time frequency distributions with complex argument a distribution highly concentrated along the group delay or the instantaneous frequency if is presented it has been defined by introducing a signal with a complex argument in time frequency tf analysis realization of a signal with a complex argument using a signal with a real argument is described the reduced interference realization of the complex argument distribution in the case of multicomponent signals is presented the proposed distribution is used for the if estimation it is shown that the estimation can be improved with respect to the wigner distribution based one since the bias can be significantly reduced with only a slight increase of the variance the theory is illustrated by examples

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

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

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

a function of time frequency lag and doppler in signal processing four functions of one variable are commonly used they are the signal in time the spectrum the auto correlation function of the signal and the auto correlation function of the spectrum the variables of these functions are denoted respectively as time frequency lag and doppler in time frequency analysis these functions of one variable are extended to quadratic functions of two variables in this paper we investigate a method for creating quartic functions of three of these variables as well as a quartic function of all four variables these quartic functions provide a meaningful representation of the signal that goes beyond the well known quadratic functions the quartic functions are applied to the design of signal adaptive kernels for the cohen class and shown to provide improvements over previous methods

cone kernel representation versus instantaneous power spectrum the cone kernel representation ckr and the instantaneous power spectrum ifs are two time frequency representations tfrs where the cross terms are localized in the region of auto terms exploring their relationship for discrete sequences we show that the ifs kernel is only two extreme terms of the ckr kernel further comparing other properties of these two tfrs viz invertibility aliasing and noise robustness we show that there is aliasing in some transform domains for signals sampled at nyquist rate and that the ckr has better noise robustness than the ifs

a simple scheme for adapting time frequency representations we propose a simple efficient technique for continuously adapting time frequency and time scale representations over time the procedure computes a short time quality measure of the representation for a range of values of a free parameter and estimates the optimal parameter value maximizing the quality measure via interpolation

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

the integer cohomology algebra of toric arrangements abstract we compute the cohomology ring of the complement of a toric arrangement with integer coefficients and investigate its dependency from the arrangement s combinatorial data to this end we study a morphism of spectral sequences associated to certain combinatorially defined subcomplexes of the toric salvetti category in the complexified case and use a technical argument in order to extend the results to full generality as a byproduct we obtain a combinatorial version of brieskorn s lemma in terms of salvetti complexes of complexified arrangements a uniqueness result for realizations of arithmetic matroids with at least one basis of multiplicity 1

modification of the ici rule based if estimator for high noise environments the nonparametric algorithm for instantaneous frequency if estimation based on the intersection of the confidence intervals ici rule and the wigner distribution wd is modified in order to produce an accurate if estimate for a high noise environment the original approach is developed under the assumption that a small amount of noise can move the wd maxima only within the auto term this is not true for high noise environments the probability that the wd maxima are outside the auto term is high for narrow windows used in the wd calculation estimates obtained with these windows are used as an initial guess in the adaptive algorithm in this paper we set the initial estimate as the one produced by the narrowest window for which the probability of error due to high noise is smaller than a threshold the error probability is estimated based on the estimates of signal amplitude and noise variance the if estimates for some windows are additionally improved by applying a median filter directly to the if estimate

double dimensional distributions another approach to quartic distributions we present the definitions of double dimensional time frequency distributions especially the wigner distribution wd of the wd and the ambiguity function of the ambiguity function the first coincides with the notion of the local ambiguity function presented in the papers of o neill and williams 1999 and o neill and flandrin 2000 and are classified as the quartic time frequency distribution the second turned out to be the fourier transform of the quartic ambiguous ambiguity function using the above notions the double dimensional extension of the cohen s class distributions is defined and illustrated by 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

enhancing the resolution of the spectrogram based on a simple adaptation procedure this work is concerned with improving the quality of signal localization for the short time fourier transform by properly adjusting the size of its analysis window over time the adaptation procedure involves the estimation of an area in the time frequency plane which is more compact than the support of the fixed window spectrogram then at each time instant the optimal window is selected such that the signal energy is maximized within the identified area the proposed method achieves its objectives and can compare favorably with alternative time adaptive spectrograms as well as with advanced quadratic representations

spectral decomposition of seismic data with continuous wavelet transform this paper presents a new methodology for computing a time frequency map for nonstationary signals using the continuous wavelet transform cwt the conventional method of producing a time frequency map using the short time fourier transform stft limits time frequency resolution by a predefined window length in contrast the cwt method does not require preselecting a window length and does not have a fixed time frequency resolution over the timefrequency space cwt uses dilation and translation of a wavelet to produce a time scale map a single scale encompasses a frequency band and is inversely proportional to the time support of the dilated wavelet previous workers have converted a time scale map into a time frequency map by taking the center frequencies of each scale we transform the time scale map by taking the fourier transform of the inverse cwt to produce a time frequency map thus a time scale map is converted into a time frequency map in which the amplitudes of individual frequencies rather than frequency bands are represented we refer to such a map as the time frequency cwt tfcwt we validate our approach with a nonstationary synthetic example and compare the results with the stft and a typical cwt spectrum two field examples illustrate that the tfcwt potentially can be used to detect frequency shadows caused by hydrocarbons and to identify subtle stratigraphic features for reservoir characterization

a quantitative study on the stability of quadratic delay difference systems this paper estimates the size of the stability region around zero for quadratic delay difference systems when the initial disturbance is in the asymptotic stability region the solution of the initial value problem of the quadratic delay difference system approaches zero examples are given for a three dimensional system and three one dimensional equations to demonstrate both stability and instability examples 2 4 show that when parameters in the systems do not satisfy the stability conditions the zero solutions can be unstable three evolution features of initial disturbances are shown numerically decaying to zero being amplified but bounded and growing to infinity example 3 further shows that the stable zero solution may not be a global attractor numerical results confirm the conclusions of the main theorem in this paper and imply that our estimation of the size of the stability region are of reasonable accuracy 1998 elsevier science ltd all rights reserved

superposition frames for adaptive time frequency analysis and fast reconstruction in this paper we introduce a broad family of adaptive linear time frequency representations termed superposition frames and show that they admit desirable fast overlap add reconstruction properties akin to standard short time fourier techniques this approach stands in contrast to many adaptive time frequency representations in the existing literature which while more flexible than standard fixed resolution approaches typically fail to provide for efficient reconstruction and often lack the regular structure necessary for precise frame theoretic analysis our main technical contributions come through the development of properties which ensure that our superposition construction provides for a numerically stable invertible signal representation our primary algorithmic contributions come via the introduction and discussion of two signal adaptation schemes based on greedy selection and dynamic programming respectively we conclude with two short enhancement examples that serve to highlight potential applications of our approach

a high resolution data adaptive time frequency representation a data adaptive time frequency representation is developed that overcomes some limitations of the short time fourier transform while avoiding the cross terms that make the wigner distribution and other bilinear representations difficult to interpret the adaptive time frequency representations uses gaussian basis functions but varies their time width and chirp rate with time and frequency to achieve high signal concentration everywhere a measure of local signal concentration allows fully automated determination of the optimal basis parameters the adaptive method is computationally expensive but may provide much better performance than any currently known technique

an adaptive optimal kernel time frequency representation time frequency representations with fixed windows or kernels figure prominently in many applications but perform well only for limited classes of signals representations with signal dependent kernels can overcome this limitation however while they often perform well most existing schemes are block oriented techniques unsuitable for on line implementation or for tracking signal components with characteristics that change with time the time frequency representation developed in the present paper based on a signal dependent radially gaussian kernel that adapts over time surmounts these difficulties the method employs a short time ambiguity function both for kernel optimization and as an intermediate step in computing constant time slices of the representation careful algorithm design provides reasonably efficient computation and allows on line implementation certain enhancements such as cone kernel constraints and approximate retention of marginals are easily incorporated with little additional computation while somewhat more expensive than fixed kernel representations this new technique often provides much better performance several examples illustrate its behavior on synthetic and real world signals

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

tomography time frequency transform the paper shows that the fractional fourier transform frft of a signal is the radon transform of the time frequency distribution of the same signal therefore a time frequency distribution known as the tomography time frequency transform ttft is defined as the inverse radon transform of the frft of the signal because the computation of the ttft does not explicitly require any window or kernel function high resolutions in both the frequency and time domains can be achieved when the signal contains multiple components the cross terms can be effectively removed by an adaptive filtering process that is applied on the frft rather than the final result therefore distortions made by the filtering process on the desired signal components can be minimized

hybrid linear quadratic time frequency attributes we present an efficient method for robustly calculating time frequency attributes of a signal including instantaneous mean frequency bandwidth kurtosis and other moments most current attribute estimation techniques involve a costly intermediate step of computing a highly oversampled two dimensonal 2 d quadratic time frequency representation tfr which is then collapsed to the one dimensonal 1 d attribute using the principles of hybrid linear quadratic time frequency analysis time frequency distribution series we propose computing attributes as nonlinear combinations of the slightly oversampled linear gabor coefficients of the signal the method is both computationally efficient and accurate it performs as well as the best techniques based on adaptive tfrs to illustrate we calculate an attribute of a seismic cross section

parameter identification using mollification for predator prey models in spatially heterogeneous environments a stable numerical method is introduced for the identification of population densitiesand spatially dependent diffusion coefficients in a lotka volterra predator prey model in which bias dispersal is present in this problem noisy data at the active boundary and initial measured population densities are given this algorithm makes no assumption on the amount and or character of the noise in the data

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

analysis and synthesis of multicomponent signals using positive time frequency distributions a new approach to the analysis and reconstruction of multicomponent nonstationary signals from their time frequency distribution tfd is presented specifically we consider a tfd based on the recently introduced minimum cross entropy principle mce this positive tfd is cross terms free and hence has an advantage over the family of bilinear distributions based on the mce tfd a new algorithm for reconstructing the phase and amplitude parameters of each component of the signal is developed to evaluate the accuracy of the algorithm monte carlo simulations are presented and compared with the corresponding cramer rao bound it is shown that the new algorithm is superior to presently available methods in both efficiency and performance it is concluded that together with the mce tfd representation the proposed approach provides a powerful tool for analysis of nonstationary multicomponent signals embedded in additive gaussian noise

parameter estimation for a drying system in a porous medium we introduce a stable numerical method for the recovery of the temperature and moisture distributions in a liukov system with space dependent diffusion coefficients in this problem only cauchy noisy data at the active boundary is given and no information about the amount and or character of the noise in the data is assumed the error analysis for the algorithm is discussed and a numerical example is presented

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

computationally lag invariant recursive spectrum estimators this paper is concerned with fourier based nonparametric spectral estimation methods for time varying environments it presents a class of interactive spectrum estimators in which the number of computations required to update the estimate at every data sample is independent of the number of the autocorrelation lags employed to produce this estimate these estimators are termed lag invariant and form an important class due to their computational simplicity because of the lag invariant property resolution enhancement is achieved without additional cost in computations per frequency sample the proposed class of recursive spectral estimators is based on specific selection of the lagged product windows used in the autocorrelation function estimation in this paper the conditions necessary to produce the lag invariance property are delineated

characteristic polynomials of subspace arrangements and finite fields let a be any subspace arrangement in rndefined over the integers and let fqdenote the finite field withqelements letqbe a large prime we prove that the characteristic polynomial a q of a counts the number of points in fnqthat do not lie in any of the subspaces ofa viewed as subsets of fnq this observation which generalizes a theorem of blass and sagan about subarrangements of the bnarrangement reduces the computation of a q to a counting problem and provides an explanation for the wealth of combinatorial results discovered in the theory of hyperplane arrangements in recent years the basic idea has its origins in the work of crapo and rota 1970 we find new classes of hyperplane arrangements whose characteristic polynomials have simple form and very often factor completely over the nonnegative integers

seismic sequence analysis and attribute extraction using quadratic time frequency representations the variation of frequency content of a seismic trace with time carries information about the properties of the subsurface reflectivity sequence as a result analysis of the data in terms of the local frequency content can provide a worthwhile addition to the standard procedures that are used in seismo stratigraphic interpretation the theory of quadratic time frequency representations provides a solid foundation for local frequency analysis of seismic data and seismic attribute extraction two applications of the quadratic time frequency representations are demonstrated seismic sequence analysis and seismic attribute extraction the joint time frequency representation of a seismic reflection pattern is often much more easily interpreted in terms of subsurface stratification than the time or frequency domain description alone we show how the time frequency representation can be used to delineate seismic sequences on the basis of the time frequency characteristics of the signal there exists a close rel

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

a four parameter atomic decomposition of chirplets a new four parameter atomic decomposition of chirplets is developed for compact and precise representation of signals with chirp components the four parameter chirplet atom is obtained from the unit gaussian function by successive applications of scaling fractional fourier transform frft and time shift and frequency shift operators the application of the frft operator results in a rotation of the wigner distribution of the gaussian in the time frequency plane by a specified angle the decomposition is realized by using the matching pursuit algorithm for this purpose the four parameter space is discretized to obtain a small but complete subset in the hilbert space a time frequency distribution tfd is developed for clear and readable visualization of the signal components it is observed that the chirplet decomposition and the related tfd provide more compact and precise representation of signal inner structures compared with the commonly used time frequency representations

time frequency representation based on an adaptive short time fourier transform in this paper a new concise algorithm about time frequency representation tfr based on an adaptive short time fourier transform astft is presented in this algorithm the analysis window width is equal to the local stationary length which is measured by the instantaneous frequency gradient ifg of the signal and the instantaneous frequency if of the signal is obtained by detecting the ridge of wavelet transform wt the astft provides much better performance than conventional tfr algorithms furthermore the algorithm is simpler and more computational efficient than some of other adaptive tfr algorithms proposed previously several examples are presented to illustrate its behavior on different kinds of signals and demonstrate its validity

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

lower bound on snr of the nth order hyperbolic time frequency kernel abstract an estimate of the lower bound on signal to noise ratio snr of the nth order hyperbolic time frequency kernel is given the effects of kernel parameters such as n and a on the snr are discussed in particular the direct relationship between the snr and auto term slope a is studied in detail conditions under which the lower bound on snr is obtained are derived preliminary observations on a transfer function model with the auto term slope a and as inputs and lower bound on snr as output are given possible further work is outlined

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

nonparametric detection of fm signals using time frequency ridge energy in many practical applications signals to be detected are unknown nonlinear frequency modulated fm and are corrupted by strong noise the phase histories of signals are assumed to be unknown smooth functions of time and these functions are poorly modeled or unmodeled by a small number of parameters thus the conventional parametric based detection methods are invalid in these cases this paper proposes a nonparametric detection method using the ridge energy of observations the detection process consists of three steps tf ridge detection ridge energy extraction and decision first the directionally smoothed pseudo wigner ville distribution dspwvd is introduced to highlight the instantaneous frequency if points along a special direction on the if curve of a signal from noise further an angular maximal distribution amad is constructed from a set of dspwvds to highlight the entire if curve as a result the tf ridge of an observation can be estimated well from its amad by the maxima position detector second the ridge energy the total energy along the tf ridge on the pseudo wigner ville distribution pwvd is extracted a noisy signal has larger ridge energy than a pure noise does with a large probability because pure noise energy is randomly distributed throughout the tf plane while the signal energy in a noisy signal is concentrated along the estimated tf ridge third the ridge energy of an observation is used as the test statistic to decide whether or not a signal of interest is present in the observation where the decision threshold is determined by a large number of monte carlo simulations using pure noise finally the simulation experiments to two test signals are made to verify the effectiveness of the proposed method

discrete mollification and automatic numerical differentiation abstract an automatic method for numerical differentiation based on discrete mollification and the principle of generalized cross validation is presented with data measured at a discrete set of points of a given interval the method allows for the approximate recovery of the derivative function on the entire domain no information about the noise is assumed error estimates are included together with several numerical examples of interest

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

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

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

a signal dependent time frequency representation fast algorithm for optimal kernel design a time frequency representation based on an optimal signal dependent kernel has been previously been proposed in an attempt to overcome one of the primary limitations of bilinear time frequency distributions that the best kernel and distribution depend on the signal to be analyzed the optimization formulation for the signal dependent kernel results in a linear program with a unique feature a tree structure that summarizes a set of constraints on the kernel the authors present a fast algorithm based on sorting to solve a special class of linear programs that includes the problem of interest for a kernel with q variables the running time of the algorithm is o q log q which is several orders of magnitude less than any other known method for solving this class of linear program this efficiency enables the computation of the signal dependent optimal kernel time frequency representation at a cost that is on the same order as a fixed kernel distribution an important property of the optimal kernel is that it takes on essentially only the values of 1 and 0

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

robust l estimation based forms of signal transforms and time frequency representations the l estimation based signal transforms and time frequency tf representations are introduced by considering the corresponding minimization problems in the huber 1981 1998 estimation theory the standard signal transforms follow as the maximum likelihood solutions for the gaussian additive noise environment for signals corrupted by an impulse noise the median based transforms produce robust estimates of the non noisy signal transforms when the input noise is a mixture of gaussian and impulse noise the l estimation based signal transforms can outperform other estimates in quadratic and higher order tf analysis the resulting noise is inherently a mixture of the gaussian input noise and an impulse noise component in this case the l estimation based signal representations can produce the best results these transforms and tf representations give the standard and the median based forms as special cases a procedure for parameter selection in the l estimation is proposed the theory is illustrated and checked numerically

fast computation of the ambiguity function and the wigner distribution on arbitrary line segments by using the fractional fourier transformation of the time domain signals closed form expressions for the projections of their auto or cross ambiguity functions are derived based on a similar formulation for the projections of the auto and cross wigner distributions and the well known two dimensional 2 d fourier transformation relationship between the ambiguity and wigner domains closed form expressions are obtained for the slices of both the wigner distribution and the ambiguity function by using discretization of the obtained analytical expressions efficient algorithms are proposed to compute uniformly spaced samples of the wigner distribution and the ambiguity function located on arbitrary line segments with repeated use of the proposed algorithms samples in the wigner or ambiguity domains can be computed on non cartesian sampling grids such as polar grids

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

the multivariate arithmetic tutte polynomial we introduce an arithmetic version of the multivariate tutte polynomial recently studied by sokal and a quasi polynomial that interpolates between the two we provide a generalized fortuin kasteleyn representation for representable arithmetic matroids with applications to arithmetic colorings and flows we give a new proof of the positivity of the coefficients of the arithmetic tutte polynomial in the more general framework of pseudo arithmetic matroids in the case of a representable arithmetic matroid we provide a geometric interpretation of the coefficients of the arithmetic tutte polynomial

theory and applications of adaptive constant q distributions a new class of signal adaptive time frequency representations called the adaptive constant q distribution aqd is introduced the aqd exploits a priori knowledge about a signal s instantaneous frequency and bandwidth to perform signal dependent smoothing of the wigner distribution the objective is to achieve a good tradeoff between reducing the variance and preserving the resolution by means of time frequency dependent smoothing specifically for use in medical doppler ultrasound a numerical alias free implementation of the aqd is presented deterministic multicomponent signals as well as synthetic doppler ultrasound signals were analyzed with the aqd the performance of the aqd was compared with the power spectrogram the exponential distribution and the adaptive optimum kernel representation as well as with theory the error was consistently lowest for the aqd in conclusion a new signal adaptive class of time frequency distributions has been developed and its potential in nonstationary signal analysis has been demonstrated

multiple window time varying spectral analysis a multiwindow method for generating a time varying spectrum of nonstationary signals is presented the time varying spectrum is computed from an optimally weighted average of multiple orthogonal windowed spectrograms the weights are determined using linear least squares estimation with respect to a reference time frequency distribution examples are provided with performance criteria measures to demonstrate and quantify the effectiveness of the method

time frequency based detection using discrete time discrete frequency wigner distributions during the last decade a comprehensive theory for optimum time frequency tf based detection has been developed this was originally proposed in the continuous time continuous frequency case this paper deals with detectors operating on discrete time discrete frequency wigner distributions wds the purpose is to discuss some existing definitions of this distribution within the context of tf based detection and selecting those that do not affect the performance of the decision device with which they are associated this question is of interest since there exist several approaches for discretizing the wd sometimes resulting in a loss of fundamental properties first the discrete time discrete frequency formulations of optimum detection are investigated next the problem of the design of tf based detectors from training data keeping in mind severe effects of the curse of dimensionality is considered

polyhedra and parameter spaces for matroids over valuation rings abstract in this paper we address two of the major foundational questions in the theory of matroids over rings first we provide a cryptomorphic axiomatisation by introducing an analogue of the base polytope for matroids second we describe a parameter space for matroids over a valuation ring which turns out to be a tropical version of the bott samelson varieties for the full flag variety thus a matroid over a valuation ring is a sequence of flags of tropical linear spaces a k a valuated matroids

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 comparison between different discrete ambiguity domain definitions in stochastic time frequency analysis the ambiguity domain plays a central role in estimating the time varying spectrum and in estimating the covariance function of nonstationary random processes in continuous time for processes in discrete time there exist different definitions of the ambiguity domain but it is well known that neither of these definitions perfectly resembles the usefulness of the continuous ambiguity domain in this paper we present some of the most frequently used definitions of the ambiguity domain in discrete time the claasen mecklenbrauker the jeong williams and the nuttall definitions for the first time we prove their equivalence within some necessary conditions and we present theorems that justify their usage

on noise robustness of the mexican hat and hyperbolic wavelets with an experiment on chaos detection this paper studies the noise robustness of laplace of hyperbolic loh and laplace of gaussian log wavelets a typical example of using the loh and log to study duffing oscillation under normal and noisy conditions is given relative merits of each method are discussed

use of the cross wigner ville distribution for estimation of instantaneous frequency an iterative instantaneous frequency if estimation scheme is presented in which successive if estimates are obtained from the peak of the cross wigner ville distribution xwvd using a reference signal synthesized from an initial if estimate theoretical and practical aspects of performance are discussed and the performance is compared with that of other methods

arithmetic matroids the tutte polynomial and toric arrangements abstract we introduce the notion of an arithmetic matroid whose main example is a list of elements of a finitely generated abelian group in particular we study the representability of its dual providing an extension of the gale duality to this setting guided by the geometry of generalized toric arrangements we provide a combinatorial interpretation of the associated arithmetic tutte polynomial which can be seen as a generalization of crapo s formula for the classical tutte polynomial

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

finite word length effects in implementation of distributions for time frequency signal analysis this correspondence presents an analysis of the finite register length influence on the accuracy of results obtained by the time frequency distributions tfds in order to measure quality of the obtained results the variance of the proposed model is found signal to quantization noise ratio snr is defined and appropriate expressions are derived floating and fixed point arithmetic are considered with the analysis of discrete random and discrete deterministic signals it is shown that commonly used reduced interference distributions rids exhibit similar performance with respect to the snr we have also derived the expressions establishing the relationship between the number of bits and the required quality of representation which is defined by the snr which may be used for register length design in hardware implementation of time frequency algorithms

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