In close connection with time-frequency uncertainty relations, spectrograms are known to have some built-in redundancy which constrains the landscape of their surface, thus calling for simplified descriptions based on a reduced number of salient features. This is investigated here in some detail for spectrogram local maxima in the generic case of white Gaussian noise. A simple model, based on a randomized hexagonal lattice structure, is proposed for the distribution of their time-frequency locations considered as realizations of a 2D point process in the plane. While the rationale of the model is discussed and its relevance tested, further consideration is also given to the distribution of maximal values as well as to that of zeros that can be inferred from the proposed model.

The success of Empirical Mode Decomposition (EMD) resides in its practical approach to dissect non-stationary data. EMD repetitively goes through the entire data span to iteratively extract Intrinsic Mode Functions (IMFs). This approach, however, is not suitable for data stream as the entire data set has to be reconsidered every time a new point is added. To overcome this, we propose Online EMD, an algorithm that extracts IMFs on the fly. The two key elements of Online EMD are a sliding window to compute local IMFs, and a stitching procedure to gradually append local IMFs to the final result. Using synthetic data we show that the decomposition quality of Online EMD is similar to classical EMD. We also present results obtained with a real data set to expose the practical advantages of Online EMD when dealing with data stream or large data set.

This letter introduces new chirp rate and instantaneous frequency estimators designed for frequency-modulated signals. These estimators are first investigated from a deterministic point of view, then compared together in terms of statistical efficiency. They are also used to design new recursive versions of the vertically synchrosqueezed short-time Fourier transform, using a previously published method (D. Fourer, F. Auger, and P. Flandrin, “Recursive versions of the Levenberg-Marquardt reassigned spectrogram and of the synchrosqueezed STFT,” in Proc. IEEE Int. Conf. Acoust., Speech Signal Process., Mar. 2016, pp. 4880-4884). This study paves the way to the real-time computation of a time-frequency representation, which is both invertible and sharply localized in frequency.

In this paper, we first present a recursive implementation of a recently proposed reassignment process called the Levenberg Marquardt reassignment, which allows a user to adjust the slimness of the signal components localization in the time-frequency plane. Thanks to a generalization of the signal reconstruction formula, we also present a recursive implementation of the synchrosqueezed short-time Fourier transform. This approach paves the way for a real-time computation of a reversible and adjustable almost-ideal time-frequency representation.

For a proper choice of the analysis window, a short-time Fourier transform is known to be completely characterized by its zeros, which coincide with those of the associated spectrogram. A simplified representation of the time-frequency structure of a signal can therefore be given by the Delaunay triangulation attached to spectrogram zeros. In the case of multicomponent nonstationary signals embedded in white Gaussian noise, it turns out that each time-frequency domain attached to a given component can be viewed as the union of adjacent Delaunay triangles whose edge length is an outlier as compared to the distribution in noise-only regions. Identifying such domains offers a new way of disentangling the different components in the time-frequency plane, as well as of reconstructing the corresponding waveforms.

Temporal networks describe a large variety of systems having a temporal evolution. Characterization and visualization of their evolution are often an issue especially when the amount of data becomes huge. We propose here an approach based on the duality between graphs and signals. Temporal networks are represented at each time instant by a collection of signals, whose spectral analysis reveals connection between frequency features and structure of the network. We use nonnegative matrix factorization (NMF) to find these frequency features and track them over time. Transforming back these features into subgraphs reveals the underlying structures which form a decomposition of the temporal network.

This paper presents a 2D transposition of the Hilbert-Huang Transform (HHT), an empirical data analysis method designed for studying instantaneous amplitudes and phases of non-stationary data. The principle is to adaptively decompose an image into oscillating parts called Intrinsic Mode Functions (IMFs) using an Empirical Mode Decomposition method (EMD), and then to perform Hilbert spectral analysis on the IMFs in order to recover local amplitudes and phases. For the decomposition step, we propose a new 2D mode decomposition method based on non-smooth convex optimization, while for the instantaneous spectral analysis, we use a 2D transposition of Hilbert spectral analysis called monogenic analysis, based on Riesz transform and allowing to extract instantaneous amplitudes, phases, and orientations. The resulting 2D-HHT is validated on simulated data.

This paper presents a speech enhancement technique for signals corrupted by nonstationary acoustic noises. The proposed approach applies the empirical mode decomposition (EMD) to the noisy speech signal and obtains a set of intrinsic mode functions (IMF). The main contribution of the proposed procedure is the adoption of the Hurst exponent in the selection of IMFs to reconstruct the speech. This EMD and Hurst-based (EMDH) approach is evaluated in speech enhancement experiments considering environmental acoustic noises with different indices of nonstationarity. The results show that the EMDH improves the segmental signal-to-noise ratio and an overall quality composite measure, encompassing the perceptual evaluation of speech quality (PESQ). Moreover, the short-time objective intelligibility (STOI) measure reinforces the superior performance of EMDH. Finally, the EMDH is also examined in a speaker identification task in noisy conditions. The proposed technique leads to the highest speaker identification rates when compared to the baseline speech enhancement algorithms and also to a multicondition training procedure.

This paper provides an extension of the 1D Hilbert Huang transform for the analysis of images using recent optimization techniques. The proposed method consists of: 1) adaptively decomposing an image into oscillating parts called intrinsic mode functions (IMFs) using a mode decomposition procedure and 2) providing a local spectral analysis of the obtained IMFs in order to get the local amplitudes, frequencies, and orientations. For the decomposition step, we propose two robust 2D mode decompositions based on nonsmooth convex optimization: 1) a genuine 2D approach, which constrains the local extrema of the IMFs and 2) a pseudo-2D approach, which separately constrains the extrema of lines, columns, and diagonals. The spectral analysis step is an optimization strategy based on Prony annihilation property and applied on small square patches of the IMFs. The resulting 2D Prony–Huang transform is validated on simulated and real data.

Sensor deployments in large buildings allow the administrators to supervise the building infrastructure and identify abnormalities. Nevertheless, the numerous data streams reported by the increasing number of sensors overwhelm the building administrators. We propose a methodology that assists them to identify abnormal devices usages. The proposed method takes advantage of Ensemble Empirical Mode Decomposition (E-EMD) to uncover the patterns of power-draw signals, thereby enabling us to estimate the intrinsic inter-device correlations. By monitoring the devices correlations over time we compute the usual usage of the devices and report the devices that deviate from their normal usage. Our evaluation with 10 weeks of real data shows the efficiency of the proposed method to uncover the devices intrinsic relationships and detect peculiar events that require the administrators attention.

A typical large building contains thousands of sensors, monitoring the HVAC system, lighting, and other operational sub-systems. With the increased push for operational efficiency, operators are relying more on historical data processing to uncover opportunities for energy-savings. However, they are overwhelmed with the deluge of data and seek more efficient ways to identify potential problems. In this paper, we present a new approach called the Strip, Bind and Search (SBS); a method for uncovering abnormal equipment behavior and in-concert usage patterns. SBS uncovers relationships between devices and constructs a model for their usage pattern relative to other devices. It then flags deviations from the model. We run SBS on a set of building sensor traces; each containing hundred sensors reporting data flows over 18 weeks from two separate buildings with fundamentally different infrastructures. We demonstrate that, in many cases, SBS uncovers misbehavior corresponding to inefficient device usage that leads to energy waste. The average waste uncovered is as high as 2500~kWh per device.

This paper presents a new time-frequency reassignment process for the spectrogram, called the Levenberg-Marquardt reassignment. Compared to the classical one, this new reassignment process uses the second-order derivatives of the phase of the short-time Fourier transform, and provides the user with a setting parameter. This parameter allows him to produce either a weaker or a stronger localization of the signal components in the time-frequency plane.

Surrogates are investigated as procedures of synthesis for multi-variate time series with prescribed properties. First it is shown how to prescribe a multivariate covariance function jointly with the (possibly non-Gaussian) marginal distributions. Second, using histogram matching by approximate optimal transport with the Sliced Wasserstein Distance, the surrogate synthesis is extended to prescribe covariance function and joint-distribution of the components. Algorithms are described and justified, and numerical examples are shown. MATLAB codes are publicly available online.

We propose a novel gap-filling technique, based on the empirical mode decomposition (EMD). The idea is that a signal with missing data can be decomposed into a set of intrinsic mode functions (IMFs) with missing data. Filling the gaps in each IMF should be easier than filling the gaps in the original signal. This is because each IMF varies much more slowly than the original signal, and also because the IMFs are known to have useful regularity properties. We demonstrate the performance of our technique on environmental pollutant data.

A complete evaluation of first-order, second-order and mixed derivatives is proposed for both the (log-)magnitude and the phase of a given Short-Time Fourier Transform (STFT), leading to equivalent expressions based on additional STFTs with specific windows. Consequences are drawn in terms of phase-magnitude relationships, resulting in new formulations of time-frequency techniques such as reassignment, as well as new insights in the structure of admissible STFTs in some special cases.

While an exact stationarization of a process with a given spectrum magnitude can be obtained via a complete randomization of the spectrum phase ("surrogates" technique), we pro pose here a softened version in which the degree of stationarization can be controlled by a perturbation of the actual phase. A basic theory for such "transitional surrogates" is first discussed, with emphasis on two effective constructions based on either white Gaussian noise or random walks. Some typical examples are considered for illustration, and performance evaluations are provided for supporting the usefulness of the approach in the context of stationarity testing.

In this paper an algorithm based on the ensemble empirical mode decomposition (EEMD) is presented. The key idea on the EEMD relies on averaging the modes obtained by EMD applied to several realizations of Gaussian white noise added to the original signal. The resulting decomposition solves the EMD mode mixing problem, however it introduces new ones. In the method here proposed, a particular noise is added at each stage of the decomposition and a unique residue is computed to obtain each mode. The resulting decomposition is complete, with a numerically negligible error. Two examples are presented: a discrete Dirac delta function and an electrocardiogram signal. The results show that, compared with EEMD, the new method here presented also provides a better spectral separation of the modes and a lesser number of sifting iterations is needed, reducing the computational cost.

We propose a new estimate of the time-varying spectra of uniformly modulated processes. The estimate is based on a resampling scheme which incorporates empirical mode decompositions and surrogate data techniques. The performance of the method is studied via simulations.

An operational framework is developed for testing stationarity relatively to an observation scale. The proposed method makes use of a family of stationary surrogates for defining the null hypothesis of stationarity. As a further contribution to the field, we demonstrate the strict-sense stationarity of surrogate signals and we exploit this property to derive the asymptotic distributions of their spectrogram and power spectral density. A statistical hypothesis testing framework is then proposed to check signal stationarity. Finally, some results are shown on a typical model of signals that can be thought of as stationary or nonstationary, depending on the observation scale used.

Usually the mel-frequency cepstral coefficients are estimated either from a periodogram or from a windowed periodogram. We state a general estimator which also includes multitaper estimators. We propose approximations of the variance and bias of the estimate of each coefficient. By using Monte Carlo computations, we demonstrate that the approximations are accurate. Using the proposed formulas, the peak matched multitaper estimator is shown to have low mean square error (squared bias + variance) on speech-like processes. It is also shown to perform slightly better in the NIST 2006 speaker verification task as compared to the Hamming window conventionally used in this context.

In the case of multicomponent AM-FM signals, the idealized representation which consists of weighted trajectories on the time-frequency (TF) plane, is intrinsically sparse. Recent advances in optimal recovery from sparsity constraints thus suggest to revisit the issue of TF localization by exploiting sparsity, as adapted to the specific context of (quadratic) TF distributions. Based on classical results in TF analysis, it is argued that the relevant information is mostly concentrated in a restricted subset of Fourier coefficients of the Wigner-Ville distribution neighbouring the origin of the ambiguity plane. Using this incomplete information as the primary constraint, the desired distribution follows as the minimum l1-norm solution in the transformed TF domain. Possibilities and limitations of the approach are demonstrated via controlled numerical experiments, its performance is assessed in various configurations and the results are compared with standard techniques. It is shown that improved representations can be obtained, though at a computational cost which is significantly increased.

We address the problem of fractal dimension estimation of a discrete sample path. After recalling the multiplicity of possible definitions, we focus on the regularity dimension and on the regularization dimension, and report on the common ingredients that underlie these definitions: a scale transform of the signal, and a geometric or statistical measure on the scaled signal. Then, we propose to interchange wavelet transforms, ordinarily used as the scale transform, with empirical mode decomposition (EMD), a recently proposed signal-adaptive transform. The adaptivity of this latter yields estimation performance that overhauls usual wavelet-based techniques. To support our claim, we obtain comprehensive results from a Monte Carlo simulation on fractional Brownian motions.

The empirical mode decomposition (EMD) has been introduced quite recently to adaptively decompose nonstationary and/or nonlinear time series. The method being initially limited to real-valued time series, we propose here an extension to bivariate (or complex-valued) time series that generalizes the rationale underlying the EMD to the bivariate framework. Where the EMD extracts zero-mean oscillating components, the proposed bivariate extension is designed to extract zero-mean rotating components. The method is illustrated on a real-world signal, and properties of the output components are discussed. Free Matlab/C codes are available at http://perso.ens-lyon.fr/patrick.flandrin.

In this paper, we propose a method for selecting time-frequency distributions appropriate for given learning tasks. It is based on a criterion that has recently emerged from the machine learning literature: the kernel-target alignment. This criterion makes possible to find the optimal representation for a given classification problem without designing the classifier itself. Some possible applications of our framework are discussed. The first one provides a computationally attractive way of adjusting the free parameters of a distribution to improve classification performance. The second one is related to the selection, from a set of candidates, of the distribution that best facilitates a classification task. The last one addresses the problem of optimally combining several distributions

The rationale underlying the nonlinear empirical mode decomposition method is intrinsically a continuous-time approach. The method can however only be applied in practice to discrete-time signals. EMD is obtained through iterating a basic nonlinear operator for which we derive an upper bound for the effects of sampling. Finally the effects of sampling for a complete EMD are assessed using the knowledge on the basic operator

Huang's data-driven technique of empirical mode decomposition (EMD) is applied to the versatile, broadband, model of fractional Gaussian noise (fGn). The spectral analysis and statistical characterization of the obtained modes reveal an equivalent filter bank structure together with gamma distributed variances, both sharing some properties with wavelet decompositions. These common features are then used to mimic wavelet based techniques aimed at estimating the Hurst exponent.

Empirical mode decomposition (EMD) has recently been pioneered by Huang et al. for adaptively representing nonstationary signals as sums of zero-mean amplitude modulation frequency modulation components. In order to better understand the way EMD behaves in stochastic situations involving broadband noise, we report here on numerical experiments based on fractional Gaussian noise. In such a case, it turns out that EMD acts essentially as a dyadic filter bank resembling those involved in wavelet decompositions. It is also pointed out that the hierarchy of the extracted modes may be similarly exploited for getting access to the Hurst exponent.

A definition of stochastic discrete scale invariance (DSI) is proposed and its properties studied. It is shown how the Lamperti (1962) transformation, which transforms stationarity in self-similarity, is also a means to connect processes deviating from stationarity and processes which are not exactly scale invariant: in particular we interpret DSI as the image of cyclostationarity. This theoretical result is employed to introduce a multiplicative spectral representation of DSI processes based on the Mellin transform, and preliminary remarks are given about estimation issues.

The scaling properties of Internet data are analysed in detail through the unifying viewpoint of infinitely divisible cascades (IDC). From exceptionally precise TCP/IP traffic traces are extracted time series including arrival rate, durations, and interarrival times of TCP connections. We show that IDC offer a pertinent description of these series. Relations between them are investigated, yielding insights on the sources of the scaling and possible modelling approaches.

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.

Theoretical results have been established in non-parametric entropy estimation, based on asymptotic properties of minimal spanning trees (MST). A new application is proposed for the automatic extraction of time-frequency skeletons in the case of multicomponent chirp-like signals. The proposed method makes use of local maxima of a time-frequency distribution (considered as realizations of a 2D or 3D process), and exploits the efficiency of MSTs for density discrimination and clustering.

Infinitely divisible cascades are a model class previously introduced in the field of turbulence to describe the statistics of velocity fields. In this paper, using a wavelet reformulation of the cascades, we investigate their ability to analyze band model scaling properties of data and compare their fundamental ingredients to those of other scaling model classes such as self-similar and multifractal processes. We also propose an estimation procedure for the propagator or kernel of the cascades. Finally the cascade model is successfully applied to describe Internet TCP network traffic data, bringing new insights into their scaling properties and revealing a pitfall in existing techniques.

We, study self-similar processes with possibly infinite second order statistics and long-range dependence. To do so, we detail the statistical properties of the wavelet coefficients of /spl alpha/-stable self similar processes, used as a paradigm for those situations. We, then, propose a wavelet based estimator for the self-similarity parameter and analyse its statistical performance both theoretically and numerically. We show that it is unbiased, that its variance decreases as the inverse of the length of the data and that it can be easily implemented.

In a previous paper we presented a method for atomic decomposition with chirped, Gabor functions based on maximum likelihood estimation. In this paper we present the Cramer-Rao lower bounds for estimating the seven chirp parameters, and the results of a simulation showing that our sub-optimal, but computationally tractable, estimators perform well in comparison to the bound at low signal-to-noise ratios. We also show that methods based on signal dictionaries will require much higher computations to perform well in low signal-to-noise ratios.

Among the myriad of time-frequency distributions, the Wigner distribution stands alone in satisfying many desirable mathematical properties. Attempts to extend definitions of the Wigner distribution to discrete signals have not been completely successful. In this letter, we propose an alternative definition for the Wigner distribution, which has a clear extension to discrete signals. Under this definition, we show that the Wigner distribution does not exist for certain classes of discrete signals.

A geometrical description is given for reassignment vector fields of spectrograms. These vector fields are shown to be connected with both an intrinsic phase characterization and a scalar potential. This allows for the generalization of the original reassignment process to a differential version based on a dynamical evolution of time-frequency particles.

Three new methods of instantaneous frequency estimation are introduced and compared in view of characterizing gravitational waves. Two methods are Bayesian and can be formulated as solutions of an ill-posed inverse problem with two different stochastic regularizations. Using either a state-space model for the time-frequency data or a compound non-uniform Bernoulli-Gauss model for the instantaneous frequency. The third method uses a reassignment technique applied to a spectrogram. In each case, averages based on different windowings permit to enhance the signal-to-noise ratio, leading to accurate results even below 0 dB.

Reassigning each value of a time-frequency representation to a different location in the plane can produce a better localization of the signal components. This idea, pioneered by Kodera et al. (1976, 1978), was only applied to the sole spectrogram. We present a new formulation of this method which allows a generalization of its use for any bilinear time-frequency or time-scale representation. The resulting reassigned distributions are easily computable versatile tools which highlight the signal features and preserve many theoretical properties.>

Many functions have been proposed for estimating signal information content and complexity on the time-frequency plane, including moment-based measures such as the time-bandwidth product and the Shannon and Renyi(see 4th Berkeley Symp. Math., Stat., Prob., vol.1) entropies. When applied to a time-frequency representation from Cohen's (1989) class, the Renyi entropy conforms closely to the visually based notion of complexity that we use when inspecting time-frequency images. A detailed discussion reveals many of the desirable properties of the Renyi information measure for both deterministic and random signals.>

The authors show that making use of the discrete wavelet transform to analyse data implies performing a preliminary initialization of the fast pyramidal algorithm. An approximation enabling easy performance of such an initialization is proposed.>

It is shown using some examples that the problem of estimating the evolution of scaling exponents characterizing locally a self-similar process can be efficiently handled within the general framework of time-scale energy distributions related to the wavelength transform. As is implicit from the structure of the estimators considered, the proposed methodology is dependent on the degree of nonstationarity of such evolutions, with fast changes leading to bias-variance tradeoffs.>

Discrimination between chaotic and stochastic processes is usually approached with second-order algorithms such as correlation integral or local intrinsic dimensionality. However, if these methods behave as expected for white Gaussian noise, they may fail for more structured processes. This fact is investigated in the case of AR(1) processes. Improvements to second-order algorithms are proposed by incorporating fourth-order informations, the idea being to track statistical independence beyond uncorrelation. The effectiveness of this new approach is illustrated on the same AR(1) processes.>

Fractional Brownian motion (FBM) offers a convenient modeling for nonstationary stochastic processes with long-term dependencies and 1/f-type spectral behavior over wide ranges of frequencies. Statistical self-similarity is an essential feature of FBM and makes natural the use of wavelets for both its analysis and its synthesis. A detailed second-order analysis is carried out for wavelet coefficients of FBM. It reveals a stationary structure at each scale and a power-law behavior of the coefficients' variance from which the fractal dimension of FBM can be estimated. Conditions for using orthonormal wavelet decompositions as approximate whitening filters are discussed, consequences of discretization are considered, and some connections between the wavelet point of view and previous approaches based on length measurements (analysis) or dyadic interpolation (synthesis) are briefly pointed out.>

Five different fractal dimension estimators are chosen which operate either in the frequency domain (identification of a spectral exponent via spectrum analysis), in the time domain (maximum likelihood on one hand, methods based on length measurements of fractional Brownian motion samples at different observation scales on the other hand), or in a mixed time-scale domain (identification of a self-similarity parameter via the variance of wavelets coefficients). The relevance of these different estimators is discussed, and their performance is compared on simulated and real data. Performance evaluation of analysis is made difficult by the fact that there exists no unique and satisfactory synthesis method for generating such processes.>

A formalism of signal energy representations depending on time and scale is presented. Precise links between time-frequency and time-scale energy distributions are provided. It is known that a full description of the former is given by Cohen's class, which can be described as a generalization of the spectrogram appropriately parameterized by a smoothing function acting on the Wigner-Ville distribution. A full description of the latter is given, resulting in a class of representations in which the smoothing of the Wigner-Ville distribution is scale-dependent. Through proper choice of the smoothing function, interesting properties may be imposed on the representation, which makes it a versatile tool for the analysis of nonstationary signals. Also, specific choices allow known definitions to be recovered (including the Bertrands' and the energetic version of the wavelet transform, referred to as the scalogram). Another very flexible choice uses separable smoothing functions. It is shown, in particular, that Gaussian kernels provide a continuous transition between spectrograms and scalograms by means of the Wigner-Ville distribution.>

The purpose of this study is to demonstrate what can be gained from a time-frequency analysis in the fields of machine diagnosis and monitoring. The problem consists of detecting a possible abnormal state of the in-core instrumentation thimbles in PWR power plants, due to vibration phenomena, and the objective is to increase the automatization of the monitoring. It is shown that a Wigner-Ville time-frequency analysis is well-suited not only to confirm the diagnosis, but also to provide a better understanding of the involved physical phenomena.>

Fractional Brownian motions (FBMs) provide useful models for a number of physical phenomena whose empirical spectra obey power laws of fractional order. However, due to the nonstationary nature of these processes, the precise meaning of such spectra remains generally unclear. Two complementary approaches are proposed which are intended to clarify this point. The first one, based on a time-frequency analysis, takes into account the nonstationary nature of FBM and puts emphasis on time-averaged measurements; the second one, based on a time-scale analysis, is matched to self-similarity properties of FBM and reveals an underlying stationary structure relative to each time-scaling.>

An approach which operates directly in the time-frequency plane is proposed to address the problem of finding the signal whose time-frequency energy distribution is most concentrated in a given time-frequency domain of any shape. The solution is shown to be that of an eigenequation depending on both the considered domain and the chosen distribution. Explicit solutions are derived for elliptic domains in the Wigner-Ville case, and connections with classical approaches are pointed out.>

A time-frequency formulation is proposed for the optimum detection of nonstationary Gaussian signals in Gaussian noise, with emphasis on low-SNR (signal-to-noise ratio) situations. Time-frequency correlators are shown to match mathematical optimality with a physically meaningful interpretation when they are based on the Wigner-Ville transform. Two illustrations of the properties of the corresponding receivers are discussed: the first one is related to random time-variant channels and the second one to detection in presence of clutter.>

It has been recently shown [1] that the Wigner-Ville distribution can be used for optimum detection, leading to a very simple time-frequency formulation in the case of chirp signals : this paper is intended to extend and illustrate this approach. A generalization of Moyal's formula is first discussed for determining candidates to optimum detection in Cohen's class of bilinear time-frequency distributions. Taking into account practical requirements, a modified strategy is then proposed, which both admits a physically very meaningful interpretation and defines a fairly general class of receivers in between semi-coherent and non-coherent ones. This modified approach is illustrated on real data by a time delay measurement problem and possible extensions are suggested.

The Wigner-Ville Distribution (WVD) is now known to be a convenient tool for the time-frequency analysis of non-stationary signals, and especially monocomponent ones. However, in the case of multicomponent signals, its bilinear structure is also known to create cross-terms without any physical significance. Starting with the general formulation of time-frequency representations, which only depend on an arbitrary kernel function, we first characterize properties of such cross-terms and then propose appropriate smoothings of the WVD in order to reduce their influence. Such suitable and versatile approximations are compared on synthetic and natural signals and an extension to time-frequency filtering is proposed.

Among the various methods of analysing time-varying signals, one of the most interesting is WIGNER-VILLE's. - In the case of signals with a large bandwidth-duration BT product, we recall that their behaviour in the time-frequency domain can be reduced to just two typical cases. The numerical studies performed on different examples are in good agreement with the analytical ones and lead to the possibility of modeling signals in the time-frequency domain. -In the case of signals with a BT product of any magnitude, we study in detail, analytically, and numerically the WIGNER-VILLE representation of linearly frequency modulated signals. These results are then compared with those obtained by usual methods such as the "Moving Window Method" (MWM). The performance of each method is evaluated and ways of optimizing MWM are proposed.