a method for improved distribution concentration in the time frequency analysis of multicomponent signals using the l wigner distribution the energy location in the cohen class of time frequency distributions is analyzed if the instantaneous frequency is linear then only the wigner distribution produces the ideal energy concentration the scaled version of the wigner distribution l wigner distribution is used to improve the time frequency representation of signals with nonlinear instantaneous frequencies in the case of multicomponent signals the cross terms appearing in the wigner distribution and in the l wigner distribution can be easily removed or reduced in a computationally very efficient way the theory is illustrated on the numerical examples with multicomponent noisy signals

the beginning of the fucik spectrum for the p laplacian has a nontrivial solution u here 1 p 2p u div u p 2 u u max u 0 u uu in 1 1 denoting by 1 2 the first two eigenvalues of 2p on w 1 p 0 0 cf the end of this introduction 7p clearly contains in particular 1 1 2 2 and the two lines 1 r and r 1 this generalized notion of spectrum was introduced for p 2 i e in the linear case in the 1970s by fuc ik 21 and dancer 15 in connection article id jdeq 1999 3645 available online at http www idealibrary com on

multiple periodic solutions of ordinary differential equations with double resonance abstract in this paper we study by morse theory the existence of multiple periodic solutions of a class of ordinary differential equation with double resonance at infinity between two consecutive eigenvalues and with resonance at origin

instantaneous frequency estimation using the wigner distribution with varying and data driven window length the estimation of the instantaneous frequency if of a harmonic complex valued signal with an additive noise using the wigner distribution is considered if the if is a nonlinear function of time the bias of the estimate depends on the window length the optimal choice of the window length based on the asymptotic formulae for the variance and bias can be used in order to resolve the bias variance tradeoff however the practical value of this solution is not significant because the optimal window length depends on the unknown smoothness of the if the goal of this paper is to develop an adaptive if estimator with a time varying and data driven window length which is able to provide quality close to what could be achieved if the smoothness of the if were known in advance the algorithm uses the asymptotic formula for the variance of the estimator only its value may be easily obtained in the case of white noise and relatively high sampling rate simulation shows good accuracy for the proposed adaptive algorithm

dirichlet problems with double resonance and an indefinite potential abstract we consider semilinear dirichlet problems with an unbounded and indefinite potential and with a caratheodory reaction we assume that asymptotically at infinity the problem exhibits double resonance using variational methods together with morse theory and flow invariance arguments we prove multiplicity theorems producing three five six or seven nontrivial smooth solutions in most multiplicity theorems we provide precise sign information for all the solutions established

wigner distribution of noisy signals the effect of noise on the wigner distribution is analyzed the mean and variance of the wigner distribution of signals contaminated by additive noise are derived it is shown that in the case of white noise the wigner distribution calculated by definition calculated by definition cannot be used for estimation for estimation windowing in the time domain is not sufficient smoothing in the time domain is necessary very simple expressions for variance and bias estimation are obtained if the signal and noise are analytic only one window is sufficient to make the variance finite the results are demonstrated by numerical examples with linear and sinusoidal chirp pulses

periodic integrable systems with delta potentials in this paper we study root system generalizations of the quantum bose gas on the circle with pair wise delta function interactions the underlying symmetry structures are shown to be governed by the associated graded algebra of cherednik s suitably filtered degenerate double affine hecke algebra acting by dunkl type differential reflection operators we use gutkin s generalization of the equivalence between the impenetrable bose gas and the free fermi gas to derive the bethe ansatz equations and the bethe ansatz eigenfunctions

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

wigner ville spectral analysis of nonstationary processes the wigner ville spectrum has been recently introduced as the unique generalized spectrum for time varying spectral analysis its properties are revised with emphasis on its central role in the analysis of second order properties of nonstationary random signals we propose here a general class of spectral estimators of the wigner ville spectrum this class is based on arbitrarily weighted covariance estimators and its formal description corresponds to the general class of conjoint time frequency representations of deterministic signals with finite energy classical estimators like short time periodograms and the recently introduced pseudo wigner estimators are shown to be special cases of the general class the generalized framework allows the calculation of the moments of general spectral estimators and comparing the results emphasizes the versatility of the new pseudo wigner estimators the effective numerical implementation by an n point fft of pseudo wigner estimators of 2n points is indicated and various examples are given

nonparametric estimation of instantaneous frequency the local polynomial approximation of time varying phase is used in order to estimate the instantaneous frequency and its derivatives for a complex valued harmonic signal given by discrete time observations with a noise the considered estimators are high order nonparametric generalizations of the short time fourier transform and the wigner ville distribution the asymptotic variance and bias of the estimates are obtained

on the wigner distribution of discrete time noisy signals with application to the study of quantization effects the influence of noise on the wigner distribution wd of discrete time signals is analyzed signals contaminated by white noise are considered the expressions for the mean and variance of the wd estimator are derived it is then shown that the application of only one window is sufficient to make the variance of the estimator finite this is a significant difference from the analog case where two windows are needed the expression for the optimal window width is derived the results are illustrated by several numerical examples using the derived expressions the effects of finite register length on the wd are analyzed fixed and floating point arithmetic are considered

roundoff error analysis of the discrete wigner distribution using fixed point arithmetic the issue of roundoff noise effects in the implementation of the discrete wigner distribution using fixed point arithmetic is addressed the sign magnitude number representation is assumed throughout the analysis the measure of roundoff noise effects in an algorithm is the output noise to signal ratio using a statistical model an analytical expression of the noise to signal ratio is derived as a function of the wordlength b and the transform length n the noise to signal ratio is obtained by evaluating the signal and noise powers at different points in the algorithm then reflecting to the output both signal and noise powers based on the derived noise to signal ratio is is noted that if the transform length is doubled then one additional bit is required in the wordlength to maintain a constant noise to signal ratio it is demonstrated through the software simulations that the predicted noise to signal ratio is a good closed form estimate of the true roundoff error it is also found from the simulation that the wordlength b and the transform length n 2 sup v must satisfy the condition b v or 4

performance analysis of the adaptive algorithm for bias to variance tradeoff an algorithm for the mean squared error mse minimization through the bias to variance ratio optimization has been recently proposed and used in the literature this algorithm is based on the analysis of the intersection of confidence intervals icis the algorithm does not require explicit knowledge of the estimation bias for a near to optimal parameter estimation this paper presents a detailed analysis of the algorithm performances including procedures and relations that can be used for a fine adjustment of the algorithm parameters reliability of the algorithm is studied for various kinds of estimation noise results are confirmed on a simulated example with uniform gaussian and laplacian noise an illustration of the algorithm application on a simple filtering example is given

variational and non variational eigenvalues of the p laplacian it is well known that all the eigenvalues of the linear eigenvalue problem u q r u in rn can under appropriate conditions on q r and be characterized by minimax principles but it has been a long standing question whether that remains true for analogous equations involving the p laplacian p it will be shown that there are corresponding nonlinear eigenvalue problems pu q r u p 1sgnu in rn with 1 0 on for which not all eigenvalues are of variational type as far as we know this is the first observation of such a phenomenon and examples will be given for one and higher dimensional equations the question of exactly which eigenvalues are variational is also discussed when n 1

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

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

existence and multiplicity of solutions for nonlinear periodic problems with the scalar p laplacian and double resonance abstract we consider nonlinear periodic problems driven by the scalar p laplacian and with a caratheodory reaction which can exhibit double resonance at combining variational methods based on the critical point theory with morse theoretic techniques we show that we have existence when the double resonance occurs at any spectral interval and we have multiplicity with at least three nontrivial solutions when the double resonance occurs at any spectral interval distinct from the principal one 0 0 1

the spectrum of the periodic p laplacian we consider one dimensional p laplacian eigenvalue problems of the form pu q u p 1sgnu on 0 b together with periodic or separated boundary conditions where p 1 p is the p laplacian q c1 0 b and b 0 r it will be shown that when p 2 the structure of the spectrum in the general periodic case that is with q 0 and periodic boundary conditions can be completely different from those of the following known cases i the general periodic case with p 2 ii the periodic case with p 2 and q 0 and iii the general separated case with any p 1

minimum variance time frequency distribution kernels for signals in additive noise we derive the minimum variance time frequency distribution kernel for signals in additive circular complex gaussian white noise processes it is shown that the kernel that minimizes the average variance and simultaneously satisfies the time frequency constraints for the noise only ease remains optimum in the presence of frequency modulated fm signals

periodic solutions for second order differential inclusions with the scalar p laplacian we study periodic problems driven by the scalar p laplacian with a multivalued right hand side nonlinearity we prove two existence theorems in the first we assume nonuniform nonresonance conditions between two successive eigenvalues of the negative p laplacian with periodic boundary conditions in the second we employ certain landesman lazer type conditions our approach is based on degree theory

notes on the structure of the function interacting gas intertwining operator in the degenerate affine hecke algebra the wavefunction of the function interacting bose gas on the infinite interval is studied by introducing the intertwining operator of the degenerate affine hecke algebra the non symmetric eigenfunction of the dunkl operator is constructed

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

the wigner distribution of noisy signals with adaptive time frequency varying window time frequency representations using the wigner distribution wd may be significantly obscured by the noise in the observations the analysis performed for the wd of discrete time noisy signals shows that this time frequency representation can be optimized by the appropriate choice of the window length however the practical value of this analysis is not significant because the optimization requires knowledge of the bias which depends on the unknown derivatives of the wd a simple adaptive algorithm for the efficient time frequency representation of noisy signals is developed in this paper the algorithm uses only the noisy estimate of the wd and the analytical formula for the variance of this estimate the quality of this adaptive algorithm is close to the one that could be achieved by the algorithm with the optimal window length provided that the wd derivatives were known in advance the proposed algorithm is based on the idea that has been developed in our previous work for the instantaneous frequency if estimation here a direct addressing to the wd itself rather than to the instantaneous frequency resulted in a time and frequency varying window length and showed that the assumption of small noise and bias is no longer necessary a simplified version of the algorithm using only two different window lengths is presented it is shown that the procedure developed for the adaptive window length selection can be generalized for application on multicomponent signals with any distribution from the cohen 1989 1990 1992 class simulations show that the developed algorithms are efficient even for a very low value of the signal to noise ratio

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

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

further results on the minimum variance time frequency distribution kernels results for the minimum variance kernel presented by hearon and amin see ibid vol 43 p 1258 1995 for the complex gaussian white noise with independent real and imaginary parts remain valid for real noise and approximately valid for analytic noise these results are extended to the real and analytic noisy signals cases

a method for time frequency analysis a method for time frequency signal analysis is presented the proposed method belongs to the general class of smoothed pseudo wigner distributions it is derived from the analysis of the wigner distribution defined in the frequency domain this method provides some substantial advantages over the wigner distribution the well known cross term effects are reduced or completely removed the oversampling of signal is not necessary in addition the computation time can be significantly shorter the results are demonstrated on two numerical examples with frequency modulated signals

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