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 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

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

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

noncommuting contractions of oscillators with constant force we find two noncommuting contractions of the lie algebras u n and gl n realized as the symmetry algebras of n dimensional isotropic harmonic and repulsive oscillators of spring constant k with a constant force of magnitude f the contraction limit to the symmetry algebra of the ndimensional free system is k f 0 0 we take two paths in this plane determined by the order of contraction of the two parameters and show that they yield two closely related but distinct euclidean type symmetry algebras for the common contracted system we also show briefly how the wavefunctions of the one dimensional harmonic oscillator reduce to plane waves along the above two paths

gradings and contractions of affine kac moody algebras we study contractions of affine kac moody algebras kmas with respect to their twisted and untwisted kac moody subalgebras and also contract twisted kma with respect to their affine subalgebras our work generalizes the inonu wigner contraction of kma by majumdar j math phys 34 2059 1993 in several directions we compare inonu wigner contractions with respect to twisted subalgebras to another kind of contractions which we call modulo contractions we also introduce a novel type of contractions based on z gradings of kma which we call level contractions along the way we use an algorithm for constructing kac moody like algebras g for any and not just semisimple finite dimensional lie algebra g

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

the development of a discrete transform for the wigner distribution and ambiguity function a theoretical basis for the development of the discrete wigner distribution and ambiguity function is presented that is based on the temporal and spectral properties of the two dimensional wigner kernel function this two dimensional description is helpful in visualizing the various functions and shows that the increase in the number of points required to compute the discrete wigner distribution is a natural consequence of the definition of the kernel and that it does not contradict the nyquist sampling theorem a generalized set of pseudofunctions is introduced that involves applying a two dimensional weighting function to the wigner kernel or its two dimensional fourier transform resulting in a wigner distribution that is weighted or smoothed in time or frequency two forms of the wigner distribution are derived the first is a pure wigner distribution without any windowing of the kernel that may be computed over the entire data set the second involves the general set of pseudo wigner distribution that may be calculated over a subset of the data set the discrete wigner distribution of the analytic signal is also developed and it is shown that the number of points may be reduced by a factor of 2 in the same way that the number of points required to represent the discrete analytic signal may be reduced the discrete ambiguity function is developed along with the discrete wigner distribution and it is shown that the four two dimensional functions are interrelated by discrete one and two dimensional fourier transforms

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

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

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

discrete time implementation of the cone kernel time frequency representation subtle differences exist between the continuous time and discrete time formulations of the cone kernel time frequency representation that should be taken into account prior to implementation the article derives the discrete formulation from equivalence in the ambiguity domain yielding a kernel different from that obtained by directly sampling the continuous time kernel yet preserving the important properties of the continuous time representation

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

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

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

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

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

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

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

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

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

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

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 unified definition for the discrete time discrete frequency and discrete time frequency wigner distributions various discrete definitions of the wigner distribution wd for discrete time signals have been proposed in previous works the formulation developed in this paper leads to natural and unified definitions of discrete versions of the wd they are directly related to the continuous and preserve most of its properties the discretization is first considered in the time domain dtwd in the frequency domain dfwd and then in both domains simultaneously dtfwd in each case the aliasing problem is studied and generalized interpolation formulas allowing the reconstruction of the continuous wd are derived the dtfwd is particulary relevant for computer implementation of the wd

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

shift covariant time frequency distributions of discrete signals many commonly used time frequency distributions are members of the cohen 1989 class this class is defined for continuous signals and since time frequency distributions in the cohen class are quadratic the formulation for discrete signals is not straightforward the cohen class can be derived as the class of all quadratic time frequency distributions that are covariant to time shifts and frequency shifts we extend this method to three types of discrete signals to derive what we call the discrete cohen classes the properties of the discrete cohen classes differ from those of the original cohen class to illustrate these properties we also provide explicit relationships between the classical wigner distribution and the discrete cohen classes

high spectral resolution time frequency distribution kernels a new class of time frequency distribution tfd kernels is introduced members in this class satisfy the desirable tfd properties and simultaneously provide local autocorrelation functions laf that are amenable to high frequency resolution modeling techniques it is shown that members of the proposed class are product kernels fast implementation multiplication free kernels recursive kernels and optimum kernels with respect to autoterm localization

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

graph colorings spaces of edges and spaces of circuits abstract by lovasz proof of the kneser conjecture the chromatic number of a graph g is bounded from below by the index of the z 2 space hom k 2 g plus two we show that the cohomological index of hom k 2 g is also greater than the cohomological index of the z 2 space hom c 2 r 1 g for r 1 this gives a new and simple proof of the strong form of the graph coloring theorem by babson and kozlov which had been conjectured by lovasz and at the same time shows that it never gives a stronger bound than can be obtained by hom k 2 g the proof extends ideas introduced by ivaljevic in a previous elegant proof of a special case we then generalize the arguments and obtain conditions under which corresponding results hold for other graphs in place of c 2 r 1 this enables us to find an infinite family of test graphs of chromatic number 4 among the kneser graphs our main new result is a description of the z 2 homotopy type of the direct limit of the system of all the spaces hom c 2 r 1 g in terms of the z 2 homotopy type of hom k 2 g a corollary is that the coindex of hom k 2 g does not exceed the coindex of hom c 2 r 1 g by more then one if r is chosen sufficiently large thus the graph coloring bound in the theorem by babson and kozlov is also never weaker than that from lovasz proof of the kneser conjecture

random denoising and signal nonlinearity approach by time frequency peak filtering using weighted frequency reassignment abstracttime frequency peak filtering tfpf may efficiently suppress random noise and hence improve the signal to noise ratio however the errors are not always satisfactory when applying the tfpf to fast varying seismic signals we begin with an error analysis for the tfpf by using the spread factor of the phase and cumulants of noise this analysis shows that the nonlinear signal component and non gaussian random noise lead to the deviation of the pseudo wigner ville distribution pwvd peaks from the instantaneous frequency the deviation introduces the signal distortion and random oscillations in the result of the tfpf we propose a weighted reassigned smoothed pwvd with less deviation than pwvd the proposed method adopts a frequency window to smooth away the residual oscillations in the pwvd and incorporates a weight function in the reassignment which sharpens the time frequency distribution for reducing the deviation because the weight function is determined by the lateral coherence of seismic

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 aliasing problem in discrete time wigner distributions there is no straightforward way to proceed from the continuous time wigner distribution to a discrete time version of this time frequency signal representation a previously given definition of such a function turned out to yield a distribution that was periodic with period instead of 2 and this caused aliasing to occur various alternative definitions are considered and compared with respect to aliasing and computational complexity from this comparison it appears that no definition leads to a function that is optimum in all respects this is illustrated by an example

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

proof of the lovasz conjecture to any two graphs g and h one can associate a cell complex horn g h by taking all graph multihomomorphisms from g to h as cells in this paper we prove the lovasz conjecture which states that if horn c 2r 1 g is k connected then g k 4 where r k z r 1 k 1 and c 2r 1 denotes the cycle with 2r 1 vertices the proof requires analysis of the complexes horn c 2r 1 k n for even n the obstructions to graph colorings are provided by the presence of torsion in h hom c 2r 1 k n z for odd n the obstructions are expressed as vanishing of certain powers of stiefel whitney characteristic classes of horn c 2r 1 k n where the latter are viewed as z 2 spaces with the involution induced by the reflection of c 2r 1

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

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

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

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

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

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

inonu wigner contraction of kac moody algebras inonu wigner contractions of affine kac moody algebras are discussed here the sugawara construction for the contracted affine algebra exists only for a fixed value of the level k which is determined in terms of the dimension of the uncontracted part of the starting lie algebra and the quadratic casimir in the adjoint representation further contractions of the g h coset spaces are discussed and an affine translation algebra is obtained which yields a virasoro algebra via a goddard kent olive gko construction with a central charge given by dim g h

note on the use of the wigner distribution for time frequency signal analysis it is shown that a correct use of the wigner distribution wd for time frequency signal analysis requires use of the analytic signal this version often referred to as the wigner ville distribution wvd is straightforward to compute does not exhibit any aliasing problem and introduces no frequency artifacts the problems introduced by the use of the wigner distribution with a real signal are clarified

an overview of aliasing errors in discrete time formulations of time frequency representations discrete time time frequency representation tfr algorithms claim to provide alias free approximations to their continuous time tfr counterparts without requiring oversampling of the signal by counterexamples we demonstrate that some of these claims are invalid we give new necessary conditions for reducing aliasing errors in these discrete time tfr algorithms

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

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

cohomology of colorings of cycles we compute the cohomology groups of the spaces of colorings of cycles i e of the prodsimplicial complexes tt hom c m k n we perform the computation first with bbb z 2 and then with integer coefficients the main technical tool is to use spectral sequences in conjunction with a detailed combinatorial analysis of a family of cubical complexes which we call it torus front complexes as an application of our method we demonstrate how to collapse each connected component of tt hom c m c n onto a garland of cubes