beyond time frequency analysis energy densities in one and many dimensions given a unitary operator a representing a physical quantity of interest we employ concepts from group representation theory to define two natural signal energy densities for a the first is invariant to a and proves useful when the effect of a is to be ignored the second is covariant to a and measures the a content of signals we also consider joint densities for multiple operators and in the process provide an alternative interpretation of cohen s see englewood cliffs nj prentice hall 1995 general construction for joint distributions of arbitrary variables

the scale representation the authors considers scale a physical attribute of a signal and develop its properties he presents an operator which represents scale and study its characteristics and representation this allows one to define the scale transform and the energy scale density spectrum which is an indication of the intensity of scale values in a signal he obtains explicit expressions for the mean scale scale bandwidth instantaneous scale and scale group delay furthermore he derives expressions for mean time mean frequency duration frequency bandwidth in terms of the scale variable the short time transform is defined and used to obtain the conditional value of scale for a given time he shows that as the windows narrows one obtains instantaneous scale convolution and correlation theorems for scale are derived a formulation is devised for studying linear scale invariant systems he derives joint representations of time scale and frequency scale general classes for each are presented using the same methodology as for the time frequency case as special cases the joint distributions of marinovich altes 1978 1986 and bertrand bertrand 1984 are recovered also joint representations of the three quantities time frequency scale are devised a general expression for the local scale autocorrelation function is given uncertainty principles for scale and time and scale and frequency are derived

kernel synthesis for generalized time frequency distributions using the method of alternating projections onto convex sets cohen s generalized time frequency distribution gtfr requires the choice of a two dimensional kernel the kernel directly affects many performance attributes of the gtfr such as time resolution frequency resolution realness and conformity to time and frequency marginals a number of different kernels may suffice for a given performance constraint high frequency resolution for example interestingly most sets of kernels satisfying commonly used performance constraints are convex we describe a method whereby kernels can be designed that satisfy two or more of these constraints if there exists a nonempty intersection among the constraint sets then the theory of alternating projection onto convex sets pocs guarantees convergence to a kernel that satisfies all of the constraints if the constraints can be partitioned into two sets each with a nonempty intersection then pocs guarantees convergence to a kernel that satisfies the inconsistent constraints with minimum mean square error we apply kernels synthesized using pocs to the generation of some example gtfrs and compare their performance to the spectrogram wigner distribution and cone kernel gtfr

a general approach for obtaining joint representations in signal analysis i characteristic function operator method a general approach for obtaining joint representations for arbitrary physical quantities is presented the characteristic function operator method of moyal 1949 and ville 1960 and generalized by cohen 1966 1976 and scully and cohen 1987 for the time frequency case is developed for arbitrary variables

equivalence of generalized joint signal representations of arbitrary variables joint signal representations jsrs of arbitrary variables generalize time frequency representations tfrs to a much broader class of nonstationary signal characteristics two main distributional approaches to jsrs of arbitrary variables have been proposed by cohen see time frequency analysis englewood cliffs nj prentice hall 1995 and proc spie 1566 san diego 1991 and baraniuk see proc ieee int conf acoust speech signal processing icassp 94 vol 3 p 357 60 1994 cohen s method is a direct extension of his original formulation of tfrs and baraniuk s approach is based on a group theoretic formulation both use the powerful concept of associating variables with operators one of the main results of the paper is that despite their apparent differences the two approaches to generalized jsrs are completely equivalent remarkably the jsrs of the two methods are simply related via axis warping transformations with the broad implication that jsrs with radically different covariance properties can be generated efficiently from jsrs of cohen s method via simple pre and post processing the development in this paper which is illustrated with examples also illuminates other related issues in the theory of generalized jsrs in particular we derive an explicit relationship between the hermitian operators in cohen s method and the unitary operators in baraniuk s approach thereby establishing the relationship between the two types of operator correspondences

optimal quadratic detection and estimation using generalized joint signal representations time frequency analysis has significant advances in two main directions statistically optimized methods that extend the scope of time frequency based techniques from merely exploratory data analysis to more quantitative application and generalized joint signal representations that extend time frequency based methods to a richer class of nonstationary signals this paper fuses the two advances by developing optimal detection and estimation techniques based on generalized joint signal representations by generalizing the statistical methods developed for time frequency representations to arbitrary joint signal representations this paper develops a unified theory applicable to a wide variety of problems in nonstationary statistical signal processing

a statistical formant pattern model for segregating vowel type and vocal tract length in developmental formant data this paper investigates the theoretical basis for estimating vocal tract length vtl from the formant frequencies of vowel sounds a statistical inference model was developed to characterize the relationship between vowel type and vtl on the one hand and formant frequency and vocal cavity size on the other the model was applied to two well known developmental studies of formant frequency the results show that vtl is the major source of variability after vowel type and that the contribution due to other factors like developmental changes in oral pharyngeal ratio is small relative to the residual measurement noise the results suggest that speakers adjust the shape of the vocal tract as they grow to maintain a specific pattern of formant frequencies for individual vowels this formant pattern hypothesis motivates development of a statistical inference model for estimating vtl from formant frequency data the technique is illustrated using a third developmental study of formant frequencies the vtls of the speakers are estimated and used to provide a more accurate description of the complicated relationship between vtl and glottal pulse rate as children mature into adults

preservation of prox regularity of sets with applications to constrained optimization in this paper we first provide counterexamples showing that sublevels of prox regular functions and levels of differentiable mappings with lipschitz derivatives may fail to be prox regular then we prove the uniform prox regularity of such sets under usual verifiable qualification conditions the preservation of uniform prox regularity of intersection and inverse image under usual qualification conditions is also established applications to constrained optimization problems are given

a numerical scheme for impact problems i the one dimensional case we consider a mechanical system with impact and one degree of freedom the system is not necessarily lagrangian the representative point is subject to the constraint u t in er for all t we assume that at impact the velocity is reversed and multiplied by a given coefficient of restitution e in 0 1 we define a numerical scheme which enables us to approximate the solutions of the cauchy problem this is an ad hoc scheme which does not require a systematic search for the times of impact we prove the convergence of this numerical scheme to a solution many of the features of this proof will be reused in the nonconvex multidimensional case written in generalized coordinates given in the companion paper l paoli and m schatzman siam j numer anal 40 2002 pp 734 768 we present some numerical results obtained with the scheme for a spring dashpot system and we compare them to the results obtained by impact detection and penalization

closed geodesics with lipschitz obstacle abstract geodesics on a general subsetmof r nare considered whenmis the closure of an open subset with lipschitz boundary and suitable topological conditions are fulfilled the existence of a nonconstant closed geodesic is proved

on generalized marginal time frequency distributions we introduce a family of time frequency tf distributions with generalized marginals i e beyond the time domain and the frequency domain marginals in the sense that the projections of a tf distribution along one or more angles are equal to the magnitude squared of the fractional fourier transforms of the signal we present a necessary and sufficient condition for a tf distribution in cohen s class to satisfy generalized marginals we then modify the existing well known tf distributions in cohen s class such as choi williams 1989 and page distributions so that the modified ones have generalized marginals numerical examples are presented to show that the proposed tf distributions have the advantages of both wigner ville and other quadratic tf distributions which only have the conventional marginals moreover they also indicate that the generalized marginal tf distributions with proper marginals are more robust than the wigner ville and the choi williams distributions when signals contain additive noise

time scale energy density functions scale like frequency is a physical characteristic of a signal to measure the scale content of a signal the signal must be appropriately transformed a theory for joint time scale energy density functions is presented and a method for generating such functions for any signal is given examples for synthetic signals and real data are presented the theory and method can be extended to arbitrary joint densities of any variables for example frequency and scale

duality of log polar image representations in the space and spatial frequency domains in this paper we study the result of applying a lowpass variant filtering using scaling rotating kernels to both the spatial and spatial frequency representations of a two dimensional 2 d signal image it is shown that if we apply this transformation to a fourier pair the two resulting signals can also form a fourier pair when the filters used in each domain maintain a dual relationship for a large class of self dual filters a perfect symmetry exists so that the lowpass scaling rotating variant filtering srvf is the same in both domains thus commuting with the fourier transform operator the lowpass srvf of an image is often referred to as a foveated image whereas its fourier pair the lowpass srvf of its spectrum can be realized as a local spectrum estimation around the point of attention this lowpass srvf is equivalent to a log polar warping of the image representation followed by a lowpass invariant filtering and the corresponding inverse warping the use of the log polar warped representation allows us to extend the one dimensional 1 d scale transform to higher dimensions in particular to images for which we have defined a scale rotation invariant representation we also present an efficient implementation using steerable filters to compute both the foveated image and the local spectrum

wide band proportional bandwidth wigner ville analysis the wigner ville w v distribution is a time frequency representation that yields a highly accurate estimate of instantaneous frequency it is related to the narrow band ambiguity function by an internal transform and it can be used in a variety of detection and estimation problems a spectrogram constructed with constant bandwidth filters can be obtained by convolving two w v distributions or by forming a magnitude squared narrow band cross ambiguity function the wide band ambiguity function represents the doppler effect with dilation or compression rather than with frequency shift as in the narrow band approximation the q distributions a modified w v representation that is related to the wide band ambiguity function by an integral transform is defined a spectrogram constructed with proportional bandwidth or constant q filters can be obtained by a convolution like operation involving two q distributions or by forming a magnitude squared wide band cross ambiguity function the q distribution is thus a wide band version of the w v distribution properties of the q distribution indicate that it may prove useful for detection and parameter estimation as well as for tomographic measurement of wideband scattering functions with relatively few transmitted waveforms

kernel design for reduced interference distributions the authors present a class of time frequency signal representations tfrs called the reduced interference distribution rid an overview of commonly used tfrs is given and desirable distribution properties are introduced particular attention is paid to the interpretation of cohen s class of time frequency distributions of tfrs in the ambiguity temporal correlation spectral correlation and time frequency domains based on the desirable kernel requirements the rid is discussed and further defined a systematic procedure to create rid kernels or equivalently compute rids is proposed some aspects and properties of the rid are discussed the authors estimate design considerations for rids and compare various selections of the primitive window some experimental results demonstrating the performance of the rid are presented

integral transforms covariant to unitary operators and their implications for joint signal representations fundamental to the theory of joint signal representations is the idea of associating a variable such as time or frequency with an operator a concept borrowed from quantum mechanics each variable can be associated with a hermitian operator or equivalently and consistently as we show with a parameterized unitary operator it is well known that the eigenfunctions of the unitary operator define a signal representation which is invariant to the effect of the unitary operator on the signal and is hence useful when such changes in the signal are to be ignored however for detection or estimation of such changes a signal representation covariant to them is needed using well known results in functional analysis we show that there always exists a translationally covariant representation that is an application of the operator produces a corresponding translation in the representation this is a generalization of a recent result in which a transform covariant to dilations is presented using stone s theorem the covariant transform naturally leads to the definition of another unique dual parameterized unitary operator this notion of duality which we make precise has important implications for joint distributions of arbitrary variables and their interpretation in particular joint distributions of dual variables are structurally equivalent to cohen s class of time frequency representations and our development shows that for two variables the hermitian and unitary operator correspondences can be used consistently and interchangeably if and only if the variables are dual

on p convex sets and geodesics on etudie les geodesiques sur une variete contrainte m a bord irregulier on considere une classe d ensembles irreguliers m qui permet d obtenir des resultats d existence et de multiplicite pour les geodesiques

sweeping by a continuous prox regular set abstract the evolution problem known as sweeping process is considered for a class of nonconvex sets called prox regular or convex assuming essentially that such sets contain in the interior a suitable subset and move continuously w r t the hausdorff distance we prove local and global existence as well as uniqueness of solutions which are continuous functions with bounded variation some examples are presented

a class of second order stationary self similar processes for 1 f phenomena we propose a class of statistically self similar processes and outline an alternative mathematical framework for the modeling and analysis of 1 f phenomena the foundation of the proposed class is based on the extensions of the basic concepts of classical time series analysis in particular on the notion of stationarity we consider a class of stochastic processes whose second order structure is invariant with respect to time scales i e e x t x spl lambda t t sup 2h spl lambda sup h r spl lambda t 0 for some x h spl infin for h 0 we refer to these processes as wide sense scale stationary we show that any self similar process can be generated from scale stationary processes we establish a relationship between linear scale invariant system theory and the proposed class that leads to a concrete analysis framework we introduce new concepts such as periodicity autocorrelation and spectral density functions by which practical signal processing schemes can be developed we give several examples of scale stationary processes including gaussian non gaussian covariance and generative models as well as fractional brownian motion as a special case in particular we introduce a class of finite parameter self similar models that are similar in spirit to the ordinary arma models by which an arbitrary self similar process can be approximated results from our study suggest that the proposed self similar processes and the mathematical formulation provide an intuitive general and mathematically simple approach to 1 f signal processing

iterative algorithms for a new class of extended general nonconvex set valued variational inequalities abstract some new classes of extended general nonconvex set valued variational inequalities and the extended general wiener hopf inclusions are introduced by the projection technique equivalence between the extended general nonconvex set valued variational inequalities and the fixed point problems as well as the extended general nonconvex wiener hopf inclusions is proved then by using this equivalent formulation we discuss the existence of solutions of the extended general nonconvex set valued variational inequalities and construct some new perturbed finite step projection iterative algorithms with mixed errors for approximating the solutions of the extended general nonconvex set valued variational inequalities we also verify that the approximate solutions obtained by our algorithms converge to the solutions of the extended general nonconvex set valued variational inequalities the results presented in this paper extend and improve some known results from the literature

periodic solutions with prescribed minimal period for convex autonomous hamiltonian systems clarke has shown that the problem of findingt periodic solutions for a convex hamiltonian system is equivalent to the problem of finding critical points to a certain functional dual to the classical action functional in this paper we relate the morse index of the critical point to the minimal period of the correspondingt periodic solution in particular we show that if the critical point is obtained by the ambrosetti rabinowitz mountain pass theorem the corresponding solution has minimal periodt that is it cannot bet k periodic withk integer k 2 as a consequence we prove that if the hamiltonian is flat near an equilibrium and superquadratic near infinity then for anyt 0 the corresponding hamiltonian system has a periodic solution with minimal periodt

periodic bounce solutions of dynamical conservative systems and their minimal period several important studies have been made on the problem of finding periodic solutions of a conservative dynamical system an important part of such a study is concerned with the case in which the minimal period of the trajectory is given see e g l 3 6 9 14 in this paper we consider the problem of existence of periodic trajectories with prescribed minimal period when the material point moves in a convex billiard in i under the action of a conservative field and it bounces in a completely elastic way against the boundary hitting it orthogonally if q is a bounded open convex subset of i not necessarily regular t 0 and v is a real cz function in a neighbourhood of 0 we consider the nonconstant periodic orbits q r q which are called principal bounce trajectories b t having period 2t such that

a sampling theorem associated with quasi fourier transform we propose in this correspondence a unified generalized bandlimited sampling theorem associated with quasi fourier transform of the form f sub spl phi f spl xi spl int sub d f x exp i spl epsiv spl phi x dx where the phase function satisfies spl phi x 0 in its domain our result contains classical shannon s sampling theorem and the sundaram et al 1997 scale limited sampling theorem as a special case new results such as the chirp limited sampling theorem can be established

p np co np and weak systems of arithmetic abstract we study the statements np co np and p np co np and their possible interpretations in weak systems of arithmetic in order to relate them to nonstandard models of arithmetic

scale periodicity and its sampling theorem the scalar transform is a new representation for signals offering a perspective that is different from the fourier transform we introduce the notion of a scalar periodic function these functions are then represented through the discrete scale series we also define the notion of a strictly scale limited signal analogous to the shannon interpolation formula we show that such signals can be exactly reconstructed from exponentially spaced samples of the signal in the time domain as an interesting practical application we show how properties unique to the scale transform make it very useful in computing depth maps of a scene

multiplicity of solutions for the bounce problem abstract the bounce trajectories in a convex set which assume assigned positions in two fixed time instants are sought sufficient conditions in order to obtain the existence of infinitely many bounce trajectories are found

unitary equivalence a new twist on signal processing unitary similarity transformations furnish a powerful vehicle for generating infinite generic classes of signal analysis and processing tools based on concepts different from time frequency and scale implementation of these new tools involves simply preprocessing the signal by a unitary transformation performing standard processing on the transformed signal and then in some cases transforming the resulting output the resulting unitarily equivalent systems can focus on the critical signal characteristics in large classes of signals and hence prove useful for representing and processing signals that are not well matched by current techniques as specific examples of this procedure we generalize linear time invariant systems orthonormal basis and frame decompositions and joint time frequency and time scale distributions these applications illustrate the utility of the unitary equivalence concept for uniting seemingly disparate approaches proposed in the literature

separability positivity and minimum uncertainty in time frequency energy distributions gaussian signals play a very special role in classical time frequency analysis because they are solutions of apparently unrelated problems such as minimum uncertainty and positivity and separability of wigner ville distributions we investigate here some of the logical connections which exist between these different features and we discuss some examples and counterexamples of their extension to more general joint distributions within cohen s class and the affine class

the use of cone shaped kernels for generalized time frequency representations of nonstationary signals generalized time frequency representations gtfrs which use cone shaped kernels for nonstationary signal analysis are presented the cone shaped kernels are formulated for the gtfrs to produce good resolution simultaneously in time and frequency specifically for a gtfr with a cone shaped kernel finite time support is maintained in the time dimension along with an enhanced spectrum in the frequency dimension and the cross terms are smoothed out experimental results on simulated data and real speech show the advantages of the gtfrs with the cone shaped kernels through comparisons to the spectrogram and the pseudo wigner distribution

uncertainty inequalities for the continuous wavelet transform uncertainty inequalities of heisenberg type are established for the continuous wavelet transform resting on classical heisenberg and heisenberg von keumann principles

tomograms and other transforms a unified view a general framework is presented which unifies the treatment of wavelet like quasidistribution and tomographic transforms explicit formulae relating the three types of transforms are obtained the case of transforms associated with the symplectic and affine groups is treated in some detail special emphasis is given to the properties of the scale time and scale frequency tomograms tomograms are interpreted as a tool to sample the signal space by a family of curves or as the matrix element of a projector

improved time frequency representation of multicomponent signals using exponential kernels the authors introduce a time frequency distribution of l cohen s 1966 class and examines its properties this distribution is called exponential distribution ed after its exponential kernel function first the authors interpret the ed from the spectral density estimation point of view they then show how the exponential kernel controls the cross terms as represented in the generalized ambiguity function domain and they analyze the ed for two specific types of multicomponent signals sinusoidal signals and chirp signals next they define the ed for discrete time signals and the running windowed exponential distribution rwed which is computationally efficient finally the authors present numerical examples of the rwed using the synthetically generated signals it is found that the ed is very effective in diminishing the effects of cross terms while retaining most of the properties which are useful for a time frequency distribution

modeling and control of impact in mechanical systems theory and experimental results considers the equations of motion of mechanical systems subject to inequality constraints which can be obtained by looking for the stationary value of the action integral two different methods are used to take into account the inequality constraints in the computation of the stationary value of the action integral the valentine variables method and the penalty functions method the equations of motion resulting from the application of the valentine variables method which introduces the concept of nonsmooth impacts constitute the exact model of the constrained mechanical system such a model is suitable to be employed when the impacting parts of the actual mechanical system are very stiff the equations of motion resulting from the application of the penalty functions method which introduces the concept of smooth impacts constitute an approximate model of the constrained mechanical system such a model is suitable to be employed when the impacting parts of the actual mechanical system show some flexibility various feedback control laws from the natural outputs and from their time derivatives are studied with reference to both models of impact the closed loop systems resulting from the application of the same control law to both models show pretty much the same global asymptotic stability properties the proposed control laws are only concerned with regulation problems in the presence of possible contacts and impacts among parts of the mechanical system or with the external environment the effectiveness of the proposed control structure has been tested experimentally with reference to a single link robot arm showing a valuable behavior

np co np and models of arithmetic abstract a mate proposed a semantic approach of the complexity question np co np he showed that the existence of two models of arithmetic with a particular property of extensions implied np co np we prove here the converse of his result which yields an exact characterization of np co np in terms of models of arithmetic

uniqueness in the elastic bounce problem ii in a recent paper uniqueness for the elastic bounce problem has been studied in a very general framework the aim of this paper is to show that if u is assumed to be real analytic and u equiv 0 then these phenomena disappear and the solution to the elastic bounce problem is unique for every choice of the cauchy data

bilinear time frequency representations new insights and properties an analysis of the interference terms of cohen class bilinear time frequency representations tfr of multicomponent signals is presented constraints for achieving new interference properties are derived imposing these interference time and interference frequency concentration constraints on a tfr guarantees that the tfr will be zero everywhere the signal s t is zero and the tfr will contain only those frequencies that occur in the signal thus these new constraints guarantee strong finite support in a tfr when these interference concentration properties are combined with interference attenuation tradeoffs between finite support the marginals and the interference properties are shown to be unavoidable the useful class of product kernels are considered and generalized further to allow tfr with potentially superior interference properties the interference frequency concentration and attenuation properties allow tfr with spectrogram like interference suppression but without the spectrogram s inherent time frequency resolution tradeoff other useful combinations of properties are discussed and analyzed and properties and tradeoffs are illustrated by examples

on the metamathematics of the p vs np question abstract we review the work from 1976 to 1996 by several researchers on the metamathematics of p vs np that work points towards the possibility that given some strong consistent axiomatic system s with a recursively enumerable set of theorems which includes arithmetic for the 2 sentence p np that formalizes the p np hypothesis s p np is consistent we consider the work of several authors like for instance j hartmanis j hopcroft independence results in computer science sigact news 13 1976 m o donnell a programming language theorem which is independent of peano arithmetic in proceedings of the 11th annual acm symposium on the theory of computation 1979 pp 176 188 r a demillo r j lipton the consistency of p np and related problems with fragments of number theory in proceedings of the 12th annual acm symposium on the theory of computing 1980 pp 45 57 d joseph p young fast programs for initial segments and polynomial time computation in weak models of arithmetic stoc milwaukee 1981 1981 pp 55 61 w kowalczyk a sufficient condition for the consistency of p np with peano arithmetic fund inform 5 1982 233 245 we then relate all those results to the n c a da costa f a doria consequences of an exotic formulation for p np appl math comput 145 2003 655 665 also addendum 172 2006 1364 1367 conditional consistency result for zfc p np and elaborate on it

some results on optimal control with unilateral state constraints in this paper we study the problem of quadratic optimal control with state variables unilateral constraints for linear time invariant systems the necessary conditions are formulated as a linear invariant system with complementary slackness conditions some structural properties of this system are examined then it is shown that the problem can benefit from the higher order moreau s sweeping process that is a specific distributional differential inclusion and from ten dam s geometric theory for the partitioning of the admissible domain boundary in fact the first step may be also seen as follows does the higher order moreau s sweeping process developed in cite va bb dg jj2003 correspond to the necessary conditions of some optimal control problem with an adapted integral action the knowledge of the qualitative behaviour of optimal trajectories is improved with the approach which also paves the way towards efficient time stepping numerical algorithms to solve the optimal control boundary value problem

penalty approximation for non smooth constraints in vibroimpact abstract we examine the penalty approximation of the free motion of a material point in an angular domain we choose an over damped penalty approximation and we prove that if the first impact point is not at the vertex then the limit of the approximation exists and is described by moreau s rule for inelastic impacts the proofs rely on validated asymptotics and use some classical tools of the theory of dynamical systems

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

nondeterministic polynomial time computations and models of arithmetic a semantic or model theoretic approach is proposed to study the problems p np and np co np this approach seems to avoid the difficulties that recursion theoretic approaches appear to face in view of the result of baker et al on relativizations of the p np question moreover semantical methods are often simpler and more powerful than syntactical ones the connection between the existence of certain partial extensions of nonstandard models of arithmetic and the question np co np is discussed several problems are stated about nonstandard models and a possible link between the davis matijasevi