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

linear estimation of boundary value stochastic processes part ii 1 d smoothing problems this paper addresses the fixed interval smoothing problem for linear two point boundary value stochastic processes of the type introduced by krener 5 as these models are not markovian kalman filtering and associated smoothing algorithms are not applicable the smoothing problem for this class of noncausal processes is solved here by an application of the estimator solution which is developed in part i of this paper 3 via the method of complementary models for an n th order model this approach yields the smoother as a 2 n th order two point boundary value problem it is shown that this smoother can be realized in a stable two filter form which is remarkably similar to two filter smoothers for causal processes in addition expressions for the smoothing error and smoothing error covariance are developed these equations are employed to perform a covariance analysis of estimating the temperature and heat flow in a cooling fin

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

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

linear estimation of boundary value stochastic processes part i the role and construction of complementary models this paper presents a substantial extension of the method of complementary models for minimum variance linear estimation introduced by weinert and desai in their important paper 1 specifically the method of complementary models is extended to solve estimation problems for both discrete and continuous parameter linear boundary value stochastic processes in one and higher dimensions a major contribution of this paper is an application of green s identity in deriving a differential operator representation of the estimator to clarify the development and to illustrate the range of applications of our approach two brief examples are provided one is a 1 d discrete two point boundary value process and the other is a 2 d process governed by poisson s equation on the unit disk

boundary value descriptor systems well posedness reachability and observability in this paper we introduce the class of two point boundary value descriptor systems tpbvds discrete time systems described by possibly linear dynamics and a set of boundary conditions constraining the values of the system state at the two endpoints of the system s interval of definition by introducing a standard form for regular pencils we obtain a new and simple generalized cayley hamilton theorem that simplifies our investigation of well posedness green s function solutions and reachability and observability for tpbvds there are two distinct notions of reachability and observability that one can define for tpbvds associated with processes that propagate inward from and outward toward the boundaries we investigate each of these in detail obtaining among other things far simpler forms for the reachability and observability results than found previously in the literature in addition we describe several methods for the efficient solution of tpbvds one involving recursions from each end of t

exponential stability and global stability of cellular neural networks in this paper we present two types of matrix stability complete stability and strong stability and some properties of them by using these properties we discuss stability of cellular neural networks and obtain some conditions ensuring uniqueness exponential stability and global asymptotic stability of the equilibrium point for cellular neural networks in particular in the two cell case the network is globally exponentially stable if and only if the feedback matrix minus the unit matrix is a complete stable matrix

consistency of modified ls estimation method for identifying 2 d noncausal sar model parameters least squares ls and maximum likelihood ml are the two main methods for parameter estimation of two dimensional 2d noncausal simultaneous autoregressive sar models ml is asymptotically consistent and unbiased but computationally unattractive on the other hand conventional ls is computationally efficient but does not produce accurate parameter estimates for noncausal models recently zhao yu 1993 proposed a modified ls estimation method and was shown to be unbiased in this paper we prove that under certain assumptions the method introduced by zhao yu is also consistent

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

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

wigner based formulation of the chirplet transform using the wigner distribution we derive and analyze a matrix formulation for the chirplet transform a signal analysis tool that generalizes the wavelet and short time fourier transforms the formulation expresses the translations scalings and shears of the chirplet transform in terms of affine matrix transformations on the time frequency plane our approach leads naturally to several new signal transforms which we derive analyze and extend

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

recent directions in matrix stability abstract matrix stability has been intensively investigated in the past two centuries we review work that has been done in this topic focusing on the great progress that has been achieved in the last decade or two we start with classical stability criteria of lyapunov routh and hurwitz and lienard and chipart we then study recently proven sufficient conditions for stability with particular emphasis on p matrices we investigate conditions for the existence of a stable scaling for a given matrix we review results on other types of matrix stability such as d stability additive d stability and lyapunov diagonal stability we discuss the weak principal submatrix rank property shared by lyapunov diagonally semistable matrices we also discuss the uniqueness of lyapunov scaling factors maximal lyapunov scaling factors cones of real positive semidefinite matrices and their applications to matrix stability and inertia preserving matrices

delayed standard neural network models for control systems in order to conveniently analyze the stability of recurrent neural networks rnns and successfully synthesize the controllers for nonlinear systems similar to the nominal model in linear robust control theory the novel neural network model named delayed standard neural network model dsnnm is presented which is the interconnection of a linear dynamic system and a bounded static delayed or nondelayed nonlinear operator by combining a number of different lyapunov functionals with s procedure some useful criteria of global asymptotic stability and global exponential stability for the continuous time dsnnms cdsnnms and discrete time dsnnms ddsnnms are derived whose conditions are formulated as linear matrix inequalities lmis based on the stability analysis some state feedback control laws for the dsnnm with input and output are designed to stabilize the closed loop systems most rnns and neurocontrol nonlinear systems with or without time delays can be transformed into the dsnnms to be stability analyzed or stabilization synthesized in a unified way in this paper the dsnnms are applied to analyzing the stability of the continuous time and discrete time rnns with or without time delays and synthesizing the state feedback controllers for the chaotic neural network system and discrete time nonlinear system it turns out that the dsnnm makes the stability conditions of the rnns easily verified and provides a new idea for the synthesis of the controllers for the nonlinear systems

a comparison of the squared energy and teager kaiser operators for short term energy estimation in additive noise time frequency distributions that evaluate the signal s energy content both in the time and frequency domains are indispensable signal processing tools especially for nonstationary signals various short time energy computation schemes are used in practice including the mean squared amplitude and teager kaiser energy approaches herein we focus primarily on the short and medium term properties of these two energy estimation schemes as well as on their performance in the presence of additive noise to facilitate this analysis and generalize the approach we use a harmonic noise model to approximate the noise component the error analysis is conducted both in the continuous and discrete time domains deriving similar conclusions the estimation errors are measured in terms of normalized deviations from the expected signal energy and are shown to greatly depend on both the signals spectral content and the analysis window length when medium and long term analysis windows are employed the teager kaiser energy operator is proven superior to the common squared energy operator provided that the spectral content of the noise is more lowpass than the corresponding signal content and vice versa however for shorter window lengths the teager kaiser operator always outperforms the squared energy operator the theoretical results are experimentally verified for synthetic signals finally the performance of the proposed energy operators is evaluated for short term analysis of noisy speech signals and the implications for speech processing applications are outlined

a general approach for obtaining joint representations in signal analysis ii general class mean and local values and bandwidth for pt i see ibid vol 44 no 5 p 1080 90 1996 using the method developed in cohen 1996 part i the concepts of instantaneous frequency and group delay are generalized to arbitrary variables in addition new expressions for mean values and bandwidths are obtained the kernel method is used to define a general class for arbitrary variables as in the time frequency case the general class generates all possible distributions the method is also formulated in terms of the local autocorrelation function

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

on the control of linear systems whose coefficients are functions of parameters the stabilization and regulation of linear discrete time systems whose coefficients depend on one or more parameters is studied for linear systems whose coefficients are continuous functions of real or complex parameters respectively analytic or rational functions of real parameters it is shown that reachability of the system for all values of the parameters implies that the system can be stabilized using gains that are also continuous analytic rational functions of the parameters closed form expressions for a collection of stabilizing gains are given in terms of the reachability gramian for systems which are stabilizable for all values of the parameters it is shown that continuous analytic rational stabilizing gains can be computed from a finite time solution to a riccati difference equation whose coefficients are functions of the parameters these results are then applied to the problem of tracking and disturbance rejection in the case when both the plant and the exogenous signals contain parameters

two filter smoothing formulae by diagonalization of the hamiltonian equations we present a new approach to two filter smoothing formulae via diagonalization of the general time variant hamiltonian equations of the linear estimation problem this approach shows the special role of the famous mayno fraser two filter formulae and also provides insight into certaini nvariance properties of backwards kalman filter estimates

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

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

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

optimal smoothing for noncausal 2 d systems based on a descriptor model an optimal smoothing problem for noncausal two dimensional systems is formulated and solved based on the descriptor model with coefficient functions using this model a forward and backward recursion procedure is obtained it is shown that for the existence of solutions pointwise reachability and observability of the descriptor model at infinite poles are important

on complementary models and fixed interval smoothing a new algorithm is derived for the standard fixed interval linear smoothing problem in which the signal is generated by a state model the structure of this new algorithm allows the smoothed estimate to be easily updated in response to a change in the initial state covariance matrix pi 0 since unlike in existing algorifiuns the relevant riccati equation is entirely independent of pi 0 the derivation of the algorithm is based on properties of complementary models