a fast refinement for adaptive gaussian chirplet decomposition the chirp function is one of the most fundamental functions in nature many natural events for example most signals encountered in seismology and the signals in radar systems can be modeled as the superposition of short lived chirp functions hence the chirp based signal representation such as the gaussian chirplet decomposition has been an active research area in the field of signal processing a main challenge of the gaussian chirplet decomposition is that gaussian chirplets do not form an orthogonal basis a promising solution is to employ adaptive type signal decomposition schemes such as the matching pursuit the general underlying theory of the matching pursuit method has been well accepted but the numerical implementation in terms of computational speed and accuracy of the adaptive gaussian chirplet decomposition remains an open research topic we present a fast refinement algorithm to search for optimal gaussian chirplets with a coarse dictionary the resulting adaptive gaussian chirplet decomposition is not only fast but is also more accurate than other known adaptive schemes the effectiveness of the algorithm introduced is demonstrated by numerical simulations

some comments on fourier analysis uncertainty and modeling investigation of the problem of simultaneously concentrating a function and its fourier transform has led to some interesting special functions that have widespread applications in engineering they provide a means of proving a rigorous version of an engineering folk theorem called the 2wt theorem many generalizations of these ideas seem to possess a similar elegant mathematical structure a brief descriptive review is given of these developments

nonsmooth analysis and hamilton jacobi equations on riemannian manifolds we establish some perturbed minimization principles and we develop a theory of subdifferential calculus for functions defined on riemannian manifolds then we apply these results to show existence and uniqueness of viscosity solutions to hamilton jacobi equations defined on riemannian manifolds

efficient mixed spectrum estimation with applications to target feature extraction we present a decoupled parameter estimation dpe algorithm for estimating sinusoidal parameters from both 1 d and 2 d data sequences corrupted by autoregressive ar noise in the first step of the dpe algorithm we use a relaxation relax algorithm that requires simple fast fourier transforms ffts to obtain the estimates of the sinusoidal parameters we describe how the relax algorithm may be used to extract radar target features from both 1 d and 2 d data sequences in the second step of the dpe algorithm a linear least squares approach is used to estimate the ar noise parameters the dpe algorithm is both conceptually and computationally simple the algorithm not only provides excellent estimation performance under the model assumptions in which case the estimates obtained with the dpe algorithm are asymptotically statistically efficient but is also robust to mismodeling errors

the weyl correspondence and time frequency analysis describes the weyl correspondence and its properties showing how it gives a window independent definition of time frequency concentration for use in models in signal detection the definition of concentration is justified by showing that it gives reasonable answers in certain intuitive cases the weyl correspondence expresses a linear transformation as a weighted superposition of time frequency shifts of the signal and then authors explain why this is not the same as transforming a signal into the time frequency domain multiplying by a weight in the transform domain and taking the inverse the investigation into time frequency concentration and the weyl correspondence is justified by a new result the authors show that convolving the wigner distribution with a general smoothing function is equivalent to evaluating a weighted sum of spectrograms this is a new interpretation of the process of smoothing the wigner distribution to reduce cross terms it relates smoothing of the wigner distribution to the multiple window technique pioneered by thomson 1982

the linear dynamic complementarity problem is a special case of the extended linear complementarity problem in this paper we consider some extensions of the linear complementarity problem which is one of the fundamental problems in mathematical programming more specifically we consider the linear dynamic complementarity problem ldcp the generalized linear complementarity problem glcp and the extended linear complementarity problem elcp in this note we show that the ldcp is a special case of the elcp and the glcp

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

a four parameter atomic decomposition of chirplets a new four parameter atomic decomposition of chirplets is developed for compact and precise representation of signals with chirp components the four parameter chirplet atom is obtained from the unit gaussian function by successive applications of scaling fractional fourier transform frft and time shift and frequency shift operators the application of the frft operator results in a rotation of the wigner distribution of the gaussian in the time frequency plane by a specified angle the decomposition is realized by using the matching pursuit algorithm for this purpose the four parameter space is discretized to obtain a small but complete subset in the hilbert space a time frequency distribution tfd is developed for clear and readable visualization of the signal components it is observed that the chirplet decomposition and the related tfd provide more compact and precise representation of signal inner structures compared with the commonly used time frequency representations

viscosity solutions to second order partial differential equations on riemannian manifolds we prove comparison uniqueness and existence results for viscosity solutions to a wide class of fully nonlinear second order partial differential equations f x u du d 2 u 0d ef ined on afi nite dimensional riemannian manifold m finest results with hypothesis that require the function f to be degenerate elliptic that is nonincreasing in the second order derivative variable and uniformly continuous with respect to the variable x are obtained under the assumption that m has nonnegative sectional curvature while if one additionally requires f to depend on d 2 u in a uniformly continuous manner then comparison results are established with no restrictive assumptions on curvature

chirplet approximation of band limited real signals made easy in this paper we present algorithms for approximating real band limited signals by multiple gaussian chirps these algorithms do not rely on matching pursuit ideas they are hierarchical and at each stage the number of terms in a given approximation depends only on the number of positive valued maxima and negative valued minima of a signed amplitude function characterizing part of the signal like the algorithms used in j m greenberg z wang and j li ieee trans signal process 55 2007 pp 734 741 and unlike previous methods our chirplet approximations require neither a complete dictionary of chirps nor complicated multidimensional searches to obtain suitable choices of chirp parameters

the chirplet transform physical considerations we consider a multidimensional parameter space formed by inner products of a parameterizable family of chirp functions with a signal under analysis we propose the use of quadratic chirp functions which we will call q chirps for short giving rise to a parameter space that includes both the time frequency plane and the time scale plane as 2 d subspaces the parameter space contains a time frequency scale volume and thus encompasses both the short time fourier transform as a slice along the time and frequency axes and the wavelet transform as a slice along the time and scale axes in addition to time frequency and scale there are two other coordinate axes within this transform space shear in time obtained through convolution with a q chirp and shear in frequency obtained through multiplication by a q chirp signals in this multidimensional space can be obtained by a new transform which we call the q chirplet transform or simply the chirplet transform the proposed chirplets are generalizations of wavelets related to each other by 2 d affine coordinate transformations translations dilations rotations and shears in the time frequency plane as opposed to wavelets which are related to each other by 1 d affine coordinate transformations translations and dilations in the time domain only

viscosity solutions of second order partial differential equations with boundary conditions on riemannian manifolds abstract the comparison uniqueness and existence for viscosity solution to a class of fully nonlinear second order partial differential equations with boundary condition on a finite dimensional riemannian manifold are studied some results regarding the regularity of the boundary are obtained these results show that every compact manifold with c 1 1 boundary satisfies the regularity condition

wavelets and dilation equations a brief introduction wavelets are new families of basis functions that yield the representation f x sum b jk w 2 j x k their construction begins with the solution phi x to a dilation equation with coefficients c k then w comes from phi and the basis comes by translation and dilation of w it is shown in part 1 how conditions on the c k lead to approximation properties and orthogonality properties of the wavelets part 2 describes the recursive algorithms also based on the c k that decompose and reconstruct f the object of wavelets is to localize as far as possible in both time and frequency with efficient algorithms

the generalized linear complementarity problem and an algorithm to find all its solutions motivated by a number of typical applications a generalization of the classicallinear complementarity problem is presented together with an algorithm to determine the complete solution set the algorithm is based on the double description method for solving linear inequalities and succeeds in describing continuous as well as unbounded solution sets

new approaches for chirplet approximation this correspondence proposes efficient algorithms for approximating complex valued and real valued signals as a weighted sum of multiple chirplet atoms which are characterized by four parameters namely the scale time frequency and chirp rate direct sequential estimation of the parameters of multiple chirplets causes error propagations i e the estimated parameters of the initial chirplets significantly affect the parameter estimation of the subsequent chirplets and may result in large chirplet approximation errors to deal with this problem we further exploit a relaxation relax method for recursive chirplet parameter estimation relax can be used in conjunction with time domain or frequency domain algorithms to improve the parameter estimation accuracy for multiple chirplets unlike previous methods our chirplet approximations require neither any a priori complete dictionary of chirplets nor complicated multidimensional searches to obtain suitable choices of chirplet parameters the effectiveness of the proposed approaches is demonstrated via a number of simulated examples

linear complementarity systems we introduce a new class of dynamical systems called linear complementarity systems the time evolution of these systems consists of a series of continuous phases separated by events which cause a change in dynamics and possibly a jump in the state vector the occurrence of events is governed by certain inequalities similar to those appearing in the linear complementarity problem of mathematical programming the framework we describe is suitable for certain situations in which both differential equations and inequalities play a role for instance in mechanics electrical networks piecewise linear systems and dynamic optimization we present a precise definition of the solution concept of linear complementarity systems and give sufficient conditions for existence and uniqueness of solutions

the wavelet transform time frequency localization and signal analysis two different procedures for effecting a frequency analysis of a time dependent signal locally in time are studied the first procedure is the short time or windowed fourier transform the second is the wavelet transform in which high frequency components are studied with sharper time resolution than low frequency components the similarities and the differences between these two methods are discussed for both schemes a detailed study is made of the reconstruction method and its stability as a function of the chosen time frequency density finally the notion of time frequency localization is made precise within this framework by two localization theorems

time scale energy distributions a general class extending wavelet transforms the theory of a new general class of signal energy representations depending on time and scale is developed time scale analysis has been introduced recently as a powerful tool through linear representations called continuous wavelet transforms wts a concept for which an exhaustive bilinear generalization is given although time scale is presented as an alternative method to time frequency strong links relating the two are emphasized thus combining both descriptions into a unified perspective the authors provide a full characterization of the new class the result is expressed as an affine smoothing of the wigner ville distribution on which interesting properties may be further imposed through proper choices of the smoothing function parameters not only do specific choices allow recovery of known definitions but they also provide via separable smoothing a continuous transition from wigner ville to either spectrograms or scalograms squared modulus of the wt this property makes time scale representations a very flexible tool for nonstationary signal analysis

minimal realization in the max algebra is an extended linear complementarity problem we demonstrate that the minimal state space realization problem in the max algebra can be transformed into an extended linear complementarity problem elcp we use an algorithm that finds all solutions of an elcp to find all equivalent minimal state space realizations of a single input single output siso discrete event system we also give a geometrical description of the set of all minimal realizations of a siso max linear discrete event system

fast matching pursuit with a multiscale dictionary of gaussian chirps we introduce a modified matching pursuit algorithm called fast ridge pursuit to approximate n dimensional signals with m gaussian chirps at a computational cost o mn instead of the expected o mn sup 2 logn at each iteration of the pursuit the best gabor atom is first selected and then its scale and chirp rate are locally optimized so as to get a good chirp atom i e one for which the correlation with the residual is locally maximized a ridge theorem of the gaussian chirp dictionary is proved from which an estimate of the locally optimal scale and chirp is built the procedure is restricted to a sub dictionary of local maxima of the gaussian gabor dictionary to accelerate the pursuit further the efficiency and speed of the method is demonstrated on a sound signal