APPLICATION TO LOCALLY OPTIMUM DETECTION OF A NEW NOISE MODEL Alessandra Tesei, and Carlo S. Regazzoni Department of Biophysical and Electronic Engineering (DIBE) University of Genoa, Via Opera Pia 11A - 16145 Genova, Italy
[email protected]
ABSTRACT The work is addressed to provide realistic modelling of a generic noise probability density function (pdf), in order to optimize signal detection in non-Gaussian environments. The target is to obtain a model depending on few parameters (quick and easy to estimate), and so general to be able to describe many kinds of noise (e.g., symmetric or asymmetric, with variable sharpness). To this end, a new HOS-based model is introduced, which derives from the generalized Gaussian function, and depends on three parameters: kurtosis, for representing variable sharpness, and left and right variances (whose combination provides the same information of skewness) for describing deviation from symmetry. This model is applied in the design of a Locally Optimum Detection (LOD) test. Promising experimental results are presented which derive from the application of the test for detecting signals corrupted by real underwater acoustic noise.
1. INTRODUCTION This work is addressed to modelling in a realistic way generic background noise, in order to optimize signal detection in non-Gaussian environments. Detection is dealt with as binary hypothesis testing in the context of statistical inference [1]: the purpose is to decide between the two hypotheses of the presence (H1) or the absence (H0) of a transmitted deterministic signal {sk, k=1, .., K} (the approach can be extended to the stochastic case), on the basis of acquired observations {yk, k=1, .., K} [1]; the noise, {nk, k=1, .., K} corrupting the signal during the propagation is assumed additive, independent and identically distributed, stationary, and generally nonGaussian. The addressed main target is to design a detector characterized by: (a) high performances in the case of weak signals; (b) easy applicability to real cases (in
particular, easy and realistic estimation of needed parameters, realistic noise modelling, and robustness to variable boundary conditions); (c) algorithmical simplicity. In order to detect signals in the case of low values of the Signal-to-Noise Ratio (SNR) (in the range -20÷0 dB) (property (a)), the statistical testing approach selected is a Locally Optimum Detector (LOD) [1]. For satisfying conditions (b) and (c), the investigation is addressed to express generalized noise pdf models, usually depending on parameters difficult to be estimated from real data samples, in terms of Higher-OrderStatistics (HOS) parameters [2]. HOS parameters are very easy and quick to be extracted from data and are particularly suitable for quantifying deviation from Gaussianity in terms of asymmetry (with third-order parameters) and variable sharpness (with fourth-order parameters). As conventional signal processing algorithms based on the Second Order Statistics, optimized in presence of Gaussian noise, may decay in non-Gaussian environments, 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 can be applied only under certain assumptions of noise distributions [8][9]; finally some algorithms are complicated [8]. In this paper, the asymmetric generalized Gaussian function is introduced. It derives from the combination of the well-known generalized Gaussian pdf [10] and of the asymmetric Gaussian model, presented in [11]. The first model is symmetric and 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, β2. Hence the analytical relationship between c and β2 is introduced. The resulting symmetric function based on kurtosis has the same characteristics of the
generalized Gaussian pdf, and is a realistic noise-pdf model for β2≥1.87. In order to introduce also possible deviation from symmetry, the resulting kurtosis-based function is modified by taking into account the asymmetric Gaussian model introduced in [11]. It directly derives from the Gaussian shape, but is asymmetric and depends on two second-order parameters, the left and right variances [11]. By introducing these two parameters in the kurtosis-based generalized Gaussian function, the "asymmetric generalized Gaussian" model can be obtained. The new model is compared with the generalized Gaussian and the asymmetric Gaussian pdfs, which result as particular cases of it. It is applied in the design of a LOD test, used for detecting deterministic known signals corrupted by real underwater acoustic noise radiated by ship traffic [12]. 2. THE ASYMMETRIC GENERALIZED GAUSSIAN MODEL AND ITS APPLICATION IN A 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 β2, i.e., the ratio of the fourth and the square of the second central moments. It is equal to 3 in the Gaussian case; the sharpness of a pdf shape is higher (lower) than the corresponding Gaussian function as β2 is larger (smaller) than 3. A good model for general pdfs has variable sharpness. One of the well-known symmetric pdf models having variable sharpness is the generalized Gaussian, which depends on the parameter c: c γc − γ (n −µ ) pgG ( n ) = e (1) 2 Γ (1 / c ) where {n} is generic noise with mean value µ and
variance σ2, γ =
Γ(3 / c) σ 2Γ ( 1 / c )
, Γ(k ) =
+∞
z
e− x x k −1dx .
0
c cannot be directly estimated from data samples; hence the relationship between c and β2 is introduced. It derives from the β2 definition and is expressed by the following formula: m4x Γ ( 5 / c ) Γ (1 / c ) x β2 = = (2) x 2 ( m2 ) ( Γ ( 3 / c ))2 Expressing exactly c in terms of β2 is not possible because of the presence of the Γ(⋅) function. That's why a good analytical approximation of the function c=c(β2) was found:
c = c ( β2 ) ≈
5 − 0. 12 for 1.87≤β ≤30. 2 β2 − 1. 865
(3)
This formula allows one to express pgG(n) in terms of β2. Its validity is confirmed by observing that for β2>3 the resulting pdf has heavy tails, as expected [10]. Figure 1 shows a family of generalized Gaussian functions as β2 varies. In order to generalize this model so that it can be also asymmetric, the asymmetric Gaussian model presented in [11] is taken into account. It depends on two second-order parameters (deriving from the definition of variance), σ2l and σ2r , called respectively "left" and "right variances" and estimated from finite sequences of the process {n} according to the following formulas:
I F N J G J G K H I F Nr 1 2J n σ2r = − µ G ( ) ∑ k J Nr − 1 G K Hk =1,nk >µ σ2l =
l 1 ( nk − µ )2 ∑ Nl − 1 k =1, n
