IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 48, NO. 4, APRIL 2000 935 Robust Parameter Estimation of a Deterministic Signal in Impulsive Noise Jonathan Friedmann, Hagit Messer, Senior Member, IEEE, and Jean-François Cardoso, Member, IEEE Abstract—This paper presents a robust class of estimators for the parameters of a deterministic signal in impulsive noise. The proposed technique has the structure of the maximum likelihood estimator (MLE) but has an extra degree of freedom: the choice of a nonlinear function (which is different from the score function suggested by the MLE) that can be adjusted to improve robust- ness. The effect of this nonlinear function is studied analytically via an asymptotic performance analysis. We investigate the covari- ance of the estimates and the loss of efficiency induced by nonop- timal choices of the nonlinear function, giving special attention to the case of α-stable noise. Finally, we apply the theoretical results to the problem of estimating parameters of a sinusoidal signal in impulsive noise. Index Terms— -stable distribution, impulsive noise, parameter estimation, robust estimation. I. INTRODUCTION T HIS PAPER addresses the problem of estimating some un- known parameters of a deterministic signal received in ad- ditive impulsive noise. The received discrete data sequence is modeled as (1) where is the deterministic signal that is known to within a parameter vector (e.g., the received signal in a radar system that is an attenuated and delayed version of the known, trans- mitted waveshape), is a noise sequence, and quanti- fies the amplitude of the additive noise. The theory and techniques for estimating a parametric signal in white Gaussian noise are well established. Even with non- Gaussian noise, the problem has been studied, especially in the radar community. In this paper, we are interested in the param- eter estimation problem in presence of impulsive noise, possibly modeled as having an stable distribution. Modeling of impulsive random processes by the -stable statistics has been shown to be an effective tool in modern statistical signal processing [1]. In particular, in many important applications (e.g., HF and cellular communication, underwater acoustics, etc.), the additive noise has been shown to be of Manuscript received October 6, 1998; revised August 18, 1999. The associate editor coordinating the review of this paper and approving it for publication was Dr. Lal C. Godara. J. Friedmann and H. Messer are with the Department of Electrical Engineering—Systems, Tel Aviv University, Tel Aviv, Israel (e-mail: jonaf@eng.tau.ac.il; messer@eng.tau.ac.il). J.-F. Cardoso is with the Centre National de la Recherche Scientifique, Département TSI, École Nationale Supérieure des Télécommunications, Paris, France. Publisher Item Identifier S 1053-587X(00)02375-8. an impulsive nature (see [2] and references therein). Unfor- tunately, standard techniques for parameter estimation [such as the maximum likelihood (ML)] are not easily implemented in the -stable case. It is possible to estimate the parameters of the -stable distribution [3], [4] and use these estimates for approximating the likelihood, but this approach leads to complex estimators. The approach followed in this paper is to consider “ -es- timates” [5] for that are obtained as solutions of estimating equations having the same structure as those of the ML ap- proach. These estimators use a fixed nonlinear function that is not necessarily the score function associated with the distribu- tion of the noise. This leads to suboptimal but possibly robust estimates. In Section II, we discuss the Cramér–Rao lower bound on the achievable estimation error of This bound serves as a ref- erence to the optimality of any unbiased estimator In Sec- tion III, we present the appropriate -estimates, using the ML estimator as a starting point. Section IV gives a detailed asymp- totic analysis of the procedure and investigates the efficiency of the proposed technique. Simulation results are also presented for the estimation of the parameters of a sinusoidal signal in sym- metric -stable noise. II. CRAMÉR–RAO BOUND The Cramér–Rao inequality sets a lower bound on the covari- ance of any unbiased estimator of (2) where is the Fisher information matrix (FIM) at The problem of estimating parameters of a deterministic signal in white noise of known distribution has already been addressed; see, e.g., [6]. The Cramér-Rao bound (CRB) (the inverse of the FIM) takes the form (3) where is the “Fisher information for location shift” for the (unscaled) noise density (4) and matrix is defined as (5) 1053–587X/00$10.00 © 2000 IEEE