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International audience; The detrended fluctuation analysis (DFA) and its variants are popular methods to analyze the self-similarity of a signal. Two steps characterize them: firstly, the trend of the centered integrated signal is estimated and removed. Secondly, the properties of the so-called fluctuation function which is an approximation of the standard deviation of the resulting process is analyzed. However, it appears that the statistical mean was assumed to be equal to zero to obtain it. As there is no guarantee that this assumption is true a priori, this hypothesis is debatable. The purpose of this paper is to propose two alternative definitions of the fluctuation function. Then, we compare all of them based on a matrix formulation and the filter-based interpretation we recently proposed. This analysis will be useful to show that the approach proposed in the original paper remains a good compromise in terms of accuracy and computational cost.

In signal processing, ARMA processes are widely used to model short-memory processes. In various applications, comparing or classifying ARMA processes is required. In this paper, our purpose is to provide analytical expressions of the divergence rates of the Kullback-Leibler divergence, the Renyi divergence (RD) of order α and their symmetric versions for two Gaussian ARMA processes, by taking advantage of results such as the Yule-Walker equations and notions such as inverse filtering. The divergence rates can be interpreted as the sum of different quantities: power of one ARMA process filtered by the inverse filter associated with the second ARMA process, cepstrum, etc. Finally, illustrations show that the ranges of values taken by the divergence rates of the RD are sensitive to α, especially when the latter is close to 1.

The purpose of this paper is first to derive the expressions of the Chernoff, Bhattacharyya, Renyi and Sharma-Mittal divergences when comparing two probability density functions of vectors storing k consecutive samples of Gaussian ARMA processes. This can be useful for process comparison or statistical change detection issue. Then, we analyze the behaviors of the divergences when k increases and tends to infinity by using some results related to ARMA processes such as the Yule-Walker equations. Comments and illustrations are given.

International audience; To estimate the Hurst exponent of a mono-fractal process, the detrended fluctuation analysis (DFA) is based on the estimation of the trend of the integrated process. The latter is subtracted from the integrated process. The power of the residual is then computed and corresponds to the square of the fluctuation function. Its logarithm is proportional to the Hurst exponent. In the last few years, a few variants of this method have been proposed and differ in the way of estimating the trend. Our contribution in this paper is threefold. First, we introduce a new variant of the DFA, based on a regularized least-square criterion to estimate the trend. Then, the influence of the regularization parameter on the fluctuation function is analyzed in two cases: when the process is wide sense stationary and when it is not. Finally, an application is presented in the field of aeronautics to characterize an attentional impairment: the visual tunneling.

The symmetric Kullback–Leibler divergence known as Jeffrey's divergence (JD) has found applications in signal and image processing, from radar clutter modeling to texture analysis. Recently, several studies were done on the JD between ergodic wide-sense stationary autoregressive (AR) and/or moving average (MA) processes. It was shown that the so-called asymptotic JD increment can be useful to compare ergodic wide-sense stationary ARMA processes. An interpretation of the asymptotic JD increment was also proposed. It consists in calculating the power of the first process filtered by the inverse filter associated with the second process, and conversely. However, in some biomedical applications, econometrics and other areas, long-memory processes have rather to be studied. Therefore, this paper aims at addressing the JD between ergodic wide-sense stationary autoregressive fractionally integrated moving average (ARFIMA) processes. More particularly, we study the influence of the ARFIMA parameters on the value of the asymptotic JD increment. Then, we analyze if the interpretation of the asymptotic JD increment based on inverse filtering is still valid for this type of process. Finally, some simulation results illustrate the theoretical analysis.

International audience; Jeffrey’s divergence (JD), which is the symmetric version of the Kullback-Leibler divergence, has been used in a wide range ofapplications, from change detection to clutter homogeneity analysis in radar processing. It has been calculated between the jointprobability density functions of successive values of autoregressive (AR) processes. In this case, the JD is a linear function ofthe variate number to be considered. Knowing the derivative of the JD with respect to the number of variates is hence enough tocompare noise-free AR processes. However, the processes can be disturbed by additive uncorrelated white noises. In this paper,we suggest comparing two noisy 1st-order AR processes. For this purpose, the JD is expressed from the JD between noise-freeAR processes and the bias the noises induce. After a transient period, the derivative of this bias with respect to the variate numberbecomes constant as well as the derivative of the JD. The resulting asymptotic JD increment is then used to compare noisy ARprocesses. Some examples illustrate this theoretical analysis.

In signal and image processing, Jeffrey's divergence (JD) is used in many applications for classification, change detection, etc. The previous studies done on the JD between ergodic wide-sense stationary (WSS) autoregressive (AR) and/or moving average (MA) processes state that the asymptotic JD increment, which is the difference between two JDs based on k and ( k − 1 ) -dimensional random vectors when k becomes high, tends to a constant value, except JDs which involve a 1st-order MA process whose power spectral density (PSD) is null for one frequency. In this paper, our contribution is threefold. We first propose an interpretation of the asymptotic JD increment for ergodic WSS ARMA processes: it consists in calculating the power of the first process filtered by the inverse filter associated with the second process and conversely. This explains the atypical cases identified in previous works and generalizes them to any ergodic WSS ARMA process of any order whose PSD is null for one or more frequencies. Then, we suggest comparing other random processes such as noisy sums of complex exponentials (NSCE) by using the JD. In this case, the asymptotic JD increment and the convergence speed towards the asymptotic JD are useful to compare the processes. Finally, NSCE and pth-order AR processes are compared. The parameters of the processes, especially the powers of the processes, have a strong influence on the asymptotic JD increment.

Even if they can be outperformed by other methods, the detrended fluctuation analysis (DFA) and the detrended moving average (DMA) are widely used to estimate the Hurst exponent because they are based on basic notions of signal processing. For the last years, a great deal of interest has been paid to compare them and to better understand their behaviors from a mathematical point of view. In this paper, our contribution is the following: we first propose to express the square of the so-called fluctuation function as a 2D Fourier transform (2D-FT) of the product of two matrices. The first one is defined from the instantaneous correlations of the signal while the second, called the weighting matrix, is representative of each method. Therefore, the 2D-FT of the weighting matrix is analyzed in each case. In this study, differences between the DFA and the DMA are pointed out when the approaches are applied on non-stationary processes.

The detrended fluctuation analysis (DFA) is widely used to estimate the Hurst exponent. Although it can be outperformed by wavelet based approaches, it remains popular because it does not require a strong expertise in signal processing. Recently, some studies were dedicated to its theoretical analysis and its limits. More particularly, some authors focused on the so-called fluctuation function by searching a relation with an estimation of the normalized covariance function under some assumptions. This paper is complementary to these works. We first show that the square of the fluctuation function can be expressed in a similar matrix form for the DFA and the variant we propose, called Continuous-DFA (CDFA), where the global trend is constrained to be continuous. Then, using the above representation for wide-sense-stationary processes, the statistical mean of the square of the fluctuation function can be expressed from the correlation function of the signal and consequently from its power spectral density, without any approximation. The differences between both methods can be highlighted. It also confirms that they can be seen as ad hocwavelet based techniques.

In this paper, we propose to analyze the properties of the cross entropy between the probability density functions of k consecutive samples of wide-sense stationary Gaussian ARMA processes. It is shown that, when k increases, the cross entropy has a transient behavior before tending to an affine function of k whose slope is defined by the so-called asymptotic increment. The latter depends on the parameters of the processes and can be a reasonable choice to characterize the cross entropy. Some illustrations are then given.

In this paper, we first propose to analyze the properties of the Kullback-Leibler divergence between wide-sense stationary random processes that can be modeled by ARMA or ARFIMA processes. This study makes it possible to introduce a new feature useful to compare two random processes and called "the asymptotic KL increment". The latter depends on various parameters such as the inverse filters associated to the random processes. An interpretation of "the asymptotic KL increment" is also given. Then, we propose to use it in order to compare the inter-beat intervals, which characterize the cardiac rhythm, when the subjects are either in a calm and soothing situation or under stress.

When using Bayesian estimation techniques, the algorithm is strongly sensitive to the system evolution model and more particularly to the setting of the state-noise covariance matrix. Recently, Bayesian non-parametric models and in particular Dirichlet processes (DPs) have been proposed as a scalable solution to this issue. They assume that the system can switch between an infinite number of state-space representations corresponding to different values of the state-noise covariance matrix. In this framework, jointly estimating the state vector and the covariance matrix is a non-linear non-Gaussian problem. The inference is thus classically carried out using particle filtering techniques. In this case, the choice of the proposal distribution for the particles is of paramount importance regarding the estimation accuracy. A first contribution of this paper is to derive an approximation of the optimal proposal distribution of the particle filter when a DP prior is placed on the distribution of the state-noise covariance matrix. Then, an alternative DP-based formulation of the inference problem is proposed to reduce its dimensionality. It takes advantage that the possible functional forms of the state-noise covariance matrices are known up to a reduced number of time-switching hyperparameters in many applications. An approximation of the optimal proposal distribution is also derived. Finally, the relevance of both proposed approaches is analyzed in the framework of target tracking and a comparative study with existing methods is carried out. HighlightsWe address the joint estimations of the state vector and the state-noise covariance matrix.We use Bayesian non-parametric methods implemented by particle filtering.Two approaches are presented.In both cases, the optimal proposal distribution is derived.Both proposed algorithms are applied to target tracking.

Jeffrey’s divergence (JD) is used in many applications, from change detection to classification. Several studies were done on the JD between ergodic wide-sense stationary autoregressive and moving average (ARMA) processes. It was shown that the derivate of the JD between the probability density functions of k consecutive samples of two ARMA processes tends to the so-called asymptotic JD increment. This latter is enough to compare the processes and amounts to calculating the power of the first process filtered by the inverse filter associated with the second process and conversely. In this paper, our purpose is to study if this result can be extended to ARFIMA processes. As a first step, a special case, namely the JD between wide-sense stationary fractionally integrated white noises, is addressed. The influences of the process parameters on the asymptotic JD increment are analyzed. Our investigations validate the inverse filtering interpretation of the JD.

Like other divergences, Jeffrey's divergence (JD) is used for change detection, for model comparison, etc. Recently, a great deal of interest has been paid to this symmetric version of the Kullback-Leibler (KL) divergence. This led to analytical expressions of the JD between autoregressive (AR) processes, moving-average (MA) processes, either noise-free or disturbed by additive white noises, as well as ARMA processes. In this paper, we propose to study the JD between processes that are defined as sums of complex-valued sinusoidal processes disturbed by additive white noises. We show that the JD tends to a stationary behavior when the number of variates becomes large. The derivative of the JD becomes a constant that depends on the parameters defining the processes. The convergence speed towards this stationary regime depends on the differences between the normalized angular frequencies. The smaller the difference, the slower the convergence. This result can be obtained by interpreting some steps to compute the JD as orthogonal projections. Some examples illustrate the theoretical analysis.

Various works have been carried out about the Jeffrey's divergence (JD) which is the symmetric version of the Kullback-Leibler (KL) divergence. An expression of the JD for Gaussian processes can be deduced from the definition of the KL divergence and the expression of the Gaussian-multivariate distributions of k-dimensional random vectors. It depends on the k × k Toeplitz covariance matrices of the stationary processes. However, the resulting computational cost may be high as these matrices must be inverted and it is all the higher as k increases. To circumvent this problem, a recursive expression can be obtained for real 1st-order autoregressive (AR) processes. When they are disturbed by additive uncorrelated white noises, we showed that when k becomes large, the derivative of the JD with respect to k tends to be constant. This constant is sufficient to compare the noisy AR processes. In this paper, we propose to extend our work to AR moving-average (MA) processes with one AR term and one MA term. Some examples illustrate the theoretical analysis.

Comparing processes or models is of interest in various applications. Among the existing approaches, one of the most popular methods is to use the Kullback-Leibler (KL) divergence which is related to Shannon’s entropy. Similarly, the Renyi divergence of order α can be deduced from the Renyi entropy. When α tends to 1, it leads to the KL divergence. In this paper, our purpose is to derive the expression of the Renyi divergence between the probability density functions of k consecutive samples of two real first-order moving average (MA) processes by using the eigen-decompositions of their Toeplitz correlation matrices. The resulting expression is compared with the expressions of the Rao distance and the Jeffrey’s divergence (JD) based on the eigenvalues. The way these quantities evolve when k increases is then presented. When dealing with unit-zero MA processes, the derivate is infinite for the JD and finite for the others. The influence of α is also studied.

Autoregressive (AR) and time-varying AR (TVAR) models are widely used in various applications, from speech processing to biomedical signal analysis. Various dissimilarity measures such as the Itakura divergence have been proposed to compare two AR models. However, they do not take into account the variances of the driving processes and only apply to stationary processes. More generally, the comparison between Gaussian processes is based on the Kullback-Leibler (KL) divergence but only asymptotic expressions are classically used. In this letter, we suggest analyzing the similarities of two TVAR models, sample after sample, by recursively computing the Jeffrey’s divergence between the joint distributions of the successive values of each TVAR model. Then, we show that, under some assumptions, this divergence tends to the Itakura divergence in the stationary case.

This paper deals with the analysis of the Jeffrey's divergence (JD) between an autoregressive process (AR) and a sum of complex exponentials (SCE), whose magnitudes are Gaussian random values, which is then disturbed by an additive white noise. As interpreting the value of the JD may not be necessarily an easy task, we propose to give an expression of the JD and to analyze the influence of each process parameter on it. More particularly, we show that the ratios between the variance of the additive white noise and the variance of the AR-process driving process on the one hand, and the sum of the ratios between the SCE process power and the AR-process PSD at the normalized angular frequencies on the other hand, has a strong impact on the JD. The 2-norm of the AR-parameter has also an influence. Illustrations confirm the theoretical part.

In this paper, single-target tracking using radar measurements is addressed. Recently, algorithms based on Bernoulli random finite sets have proved efficient in a cluttered environment. However, in Bayesian approaches, the choice of the motion model impacts the trajectory estimation accuracy. To select an appropriate set of motion models, a joint tracking and classification (JTC) algorithm can be used. The principle is to consider different target classes depending on their maneuvrability, each of them being associated to a set of motion models. In this context, additional information such as a target length extent measurement can improve both classification and trajectory estimation. Therefore, we propose a multiple-model Bernoulli filter to perform JTC. To jointly estimate the trajectory and the target length which is constant, a Rao-Blackwellized approach is considered. Another contribution is that a bank of probabilistic data association filters is run instead of Kalman filters to account for false detections.

Time series models play a key-role in many applications from biomedical signal analysis to applied econometric. The purpose of this paper is to compare 1 st -order moving-average (MA) models by using dissimilarity measures such as the Jeffrey's divergence (JD), which is the symmetric version of the Kullback-Leibler divergence (KL). The MA models can be real or complex. They can also be disturbed by additive white noises or not. Analytical expressions are first proposed and analyzed. Then, the JD is used to compare more than two 1 st -order MA models in order to extract MA model subsets. In the latter case, we suggest analyzing the higher-order singular values of a tensor defined from the JDs between models over time to deduce the number of subsets and their cardinals. Simulation results illustrate the theoretical analysis. HighlightsAnalytical Jeffrey's Divergence expressions are provided for 1st-order MA models.Real or complex, noise-free or noisy 1st-order MA model cases are studied.Time evolution of the Jeffrey's Divergence depends on the 1st-order MA parameters.Comparative study with other dissimilarity measures (Rao distance, etc.) is done.Two methods are proposed to compare more than two 1st-order MA models.

Jeffrey’s divergence (JD), the symmetric Kullback-Leibler (KL) divergence, has been used in a wide range of applications. In recent works, it was shown that the JD between probability density functions of k successive samples of autoregressive (AR) and/or moving average (MA) processes can tend to a stationary regime when the number k of variates increases. The asymptotic JD increment, which is the difference between two JDs computed for k and \(k-1\) successive variates tending to a finite constant value when k increases, can hence be useful to compare the random processes. However, interpreting the value of the asymptotic JD increment is not an easy task as it depends on too many parameters, i.e. the AR/MA parameters and the driving-process variances. In this paper, we propose to compute the asymptotic JD increment between the processes that have been normalized so that their powers are equal to 1. Analyzing the resulting JD on the one hand and the ratio between the original signal powers on the other hand makes the interpretation easier. Examples are provided to illustrate the relevance of this way to operate with the JD.

This paper deals with model comparison based on the Jeffrey's divergence (JD). More particularly, after providing the JD between the joint distributions of k consecutive values of a white noise and the ones of a real moving-average or autoregressive model, the JD between real 1st-order MA and real 1st-order AR models is studied. Except when the 1st MA parameter is equal to 1, we show that, after a transient period, the JD between both models is incremented by a constant value that depends on the model parameters while k is incremented by 1. The JD is hence characterized by this increment and it is not necessary to consider a lot of samples.

The autoregressive models (AR) and moving-average models (MA) are regularly used in signal processing. Previous works have been done on dissimilarity measures between AR models by using a Riemannian distance, the Jeffrey's divergence (JD) and the spectral distances such as the Itakura-Saito divergence. In this paper, we compare the Rao distance and the JD for MA models and more particularly in the case of 1st-order MA models for which an analytical expression of the inverse of the covariance matrix is available. More particularly, we analyze the advantages of the Rao distance use. Secondly, the simulation part compares both dissimilarity measures depending on the MA parameters but also on the number of data available.

For a high-resolution radar, an extended target is characterized by a few main scatterers spread over several range gates not necessarily consecutive. The joint detections and localizations of these scatterers are of particular interest in order to estimate the range profile and identify the target. In this paper, we study a detector based on the Generalized Likelihood Ratio Test considering the unknown locations of the scatterers. Given the number of scatterers and using order statistics results, we derive approximations of the probability of false alarm and the probability of detection and localization which can be a relevant measure of performance in this context. Finally, a comparative study is carried out with other existing detectors.

This paper deals with the identification of a nonlinear SISO system modelled by a second-order Volterra series expansion when both the input and the output are disturbed by additive white Gaussian noises. Two methods are proposed. Firstly, we present an unbiased on-line approach based on the LMS. It includes a bias correction scheme which requires the variance of the input additive noise. Secondly, we suggest solving the identification problem as an errors-in-variables issue, by means of the so-called Frisch scheme. Although its computational cost is high, this approach has the advantage of estimating the Volterra kernels and the variances of both the additive noises and the input signal, even if the signal-to-noise ratios at the input and the output are low.

A great deal of interest has been paid to autoregressive parameter estimation in the noise-free case or when the observation data are disturbed by random noise. Tracking time-varying autoregressive (TVAR) parameters has been also discussed, but few papers deal with this issue when there is an additive zero-mean white Gaussian measurement noise. In this paper, one considers deterministic regression methods (or evolutive methods) where the TVAR parameters are assumed to be weighted combinations of basis functions. However, the additive white measurement noise leads to a weight-estimation bias when standard least squares methods are used. Therefore, we propose two alternative blind off-line methods that allow both the variance of the additive noise and the weights to be estimated. The first one is based on the errors-in-variable issue whereas the second consists in viewing the estimation issue as a generalized eigenvalue problem. A comparative study with other existing methods confirms the effectiveness of the proposed methods.

The paper deals with the design of Rayleigh fading channel simulators based on the inner-outer factorization. The core of the approach is to approximate the outer spectral factor of the channel power spectral density (PSD) by either finite-order polynomials or rational functions. This, respectively, leads to MA or AR/ARMA models. The parameter estimation operates in two steps: the outer factor, which leads to a minimum-phase filter, is first evaluated inside the unit disk of the z-plane. Then, we propose to compute the Taylor expansion coefficients of the outer factor because they coincide with the model parameters. Unlike other simulation techniques, this has the advantage that the first p parameters remain unchanged when one increases the model order from p to p+1. A comparative study with existing channel simulation approaches points out the relevance of our ARMA model-based method. Moreover, the ARMA model weakens the oscillatory deviations from the theoretical PSD in the case of AR models, or low peaks at the Doppler frequencies for MA models.

This letter deals with a new Rayleigh channel simulator. In many cases, the channel is modeled by an autoregressive process, but this choice is not well suited due to the band limitation of the theoretical Rayleigh channel power spectrum density. Therefore, we suggest using a low-pass filtered version of the so-called stochastic sinusoidal process. It consists of sinusoids in quadrature with random magnitudes modeled by autoregressive processes. Unlike the autoregressive channel modeling, this simulator has the advantage of exhibiting the power spectrum density (PSD) peaks at the maximum Doppler frequency. Since estimating the autoregressive parameters of the amplitudes directly using a matrix approach from the theoretical Rayleigh channel autocorrelation leads to numerical errors, we propose two alternative methods using either the asymptotic behavior of the Bessel function or genetic algorithms.

This paper deals with the joint signal and parameter estimation for linear state-space models. An efficient solution to this problem can be obtained by using a recursive instrumental variable technique based on two dual Kalman filters. In that case, the driving process and the observation noise in the state-space representation for each filter must be white with known variances. These conditions, however, are too strong to be always satisfied in real cases. To relax them, we propose a new approach based on two dual Hinfin filters. Once a new observation of the disturbed signal is available, the first Hinfin algorithm uses the latest estimated parameters to estimate the signal, while the second Hinfin algorithm uses the estimated signal to update the parameters. In addition, as the Hinfin filter behavior depends on the choice of various weights, we present a way to recursively tune them. This approach is then studied in the following cases: (1) consistent estimation of the AR parameters from noisy observations and (2) speech enhancement, where no a priori model of the additive noise is required for the proposed approach. In each case, a comparative study with existing methods is carried out to analyze the relevance of our solution.

This work aims at improving the power amplifier (PA) efficiency in uplink OFDM-based cognitive radio (CR) communications. Unlike the traditional approaches, we suggest transmitting a non-linearily ampliied signal without any il-tering and addressing the OFDM sample estimation from the distorted signal at the receiver. The proposed post-distortion and detection technique is based on a Volterra model for the PA and the channel. As the transmission can switch from one sub-band to another, the CR-PA behavior varies over time and the Volterra kernels can be constant or suddenly change. Therefore, an interactive multiple model (IMM) combining extended Kalman filters is considered. The transition probability matrix, which plays a key role in the IMM, is also sequentially estimated. The resulting uplink system has various advantages: it learns from the observations and a part of the computational load is exported to the receiver, which is not battery driven unlike the mobile terminal.

Dirichlet process (DP) mixtures were recently introduced to deal with switching linear dynamical models (SLDM). They assume the system can switch between an a priori infinite number of state-space representations (SSR) whose parameters are on-line inferred. The estimation problem can thus be of high dimension when the SSR matrices are unknown. Nevertheless, in many applications, the SSRs can be categorized in different classes. In each class, the SSRs are characterized by a known functional form but differ by a reduced set of unknown hyperparameters. To use this information, we thus propose a new hierarchical model for the SLDM wherein a discrete variable indicates the SSR class. Conditionally to this class, the distributions of the hyperparameters are modeled by DPs. The estimation problem is solved by using a Rao-Blackwellized particle filter. Simulation results show that our model outperforms existing methods in the field of target tracking.

This paper presents a spectrum-based method to estimate the integral non-linearity (INL) of an analog-to-digital converter. It relies on a previous work published in the literature that consists in expanding in Fourier series the analytical expression of a converter distorted signal output. The INL estimation process is then reduced to three operations: a discrete Fourier transform (DFT), a matrix inversion and an inverse DFT. In comparison with the initial approach, the algorithm proposed in this article reduces by two the set of parameters required to obtain the INL. This new method is validated on experimental data from the 12-bit ADC12D800RF converter run at 800MHz.

Autoregressive (AR) models are used in various applications, from speech processing to radar signal analysis. In this paper, our purpose is to extract different model subsets from a set of two or more AR models. The approach operates with the following steps: firstly the matrix composed of dissimilarity measures between AR-model pairs are created. This can be based on the symmetric Itakura divergence, the symmetric Itakura-Saito divergence, the log-spectral distance or Jeffrey's divergence (JD), which corresponds to the symmetric version of the Kullback-Leibler divergence. These matrices are then transformed to get the same properties as correlation matrices. Eigenvalue decompositions are performed to get the number of AR-model subsets and the estimations of their cardinals. Finally, K-means are used for classification. A comparative study points out the relevance of the JD-based method. Illustrations with sea radar clutter are also provided.

This paper deals with a phase-coded waveform which combines two binary phase codes, each impacting on specific properties of the radar receiving channel. After giving a detailed analysis of the expression of the received signal after processing when the Gaussian clutter is modeled by a pth-order autoregressive process, we focus our attention on the choice of the two phase codes: one aims at increasing the unambiguous range whereas the other is chosen by taking into account several criteria such as the detection performance. For this purpose, we suggest determining the Pareto fronts of 1st, 2nd and 3rd orders by means of an exhaustive search. Given the three Pareto fronts for different types of clutters simulated by making the AR parameters vary, we provide an automatic way to determine, before embedding it, the most robust phase codes using a fuzzy logic operator.

This paper deals with the identification of an autoregressive (AR) process disturbed by an additive moving-average (MA) noise. Our approach operates as follows: Firstly, the AR parameters are estimated by using the overdetermined high-order Yule–Walker equations. The variance of the AR process driving process can be deduced by means of an orthogonal projection between two types of estimates of AR process correlation vectors. Then, the correlation sequence of the MA noise is estimated. Secondly, the MA parameters are obtained by using inner–outer factorization. To study the relevance of the resulting method, we compare it with existing algorithms, and we analyze the identifiability limits. The identification approach is then combined with Kalman filtering for channel estimation in mobile communication systems.

In this paper, we propose to address the moving average (MA) parameters estimation issue based only on noisy observations and without any knowledge on the variance of the additive stationary white Gaussian measurement noise. For this purpose, the MA process is approximated by a high-order AR process and its parameters are estimated by using an errors-in-variables (EIV) approach, which also makes it possible to derive the variances of both the driving process and the additive white noise. The method is based on the Frisch scheme. One of the main difficulties in this case is to evaluate the minimal AR-process order that must be considered to have a 'good' approximation of the MA process. To this end, we propose a way based on K-means method. Simulation results of the proposed method are presented and compared to existing MA-parameter estimation approaches.

Optimal filters such as Kalman filters are used in a wide range of applications from speech enhancement to biomedical applications. They are based on an a priori state-space model describing a dynamic system. If this model is not well-suited, the accuracy of the state vector may be poor. Therefore, several estimators based on different models can be combined. However, state-space models that are dissimilar enough must be chosen. To our knowledge, there are no guidelines to select them, we hence address this issue in this paper. Given an initial model set, our aim is to determine subsets of similar models by using Jeffrey?s divergence between the distributions of the state-vector time paths based on the different models. Our approach operates with the following steps: the so-called dissimilarity matrix composed of Jeffrey?s divergences between model pairs is created. Then, this matrix is transformed to get the same properties as a correlation matrix and an eigenvalue decomposition is performed. Subsequently, we propose an interpretation of the predominant eigenvalues which is then used to deduce the number of model subsets and their cardinals. A classification algorithm can then be considered to determine which models belong to which subsets. HighlightsWe propose to compare two state models by using the Jeffrey's divergence (JD).We generalize the above approach to determine subsets in an initial model set.The eigenvalues decomposition is then interpreted to deduce the number of model subsets.Then, a classification algorithm can be used to determine which model belongs to which subset.The method is more particularly applied to a toy example and to tracking.

The estimation of the parameters of an autoregressive process (AR) from noisy observations is still a challenging problem. In this paper, we propose to sequentially estimate both the signal and the parameters, avoiding a non-linear approach such as the extended Kalman filter. The method is based on two conditionally linked Kalman filters running in parallel. Once a new observation is available, the first filter uses the latest estimated AR parameters to estimate the signal, while the second filter uses the estimated signal to update the AR parameters. This approach can be viewed as a recursive instrumental variable-based method and hence has the advantage of providing consistent estimates of the parameters from noisy observations. A comparative study with existing algorithms illustrates the performances of the proposed method when the additive noise is either white or coloured.

When enhancing a speech signal using a single microphone system, various approaches based on an autoregressive speech model are referenced in the literature. Using a Kalman filter, they operate in two steps: (1) the noise variances and the autoregressive parameters are estimated, (2) the speech signal is retrieved using standard Kalman filtering. However the existing methods are usually iterative and a voice activity detector (VAD) is often required to find the silent frames for the estimation of the variance of the white noise. To avoid these drawbacks, we propose to consider Kalman filter-based speech enhancement as a realisation issue, i.e. as the estimation of the system matrices in the state space representation using the estimation of the correlation function of the observations. For this purpose, we first present various solutions, based on works initially developed in the field of identification by Van Overschee et al. and Verhaegen. Their non-iterative extensions to coloured noise are also addressed and used with car noise. In the second part of the paper we propose an alternative approach based on Mehra et al. and Belanger's approaches dealing with the estimation of the steady Kalman gain and previously derived in the framework of identification. This approach still avoids the use of a VAD.

One of the most important challenges when applying multiple-model approaches is model-set design. However, to our knowledge, there are not general rules to choose models. For this purpose, we recently proposed an approach based on the Jeffrey's divergence to characterize dissimilarities between two state models. In this paper, our contribution is to use the Jeffrey's divergence to classify at least two models into subsets. Our approach consists in creating a dissimilarity matrix composed of Jeffrey's divergences between model pairs. Then, we transform this matrix to get a correlation-like matrix and an eigenvalue decomposition is computed. We propose an interpretation of the predominant eigenvalues and use it to deduce the number of model subsets and their cardinals. Finally, a classification algorithm can be considered to determine which models belong to which subsets. Among the applications, we focus on target tracking.

In ISAR processing, post-processing of the range Doppler image is useful to help the practitioner for ship recognition. Among the image post-processing tools, interpolation methods can be of interest especially when zooming. In this paper, we study the relevance of the Holderian regularity-based interpolation. In that case, interpolating consists in adding a new scale in the wavelet transform and the new wavelet coefficients can be estimated from others. In the original method, initially proposed by two of the authors, the image is first interpolated along the rows and then along the columns. Concerning the diagonal pixels, they are estimated as the mean of the adjacent original and interpolated pixels. Here, we propose a variant where the diagonal pixels are estimated by taking into account the local orientation of the image. It has the advantage of conserving local regularity on all interpolated pixels of the image. A comparative study on synthetic data and real range-Doppler images is then carried out with alternative interpolation techniques such as the linear interpolation, the bicubic one, the nearest neighbour interpolation, etc. The simulation results confirm the effectiveness of the approach.

A great deal of interest has been paid to the estimation of time-varying autoregressive (TVAR) parameters. However, when the observations are disturbed by an additive white measurement noise, using standard least squares methods leads to a weight-estimation bias. In this paper, we propose to jointly estimate the TVAR parameters and the measurement-noise variance from noisy observations by means of a generalized eigenvalue decomposition. It extends to the TVAR case an off-line method that was initially proposed for AR parameter estimation from noisy observations. A comparative study is then carried out with existing methods such as the recursive errors-in-variable approach and Kalman based algorithms.

Space time adaptive processing (STAP) is useful in radar processing to detect a target by filtering the clutter and the additive thermal noise. A derived version based on a multichannel autoregressive (M-AR) model of the clutter has the advantage of reducing the computational cost. Nevertheless, the estimation of the AR matrix parameters is a key issue because the clutter is not Gaussian in real cases. When dealing with an off-line solution, the multichannel least squares method (MLS) can be considered, but the estimation of the disturbance covariance matrix is required. In this paper, we suggest using the so-called fixed point method since it has “matrix- and texture-constant false alarm rate” property (matrix-CFAR and texture-CFAR) and it provides an unbiased and consistent estimate in a non-Gaussian case. A comparative study is then carried out between off-line M-AR based STAP methods and it points out the relevance of the solution we propose.

In various applications from radar processing to mobile communication systems based on CDMA or OFDM, M-AR multichannel processes are often considered and may be combined with Kalman filtering. However, the estimations of the M-AR parameter matrices and the autocorrelation matrices of the additive noise and the driving process from noisy observations are key problems to be addressed. In this paper, we suggest solving them as an errors-in-variables issue. In that case, the noisy-observation autocorrelation matrix compensated by a specific diagonal block matrix and whose kernel is defined by the M-AR parameter matrices must be positive semi-definite. Hence, the parameter estimation consists in searching every diagonal block matrix that satisfies this property, in reiterating this search for a higher model order and then in extracting the solution that belongs to both sets. A comparative study is then carried out with existing methods including those based on the Extended Kalman Filter (EKF) and the Sigma-Point Kalman Filters (SPKF). It illustrates the relevance and advantages of the proposed approaches.

In this paper, we propose a new Rayleigh channel simulator. Modeling the channel by an AR process leads to numerical problems due to the bandlimitation of the theoretical power density spectrum (PSD) of a Rayleigh channel. Therefore, we suggest modeling the channel by a low-pass filtered version of the so-called stochastic sinusoidal process. It consists of sinusoids in quadrature with random magnitudes modeled as AR processes. To estimate the AR parameters of the amplitudes, we take advantage of the asymptotic behavior of the first-kind zero-order Bessel function. We show that unlike an AR channel modeling, this simulator has the advantage of exhibiting the PSD peaks at the maximum Doppler frequency, for any AR process order.

Among parametric methods for speech enhancement, one consists in combining an autoregressive model for speech and a Kalman filter. This filtering is optimal in the H/sub 2/ sense providing the initial state vector, the input and the observation vectors in the state space representation of the system are independent, white and Gaussian. However, these assumptions do not necessarily hold when processing speech. In this paper, we propose to investigate an alternative approach, which is based on H/sub /spl infin// filtering and hence does not depend on these restrictive assumptions. In that setting, the purpose is to minimize the worst possible effects of the noises and system uncertainties on the estimation error. A comparative study between Kalman and H/sub /spl infin// filtering is carried out, when the additive colored noise can be modeled by a moving average (MA) process.

This paper deals with the identification of time-varying frequency-flat Rayleigh fading channels disturbed by an additive white Gaussian noise, using a training sequence based approach. When the channel is modeled by an Autoregressive (AR) process, it can be estimated by using a Kalman filter. However, this solution requires the preliminary unbiased estimations of the AR parameters and the variances of both the additive noise and the driving process in the state space representation of the system. Instead of using the existing noise compensated approaches which usually require a long observation window and do not necessarily provide reliable estimates when the signal to noise ratio is low, we propose an alternative approach using recent results developed for the Errors-In-Variables (EIV) issue. This method consists in estimating the kernel of specific autocorrelation matrices and has the advantage of providing both the noise variances and the channel AR parameters. Moreover, the maximum Doppler frequency can be also deduced.

This paper deals with the estimation of rapidly time- varying Rayleigh fading channels in synchronous multi-carrier direct-sequence code division multiple access (MC-DS-CDMA) systems. When the fading channel is approximated by an autoregressive (AR) process, it can be estimated by means of Kalman filtering for instance. Nevertheless, this requires the a priori estimation of the AR parameters. One standard solution consists in first fitting the AR process autocorrelation function to the Jakes one and then solving the resulting Yule-Walker equations (YWE). However, due to the band-limited nature of the Jakes Doppler spectrum, severely ill-conditioned YWE are unavoidable for all but very small AR model orders. Therefore, previous studies focused only on 1st and 2nd order AR models. To overcome the ill-conditioning problem, a very small positive bias can be added to the main diagonal of the autocorrelation matrix in the YWE. Even if the resulting process is not band-limited and corresponds to an AR process+noise model, the approximation can be of interest. Indeed, according to our simulation results, high-order AR models+noise yield significant results in terms of spectrum approximation and bit error rate (BER). However, to reduce the computational cost, a 5th order AR model can be considered.

The paper deals with the local sensitivity of the discrete-time infinite-horizon H/sup /spl infin// filtering problem. A new nonlinear perturbation bound is derived for the solution of the related matrix Riccati equation.