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

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

a chebyshev approximation for solving nonlinear integral equations of hammerstein type abstract a numerical method for solving fredholm volterra hammerstein integral equations is presented this method is based on replacement of the unknown function by truncated series of well known chebyshev expansion of functions the quadrature formula which we use to calculate integral terms can be estimated by fast fourier transform fft the numerical examples and the number of operations show the advantages of this method to some other usual methods

general population systems macroscopic limit of a class of stochastic semigroups relationships between three possible descriptions microscopic mesoscopic and macroscopic that are related to a large population composed of several subpopulations are formulated the systems describing the evolution of a large number of individuals undergoing stochastic interactions a description at microscopic level in terms of stochastic semigroups are considered the solutions of the general lotka volterra type equations without or with weak diffusion descriptions at macroscopic level are approximated by solutions of stochastic systems when the number of individuals n tends to the rate of approximation is controlled

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

design of time frequency representations using a multiform tiltable exponential kernel a novel cohen s 1981 class time frequency representation with a tiltable generalized exponential kernel capable of attaining a wide diversity of shapes in the ambiguity function plane is proposed for improving the time frequency analysis of multicomponent signals the first advantage of the proposed kernel is its ability to generate a wider variety of passband shapes e g rotated ellipses generalized hyperbolas diamonds rectangles parallel strips at arbitrary angles crosses snowflakes etc and narrower transition regions than conventional cohen s class kernels this versatility enables the new kernel to suppress undesirable cross terms in a broader variety of time frequency scenarios the second advantage of the new kernel is that closed form design equations can now be easily derived to select kernel parameters that meet or exceed a given set of user specified passband and stopband design criteria in the ambiguity function plane thirdly it is shown that simple constraints on the parameters of the new kernel can be used to guarantee many desirable properties of time frequency representations the well known choi williams 1989 exponential kernel the generalized exponential kernel and nuttall s 1990 tilted gaussian kernel are special cases of the proposed kernel

blind source separation in the time frequency domain based on multiple hypothesis testing this paper considers a time frequency t f based approach for blind separation of nonstationary signals in particular we propose a time frequency point selection algorithm based on multiple hypothesis testing which allows automatic selection of auto or cross source locations in the time frequency plane the selected t f points are then used via a joint diagonalization and off diagonalization algorithm to perform source separation the proposed algorithm is developed assuming deterministic signals with additive white complex gaussian noise a performance comparison of the proposed and existing approaches is provided

lagrange interpolation to compute the numerical solutions of differential integral and integro differential equations this paper describes new two methods for the numerical solution of linear integral equations of fredholm type and volterra type also the methods have been extended to linear integro differential and differential equations the methods are tested by different examples

adaptive instantaneous frequency estimation of multicomponent fm signals using quadratic time frequency distributions an adaptive approach to the estimation of the instantaneous frequency if of nonstationary mono and multicomponent fm signals with additive gaussian noise is presented the if estimation is based on the fact that quadratic time frequency distributions tfds have maxima around the if law of the signal it is shown that the bias and variance of the if estimate are functions of the lag window length if there is a bias variance tradeoff then the optimal window length for this tradeoff depends on the unknown if law hence an adaptive algorithm with a time varying and data driven window length is needed the adaptive algorithm can utilize any quadratic tfd that satisfies the following three conditions first the if estimation variance given by the chosen distribution should be a continuously decreasing function of the window length whereas the bias should be continuously increasing so that the algorithm will converge at the optimal window length for the bias variance tradeoff second the time lag kernel filter of the chosen distribution should not perform narrowband filtering in the lag direction in order to not interfere with the adaptive window in that direction third the distribution should perform effective cross terms reduction while keeping high resolution in order to be efficient for multicomponent signals a quadratic distribution with high resolution effective cross terms reduction and no lag filtering is proposed the algorithm estimates multiple if laws by using a tracking algorithm for the signal components and utilizing the property that the proposed distribution enables nonparametric component amplitude estimation an extension of the proposed tfd consisting of the use of time only kernels for adaptive if estimation is also proposed

kernel decomposition of time frequency distributions bilinear time frequency distributions tfds offer improved time frequency resolution over linear representations but suffer from difficult interpretation higher implementation cost and the lack of associated low cost signal synthesis algorithms in the paper the authors introduce some new tools for the interpretation and quantitative comparison of high resolution tfds these tools are used in related work to define low cost high resolution tfds and to define linear low cost signal synthesis algorithms associated with high resolution tfds first each real valued tfd is associated with a self adjoint linear operator spl psi the spectral representation of spl psi expresses the tfd as a weighted sum of spectrograms sps it is shown that the sp decomposition and weyl correspondence do not yield useful interpretations for high resolution tfds due to the fact that spl psi is not positive

large systems of interacting particles and the porous medium equation abstract we study the empirical processes of systems of interacting particles whose time evolution is given by coupled ordinary differential equations in particular as the population size tends to infinity we derive different versions of the porous medium equation as limit dynamics

seismic data analysis by adaptive sparse time frequency decomposition the variation of frequency content of a seismic trace with time carries information about the properties of the subsurface reflectivity sequence time frequency tf analysis is a significant tool to extract such information for seismostratigraphic interpretation purposes however several tf transforms have been reported in the literature higher resolution and sensitivity to local changes of the signal have always mattered we have developed an adaptive high resolution tf transform that is performed in two sequential steps first the window length is adaptively determined for each sample of the signal such that it leads to maximum compactness of energy in the resulting tf plane second the generated nonstationary windows are used to inversely decompose the signal under study via a convex constrained sparse optimization where a mixed norm of the tf coefficients is minimized subject to invertibility of the transform later on the optimized transform is used as an efficient tool for seismic data analysis such as thin bed characterization and thin bedded gas reservoir detection in the case of gas reservoir detection based on amplitude versus offset analysis in the tf domain a simple new method called the difference section was evaluated the results of various numerical examples from synthetic and field data revealed a remarkable performance of the proposed method compared with the state of the art tf transforms

simulation of the solution of a viscous porous medium equation by a particle method to solve a particular partial differential equation namely a viscous porous medium equation we discuss a particle method which is based on the concept of moderately interacting many particle systems our approach may be classified as a combination of a smoothed particle hydrodynamics method and a particle mesh method quantitatively it is assessed in terms of estimates on the approximation error

on a singular perturbation problem arising in the theory of evolutionary distributions evolution by natural selection is a process by which progeny inherit some properties from their progenitors with small variation these properties are subject to natural selection and are called adaptive traits and carriers of the latter are called phenotypes the distribution of the density of phenotypes in a population is called evolutionary distributions ed we analyze mathematical models of the dynamics of a system of ed such systems are anisotropic in that diffusion of phenotypes in each population equation remains positive in the directions of their own adaptive space and vanishes in the directions of the other s adaptive space we prove that solutions to such systems exist in a sense weaker than the usual we develop an algorithm for numerical solutions of such systems finally we conduct numerical experiments with a model in which populations compete that allow us to observe salient attributes of a specific system of ed

leak location in gas pipelines using cross time frequency spectrum of leakage induced acoustic vibrations abstract the correlation based location methods are widely used in leak detection of the pipelines assuming that the acoustic speed has been known and constant in practice the acoustic speed is frequency varying due to the dispersions of gas leak induced acoustic waves and thus the assumption is not supported in this work a location scheme based on cross time frequency spectrum ctfs is intended for the gas leak induced acoustic waves with frequency varying acoustic speed in the scheme the ctfs is obtained by the one dimensional fourier transform of the time domain convolution between the kernel function in correlation domain and the instantaneous cross correlation of the two spatially separately collected acoustic signals on either sides of a leakage then the time difference of arrival tdoa and the corresponding frequency information are extracted simultaneously when the ctfs reaches the maximum value the resulting peak frequency is used to online determine the frequency dependant acoustic speed in combination with the known dispersive curve of gas leak induced dominated mode finally the gas leakage is located by the tdoa and the frequency dependant acoustic speed of real time determination instead of constant acoustic speed consequently for the proposed scheme the constant acoustic speed is no longer a prerequisite the proposed scheme has been experimentally validated in leak detection of gas pipelines and results demonstrate that the average relative location errors are reduced by six times compared with the commonly used correlation based location method

a law of large numbers for moderately interacting diffusion processes we consider two special models of interacting diffusion processes and derive in the limit as the number of different processes tends to infinity and the interaction is rescaled in a suitable moderate way a law of large numbers for the empirical processes as limit dynamics we obtain certain nonlinear diffusion equations

time frequency kernel design by the two dimensional frequency transformation method a subclass of cohen s class of time frequency tf distributions is introduced in which the tf distribution kernels are provided via the frequency transformation method ftm used in two dimensional 2 d filter design the ftm kernels have finite extent in time and frequency and allow the tf distribution to be efficiently implemented in all four domains

vector valued time frequency representations cohen s 1989 class of time frequency distributions tfds which includes the spectrogram sp wigner distribution wd and reduced interference distributions rids has become widely known as a useful signal analysis tool it has been shown that every real valued tfd can be written as a weighted sum of sps the sp decomposition has been used to construct fast approximations to desirable tfds using the sp building block for which there exist accessible and efficient hardware and software implementations we introduce a class of linear vector valued time frequency representations tfrs that are easily related to associated bilinear tfds through the sp decomposition we solve a least squares signal synthesis problem on modified vector valued tfrs that are associated with nonnegative tfds as a weighted sum of least squares short time fourier transform stft signal synthesis schemes we extend the solution to vector valued tfrs associated with high resolution tfds in order to define a high resolution alternative to stft signal synthesis as demonstrated by desirable properties and examples the resulting signal synthesis methods can be realized as a weighted sum of stft synthesis schemes for which there exist accessible and efficient hardware and software implementations

modified cohen lee time frequency distributions and instantaneous bandwidth of multicomponent signals cohen 1989 1995 has introduced and extensively studied and developed the concept of the instantaneous bandwidth of a signal specifically instantaneous bandwidth is interpreted as the spread in frequency about the instantaneous frequency which is itself interpreted as the average frequency at each time this view stems from a joint time frequency distribution tfd analysis of the signal where instantaneous frequency and instantaneous bandwidth are taken to be the first two conditional spectral moments respectively of the distribution however the traditional definition of instantaneous frequency namely as the derivative of the phase of the signal is not consistent with this interpretation and new definitions have therefore been proposed previously we show that similar problems arise with the cohen lee 1988 1989 instantaneous bandwidth of a signal and propose a new formulation for the instantaneous bandwidth that is consistent with its interpretation as the conditional standard deviation in frequency of a tfd we give the kernel constraints for a distribution to yield this new result which is a modification of the kernel proposed by cohen and lee these new kernel constraints yield a modified cohen lee tfd whose first two conditional moments are interpretable as the average frequency and bandwidth at each time respectively

some windows with very good sidelobe behavior some of the windows presented by harris 1 are not correct in terms of their reported peak sidelobes and optimal behavior we present corrected plots of harris windows and also derive additional windows with very good sidelobes and optimal behavior under several different constraints the temporal weightings are characterized as a sum of weighted cosines over a finite duration the plots enable the reader to select a window to suit his requirements in terms of bias due to nearby sidelobes and bias due to distant sidelobes

a centrosymmetric kernel decomposition for time frequency distribution computation time frequency distributions tfds are bilinear transforms of the signal and as such suffer from a high computational complexity previous work has shown that one can decompose any tfd in cohen s class into a weighted sum of spectrograms this is accomplished by decomposing the kernel of the distribution in terms of an orthogonal set of windows in this paper we introduce a mathematical framework for kernel decomposition such that the windows in the decomposition algorithm are not arbitrary and that the resulting decomposition provides a fast algorithm to compute tfds using the centrosymmetric structure of the time frequency kernels we introduce a decomposition algorithm such that any tfd associated with a bounded kernel can be written as a weighted sum of cross spectrograms the decomposition for several different discrete time kernels are given and the performance of the approximation algorithm is illustrated for different types of signals

on the connection between hamiltonian many particle systems and the hydrodynamical equations we consider certain hamiltonian systems with many particles interacting through a potential whose range is large in comparison with the typical distance between neighbouring particles it is shown that the empirical processes of the positions and the velocities respectively converge to solutions of the continuity equation and the euler equation in the limit as the particle number tends to infinity

periodic behavior of the stochastic brusselator in the mean field limit we prove a propagation of chaos result for the mean field limit of a model for a trimolecular chemical reaction called brusselator then we show that the pair of nonlinear i e law dependent stochastic differential equations describing the evolution of the concentration of the molecules at a given site in the mean field limit has a solution with a periodic law in t finally we identify the limit and establish a central limit theorem for the periodic law in the case where the noise tends to zero

the discrete time phase derivative as a definition of discrete instantaneous frequency and its relation to discrete time frequency distributions we compare two definitions of the instantaneous frequency of a discrete signal the two point symmetric phase difference and the discrete time phase derivative the phase derivative definition avoids the pitfalls associated with the two point definition and is equal to the first moment w r t frequency of a properly defined alias free time frequency distribution of the signal which is consistent with the continuous time case

space spatial frequency analysis based filtering space invariant filtering of signals that overlap with noise in both space and frequency can be inefficient however the signal and noise may be well separated in the joint space spatial frequency domain then it is possible to benefit from the application of space spatial frequency approaches processing based on these approaches can outperform space or frequency invariant based methods to this aim the concept of nonstationary space varying filtering is introduced in this paper as an extension of the time varying filtering concept the filtering definitions are based on statistical averages although the filtering should commonly be applied knowing only a single noisy signal realization the procedures that can produce good estimates of quantities crucial for efficient filtering based on a single noisy signal realization are considered special attention has been paid to the region of support estimation and cross term effects removal the efficiency of the proposed space spatial frequency filtering concept is tested on the signal forms inspired by the interferograms in optics including real images as disturbances examples demonstrate the superiority of the proposed filtering over the space invariant one for the considered type of signals and noise

on alias free formulations of discrete time cohen s class of distributions the transition of the cohen s 1989 class of distributions from the continuous time case to the discrete time case is not straightforward because of aliasing problems we classify the aliasing problems which occur for joint time frequency representations tfrs into two categories type i and type ii aliasings type i aliasing can be avoided by properly defined discrete time versions of some members of cohen s class in particular properly defined kernels whereas type ii aliasing can be reduced and or eliminated by increasing the sampling rate a type i alias free formulation of the discrete time cohen s class af dtcc which is equivalent to the af gdtft of joeng and williams see ibid vol 40 no 2 p 1084 1992 is then introduced based on the fact that the cohen s class can be expressed as the 2 d fourier transform of the generalized ambiguity function af based on this definition two discretization schemes for kernel functions are presented in both the af domain and the time lag domain and are shown to be equivalent under certain conditions we also do the following 1 we show that a discrete time wigner ville distribution dwvd and discrete time spectrogram dspg are type i alias free and members of af dtcc 2 we use all the available correlation information from a given data sequence by using the woodward af instead of the sussman af 3 we give kernel constraints in the af domain for various distribution properties and 4 we provide a type i and type ii alias free formulation for those distributions whose kernel functions satisfy the finite frequency support constraint

the uncertainty principle global local or both we address the issue of the relation between local quantities and the uncertainty principle we approach the problem by defining local quantities as conditional standard deviations and we relate these to the uncertainty product appearing in the standard uncertainty principle we show that the uncertainty product for the average local standard deviations is always less than or equal to the standard uncertainty product and that it can be arbitrarily small we apply these results to the short time fourier transform spectrogram to explore the commonly held notion that the uncertainty principle somehow limits local quantities we show that indeed for the spectrogram there is a lower bound on the local uncertainty product of the spectrogram due to the windowing operation of this method this limitation is an inherent property of the spectrogram and is not a property of the signal or a fundamental limit we also examine the local uncertainty product for a large class of time frequency distributions that satisfy the usual uncertainty principle including the wigner distribution the choi williams distribution and many other commonly used distributions we obtain an expression for the local uncertainty product in terms of the signal and show that for these distributions the local uncertainty product is less than that of the spectrogram and can be arbitrarily small extension of our approach to an entropy formulation of the uncertainty principle is also considered

radial function based kernel design for time frequency distributions a framework based on the n dimensional fourier transform of a radially symmetric function is introduced to design kernels for cohen time frequency distributions under this framework we derive a kernel formula which generalizes and unifies margenau hill born jordan and bessel distributions using a realization based on a n dimensional radial delta function the higher order radial kernels suppress more cross term energy compared with existing lower order kernels which is illustrated by the time frequency analysis of atrial fibrillation from surface electro cardiogram data

auto term representation by the reduced interference distributions a procedure for kernel design an analysis of auto term presentation using the reduced interference distributions rid is done comparison with an ideal time frequency signal representation is taken as a basis for this analysis the following distributions are considered choi williams 1989 zao atlas marks born jordan sinc zhang sato see ibid vol 42 no 1 p 54 1994 butterworth spectrogram and the author s recently proposed s method for time frequency analysis various distributions produce different auto term shapes in all cases the condition for cross term reduction is contradictory to the condition for high auto term quality a procedure for designing a kernel that will produce the desired auto term shape is demonstrated an optimal kernel with respect to the auto term quality and cross term suppression is derived

volterra series representation of time frequency distributions this paper addresses a volterra series representation of bilinear or quadratic time frequency distributions that belong to cohen s class whereby the analogy of the bilinear class with a second order double volterra series is utilized in addition a different viewpoint for the bilinear kernel and a complementary interpretation concerning the quadratic time frequency distributions are provided

stft with adaptive window width based on the chirp rate an adaptive time frequency representation tfr with higher energy concentration usually requires higher complexity recently a low complexity adaptive short time fourier transform astft based on the chirp rate has been proposed to enhance the performance this method is substantially modified in this paper i because the wavelet transform used for instantaneous frequency if estimation is not signal dependent a low complexity astft based on a novel concentration measure is addressed ii in order to increase robustness to if estimation error the principal component analysis pca replaces the difference operator for calculating the chirp rate and iii a more robust gaussian kernel with time frequency varying window width is proposed simulation results show that our method has higher energy concentration than the other astfts especially for multicomponent signals and nonlinear fm signals also for if estimation our method is superior to many other adaptive tfrs in low signal to noise ratio snr environments

the use of a filter bank and the wigner ville distribution for time frequency representation we present a new method for time frequency representation which combines a filter bank and the wigner ville distribution wvd the filter bank decomposes a multicomponent signal into a number of single component signals before the wvd is applied cross terms as well as noise are reduced significantly whereas high time frequency concentration is attained properties of the proposed time frequency distribution tfd are investigated and the requirements for the filter bank to fulfil these are given the ability of the proposed non cohen s 1995 class tfd to reduce cross terms as well as noise as well as its ability to approximately reconstruct signals are illustrated by examples the results are compared with those from the wvd the choi williams 1989 distribution cwd and spectrogram

blind source separation based on time frequency signal representations blind source separation consists of recovering a set of signals of which only instantaneous linear mixtures are observed thus far this problem has been solved using statistical information available on the source signals this paper introduces a new blind source separation approach exploiting the difference in the time frequency t f signatures of the sources to be separated the approach is based on the diagonalization of a combined set of spatial t f distributions in contrast to existing techniques the proposed approach allows the separation of gaussian sources with identical spectral shape but with different t f localization properties the effects of spreading the noise power while localizing the source energy in the t f domain amounts to increasing the robustness of the proposed approach with respect to noise and hence improved performance asymptotic performance analysis and numerical simulations are provided

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

improving the readability of time frequency and time scale representations by the reassignment method in this paper the use of the reassignment method first applied by kodera gendrin and de villedary 1976 to the spectrogram is generalized to any bilinear time frequency or time scale distribution this method creates a modified version of a representation by moving its values away from where they are computed so as to produce a better localization of the signal components we first propose a new formulation of this method followed by a thorough theoretical study of its characteristics its practical use for a large variety of known time frequency and time scale distributions is then addressed finally some experimental results are reported to demonstrate the performance of this method

qualitative behavior of solutions to cross diffusion systems from population dynamics abstract a general class of cross diffusion systems for two population species in a bounded domain with no flux boundary conditions and lotka volterra type source terms is analyzed although the diffusion coefficients are assumed to depend linearly on the population densities the equations are strongly coupled generally the diffusion matrix is neither symmetric nor positive definite three main results are proved the existence of global uniformly bounded weak solutions their convergence to the constant steady state in the weak competition case and the uniqueness of weak solutions the results hold under appropriate conditions on the diffusion parameters which are made explicit and which contain simplified shigesada kawasaki teramoto population models as a special case the proofs are based on entropy methods which rely on convexity properties of suitable lyapunov functionals

robust wigner distribution with application to the instantaneous frequency estimation the wigner distribution wd produces highly concentrated time frequency tf representation of nonstationary signals it may be used as an efficient signal analysis tool including the cases of frequency modulated signals corrupted with the gaussian noise in some applications a significant amount of impulse noise is present then the wd fails to produce satisfactory results the robust periodogram has been introduced for spectral estimation of this kind of noisy signals it can produce good concentration for pure harmonic signals however it is not so efficient in the cases of signals with rapidly varying frequency this is the motivation for introducing the robust wd it is a reliable tf representation tool for wide class of nonstationary signals corrupted with impulse noise this distribution produces good accuracy of the instantaneous frequency if estimation using the huber 1981 loss function a generalization of the wd is presented it includes both the standard and the robust wd as special cases this distribution can be used for tf analysis of signals corrupted with a mixture of impulse and gaussian noise the presented theory is illustrated on examples including applications on the if estimation and time varying filtering of signals corrupted with a mixture of the gaussian and impulse noise the case study analysis of the if estimators accuracy based on the standard and the robust wd forms is performed in order to improve the if estimation a median filter is applied on the obtained if estimate

from a discrete model of chemotaxis with volume filling to a generalized patlak keller segel model we present a discrete model of chemotaxis whereby cells responding to a chemoattractant are seen as individual agents whose movement is described through a set of rules that result in a biased rand

the time frequency distributions of nonstationary signals based on a bessel kernel a kernel based on the first kind bessel function of order one is proposed to compute the time frequency distributions of nonstationary signals this kernel can suppress the cross terms of the distribution effectively it is shown that the bessel distribution the time frequency distribution using bessel kernel meets most of the desirable properties with high time frequency resolution a numerical alias free implementation of the distribution is presented examples of applications in time frequency analysis of the heart s sound and doppler blood flow signals are given to show that the bessel distribution can be easily adapted to two very different signals for cardiovascular signal processing by controlling a kernel parameter this distribution can be used to compute the time frequency representations of transient deterministic and random signals the study confirms the potentials of the proposed distribution in nonstationary signal analysis

time frequency distribution kernels using fir filter design techniques in this correspondence time frequency distribution tfd kernels are obtained using finite impulse response fir filter design methods namely the windowing method and the equiripple approximation method based on chebyshev criterion it is shown that the class of the window designed kernels are simple to obtain and can handle most time varying environments

comments on a high resolution quadratic time frequency distribution for multicomponent signals analysis for original paper see barkat and boashash ieee trans signal processing vol 49 p 2232 9 2001 it is shown that the time frequency distribution tfd proposed in the above paper is not well defined in the ordinary sense for power signals including the single tone sinusoid and it needs the introduction of generalized functions and transforms it is also shown that the proposed tfd does not satisfy the conditions cited by the authors of the paper to justify the claim that it has the instantaneous frequency property

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

interference suppression in the wigner distribution using fractional fourier transformation and signal synthesis a novel method for the suppression of cross terms in the time frequency domain is introduced first interference is identified using a fractional fourier transform based technique then auto terms are detected synthesized and subtracted from the original signal the process is repeated until all signal components are extracted finally the wigner distributions of pure auto terms are superimposed to yield a high readability representation

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

alias free generalized discrete time time frequency distributions a definition of generalized discrete time time frequency distribution that utilizes all of the outer product terms from a data sequence so that one can avoid aliasing is introduced the new approach provides 1 proper implementation of the discrete time spectrogram 2 correct evaluation of the instantaneous frequency of the underlying continuous time signal and 3 correct frequency marginal the formulation provides a unified framework for implementing members of cohen s class which was formulated in the continuous time domain some requirements for the discrete time kernel in the new approach are discussed in association with desirable distribution properties some experimental results are provided to illustrate the features of the proposed method

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

subspace analysis of spatial time frequency distribution matrices spatial time frequency distributions stfds have been previously introduced as the natural means to deal with source signals that are localizable in the time frequency domain previous work in the area has not provided the eigenanalysis of stfd matrices which is key to understanding their role in solving direction finding and blind source separation problems in multisensor array receivers the aim of this paper is to examine the eigenstructure of the stfd matrices we develop the analysis and statistical properties of the subspace estimates based on stfds for frequency modulated fm sources it is shown that improved estimates are achieved by constructing the subspaces from the time frequency signatures of the signal arrivals rather than from the data covariance matrices which are commonly used in conventional subspace estimation methods this improvement is evident in a low signal to noise ratio snr environment and in the cases of closely spaced sources the paper considers the music technique to demonstrate the advantages of stfds and uses it as grounds for comparison between time frequency and conventional subspace estimates

diffusion with interactions and collisions between coloured particles and the propagation of chaos one considers a system ofn particles on the real line which are of two different types colours their dynamics is given by a stochastic differential equation with constant diffusion part the drift felt by a particle of either type depends on the empirical measures of type 1 and 2 particles at every instant further a reflection condition is imposed so that particles of different type are not allowed to cross each other the article studies the vlasov mckean limit of the system asn propagation of chaos and an evolution equation for the limiting empirical measures is established from where in particular an equation for the separating front between the two types follos

numerical expansion methods for solving integral equations by interpolation and gauss quadrature rules in this paper we introduce a numerical method for solving linear integral equations the main idea based on interpolation for unknown function where is interpolated in the zeros of the chebyshev s polynomials next we use gauss quadrature rules as gauss chebyshev or clenshaw curtis the technique is very effective and simple specially for integral equations of first kind as fredholm s and volterra s types in the end for showing efficiency of this method we use numerical examples

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

time varying filtering and signal estimation using wigner distribution synthesis techniques the short time fourier transform stft the ambiguity function af and the wigner distribution wd are mixed time frequency signal representations that use fourier transform techniques to map a one dimensional function of time into a two dimensional function of time and frequency these mixed time frequency mappings have been used to analyze the local frequency characteristics of a variety of signals and systems although much work has also been done to develop stft and af synthesis algorithms that can be used to implement a variety of time varying signal processing operations no such synthesis techniques have thus far been developed for the wd in this paper a signal synthesis algorithm that works directly with the real valued high resolution wd will be derived examples of how this wd synthesis procedure can be used to perform time varying filtering operations or signal separation will be given

chebyshev expansion method for solving second and fourth order elliptic equations in this paper we introduce a new spectral method based on chebyshev polynomials for solving second and fourth order elliptic equations moreover the suggested method is applicable for a wide area of differential equations an explicit formula for the chebyshev polynomials in terms of arbitrary order of their derivatives is presented also a formula for the successive integration of chebyshev polynomials in terms of chebyshev polynomials is proved numerical results indicate that the suggested method is significantly more accurate than that based on the chebyshev tau method the present results are in satisfactory agreement with the exact solutions

performance of quadratic time frequency distributions as instantaneous frequency estimators general performance analysis of the shift covariant class of quadratic time frequency distributions tfds as instantaneous frequency if estimators for an arbitrary frequency modulated fm signal is presented expressions for the estimation bias and variance are derived this class of distributions behaves as an unbiased estimator in the case of monocomponent signals with a linear if however when the if is not a linear function of time then the estimate is biased cases of white stationary and white nonstationary additive noises are considered the well known results for the wigner distribution wd and linear fm signal and the spectrogram of signals whose if may be considered as a constant within the lag window are presented as special cases in addition we have derived the variance expression for the spectrogram of a linear fm signal that is quite simple but highly signal dependent this signal is considered in the cases of other commonly used distributions such as the born jordan and the choi williams distributions it has been shown that the reduced interference distributions outperform the wd but only in the case when the if is constant or its variations are small analysis is extended to the if estimation of signal components in the case of multicomponent signals all theoretical results are statistically confirmed

fast implementations of generalized discrete time frequency distributions cohen s class of time frequency distributions tfds have significant potential for the analysis of complex signals in order to evaluate the tfd of a signal using its samples discrete time tfds dtfds have been defined as the fourier transform of a smoothed discrete autocorrelation existing algorithms evaluate real valued dtfds using ffts of the conjugate symmetric autocorrelation although the computation required to smooth the autocorrelation is often greater than that for the fft there are no widely applicable fast algorithms for this part of the processing since the fft is relatively inexpensive downsampling is ineffective for reducing computation if the dtfd needs only to be evaluated at a few frequencies for each time instant the cost per time frequency sample can be extremely high the authors introduce two approaches for reducing the computation time of dtfds first they define approximations to real valued dtfds using spectrograms that admit fast space saving evaluations frequency downsampling reduces the computation time of these approximations next they define dtfds that admit fast evaluations over sparse sets of time frequency samples a single short time fourier transform is calculated in order for dtpd time frequency samples to be evaluated at an additional fixed cost per sample

a time frequency based approach to phase and phase synchrony estimation time varying phase synchrony is an important bivariate measure that quantifies the dynamics between nonstationary signals and has been widely used in many applications including chaotic oscillators in physics and multichannel electroencephalography recordings in neuroscience current state of the art in time varying phase estimation uses either the hilbert transform or the complex wavelet transform of the signals both of these methods have some major drawbacks such as the assumption that the signals are narrowband for the hilbert transform and the nonuniform time frequency resolution inherent to the wavelet analysis in this paper a new phase estimation method based on the rihaczek distribution and reduced interference rihaczek distribution belonging to cohen s class is proposed these distributions offer phase estimates with uniformly high time frequency resolution which can be used for defining time and frequency dependent phase synchrony properties of the phase estimator and the corresponding phase synchrony measure are evaluated both analytically and through simulations showing the effectiveness of the new measures compared to existing methods

mechanism of the cross terms in spectrograms using the ambiguity domain interpretation the authors explore the mechanism of the cross terms or interferences in spectrograms they emphasize that the spectrogram does in general have cross terms depending on the signal structure and the window used the cross terms may sometimes be conspicuous reduced or completely eliminated the cross term suppression in spectrograms is governed by the signal theoretic version of the uncertainty principle some experimental results are provided to illustrate the cross term mechanism in spectrograms

on the local frequency group shift and cross terms in some multidimensional time frequency distributions a method for multidimensional time frequency analysis presents an analysis of the representation of local frequency and group shift using multidimensional time frequency distributions in the second part of the correspondence the authors extend the analysis to the multicomponent signals and cross terms effects on the basis of that analysis an efficient method derived from the analysis of the multidimensional wigner distribution defined in the frequency domain is proposed this method provides some substantial advantages over the wigner distribution the well known cross terms effects are reduced or completely removed the oversampling of signals is shown to be unnecessary and the computation time can be significantly reduced as well the theory is illustrated by a two dimensional numerical example

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

on the derivation of reaction diffusion equations as limit dynamics of systems of moderately interacting stochastic processes we consider systems of moderately interacting particles which are divided into a finite number of different subpopulations and show that in the limit as the population size tends to infinity the empirical processes of the subpopulations converge to the solution of a system of reaction diffusion equations

about the entropic structure of detailed balanced multi species cross diffusion equations abstract this paper links at the formal level the entropy structure of a multi species cross diffusion system of shigesada kawasaki teramoto skt type cf 1 satisfying the detailed balance condition with the entropy structure of a reversible microscopic many particle markov process on a discretised space the link is established by first performing a mean field limit to a master equation over discretised space then the spatial discretisation limit is performed in a completely rigorous way this by itself provides a novel strategy for proving global existence of weak solutions to a class of cross diffusion systems

bilinear signal synthesis in array processing multiple source signals impinging on an antenna array can be separated by time frequency synthesis techniques averaging of the time frequency distributions tfds of the data across the array permits the spatial signatures of sources to play a fundamental role in improving the synthesis performance array averaging introduces a weighting function in the time frequency domain that decreases the noise levels reduces the interactions of the source signals and mitigates the crossterms this is achieved independent of the temporal characteristics of the source signals and without causing any smearing of the signal terms the weighting function may take noninteger values which are determined by the communication channel the source positions and their angular separations unlike the recently devised blind source separation methods using spatial tfds the proposed method does not require whitening or retrieval of the source directional matrix the paper evaluates the proposed method in terms of performance and computations relative to the existing source separation techniques based on quadratic tfds

a spectral collocation method based on integrated chebyshev polynomials for two dimensional biharmonic boundary value problems this paper reports a new spectral collocation method for numerically solving two dimensional biharmonic boundary value problems the construction of the chebyshev approximations is based on integration rather than conventional differentiation this use of integration allows i the imposition of the governing equation at the whole set of grid points including the boundary points and ii the straightforward implementation of multiple boundary conditions the performance of the proposed method is investigated by considering several biharmonic problems of first and second kinds more accurate results and higher convergence rates are achieved than with conventional differential methods

generation of higher order pseudospectral integration matrices a new explicit expression of the higher order pseudospectral integration matrices is presented using an explicit formula for computing iterated integrals of chebyshev polynomials applications to initial value problems boundary value problems linear integral and integro differential equations are presented the present numerical results are in satisfactory agreement with the exact solutions and show the advantages of this method to some other known methods

the self duality of discrete short time fourier transform and its applications the self duality of short time fourier transform stft is an elegant property and is useful in shedding light on the construction of stft and its resolution capability in this paper the discrete version of self duality is studied and the property is interpreted in the context of resolution capabilities of time frequency distributions in addition two applications are provided as showcases of these insights obtained from the interpretation in the first application the problem of stft synthesis is considered and self duality serves as an important indication of whether the synthesis problem at hands is properly formulated in the second application a new kind of high resolution time frequency distribution is constructed based on the understandings obtained by contrasting two of the most popular time frequency analysis tools namely the stft and the wigner distribution

estimation of time varying velocities of moving objects by time frequency representations an approach to the estimation of motion parameters of moving objects in a video sequence by using the slide subspace based line detection algorithm algorithm is considered the proposed procedure projects video frames to the coordinate axes in order to obtain synthetic images containing information about the motion parameters these synthetic images are mapped to the fm signals by using constant spl mu propagation the problem of velocity estimation is reduced to the instantaneous frequency if estimation if estimators based on time frequency tf representations are used three tf representations spectrogram spec wigner distribution wd and s method sm are used and compared to this aim a tradeoff between concentration of the tf representation velocity estimation accuracy and reduction of the cross terms possibility for estimation of the multiple objects parameters is achieved by the sm a performance analysis of the algorithm is done theoretical results are illustrated on several numerical examples

optimum smoothing of the wigner ville distribution a compromise is found between the different requirements that we would like to be fulfilled by a time frequency distribution namely positivity and obtention of a distribution close to the dirac one for the unimodular signal s t exp i phi t the fulfillment of the marginal conditions being of less interest in signal theory starting from the usual wigner ville distribution we define an optimum smoothing by minimizing the width of the different functions approximating the desired dirac distribution the smoothing is obtained by a convolution through a double gaussian of width t and such that t 1 2 two possibilities appear in the first one we do not introduce any correlation between t and in the convolution kernel and obtain a simple result in the second one extrapolating the frequency variation and still using a gaussian we obtain a better result although the smoothing process becomes more complex these results to be physically meaningful impose inequalities on the successive derivatives of which are equivalent to those used for the obtention of the classical limit for the corresponding quantum problem

uniform convergence of proliferating particles to the fkpp equation abstract in this paper we consider a system of brownian particles with proliferation whose rate depends on the empirical measure the dependence is more local than a mean field one and has been called moderate interaction by oelschlager 16 17 we prove that the empirical process converges uniformly in the space variable to the solution of the fisher kolmogorov petrowskii piskunov equation we use a semigroup approach which is new in the framework of these systems and is inspired by some literature on stochastic partial differential equations

chebyshev pseudospectral method for computing numerical solution of convection diffusion equation abstract a method for computing highly accurate numerical solutions of 1d convection diffusion equations is proposed in this method the equation is first discretized with respect to the spatial variable transforming the original problem into a set of ordinary differential equations and then the resulting system is integrated in time by the fourth order runge kutta method spatial discretization is done by using the chebyshev pseudospectral collocation method before describing the method we review a finite difference based method by salkuyeh d khojasteh salkuyeh on the finite difference approximation to the convection diffusion equation appl math comput 179 2006 79 86 and contrary to the proposal of the author we show that this method is not suitable for problems involving time dependent boundary conditions which calls for revision stability analysis based on pseudoeigenvalues to determine the maximum time step for the proposed method is also carried out superiority of the proposed method over a revised version of salkuyeh s method is verified by numerical examples

the errors in calculating the pseudospectral differentiation matrices for cebysev gauss lobatto points abstract we discuss here the errors incurred using the standard formula for calculating the pseudospectral differentiation matrices for cebysev gauss lobatto points we propose explanations for these errors and suggest more precise methods for calculating the derivatives and their matrices

a fluctuation theorem for moderately interacting diffusion processes we study the asymptotic behaviour of the fluctuations of the empirical processes of the second example of moderately interacting diffusion processes of 3 and show that in the limit as the number of different processes tends to infinity these fluctuations behave gaussian

a stochastic model of a chemical reaction with diffusion stochastic systems of brownian motions with multiple deletion of particles are introduced to model a chemical reaction with diffusion convergence to the solution of a deterministic nonlinear reaction diffusion equation is proved without high density assumptions

a high resolution quadratic time frequency distribution for multicomponent signals analysis the paper introduces a new kernel for the design of a high resolution time frequency distribution tfd we show that this distribution can solve problems that the wigner ville distribution wvd or the spectrogram cannot in particular the proposed distribution can resolve two close signals in the time frequency domain that the two other distributions cannot moreover we show that the proposed distribution is more accurate than the wvd and the spectrogram in the estimation of the instantaneous frequency of a stepped fm signal embedded in additive gaussian noise synthetic and real data collected from real world applications are shown to validate the proposed distribution

minimum variance time frequency distribution kernels when dealing with random processes reduced spectrum estimate variance becomes an important property that augments the list of desirable time frequency t f distribution properties in this correspondence we derive the t f kernel that satisfies the t f constraints and provides the minimum variance for the power spectrum estimate for gaussian white noise processes

high density limit theorems for nonlinear chemical reactions with diffusion the solutionx of a nonlinear reaction diffusion equation on then dimensional unit cubes is approximated by a space time jump markov processx v n law of large numbers lln x v n is constructed on a grids n ons ofn cells wherev is proportional to the initial number of particles in each cell the deviation ofx v n fromx is computed by a central limit theorem clt the assumptions on the parametersv n are for the lln asn and for the clt frac n upsilon to 0 asn the limity y x in the clt which is a generalized ornstein uhlenbeck process is represented as the mild solution of a linear stochastic partial differential equation spde and its best possible state spaces are described the problem of stationary solutions ofy x in dependence ofx is also investigated

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed