Search

International audience

Proposed early in the 90ies, the detrended fluctuation analysis (DFA) can be used to estimate the Hurst exponent and has been proven relevant in various applications, from economics to biomedical. For the last years, variants have been proposed. They differ in the way to estimate the trend of the centered integrated signal. In this paper, we recall the main principles of some of these methods, provide explanations on the behaviours of the algorithms and analyze the relevance of new variants based on the Savitzky-Golay filter, also known as the LOESS approach, and the LOWESS. They bridge the gap between the DFA and the detrending moving average (DMA). We hence show that the LOESS-based method is a generalization of the DMA.

A great deal of interest has been paid to enhance the radar range resolution for the last decades. One of the techniques is to construct a waveform combining inter-pulse and intra-pulse modulations. In this paper, stepped frequency (SF) waveform is used to represent the former, whereas phase coding (PC) is considered for the latter. To obtain the high-resolution range profile (HRRP) of a target induced by an SF waveform at the receiver, one of the approaches is the frequency domain (FD) algorithm. It has been successfully applied with SF linear frequency modulation waveforms. However, based on our investigations, processing the SFPC waveform with the FD algorithm does not lead to the performance, in terms of peak sidelobe ratio (PSLR) and integrated sidelobe ratio (ISLR), of the single-carrier phase coding (SCPC) waveform processed with a matched filter (MF). To overcome these drawbacks, we propose to split the spectrum of a phase coded pulse into a predetermined number of portions, and then to successively transmit the time-domain transformed versions of these various portions. The received echoes are then processed with a modified version of the FD algorithm. Our analysis and simulations show that the proposed waveform can be characterized by a PSLR and an ISLR close to those of the SCPC processed with MF in some scenarios.

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.

International audience; The detrended fluctuation analysis (DFA) and its higher-order variant make it possible to estimate the Hurst exponent and therefore to quantify the long-range dependence of a random process. These methods are popular and used in a wide range of applications where they have been proven to be discriminative to characterize or classify processes. Nevertheless, in practice, the signal may be short-memory. In addition, depending on the number of samples available, there is no guarantee that these methods provide the true value of the Hurst exponent, leading the user to draw erroneous conclusions on the long-range dependence of the signal under study. In this paper, using a matrix formulation and making no approximation, we first propose to analyze how the DFA and its higher-order variant behave with respect to the number of samples available. Illustrations dealing with short-memory data that can be modeled by a white noise, a moving-average process and a random process whose autocorrelation function exponentially decays are given. Finally, to avoid any wrong conclusions, we propose to derive abacuses linking the value provided by the DFA or its variant with the properties of the signal and the number of samples available.

International audience; Target classification can be of real interest for sea surveillance in both civil and military contexts. To address this issue, we present two approaches based on the Singer model. The latter has the advantage of covering a wide range of motions depending on the values of its parameters. Given noisy observations, the first method aims at estimating the motion model parameters by taking advantage of the properties of the correlation function of the estimated acceleration. It is based on a genetic algorithm. The second approach is on-line and consists in deriving a joint tracking and classification (JTC) method. Based on various simulations, we study their respective relevance in different operational settings. The proposed JTC corresponds to the best compromise in terms of performance and number of samples required.

International audience; The non-linear frequency modulation (NLFM) waveform is one of the existing waveforms that can be used in high range resolution radar applications. However, a high sampling frequency and consequently an expensive ADC are required. To overcome this drawback while taking advantage of the features of the NLFM waveform, we suggest approximating the wideband NLFM waveform by a piecewise linear waveform and using it in a stepped frequency (SF) framework. Thus, a variable chirp rate SF-LFM waveform is proposed where SF is combined with a train of LFM pulses having different chirp rates, durations, and bandwidths. In this paper, these parameters are derived from a tangent-based NLFM waveform. At the receiver, a generalized version of the time domain (TD) algorithm is proposed to process the received echoes. Our purpose is to obtain the high range resolution profile (HRRP) whose properties are of the same magnitude orders as those obtained using a tangent-based NLFM waveform. These properties are the peak sidelobe ratio, the integrated sidelobe, and the range resolution. Toward this goal, a multi-objective optimization issue is addressed to deduce the parameters of the proposed waveform by using two types of approaches based on evolutionary algorithms. Their relevance is compared. Our analysis and simulations show that the proposed approaches attain the targeted performance goals with a smaller sampling frequency at the receiver.

International audience; The detrended fluctuation analysis (DFA) and its variants such as the detrended moving average (DMA) are widely used to estimate the Hurst exponent. These methods are very popular as they do not require advanced skills in the field of signal processing and statistics while providing accurate results. As a consequence, 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 threefold. Firstly, we propose another variant avoiding the discontinuities between consecutive local trends of the DFA by a priori constraining them to be continuous. Secondly, we show that, in all these approaches, the square of the fluctuation function can be presented in a similar matrix form. When the process is wide-sense stationary (w.s.s.), the latter can be seen as the power of the output of a linear filtering whose frequency response depends on the given method. In the general case, an interpretation of the square of the fluctuation function is also given by expressing it as the convolution between the 2D-Fourier transform of two matrices, one whose elements correspond to the instantaneous correlation function of the signal and the other which depends on the detrending method. To end up, an illustration is provided in the field of avionics for the detection of the visual tunneling, a deleterious cognitive state.

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.

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.

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.

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.

When dealing with time-frequency analysis, each distribution belonging to the Cohen class (CC) is defined by its kernel. It introduces a time-frequency smoothing which impacts on the time-frequency resolution and the cross-term disappearance. Depending on the application, the practitioner has to find the ''best'' compromise. In this paper, we suggest combining several CC time-frequency representations (TFRs), corresponding to coarse-to-fine scales of smoothing. Taking advantage of this diversity, our approach consists in differentiating the signal, assumed to be characterized by 2-D near-linear stable trajectories in the time-frequency plane, and the cross-terms, assumed to be geometrically unstructured. For this purpose, a ''confidence map'' for each TFR is deduced from a local variational analysis of the time-frequency distribution. The set of confidence maps is then used to combine the different TFRs in order to obtain an ''enhanced'' TFR. Our approach is compared with various conventional TFRs by using synthetic data. Despite a comparatively higher computational cost, the resulting enhanced TFR exhibits a high time-frequency resolution while having limited cross-terms. As simulation results confirm the effectiveness of our method, it is then applied in the field of inverse synthetic aperture radar (ISAR) processing.

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.

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.

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 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.

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.

Whereas lecturing is the most widely used mode of instruction, we have explored small-group learning projects to make signal processing more appealing at the University and in Engineering schools in Bordeaux (France). The projects cover a wide range of applications, from audio processing to mobile communication system analysis and can be based on problems suggested by industrial partners. After a state of the art in the area, the students develop signal processing algorithms and usually deliver new softwares. At the end of the semester, they provide a final 8-page report and present their work during a one-day workshop. The projects enable the students to experience cooperative works, which is mostly done in industry. In addition, they help the students to see more easily the links between the courses they follow, including project managing. The students are hence involved in dialog inside their own group, in writing English reports as well as in problem solving, analysis and synthesis. This paper presents four examples dealing with speech processing, spectral analysis, Universal Mobile Telecommunications System (UMTS) mobile positioning and orthogonal frequency division multiple access (OFDMA) synchronization.

on line depuis novembre 2008; International audience; This paper deals with the on-line estimation of time-varying frequency-flat Rayleigh fading channels based on training sequences and using H∞ filtering. When the fading channel is approximated by an autoregressive (AR) process, the AR model parameters must be estimated. As their direct estimations from the available noisy observations at the receiver may yield biased values, the joint estimation of both the channel and its AR parameters must be addressed. Among the existing solutions to this joint estimation issue, Expectation Maximization (EM) algorithm or crosscoupled filter based approaches can be considered. They usually require Kalman filtering which is optimal in the H2 sense provided that the initial state, the driving process and measurement noise are independent, white and Gaussian. However, in real cases, these assumptions may not be satisfied. In addition, the state-space matrices and the noise variances are not necessarily accurately estimated. To take into account the above problem,we propose to use two crosscoupled H∞ filters. This method makes it possible to provide robust estimation of the fading channel and its AR parameters.

A great deal of attention has been paid to speech enhancement using a single microphone system. The various approaches, based on the Kalman filter, operate in two steps: (1) the noise variances and the parameters of the speech model are estimated, and (2) the speech signal is retrieved using standard Kalman filtering. This letter presents an alternative solution that does not require the explicit estimation of the noise and the driving process variances. This deals with a new formulation of the approach proposed within a control literature framework by Mehra (1970).

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.

This study deals with the estimation of a vector process disturbed by an additive white noise. When this process is modelled by a multivariate autoregressive (M-AR) process, optimal filters such as Kalman or H∞ filter can be used for prediction or estimation from noisy observations. However, the estimation of the M-AR parameters from noisy observations is a key issue to be addressed. Off-line or iterative approaches have been proposed recently, but their computational costs can be a drawback. Using on-line methods such as extended Kalman filter and sigma-point Kalman filter are of interest, but the size of the state vector to be estimated is quite high. In order to reduce this size and the resulting computational cost, the authors suggest using dual optimal filters. In this study, the authors propose to extend to the multi-channel case the so-called dual Kalman or H∞ filters-based scheme initially proposed for single-channel applications. The proposed methods are first tested with a synthetic M-AR process and then with an M-AR process corresponding to a mobile fading channel. The comparative simulation study the authors carried out with existing techniques confirms the effectiveness of the proposed methods.