RELIABLE PARAMETER ESTIMATION FOR GENERALISED GAUSSIAN PDF MODELS: APPLICATION TO SIGNAL DETECTION IN NONGAUSSIAN NOISY ENVIRONMENT Claudio Sacchi (*), Saverio Giulini (**), Carlo Regazzoni (*), Gianni Vernazza (*) (*) University of Genoa Department of Biophysical and Electronic Engineering (DIBE) Signal Processing and Telecommunications Group Via Opera Pia 11/A I-16145 Genoa (ITALY) Phone: +39-010-3532674, Fax: +39-010-3532134 E-mail:
[email protected] (**) University of Genoa Department of Architectural Science (DSA) Stradone S. Agostino 37 I-16123 Genoa (Italy) Phone: +39-010-2095903, Fax: +39-010-2095900 E-mail:
[email protected]
ABSTRACT In this paper, a new analytical approximated expression for the sharpness parameter of a Generalised Gaussian pdf model as a function of a higher-order statistic, namely normalised kurtosis is proposed. The approximation is based on some mathematical considerations concerning the Gamma function, and provides a very precise sharpness evaluation for a wide range of normalised kurtosis values. As a result, it allows the exploitation of the parametric Generalised Gaussian pdf model in advanced signal processing applications, e.g. detection of weak signals in non-Gaussian noise, where an accurate evaluation of noise distribution is required.
1. INTRODUCTION Many natural and man-made noises affecting the performances of communication systems are not well represented by the conventional Gaussian distribution. In particular, the noise produced by discrete identifiable sources is likely to be non-Gaussian, as it may be the response of a unique source or a few-component array of sources, so to make it impossible to apply the Central Limit Theorem. Middleton [13] showed that electromagnetic interference is basically heavy tail distributed or leptokurtic [10], as it is characterised by significant probabilities of large interference levels. Also underwater ocean noise could not be well modelled by a Gaussian distribution. In [14] some statistical evaluations about noisy sequences acquired in real underwater environment were presented. The results of this analysis, performed during the CEC-MAST SNECOW project (1990-1993), evidenced a flat-topped or platykurtic [10] pattern of the statistical distribution of underwater noise samples (i.e. more similar to a uniform distribution). For
this reasons, the provision of effective non-Gaussian distribution models is very important in the field of signal processing for telecommunication applications. Some well-known probability distribution function (pdf) models and the related parameter estimation methods have been already proposed in the literature; for instance: Symmetric-Alfa-Stable (SαS) distribution models [1] and Gaussian mixture models [11] have been developed for signal detection and estimation in impulsive noise. Other distribution models are the symmetric [2][10] and asymmetric [3] Generalised Gaussian (GG) pdfs. The main advantage of using the GG models lies in the possibility of providing simple analytical pdf expressions both for leptokurtic and platykurtic, both symmetrical and asymmetrical [3] distributed noises. Other models, like symmetric alfa-stable are not defined in a closed form and allow one to model only impulsive noises. Gaussian mixtures [11][12] provide reliable noise modelling for a wide range of applications, however the estimation of mixture parameters does not descend immediately by the statistical features of noise. The ExpectationMaximisation (EM) algorithm, usually considered for this aim [11], is computationally heavy and sometimes presents convergence problems [12]. On the contrary, GG models are quite simple from an analytical point of view, and the links between model parameters and noise statistics are immediately understandable. The most relevant problem in the effective use of the GG model is the difficulty with estimating the sharpness parameter from received data. This parameter is conceptually linked with the value of a higher-order statistic moment, namely normalised kurtosis, but the translation of this ideal linking into an explicit mathematical function is not a trivial problem. In [3] a first mathematical linking
between the GG sharpness parameter and the normalised kurtosis was proposed that provides good results only for a fixed range of normalised kurtosis values; unfortunately it lacks precision outside this range, where yields results of no statistical significance. In [4], a very precise expression for the sharpness was provided that works only in an impulsive range of normalised kurtosis values. It is worth noting that recently the measure of a kurtosis-like parameter, i.e. the global kurtosis has been introduced also for what concern the parameterisation of Gaussian mixture models [12] aimed at improving the performances of EM algorithm. The aim of the present work is to propose a normalised kurtosis-based method of sharpness estimation, for a reliable evaluation of the parameter c in generic (i.e., both leptokurtic and platykurtic) noisy environments, in order to allow the exploitation of the GG pdf model in a wide range of signal processing applications, considering as test application the detection of a weak known signal in non-Gaussian noise.
2. NORMALISED KURTOSIS-BASED SHARPNESS EVALUATION FOR A GENERALISED GAUSSIAN PDF Let X be a generic random variable described by a symmetric Generalised Gaussian (GG) pdf that is characterised by three parameters [2][10]: • The mean value m X ; • •
The variance σ ; The sharpness c>0, which controls the exponential decay rate of the pdf curve. The expression of the symmetric GG pdf is given by: cγ − γx X f GG e ( x) = (1) Γ(1 c ) where: Γ(3 c ) (2) γ =ˆ Γ(1 c )σ X2 2 X
c
It is clear that the pdf expression shown in (1) is able to describe a considerable range of noises, both impulsive heavy-tailed noises (called also leptokurtic [10] or ipergaussian [3]) that are characterised by c2. The Gaussian case is obviously considered by the expression (1) when c=2. In the present dealing, we are considering only the symmetric distribution case. It is shown in [3] that the GG model can represent without conceptual difficulties the asymmetrically distributed noises too. The main problem involved in the use of a GG pdf model is the estimation of the sharpness c from received samples. In this work a normalised kurtosis- based method of sharpness evaluation is proposed. The normalised kurtosis of the r.v. X is defined as:
β 2X =
{
E (X − m X
(σ ) 2 X
2
)
4
}
(3)
An unbiased and asymptotically normal estimation of the normalised kurtosis, which is theoretically linked with the r.v. pdf shapness [3] can be obtained by replacing the expectation with the simple average [7]. The mathematical linking of the normalised kurtosis with the sharpness parameter c implies the computation of the analytical inverse of the following function: Γ(5 c )Γ(1 c ) β 2X (c ) = (4) (Γ(3 c ))2 where Γ(• ) is the Gamma function. As the Gamma function is not defined in an analytical form, the inversion of (4) in the form c = c(β 2X ) is mathematically impossible. Hence a monotonic analytical approximation for (2) that admits an inverse function must be found. In [3] an approximation for c = c (β 2X ) was obtained by applying the Least Square Method (LSM) to a generic second-order expression for (2) that provides good results only for the range 1.95 ≤ β 2X ≤ 30 . The approximation is given by the formula: 5 c ≅ c1 (β 2X ) = − 0.12 (5) β 2X − 1.865
For strongly impulsive values of the normalised kurtosis the approximation shown in (5) provides incorrect and even negative values of the sharpness parameter, whereas for some sub-impulsive ranges (e.g.: β 2X ≤ 1.9 ), it is very rough. However some kinds of noises can be characterised by normalised kurtosis values falling outside the range of validity of the aforesaid approximation. A simple, yet effective, normalised kurtosis-based analytical evaluation of the sharpness parameter was presented in [4] in the case of heavy-tailed distributed noises. A generalisation of the method exposed in [4] can be considered in order to provide a reliable analytical expression that is valid and precise for a wide range of normalised kurtosis values. This new approximation is based on a generalised Stirling formula concerning the Gamma function [5]: 1 x→0 x − η + Θ( x ) Γ( x ) = (6) 2π 1 x x e − x 1 + Θ x → ∞ x x
where Θ(•) denotes a quantity which is infinitesimal of the same order than its argument. Starting from (6) and from some considerations reported in [4] and [6], the following approximations for (4) can be obtained:
5c − 12 2 5 + c→0 6c −1 9 X ( ) ≅ β2 c 3 9 1 2 1 + + 2 c → ∞ 5 3c c
(7)
The inversion of the two parts of (7) leads to the following approximation of c = c(β 2X ) :
3 1+ − 2 X X c(β 2 ) ≅ c 2 (β 2 ) ≅ log
40 β 2X − 71 9 - 5β 5 log 6 3
X 2 5
1.8 < β 2X < 3
5 X 2 β 2 − 3 9
β 2X ≥ 3
(8) The first part of (8) is obtained by the inversion of the second part of (7), which is a good approximation for (4) for platykurtic sharpness values (i.e., c>2), whereas the second part of (8) is obtained by the inversion of the first part of (7), which is a good approximation for (4) for leptokurtic sharpness values (i.e., c1.8. The approximation proposed in (8) works very well for both platykurtic and leptokurtic ranges of normalised kurtosis values, as shown in Figures 1 and 2. The analytic curve of the function c 2 (β 2X ) practically overlaps with the curve of the experimental true values of the sharpness parameter, whereas the state-of-the art approximation proposed in [3] often lacks of precision and can even provide negative values for very high the normalised kurtosis values.
Fig. 2. Comparison among different normalised-kurtosis based sharpness parameter evaluation methods: leptokurtic noise case
3. APPLICATION TO SIGNAL DETECTION IN NON-GAUSSIAN NOISE The method of sharpness estimation described in the previous section allows one to employ the GG pdf model for those signal-processing algorithms that require an accurate modelling of noise in terms of pdf. In [4] an application concerning multilevel signal estimation in impulsive noisy environments was described. Another possible application might concern with the locally optimum (L.O.) detection of known weak signals in generic non-Gaussian noise [2]. The description of a locally optimum detector is reported in [2] [8] and [9]. The test of presence/absence of a weak signal in generic noise (namely: LOD test [2][8]) can be completely defined in an adaptive way as a function of the noise-distribution parameters. Indeed, the non-linear locally optimum detection function in the case of Generalised Gaussiandistributed noise is given by [2]: g lo ( x ) = cγ c x
c −1
sign ( x )
(9)
whereas the LOD test threshold given a fixed false-alarm probability PFA can be expressed as [8]: Th LO = σ 0 {erfc −1 (PFA )} (10) where σ 0 is the standard deviation of the LOD decision variable (i.e.: the random variable compared with the threshold for taking a decision about presence of absence of the signal [2][8]). In the case of Generalised Gaussian distributed noise σ 0 is given by: Fig. 1. Comparison among different normalised-kurtosis based sharpness parameter evaluation methods: platykurtic noise case
σ0 =
E s c 2 Γ(3 c )Γ(2 − 1 c )
σ nθ
[Γ(1 c )]
2
(11)
where E s is the signal energy, σ n is the standard deviation of the background noise and θ is the (small)
signal amplitude [8]. The second factor of the expression [11] is related to the Fisher information and assumes finite values only for c>0.5 [8]. It is almost clear from expressions (9) and (11) that an imprecise estimation of the sharpness parameter starting from received data can involve a lack of adaptivity of the locally optimum detector with respect to the background noise. The detector adaptivity is the basic condition for achieving the expected results of the LOD test in terms both of improved correct detection of the known signal and of reduced false alarm probability.
infinity for negative arguments. For this reason an error in the estimation of the c parameter could be translated into wrong and very big values of the threshold, thus making the LOD detector completely blind, as evidenced by the numerical results on correct detection probability PDE shown in Table 1.
4. NUMERICAL RESULTS ABOUT LOD TEST APPLICATION The actual benefits involved by the proposed normalisedkurtosis based sharpness evaluation method for GG distributed noise are evidenced by the numerical results achieved about false alarm probabilities and correct detection probabilities achieved by a simulated LOD test. The test of presence/absence of a weak known signal at a SNR = –20dB has been performed by considering a background noise with Generalised Gaussian distribution, zero mean, variance σ n equal to 100, and normalised kurtosis values ranging from 1.872 to 21.88 (corresponding to c values ranging from 0.52 to 12.4). The fixed value of the false alarm probability PFA has been settled equal to 1%. A GG-distributed noise generator with arbitrary values of the sharpness parameter c has been implemented by following the guidelines exposed in [9]. The graphs of Figure 3 depicts the false alarm probability versus the normalised kurtosis of the background noise in the platykurtic (i.e. subgaussian) case, achieved by using the different mapping function: c 2 (β 2X ) (new proposed sharpness evaluation) and c 1 (β 2X ) (state-of-the art sharpness evaluation). The incorrect evaluation of the parameter c obtained by using c 1 (β 2X ) actually implies an increase of the false alarm rate up to 41%, whereas the expected false alarm rate settled a priori (i.e. 0.01) is reached in the overall range of normalised kurtosis values when the more precise sharpness evaluation function c 2 (β 2X ) is used. The increasing of the false alarm rate observed in the case of incorrect sharpness parameter evaluation is the logical consequence of the failed adaptation of the LOD detection function g lo (x) with respect to the background noise. A diametrically opposite situation can be verified in the case of highly impulsive normalised kurtosis values. In this case, the sharpness parameter is heavily underestimated by using the state-of-the-art sharpness evaluation function c 1 (β 2X ) . An incorrect value of the c parameter close to the limit value of 0.5 can involve the failure of the LOD test due to the lack of definition of the LOD threshold (defined only if c>0.5). Indeed, the gamma function assumes real values only for positive arguments and becomes equal to
Fig. 3. False alarm probability versus normalised kurtosis for a LOD test in platykurtic GG-distributed background noise β 2X 5.405 8.925 16.257 20.871 21.88 22.5
c ≅ c1 (β 2X ) c ≅ c2 (β 2X ) 1.068 0.721 0.47 0.393 0.379 0.36
1.076 0.778 0.58 0.532 0.523 0.51
PDE c ≅ c1 (β 2X )
PDE c ≅ c2 (β 2X )
0.99 0.99 0 0 0 0
0.999 0.999 0.999 0.990 0.990 0.990
Tab. 1. Correct detection probability for a LOD test in leptokurtic GG-distributed background noise
5. CONCLUSIONS In this paper, a novel analytical higher-order-statisticsbased methodology for estimating the sharpness parameter of a Generalised Gaussian (GG) pdf model has been presented and discussed. In particular, the application of the proposed method to an utilisation case of technical interest like the signal detection in generic non-Gaussian noise has been dealt, in order to evidence what actual improvements can be involved by the introduction of a sharpness parameter estimation more precise than one provided by the available state-of-the-art solutions. The proposed method makes it possible a precise estimation of the GG pdf sharpness parameter from normalised kurtosis, in an extended range of noisy environments (both impulsive and sub-impulsive). This allows one to exploit the non-Gaussian GG parametric
model as a valid alternative to other probabilistic models (e.g. α-stable, Gaussian mixtures etc.) in advanced signal processing applications requiring a reliable estimation of the background noise distribution.
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