USE OF FOURTH-ORDER STATISTICS FOR NON-GAUSSIAN NOISE MODELLING: THE GENERALIZED GAUSSIAN PDF IN TERMS OF KURTOSIS A. Tesei, and C.S. Regazzoni DIBE, University of Genoa Via all'Opera Pia 11A, 16145 Genova, Italy Tel: +39 10 3532792; fax: +39 10 3532134 e-mail: tale@dibe.unige.it ABSTRACT In this paper non-Gaussian noise modelling is addressed. HOS-based parametric pdf models are investigated in order to provide realistic noise modelling by means of easy and quick estimation of needed parameters. Attention is focused on the generalized Gaussian pdf. This model, generally depending on a real theoretical parameter c, difficult to estimate from data, is proposed expressed in terms of the fourth-order parameter kurtosis β 2 by introducing the analytical relationship between c and β 2 . The model is compared with well-known pdfs and used in the design of a LOD test. 1 INTRODUCTION This work is addressed to realistic characterization of generic background noise aimed at the optimization of signal detection in non-Gaussian environments. Detection is dealt with as binary hypothesis testing in the context of statistical inference [1]: the decision between the two hypotheses of the presence (H 1 ) or the absence (H 0 ) of a transmitted signal {s i , i=1, .., M} is made on the basis of acquired observations {y i , i=1, .., M} [1]; the noise, {n i , i=1, .., M}, is assumed additive, independent and identically distributed (iid), stationary, unimodal, generally non-Gaussian. Among the main targets addressed, easy applicability to real cases is focused, in terms of realistic noise modelling, easy and realistic estimation of model parameters, and robustness to variable boundary conditions. Symmetric probability density function (pdf) models are considered. In order to satisfy the mentioned requirements of easy applicability of a model to real cases, the investigation is addressed to express generalized noise pdfs, usually depending on parameters difficult to be estimated from real data samples, in terms of Higher- Order-Statistics (HOS) parameters, which are very easy and quick to be extracted from data and are particularly suitable for quantifying deviation from Gaussianity [2]. As conventional signal processing algorithms based on the Second Order Statistics, optimized in presence of Gaussian noise, may decay in non-Gaussian noise, various works used HOS theory [2] as signal-processing basis for noise analysis and detection optimization; however, some methods work only with non-Gaussian signals [3][4][5] or only in Gaussian noise [5][6][7]; some are not optimized for low SNR values [3]. In this paper, attention is focused on the generalized Gaussian function; it depends on a real parameter, c, which is not easy to estimate from data. Nevertheless, c presents a physical meaning, as linked with the pdf sharpness. The HOS parameter which better describes sharpness variability is the fourth-order kurtosis, β 2 . The analytical relationship between c and β 2 and the range of kurtosis in which the resulting pdf model can be applied are introduced. The resulting symmetric function has the same characteristics of the generalized Gaussian, and is a realistic noise-pdf model for 1.865<β 2 ≤30 (hence for both sub- and super-Gaussian pdfs). It is compared with another kurtosis-based generalized symmetric function, the Champernowne model [8][9], which results less general. In order to detect signals in the critical case of low SNR values (in the range [-20, 0] dB), the statistical testing approach selected is a Locally Optimum Detector (LOD) [1]. The new pdf proposed is applied in the design of a LOD test, used for detecting constant weak signals corrupted by real underwater acoustic noise [10][11]. 2 DESCRIPTION OF THE KURTOSIS-BASED MODEL AND ITS APPLICATION TO THE LOD TEST In the context of noise modelling, one of the most noticeable ways in which estimated noise distributions deviate from Gaussianity is in kurtosis β 2 , i.e., the ratio of the fourth and the square of the second moments. It is equal to 3 in the Gaussian case; the sharpness of the pdf shape is higher (lower) than the corresponding Gaussian function when β 2 is larger (smaller) than 3. A good model for generalized symmetric pdfs has variable sharpness.