the hyperbolic class of quadratic time frequency representations i constant q warping the hyperbolic paradigm properties and members the time frequency tf version of the wavelet transform and the affine quadratic bilinear tf representations can be used for a tf analysis with constant q characteristic the paper considers a new approach to constant q tf analysis a specific tf warping transform is applied to cohen s class of quadratic tf representations which results in a new class of quadratic tf representations with constant q characteristic the new class is related to a hyperbolic tf geometry and is thus called the hyperbolic class hc two prominent tf representations previously considered in the literature the bertrand p sub 0 distribution and the altes marinovic q distribution are members of the new hc the authors show that any hyperbolic tf representation is related to both the wideband ambiguity function and a hyperbolic ambiguity function it is also shown that the hc is the class of all quadratic tf representations which are invariant to hyperbolic time shifts and tf scalings operations which are important in the analysis of doppler invariant signals and self similar random processes the paper discusses the definition of the hc via constant q warping some signal theoretic fundamentals of the hyperbolic tf geometry and the description of the hc by 2d kernel functions several members of the hc are considered and a list of desirable properties of hyperbolic tf representations is given together with the associated kernel constraints

measuring time frequency information content using the renyi entropies the generalized entropies of renyi inspire new measures for estimating signal information and complexity in the time frequency plane when applied to a time frequency representation tfr from cohen s class or the affine class the renyi entropies conform closely to the notion of complexity that we use when visually inspecting time frequency images these measures possess several additional interesting and useful properties such as accounting and cross component and transformation invariances that make them natural for time frequency analysis this paper comprises a detailed study of the properties and several potential applications of the renyi entropies with emphasis on the mathematical foundations for quadratic tfrs in particular for the wigner distribution we establish that there exist signals for which the measures are not well defined

n dimensional superintegrable systems from symplectic realizations of lie coalgebras the construction of n dimensional nd integrable systems from coalgebras is reviewed in the case of poisson coalgebras a necessary condition for the integrability of the nd coalgebra hamiltonian coming from a given coalgebra is obtained in terms of the dimension of the symplectic realization and the number of nonlinear casimir functions from this viewpoint the full set of three four and five dimensional lie poisson coalgebras is analysed and many new families of multiparametric nd integrable systems coming from the cases that fulfil the integrability condition are obtained including the explicit form of the integrals of the motion the superintegrability of these hamiltonians is also emphasized and the generalization of the whole construction to the quantum mechanical case is straightforward

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

an algorithm to calculate the invariants of any lie algebra an analytical method to calculate invariants is developed based on the theory of complete systems of simultaneous linear first order partial differential equations the application of this algorithm to the general case in which the invariants are not polynomials is included for the case of lie algebras with one invariant the procedure of finding the invariant can be computerized as an example of this procedure the algorithm is applied in the calculation of the invariants of the group sa n r the explicit construction of its invariants for n 2 3 4 is given a theorem determining the order of a polynomial invariant of a lie group is proven allowing us to show that the invariant of sa n r is a homogeneous polynomial of order 1 2n n 1 in the generators from this fact an immediate consequence is that the order in the generators of translations is n the method is also applied in the calculation of invariants of the dimension six solvable lie algebras and dimension seven nilpotent

grading refinements in the contractions of lie algebras and their invariants all mutually inequivalent toroidal gradings of the lie algebra are constructed each of them provides several different graded contractions to other eight dimensional lie algebras these can be either continuous or discrete the continuous ones are related to generalized wigner inonu contractions the behaviour of casimir operators under the graded contractions is studied and the casimir and generalized casimir operators of the contracted lie algebras are presented

new design of estimators using covariance information with uncertain observations in linear discrete time systems this paper proposes recursive least squares rls filtering and fixed point smoothing algorithms with uncertain observations in linear discrete time stochastic systems the estimators require the information of the auto covariance function in the semi degenerate kernel form the variance of white observation noise the observed value and the probability that the signal exists in the observed value the auto covariance function of the signal is factorized in terms of the observation vector the system matrix and the cross variance function of the state variable that generates the signal with the signal these quantities are obtained from the auto covariance data of the signal it is shown that the semi degenerate kernel is expressed in terms of these quantities

pseudo affine wigner distributions definition and kernel formulation we introduce a new set of tools for time varying spectral analysis the pseudo affine wigner distributions based on the affine wigner distributions of j and p bertrand 1992 these new time scale distributions support efficient online operation at the same computational cost as the continuous wavelet transform moreover they take advantage of the proportional bandwidth smoothing inherent in the sliding structure of their implementation to suppress cumbersome interference components to formalize their place within the echelon of the affine class of time scale distributions tsds we introduce and study an alternative set of generators for this class

bilinear time frequency representations of signals the shift scale invariant class the authors consider the class of bilinear time frequency representations btfr s that are invariant or covariant to time shifts frequency shifts and time frequency scalings this shift scale invariant class is the intersection of the classical shift invariant cohen class and the recently defined affine class the mathematical description of shift scale invariant btfr s is in terms of a 1 d kernel and is thus particularly simple the paper concentrates on the time frequency localization properties of shift scale invariant btfr s since any shift scale invariant btfr is a superposition of generalized wigner distributions the time frequency localization of the family of generalized wigner distributions is studied first for those shift scale invariant btfr s that may be interpreted as smoothed versions of the wigner distribution e g the choi williams distribution an analysis in the fourier transform domain shows interesting peculiarities regarding time frequency concentration and interference geometry properties

recursive estimators of signals from measurements with stochastic delays using covariance information least squares linear one stage prediction filtering and fixed point smoothing algorithms for signal estimation using measurements with stochastic delays contaminated by additive white noise are derived the delay is considered to be random and modelled by a binary white noise with values zero or one these values indicate that the measurements arrive in time or they are delayed by one sampling time recursive estimation algorithms are obtained without requiring the state space model generating the signal but just using covariance information about the signal and the additive noise in the observations as well as the delay probabilities

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

h fixed lag smoothing for linear discrete time varying systems with uncertain observations this paper deals with the problem of h fixed lag smoothing for linear discrete time varying systems with uncertain observations and l 2 norm bounded noise first the design of h smoother is equivalent to the problem that a certain indefinite quadratic form with respect to a new state space constraints has a minimum and the smoother is such that the minimum is positive then by introducing a krein space stochastic system and applying re organized innovation analysis approach the minimum of indefinite quadratic form and its existence condition are derived finally through analyzing the existence condition of the minimum and guaranteeing the positivity of the minimum a sufficient and necessary condition for the existence of an h smoother is proposed and the smoother is obtained in terms of two standard riccati difference equations a numerical example is given to show the effectiveness of the proposed method

dispersion estimation from linear array data in the time frequency plane we consider the problem of estimating the dispersion of a wave field from data recorded by a linear array of geophones the fact that the data we are looking at may contain several propagating waves make this even more challenging in this paper a new algorithm is proposed to solve this issue currently there are two methods for estimating wave dispersion described in the literature the first method estimates the group delay function from the time frequency representation tfr of each sensor separately it is efficient as long as the patterns of the different waves do not overlap in the time frequency plane the second method estimates the dispersion from the two dimensional 2 d fourier transform of the profile or more generally from a velocity frequency representation this assumes that the dispersion is constant along the entire sensor array it is efficient as long as the patterns of the waves do not overlap in the frequency domain our method can be thought of as a hybrid of the above two methods as it is based on the construction of a tfr where the energy of waves that propagate at a selected velocity are amplified the primary advantage of our algorithm is the use of the velocity variable to separate the patterns of the propagating waves in the time frequency plane when applied to both synthetic and real data this new algorithm gives much improved results when compared with other standard methods

wigner analysis and casimir operators of sa 4 r in theories involving gravity including qcd generated gravitylike effects in hadrons sa 4 r plays a role its single casimir invariant and that of its sa 2 r and sa 3 r subgroups are evaluated the group orbits are studied and the unitary irreducible representations are classified

invariants of triangular lie algebras triangular lie algebras are the lie algebras which can be faithfully represented by triangular matrices of any finite size over the real complex number field in the paper invariants generalized casimir operators are found for three classes of lie algebras namely those which are either strictly or non strictly triangular and for so called special upper triangular lie algebras algebraic algorithm of boyko et al 2006 j phys a math gen 39 5749 preprint math ph 0602046 developed further in boyko et al 2007 j phys a math theor 40 113 preprint math ph 0606045 is used to determine the invariants a conjecture of tremblay and winternitz 2001 j phys a math gen 34 9085 concerning the number of independent invariants and their form is corroborated

the group theoretical treatment of aberrating systems i aligned lens systems in third aberration order the constituents of a lens system i e slabs of homogeneous transparent material and the refracting surfaces between them considered to third aberration order are associated to elements of a nine parameter aberration group three parameters correspond to gaussian systems and six to group classified aberrations the group multiplication operation through matrix cum vector algebra corresponds to their concatenation and the linear group action on an eight dimensional homogeneous space corresponds to the nonlinear action of the system on the optical phase space this leads to economical computation algorithms that may be extended to aberrating systems of higher order

coherent states and the role of the affine group in the quantum mechanics of the morse potential the coherent states of the morse potential that have been obtained earlier from supersymmetric quantum mechanics are shown to be connected with the representations of the affine group of the real line and some of its extensions this relation is similar to the one between the heisenberg weyl group and the coherent states of the harmonic oscillator the states that minimize the uncertainty product of the generators of the affine lie algebra are shown to contain all the coherent states of the morse oscillator plus the intelligent states of the morse hamiltonians with different shape parameter s the representations of the central extension of the affine group denoted by g0 and its further extension will be shown to define the phase space relevant to the problem by choosing an appropriate orbit of the coadjoint representation of this allows one to construct a generalized wigner function on this phase space which is again essentially in the same relation with the affine group as the ordinary wigner function with the heisenberg weyl group

polynomial wigner ville distributions and their relationship to time varying higher order spectra the wigner ville distribution wvd has optimal energy concentration for linear frequency modulated fm signals this paper presents a generalization of the wvd in order to effectively process nonlinear polynomial fm signals a class of polynomial wvd s pwvd s that give optimal concentration in the time frequency plane for fm signals with a modulation law of arbitrary polynomial form are defined a class of polynomial time frequency distributions ptfd s are also defined based on the class of pwvd s the optimal energy concentration of the pwvd enables it to be used for estimation of the instantaneous frequency if of polynomial fm signals finally a link between pwvd s and time varying higher order spectra tvhos is established just as the expected value of the wvd of a nonstationary random signal is the time varying power spectrum the expected values of the pwvd s have interpretations as reduced tvhos

the jordan schwinger representations of cayley klein groups ii the unitary groups the unitary cayley klein groups are defined as the groups that are obtained by the contractions and analytical continuations of the special unitary groups the jordan schwinger representations of the groups under consideration are discussed based on the mixed sets of creation and annihilation operators of boson or fermion type the matrix elements of finite group transformations are obtained in the bases of coherent states

signals on graphs transforms and tomograms abstract development of efficient tools for the representation of large datasets is a precondition for the study of dynamics on networks generalizations of the fourier transform on graphs have been constructed through projections on the eigenvectors of graph matrices by exploring mappings of the spectrum of these matrices we show how to construct more general transforms in particular wavelet like transforms on graphs for time series tomograms a generalization of the radon transforms to arbitrary pairs of non commuting operators are positive bilinear transforms with a rigorous probabilistic interpretation which provide a full characterization of the signals and are robust in the presence of noise here the notion of tomogram is also extended to signals on arbitrary graphs

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

hybrid linear bilinear time scale analysis the introduce a new method for the time scale analysis of nonstationary signals our work leverages the success of the time frequency distribution series cross term deleted representations into the time scale domain to match wideband signals that are better modeled in terms of time shifts and scale changes than in terms of time and frequency shifts using a wavelet decomposition and the bertrand time scale distribution we locally balance linearity and bilinearity in order to provide good resolution while suppressing troublesome interference components the theory of frames provides a unifying perspective for cross term deleted representations in general

recursive fixed point smoothing algorithm from covariances based on uncertain observations with correlation in the uncertainty this paper considers the least squares linear estimation problem of a discrete time signal from noisy observations in which the signal can be randomly missing the uncertainty about the signal being present or missing at the observations is characterized by a set of bernoulli variables which are correlated when the difference between times is equal to a certain value m the marginal distribution of each one of these variables specified by the probability that the signal exists at each observation as well as their correlation function are known a linear recursive filtering and fixed point smoothing algorithm is obtained using an innovation approach without requiring the state space model generating the signal but just the covariance functions of the processes involved in the observation equation

multiple multidimensional morse wavelets this paper defines a set of operators that localize a radial image in space and radial frequency simultaneously the eigenfunctions of the operator are determined and a nonseparable orthogonal set of radial wavelet functions are found the eigenfunctions are optimally concentrated over a given region of radial space and scale space defined via a triplet of parameters analytic forms for the energy concentration of the functions over the region are given the radial function localization operator can be generalised to an operator localizing any l2 ropf2 function it is demonstrated that the latter operator given an appropriate choice of localization region approximately has the same radial eigenfunctions as the radial operator based on a given radial wavelet function a quaternionic wavelet is defined that can extract the local orientation of discontinuous signals as well as amplitude orientation and phase structure of locally oscillatory signals the full set of quaternionic wavelet functions are component by component orthogonal their statistical properties are tractable and forms for the variability of the estimators of the local phase and orientation are given as well as the local energy of the image by averaging estimators across wavelets a substantial reduction in the variance is achieved

the power classes quadratic time frequency representations with scale covariance and dispersive time shift covariance we consider scale covariant quadratic time frequency representations qtfrs specifically suited for the analysis of signals passing through dispersive systems these qtfrs satisfy a scale covariance property that is equal to the scale covariance property satisfied by the continuous wavelet transform and a covariance property with respect to generalized time shifts we derive an existence representation theorem that shows the exceptional role of time shifts corresponding to group delay functions that are proportional to powers of frequency this motivates the definition of the power classes pcs of qtfr s the pcs contain the affine qtfr class as a special case and thus they extend the affine class we show that the pcs can be defined axiomatically by the two covariance properties they satisfy or they can be obtained from the affine class through a warping transformation we discuss signal transformations related to the pcs the description of the pcs by kernel functions desirable properties and kernel constraints and specific pc members furthermore we consider three important pc subclasses one of which contains the bertrand 1992 p sub k distributions finally we comment on the discrete time implementation of pc qtfrs and we present simulation results that demonstrate the potential advantage of pc qtfrs

the space of lie algebra 2 cocycles of an affine transformation group in this paper we present the intrinsic structure of the space of lie algebra 2 cocycles of an affine transformation group an equivariant morphism from this space into a space of quadratic maps on the affine space is constructed under this morphism the 2 cocycles of the galilei lie algebra correspond to certain affine tensor fields on galilean space time

variations on a theme of heisenberg pauli and weyl the parentage between weyl pairs the generalized pauli group and the unitary group is investigated in detail we start from an abstract definition of the heisenberg weyl group on the field and then switch to the discrete heisenberg weyl group or generalized pauli group on a finite ring the main characteristics of the latter group an abstract group of order d3 noted pd are given conjugacy classes and irreducible representation classes or equivalently lie algebra of dimension d3 associated with pd leaving the abstract sector a set of weyl pairs in dimension d is derived from a polar decomposition of su 2 closely connected to angular momentum theory then a realization of the generalized pauli group pd and the construction of generalized pauli matrices in dimension d are revisited in terms of weyl pairs finally the lie algebra of the unitary group u d is obtained as a subalgebra of the lie algebra associated with pd this leads to a development of the lie algebra of u d in a basis consisting of d2 generalized pauli matrices in the case where d is a power of a prime integer the lie algebra of su d can be decomposed into d 1 cartan subalgebras

integrable mappings and soliton equations ii abstract some simple solutions of discrete soliton equations are shown to satisfy 2d mappings we show that these belong to a recently introduced 18 parameter family of integrable reversible mappings of the plane thus lending weight to a previous conjecture we also give an example of an integrable mapping occuring in an exactly solvable model in statistical mechanics finally we discuss the notion of generalized reversibility

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

all solvable extensions of a class of nilpotent lie algebras of dimension n and degree of nilpotency n 1 we construct all solvable lie algebras with a specific n dimensional nilradical of degree of nilpotency n 1 and with an n 2 dimensional maximal abelian ideal we find that for given n such a solvable algebra is unique up to isomorphisms using the method of moving frames we construct a basis for the casimir invariants of the nilradical we also construct a basis for the generalized casimir invariants of its solvable extension consisting entirely of rational functions of the chosen invariants of the nilradical

the unitary irreducible representations of sl 2 r in all subgroup reductions we use the canonical transform realization of sl 2 r in order to find all matrix elements and integral kernels for the unitary irreducible representations of this group explicit results are given for all mixed bases and subgroup reductions these provide the full multiparameter set of integral transforms and series expansions associated to sl 2 r

computation of invariants of lie algebras by means of moving frames a new purely algebraic algorithm is presented for computation of invariants generalized casimir operators of lie algebras it uses the cartan method of moving frames and the knowledge of the group of inner automorphisms of each lie algebra the algorithm is applied in particular to computation of invariants of real low dimensional lie algebras a number of examples are calculated to illustrate its effectiveness and to make a comparison with the same cases in the literature bases of invariants of the real six dimensional solvable lie algebras with four dimensional nilradicals are newly calculated and listed in a table

the fundamental invariants of inhomogeneous classical groups the casimir operators of the following groups are explicitly constructed rn gl n r rn sl n r rn o p q cn u p q r2n sp n r cn gl n c cn sl n c cn o n c c2n sp n c hn u 2n hn su 2n hn o 2n hn sp p q and r r2n sp n r the method is based on a particular fiber bundle structure of the generic orbits generated by the co adjoint representation of a semidirect product

the metaplectic group within the heisenberg weyl ring the heisenberg weyl ring contains the metaplectic group of canonical transforms acting unitarily on l 2 r these ring elements are characterized through i the integral transform kernels ii coset distributions and iii classical functions under any quantization scheme the isomorphism under group composition leads to several new relations involving twisted products and quantization of gaussian classical functions the wigner inversion operator is a special central group element it is shown that the only quantization scheme invariant under metaplectic transformations is the weyl scheme the structure studied here appears to be relevant to the study of wave optics with aberration

the heisenberg weyl group in the coherent state basis and the bargmann transform it is shown that the representation matrices of the quantum mechanical group in the coherent state basis can be used as bases for the expansion of arbitrary square integrable functions on the group although this basis is nonorthogonal the expansion can be inverted in a manner similar to an orthogonal basis next a coherent state wave function is defined in analogy with the schrodinger wave function and the above matrix elements are used to determine the action of the group in the space of the coherent state wave functions the function space in this realization is bargmann s hilbert space of analytic functions it is shown that the mixed basis matrix element between the coordinate and coherent state bases is essentially the integral kernel of bargmann the transpose of this matrix element yields the kernel for the inversion of the transform in this construction the bargmann transform appears as the unitary transformation connecting the schrodinger and coherent state wave functions this considerably

an angular momentum approach to quadratic fourier transform hadamard matrices gauss sums mutually unbiased bases the unitary group and the pauli group the construction of unitary operator bases in a finite dimensional hilbert space is reviewed through a nonstandard approach combining angular momentum theory and representation theory of su 2 a single formula for the bases is obtained from a polar decomposition of su 2 and is analyzed in terms of cyclic groups quadratic fourier transforms hadamard matrices and generalized gauss sums weyl pairs generalized pauli operators and their application to the unitary group and the pauli group naturally arise in this approach

shear madness new orthonormal bases and frames using chirp functions the proportional bandwidth and constant bandwidth time frequency signal decompositions of the wavelet gabor and wilson orthonormal bases have attracted substantial interest for representing nonstationary signals however these representations are limited in that they are based on rectangular tessellations of the time frequency plane while much effort has gone into methods for designing nice wavelet and window functions for these frameworks little consideration has been given to methods for constructing orthonormal bases employing nonrectangular time frequency tilings the authors take a first step in this direction by deriving two new families of orthonormal bases and frames employing elements that shear or chirp in the time frequency plane in addition to translate and scale the new scale shear fan bases and shift shear chevron bases are obtained by operating on an existing wavelet gabor 1946 or wilson basis set with two special unitary warping transformations in addition to the theoretical benefit of broadening the class of valid time frequency plane tilings these new bases could possibly also be useful for representing certain types of signals such as chirping and dispersed signals

kalman filtering with intermittent observations motivated by navigation and tracking applications within sensor networks we consider the problem of performing kalman filtering with intermittent observations when data travel along unreliable communication channels in a large wireless multihop sensor network the effect of communication delays and loss of information in the control loop cannot be neglected we address this problem starting from the discrete kalman filtering formulation and modeling the arrival of the observation as a random process we study the statistical convergence properties of the estimation error covariance showing the existence of a critical value for the arrival rate of the observations beyond which a transition to an unbounded state error covariance occurs we also give upper and lower bounds on this expected state error covariance

construction of positive time frequency distributions a general method for constructing nonnegative definite joint time frequency distributions tfds satisfying the marginals of time s t sup 2 and frequency s f sup 2 is presented as nonnegative definite distributions with the correct marginals these tfds are members of the cohen posch class several examples illustrating properties of these tfds are presented for both synthetic and real signals

invariants of a semi direct sum of lie algebras we show that any semi direct sum l of lie algebras with levi factor s must be perfect if the representation associated with it does not possess a copy of the trivial representation as a consequence all invariant functions of l must be casimir operators when the number of invariants is given for all possible dimensions of l replacing the traditional method of solving the system of determining pdes by the equivalent problem of solving a system of total differential equations the invariants are found for all dimensions of the radical up to 5 an analysis of the results obtained is made and this leads to a theorem on invariants of lie algebras depending only on the elements of certain subalgebras

group delay shift covariant quadratic time frequency representations we propose classes of quadratic time frequency representations qtfrs that are covariant to group delay shifts gdss the gds covariance qtfr property is important for analyzing signals propagating through dispersive systems with frequency dependent characteristics this is because a qtfr satisfying this property provides a succinct representation whenever the time shift is selected to match the frequency dependent changes in the signal s group delay that may occur in dispersive systems we obtain the gds covariant classes from known qtfr classes such as cohen s 1995 class the affine class the hyperbolic class and the power classes using warping transformations that depend on the relevant group delay change we provide the formulation of the gds covariant classes using two dimensional 2 d kernel functions and we list desirable qtfr properties and kernel constraints as well as specific class members we present known examples of the gds covariant classes and we provide a new class the power exponential qtfr class we study the localized kernel subclasses of the gds covariant classes that simplify the theoretical development as the kernels reduce from 2 d to one dimensional 1 d functions and we develop various intersections between the qtfr classes finally we present simulation results to demonstrate the advantage of using qtfrs that are matched to changes in the group delay of a signal

kernel design for time frequency signal analysis using the radon transform the authors present a kernel design technique based on using the radon transform of the modulus of the ambiguity function of the signal for determination of angles and distances of radially distributed contents of the autoterms in the ambiguity domain the proposed kernel effectively reduces the cross terms and noise for linear fm signals the result is a tool for high resolution time frequency representation of nonstationary primarily linear fm signals

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

singularity complexity and quasi integrability of rational mappings we investigate global properties of the mappings entering the description of symmetries of integrable spin and vertex models by exploiting their nature of birational transformations of projective spaces we give an algorithmic analysis of the structure of invariants of such mappings we discuss some characteristic conditions for their quasi integrability and in particular its links with their singularities in the 2 plane finally we describe some of their propertiesqua dynamical systems making contact with arnol d s notion of complexity and exemplify remarkable behaviours

linear transformations and aberrations in continuous and finite systems in geometric optics there is a natural distinction between the paraxial and aberration regimes which contain respectively the linear and nonlinear canonical transformations of position and momentum in the phase space in the lie theoretical presentation linear inhomogeneous transformations are generated by linear and quadratic functions of the phase space while aberrations of increasing order are generated by homogeneous functions with higher powers of these coordinates in a way parallel but distinct from the schrodinger quantization of continuous classical systems we quantize the geometric optical model into discrete finite dimensional systems based on the lie algebra whose wavefunctions are n point signals phase space is a sphere and transformations are represented by the n n unitary matrices that form the group we factor this group into linear and nonlinear unitary transformations of phase space and classify all its n2 4 aberrations this offers a new parametrization of based on a chosen subgroup

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

optimal filtering with random sensor delay multiple packet dropout and uncertain observations this paper studies the problem of optimal filtering of discrete time systems with random sensor delay multiple packet dropout and uncertain observation the random sensor delay multiple packet dropout or uncertainty in observation is transformed to a stochastic parameter in the system representation a new formulation enables us to design an optimal filter for a system with multiple packet dropout in sensor data based on a stochastic definition of the norm of a system with a stochastic parameter new relations for stochastic norm are derived the stochastic norm of the estimation error is used as a criterion for the filter design the relations derived for the new norm definition are used to obtain a set of linear matrix inequalities lmis to solve the filter design problems simulation examples show the effectiveness of the proposed method

poincare coherent states the two dimensional massless case coherent states for the positive mass representations of the poincare group in 1 1 dimensions have been obtained previously using the fact that these representations are square integrable moduli of the subgroup of time translations here the method is extended by combining sections from the coset space into the group with homeomorphisms of the coset space these maps are called quasi sections then the generalized construction is applied to the zero mass representations of the 1 1 dimensional poincare group which are square integrable moduli of a subgroup of light like translations the resulting coherent states indexed as before by points in phase space yield a resolution of the identity in the krein space of the zero mass representations the first explicit example of such a structure and it turns out that they coincide with the familiar wavelets based on the ax b group

tomographic time frequency analysis and its application toward time varying filtering and adaptive kernel design for multicomponent linear fm signals since line integrals through the wigner spectrum can be calculated by dechirping calculation of the wigner spectrum may be viewed as a tomographic reconstruction problem in the paper the authors show that all time frequency transforms of cohen s class may be achieved by simple changes in backprojection reconstruction filtering the resolution cross term tradeoff that occurs in time frequency kernel selection is shown to be analogous to the resolution ringing tradeoff that occurs in computed tomography ct ideal reconstruction using a purely differentiating backprojection filter yields the wigner distribution whereas low pass differentiating filters produce cross term suppressing distributions such as the spectrogram or the born jordan distribution it is also demonstrated how this analogy can be exploited to tune the reconstruction filtering or time frequency kernel to improve the ringing resolution tradeoff some properties of the projection domain which is also known as the radon wigner transform are characterized including the response to signal delays or frequency shifts and projection masking or convolution last time varying filtering by shift varying convolution in the radon wigner domain is shown to yield superior results to its analogous cohen s class adaptive transform shift invariant convolution for the multicomponent linear fm signals that are investigated

the hyperbolic class of quadratic time frequency representations ii subclasses intersection with the affine and power classes regularity and unitarity for pt i see ibid vol 41 p 3425 444 1993 part i introduced the hyperbolic class hc of quadratic bilinear time frequency representations qtfrs the present paper defines and studies four subclasses of the hc 1 the focalized kernel subclass of the hc is related to a time frequency concentration property of qtfrs it is analogous to the localized kernel subclass of the affine qtfr class 2 the affine subclass of the hc affine hc consists of all hc qtfrs that satisfy the conventional time shift covariance property it forms the intersection of the hc with the affine qtfr class 3 the power subclasses of the hc consist of all hc qtfrs that satisfy a power time shift covariance property they form the intersection of the hc with the recently introduced power classes 4 the power warp subclass of the hc consists of all hc qtfrs that satisfy a covariance to power law frequency warpings it is the hc counterpart of the shift scale covariant subclass of cohen s class all of these subclasses are characterized by 1d kernel functions the affine hc is contained in both the localized kernel hyperbolic subclass and the localized kernel affine subclass and that any affine hc qtfr can be derived from the bertrand unitary po distribution by a convolution we furthermore consider the properties of regularity and unitarity in the hc the calculus of inverse kernels is developed and important implications of regularity and unitarity are summarized the results comprise a general method for least squares signal synthesis and new relations for the altes marinovich q distribution

adaptive diffusion as a versatile tool for time frequency and time scale representations processing a review inspired by the work on image processing by perona and malik diffusion based models were first investigated by goncalve spl grave s and payot to improve the readability of cohen class time frequency representations they rely on signal dependent partial differential equations that yield adaptive smoothed representations with sharpened time frequency components here we demonstrate the versatility and utility of this family of methods and we propose new adaptive diffusion processes to locally control both the orientation and the strength of smoothing our approach is an improvement on previous works as it provides a unified framework not only for the cohen class but for the affine class as well the latter is of particular interest because except for some special techniques such as the reassignment method no signal dependent smoothing technique exists to process bilinear time scale distributions nor even a transposition of the adaptive optimal kernel method proposed by baraniuk and jones

symbolic calculus on the time frequency half plane this study concerns a special symbolic calculus of interest for signal analysis this calculus associates functions on the time frequency half plane f 0 with linear operators defined on the positive frequency signals full attention is given to its construction which is entirely based on the study of the affine group in a simple and direct way the correspondence rule is detailed and the associated wigner function is given formulas expressing the basic operation star bracket of the lie algebra of symbols which is isomorphic to that of the operators are obtained in addition it is shown that the resulting calculus is covariant under a three parameter group which contains the affine group as subgroup this observation is the starting point of an investigation leading to a whole class of symbolic calculi which can be considered as modifications of the original one

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

generalized casimir operators of solvable lie algebras with abelian nilradicals a solvable complex lie algebra l of dimension n with an abelian nilradical of dimension r is shown to have precisely 2r n generalized casimir invariants we always have r or n 2 they are constructed as invariants of the coadjoint representation of l and depend only on variables dual to elements of the nilradical their form in general involves logarithms of these variables in addition to rational and irrational functions they give rise to genuine casimir operators whenever they happen to be polynomials

a generalized demodulation approach to time frequency projections for multicomponent signals in this paper we introduce a flexible approach for the time frequency analysis of multicomponent signals involving the use of analytic vectors and demodulation the demodulated analytic signal is projected onto the time frequency plane so that as closely as possible each component contributes exclusively to a different tile in a wavelet packet tiling of the time frequency plane and at each time instant the contribution to each tile definitely comes from no more than one component a single reverse demodulation is then applied to all projected components the resulting instantaneous frequency of each component in each tile is not constrained to a set polynomial form in time and is readily calculated as is the corresponding hilbert energy spectrum two examples illustrate the method in order better to understand the effect of additive noise the approximate variance of the estimated instantaneous frequency in any tile has been formulated by starting with pure noise and studying its evolving covariance structure through each step of the algorithm the validity and practical utility of the resulting expression for the variance of the estimated instantaneous frequency is demonstrated via a simulation experiment

regularity and unitarity of bilinear time frequency signal representations two structural properties of bilinear time frequency representations btfrs of signals are introduced and studied the definition of these properties is based on a linear operator description of btfrs the first property termed regularity has important implications with respect to the recovery of signals from the btfr outcome the derivation of other bilinear signal representations from a btfr the btfrs reaction to linear signal transformations and the construction of bases of induced btfr domain spaces the second property called unitarity is equivalent to validity of moyal s formula 1949 unitarity is thus necessary and sufficient for a closed form solution of optimal signal synthesis and for a btfr formulation or optimal detection estimation methods unitarity also allows the systematic construction of btfr product relations like the wigner distribution s interference formula and the ambiguity function s self transformation property unitarity permits the construction of induced orthogonal projection operators and guarantees the orthonormality of induced basis functions

wigner functions for a class of semi direct product groups following a general method proposed earlier we construct here wigner functions defined on coadjoint orbits of a class of semidirect product groups the groups in question are such that their unitary duals consist purely of representations from the discrete series and each unitary irreducible representation is associated with a coadjoint orbit the set of all coadjoint orbits hence uirs is finite and their union is dense in the dual of the lie algebra the simple structure of the groups and the orbits enables us to compute the various quantities appearing in the definition of the wigner function explicitly al argenumber of examples with potential use in image analysis is worked out

wideband weyl symbols for dispersive time varying processing of systems and random signals we extend the narrowband weyl symbol ws and the wideband p sub o weyl symbol pows for dispersive time frequency tf analysis of nonstationary random processes and time varying systems we obtain the new tf symbols using unitary transformations on the ws and the p sub o ws for example whereas the ws is matched to systems with constant or linear tf characteristics the new symbols are better matched to systems with dispersive nonlinear tf structures this results from matching the geometry of the unitary transformation to the specific tf characteristics of a system we also develop new classes of smoothed weyl symbols that are covariant to tf shifts or time shift and scaling system transformations these classes of symbols are also extended via unitary warpings to obtain classes of tf symbols covariant to dispersive shifts we provide examples of the new symbols and symbol classes and we list some of their desirable properties using simulation examples we demonstrate the advantage of using tf symbols that are matched to the changes in the tf characteristics of a system or random process we also provide new tf formulations for matched detection applications

covariant affine integral quantization s covariant affine integral quantization of the half plane is studied and applied to the motion of a particle on the half line we examine the consequences of different quantizer operators built from weight functions on the half plane to illustrate the procedure we examine two particular choices of the weight function yielding thermal density operators and affine inversion respectively the former gives rise to a temperature dependent probability distribution on the half plane whereas the latter yields the usual canonical quantization and a quasi probability distribution affine wigner function which is real marginal in both momentum p and position q

virtues and vices of quartic time frequency distributions we present results concerning three different types of quartic fourth order time frequency distributions tfds first we present new results on the previously introduced local ambiguity function and show that it provides more reliable estimates of instantaneous chirp rate than the wigner distribution second we introduce the class of quartic shift covariant time frequency distributions and investigate distributions that localize quadratic chirps finally we present a shift covariant distribution of time and chirp rate

duality and classification of bilinear time frequency signal representations the author discusses a fundamental duality principle of btfrs bilinear time frequency representations and presents a systematic classification of btfrs which is consistent with btfr duality btfrs are first grouped into two basic domains namely the energy density domain e domain with energetic interpretation and the correlation domain c domain with correlative interpretation it is shown that these domains are related by a fourier transform duality to any btfr btfr relation or btfr property of the e domain there corresponds a dual btfr btfr relation or btfr properties of the c domain and vice versa with this duality principle as a background a classification of btfrs is given this classification is based on two dual shift invariance properties and a self dual scale invariance property of btfrs the mathematical description of btfrs by means of kernel functions is simplified inside the respective btfr classes it is shown that btfrs which are both shift and scale invariant can be represented as superpositions of generalized wigner distributions e domain or generalized ambiguity function c domain

on positivity of time frequency distributions this correspondence addresses the problem of how to regard the fundamental impossibility with time frequency energy distributions of cohen s class always to be nonnegative and at the same time to have correct marginal distributions it is shown that the wigner distribution is the only member of a large class of bilinear time frequency distributions that becomes nonnegative after smoothing in the time frequency plane by means of gaussian weight functions with bt product equal to unity

on the equivalence of the operator and kernel methods for joint distributions of arbitrary variables generalizing the concept of time frequency representations cohen see englewood cliffs nj prentice hall 1995 has proposed a method based on operator correspondence rules for generating joint distributions of arbitrary variables as an alternative to considering all such rules which is a practical impossibility in general cohen has proposed the kernel method in which different distributions are generated from a fixed rule via an arbitrary kernel we derive a simple but rather stringent necessary condition on the underlying operators for the kernel method with the kernel functionally independent of the variables to generate all bilinear distributions of the specific pairs of variables that have been studied essentially only time and frequency satisfy the condition in particular the important variables of time and scale do not the results warrant further study for a systematic characterization of bilinear distributions in cohen s method

radon transformation of time frequency distributions for analysis of multicomponent signals the radon transform of a time frequency distribution produces local areas of signal concentration that facilitate interpretation of multicomponent signals the radon wigner transform can be efficiently implemented with dechirping in the time domain however only half of the possible projections through the time frequency plane can be realized because of aliasing we show here that the frequency dual to dechirping exists so that all of the time frequency plane projections can be calculated efficiently both time and frequency dechirping are shown to warp the time frequency plane rather rotating it producing an angle dependent dilation of the radon wigner projection axis we derive the discrete time equations for both time and frequency dechirping and highlight some practical implementation issues discrete dechirping is shown to correspond to line integration through the extended discrete rather than the discrete wigner ville distribution computationally dechirping is o 2n log 2n instead of o n sup 3 for direct projection and the computation is dominated by the fast fourier transform calculation the noise and cross term suppression of the radon wigner transform are demonstrated by several examples using dechirping and using direct radon wigner transformation

rational mappings arborescent iterations and the symmetries of integrability we describe a class of nonlinear birational representations of groups generated by a finite number of involutions these groups are symmetries of the yang baxter equations and their higher dimensional generalizations they provide discrete dynamical systems with a variety of behaviors from chaotic to integrable according to the number of invariants of the representation

fast computation of frequency warping transforms in this paper we introduce an analytical approach for the frequency warping transform criteria for the design of operators based on arbitrary warping maps are provided and an algorithm carrying out a fast computation is defined such operators can be used to shape the tiling of time frequency tf plane in a flexible way moreover they are designed to be inverted by the application of their adjoint operator according to the proposed model the frequency warping transform is computed by considering two additive operators the first one represents its nonuniform fourier transform approximation and the second one suppresses aliasing the first operator is fast computable by various interpolation approaches a factorization of the second operator is found for arbitrary shaped nonsmooth warping maps by properly truncating the operators involved in the factorization the computation turns out to be fast without compromising accuracy

a class of affine wigner functions with extended covariance properties affine wigner functions are phase space representations based on the affine group in place of the usual weyl heisenberg group of quantum mechanics such representations are relevant to the time frequency analysis of real signals an interesting family is singled out by the requirement of covariance with respect to each solvable three parameter group containing the affine group explicit forms are given in each case and properties such as unitarity and localization are discussed some particular distributions are recovered

invariants of solvable lie algebras with triangular nilradicals and diagonal nilindependent elements the invariants of solvable lie algebras with nilradicals isomorphic to the algebra of strongly upper triangular matrices and diagonal nilindependent elements are studied exhaustively bases of the invariant sets of all such algebras are constructed by an original purely algebraic algorithm based on cartan s method of moving frames

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

the jordan schwinger representations of cayley klein groups iii the symplectic groups the symplectic cayley klein groups are defined as the groups that are obtained by the contractions and analytical continuations of the classical symplectic groups the jordan schwinger representations of the groups under consideration are discussed based on the mixed sets of creation and annihilation operators of boson or fermion type the matrix elements of finite group transformations are obtained in the bases of coherent states

integrable mappings from matrix transformations and their singularity properties various examples of bi rational transformations having their origin in the theory of exactly solvable vertex models in lattice statistical mechanics as well as bi rational transformations originating from spin edge models are analysed using the singularity confinement method this method provides results concerning the integrable or not character of these bi rational transformations in complete agreement with the results obtained by visualization methods as well as methods based on the analysis of algebraic invariants

waveform mapping and time frequency processing of dna and protein sequences current state of the art approaches for biological sequence querying and alignment require preprocessing and lack robustness to repetitions in the sequence in addition these approaches do not provide much support for efficiently querying subsequences a process that is essential for tracking localized database matches we propose a query based alignment method for biological sequences that first maps sequences to time domain waveforms before processing the waveforms for alignment in the time frequency plane the mapping uses waveforms such as gaussian functions with unique sequence representations in the time frequency plane the proposed alignment method employs a robust querying algorithm that utilizes a time frequency signal expansion whose basis function is matched to the basic waveform in the mapped sequences the resulting wavequery approach was demonstrated for both deoxyribonucleic acid dna and protein sequences using the matching pursuit decomposition as the signal basis expansion we specifically evaluated the alignment localization of wavequery over repetitive database segments and we demonstrated its operation in real time without preprocessing we also demonstrated that wavequery significantly outperformed the biological sequence alignment method blast for queries with repetitive segments for dna sequences a generalized version of the wavequery approach with the metaplectic transform is also described for protein sequence structure prediction

a class of solvable lie algebras and their casimir invariants a nilpotent lie algebra with an n 1 dimensional abelian ideal is studied all indecomposable solvable lie algebras with as their nilradical are obtained their dimension is at most n 2 the generalized casimir invariants of and of its solvable extensions are calculated for n 4 these algebras figure in the petrov classification of einstein spaces for larger values of n they can be used in a more general classification of riemannian manifolds

invariants of triangular lie algebras with one nil independent diagonal element the invariants of solvable triangular lie algebras with one nil independent diagonal element are studied exhaustively bases of the invariant sets of all such algebras are constructed using an original algebraic algorithm based on cartan s method of moving frames and the special technique developed for triangular and closed algebras in boyko et al j phys a math theor 2007 40 7557 the conjecture of tremblay and winternitz j phys a math gen 2001 34 9085 on the number and form of elements in the bases is completed and proved