consequent part of the fuzzy rules. This study uses orthogonal polynomials and linearly independent functions in a functional expansion of the functional link neural networks. The FLNFN model can generate the consequent part of a nonlinear combination of input variables. Finally, the proposed FLNFN with CCPSO (FLNFN-CCPSO) is adopted in several predictive applications. Experimental results have demonstrated that the proposed CCPSO method performs well in predicting the time series problems.     

Scattering by non-Gaussian surfaces

A two-dimensional density is constructed from given marginals and a given correlation coefficient by using an expansion in orthogonal polynomials. The results are used to derive a general method of finding the field scattered by a rough surface by both physical and geometrical optics when the surface is generated by a stationary random process which is not necessarily normal. The example of an exponential surface is calculated in more detail, particularly for backscatter.     

Diffusion approximations to output processes of nonlinear systems with wide-band inputs and applications

Many problems in communication theory involve approximations of a Markov type to outputs of nonlinear systems (with or without feedback) often so that Fokker-Planck techniques can be used. A general and powerful method is presented for getting diffusion approximations to outputs of systems with wide-band inputs. The input is parametrized byvarepsilonand asvarepsilon rightarrow 0the bandwidth goes to infinity. It is proved that the sequence of system output processes converges weakly to a Markov diffusion process which is completely characterized. Many communication systems fit this model. The assumptions are of a type commonly used either explicitly or implicitly in either current methods of analysis of similar systems. The usefulness and relative simplicity of the method is illustrated by its application to three examples: a) a phase-locked loop (PPL), where a Markov-dfffusion approximation of the error process is developed, b) an adaptive antenna system, where an asymptotic analysis of the equations for the system is given, and c) a diffusion approximation to the output of a hard limiter followed by a bandpass filter; input-output S/N ratios are developed. Since weak convergence methods are used, the approximate "limits" yield approximations to many types of functionals of the actual systems.