Gérard Alengrin

Results on AR-modelling of nonstationary signals

Abstract Modelling of nonstationary signals can be performed using time-varying AR-models. The time-dependent AR-coefficients are assumed to be well represented by a linear combination of a small number of known time functions. This paper intends to compare two methods for the identification of such models. The first one is a blockwise method in which the parameters are estimated using the Morf-Dickinson-Kailath-Vieira algorithm for the resolution of covariance equations. In the second method, the identification is performed by a recursive least-squares algorithm. Finally, an extension of the second method for the detection of abrupt changes in AR-processes is presented.

Unbiased parameter estimation of nonstationary signals in noise

Recent approaches to the modeling of nonstationary signals by means of AR or ARMA models use a representation with time-varying parameters. The time-varying parameters are assumed to be linear combinations of a set of basis time functions so that the model is specified by constant parameters. For stationary signals disturbed by white noise, an approach based upon a modified least-squares method leads to a good unbiased estimator of the parameters. In this correspondence, a similar algorithm deriving the unbiased parameters for nonstationary signals in white noise is given. The experimental results show the good performance of the proposed estimator.

Polynomial-phase signal analysis using stationary moments

Abstract This article proposes a method for polynomial-phase signals analysis relying on their time-invariant high-order moments. The set of these moments is first characterized in the noiseless and noisy case. It is demonstrated that the only identifiable phase parameter is the highest-order coefficient, the estimation requiring moments of order at least the double of the phase degree. Next, the consistency of the estimated moments calculated by sample averages is proved. Finally, an estimation algorithm based on the moments having for lags the multiples of a given ‘root’ is developed. Performances, evaluated by computer simulations, show the influence of the different parameters and the efficiency of the method.

Comparison of two ARMA estimators

Two alternative ARMA (autoregressive moving average) estimators are compared both theoretically and through simulation analysis. The first is a dual algorithm that estimates the MA and the AR components as the solution of two linear and independent systems of equations. For the second estimator, the AR coefficients result from a system of linear equations, while the MA component is obtained from a fast filtering algorithm initialized with the previous AR estimated coefficients.<<ETX>>

High-order ergodicity of a complex harmonic process

The paper deals with high-order moment-ergodic properties of a complex process composed of superimposed sinusoids. This is efficiently achieved using a state-space representation of the signal and Kronecker algebra. First, a simple formulation of the stationarity hypothesis is given, and finally, the moment-ergodic theorem for the associated moments is derived. >

Spectral estimation methods avoiding eigenvector decomposition

The autoregressive principal component technique uses the singular value decomposition (SVD) of an augmented dimension estimated autocorrelation matrix R to provide an accurate identification of frequencies in white noise. To avoid the eigen-decomposition of the matrix R, S.M. Kay and A.K. Shaw (1988) have applied a transformation on the inverse of R that truncates the eigenvalues associated with the noise. However, this technique requires the inversion of R and of another matrix. Two transformations that are applied directly to the matrix R are proposed. One is based on the matrix exponential and the other on component matrices. Another transformation analog to the MUSIC method without calculus of the eigenvectors is also proposed.<<ETX>>

Estimation of ARMA( p,q ) parameters

Abstract The need for estimating the parameters of an ARMA( p , q ) process arises in many applications both in signal processing and in automatic control. Recently, we proposed an estimation procedure to get the ARMA parameters. The method is based on a 2-step approach: first the AR parameters are estimated using a transient Kalman gain, then the MA parameters are estimated by a fast filtering algorithm. This short paper shows the results of simulations conducted to evaluate this method. We study the consistency and the efficiency of the proposed estimator and we give some examples for ARMA(2, 2), ARMA(4, 4) and ARMA(4, 4) processes.

Parametric modeling of photometric signals

This paper studies a new model for photometric signals under high flux assumption. Photometric signals are modeled by Gaussian autoregressive processes having the same mean and variance denoted Constraint Gaussian Autoregressive Processes (CGARPs). The estimation of the CGARP parameters is discussed. The Cramer Rao lower bounds for these parameters are studied and compared to the estimator mean square errors. The CGARP is intended to model the signal received by a satellite designed for extrasolar planets detection. A transit of a planet in front of a star results in an abrupt change in the mean and variance of the CGARP. The Neyman-Pearson detector for this changepoint detection problem is derived when the abrupt change parameters are known. Closed form expressions for the Receiver Operating Characteristics (ROC) are provided. The Neyman-Pearson detector combined with the maximum likelihood estimator for CGARP parameters allows to study the generalized likelihood ratio detector. ROC curves are then determined using computer simulations.

Improvements of a state-space iterative noise reduction algorithm for harmonic retrieval

When the model of a noisy sinusoidal process is autoregressive moving average (ARMA), then the AR spectrum is biased. However, since the AR spectrum contains all the second-order information of the process, it is possible to retrieve the noiseless predictor from the noisy one. An iterative algorithm enabling the computation of the ARMA parameters from the AR parameters and a new well-suited initialization scheme are presented. Simulations of the state-space iterative noise reduction algorithm (SINA) are performed using various AR estimators. The mean-square-error graph is plotted for all these estimators and performances of the methods are discussed. >

A new state space algorithm for computing the frequencies of sinusoids in white noise

Abstract The Pisarenko decomposition of a random process consisting of a sum of sinusoids and a white noise signal is usually obtained by the computation of the minimum eigenvalue and the corresponding normalized eigenvector of the autocorrelation matrix which give the noise power and the frequencies of the sinusoids. The use of a mathematical library allows the calculation of all the eigenvalues, but, since only the minimum eigenvalue-eigenvector pair is needed, more efficient iterative algorithms can be found. The aim of this paper is to present a new algorithm, based on state-space modeling, for retrieving sinusoidal frequencies from noisy data. Moreover, the proposed algorithm establishes a connection between a state-space formulation and the Pisarenko decomposition. To illustrate the performances of the method, we have compared it with the bisection algorithm of Hayes and Clements. The simulation results on theoretical correlations have shown that the two methods exhibit identical behavior. It is worthy to be noted that the complexity of these algorithms can be reduced by using a ‘Split Levinson recursion’ in place of the traditional Levinson recursion.