SIGNAL DETECTION IN NON-GAUSSIAN NOISE BY A KURTOSIS-BASED PROBABILITY DENSITY FUNCTION MODEL A. Tesei1, and C.S. Regazzoni2 Department of Biophysical and Electronic Engineering (DIBE), University of Genoa Via all'Opera Pia 11A - 16145 Genova - ITALY phone: +39 10 3532792, fax: +39 10 3532134 1 e-mail:
[email protected] 2 e-mail:
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ABSRACT The problem of HOS-based signal detection methods applied in real communication systems is addressed. The Locally Optimum (LO) criterion is selected from a large number of detection criteria. It can be applied either under the ideal (but often not realistic) assumption of Gaussian background noise, or on the basis of realistic statistical models of channel noise. Conventional Fourier analysis (using first and secondorder statistics) allows a receiver to obtain optimum detection results in the presence of Gaussian noise. However, in many real communication applications using the ideal assumption of Gaussian noise causes the performances of a conventional approach to decay significantly. In these cases, Higher Order Statistics (HOS) has been selected as a powerful approach that allows complete signal and noise characterizations, and that optimizes detection performances. The present paper describes the applications of the LO criterion to both conventional and HOS approaches. Its performances have been evaluated in the field of underwater acoustics.
In this work, a new method for detecting signals in additive independent non-Gaussian background noise and for low values of Signal-to-Noise Ratio (SNR) has been developed and compared with conventional wellknown criteria. The proposed approach consists in a Locally Optimum Detector (LOD) [2]: it has been selected among the class of statistical binary hypothesis tests as it allows one to reach high performances in the case of very weak signals. It is applied by using a suitable analytical model of noise probability density function (pdf), introduced by Champernowne [3]. The pdf model is expressed in terms of a fourth-order statistical parameter: the normalized kurtosis. The detector has been tested and applied on an underwater acoustics experiment: known test signals have to be detected in presence of real shipping-trafficradiated low-frequency, hence non-Gaussian, noise. Noise time sequences were acquired during a sea campaign in the Southern Adriatic Sea (May 1993), in the context of MAST-I SNECOW project [4].
2. DESCRIPTION OF THE APPROACH 1. INTRODUCTION Conventional signal processing algorithms, based on the first and second order statistics and optimised in presence of Gaussian noise, may degrade their performances in non-Gaussian environments. Higher Order Statistics (HOS) [1] is a powerful means for characterizing and modelling non-Gaussian noise, and building efficient and robust signal detectors on the basis of this complete noise analysis.
The proposed method is based on a realistic statistical characterization of the channel background noise and is used under the hypothesis of stationary, independent identically distributed, additive, non-Gaussian noise. Specific tests, based on conventional and HOS-based approaches [5], can be applied on noise sequences in order to determine whether these assumptions are realistic or not. The block diagram in Figure 1 presents the two main classes of analysis approaches used
(conventiaonal and HOS-based techniques) and the tests derived by their application.
time sequence
conventional analysis
stationarity test
HOS analysis
Gaussianity test
sources with weak decreasing power are the traffic ships. Figures 2(a) and 2(b) present the scheme of the experiment area centred in the oceanographic ship in a certain time sample: in Fig. 2(a) the approximately equal distribution of ships is shown; in the scheme 2(b) the magnitude of each bubble (one for each present ship) is directly linked to the power of the source, as seen by the acoustic sensor (the numbers correspond to an example of set of N=8 values for i).
pdf modelling Fig. 1 Block diagram of the main steps for statistical noise analysis
sensor ship
Under the aforesaid conditions, in the presented application the background noise is statistically modelled by means of a generic pdf, introduced by Champernowne in economics and then employed by Webster [3] for modelling non-Gaussian ambient astronomic noise. The model can be applied if the N noise components xi have an iperbolic distribution of power, according with the following expression of the noise variable n:
n =
N
∑
i =1
xi . i
(a) 4 6 5
sensor ship
(1)
In practice, this means that noise has a small number of very strong sources (corresponding to low values of i) and a large number of very weak sources (associated to high values of i). In this underwater application background noise consists of a linear combination of ship-traffic-radiated acoustic components recorded by an hydrophone dropped down from an oceanographic ship: noise sources (the oceanographic ship and the ships transiting around it) can be considered with approximately equal engine powers. However, the oceanographic ship is closer to the sensor, while the other ships can be considered equally distributed in the surrounding area on the sea surface. So the Champernowne model is reasonable: the strong source is the oceanographic ship and the large number of
1
2 8
7 3
(b) Fig. 2 (a) Scheme of the experiment sea area; (b) scheme of powers corresponding to each noise source-ship.
The Champernowne model depends on a parameter, which can be expressed in terms of the observed noise normalized kurtosis β2, e.g., the ratio between the fourth and the square of the second moments [1], being the most noticeable empirical way for statistically quantifying the deviation from Gaussianity. The selected pdf is suitable for representing non-Gaussian distributions in the range 1.8≤β2≤4.2, under the above
hypothesis about components distribution. For β2=3, the pdf approximately has a Gaussian shape, so that the model is feasible even for Gaussian noise. The Champernowne pdf expression follows:
> locally optimum
2
and β2 =
21 − 5 β 2
detector
(4)
2
noise
Among many known established binary statistical testing criteria (e.g. LOD, Neyman Pearson, etc.) [2], the LOD approach [2] has been selected, as it is particularly suitable for the critical case of weak signals, and its non-linearity glo can be analytically expressed in terms of the above pdf [3]. The test criterion is based on the following expression linking the test stochastic variable λlo and the statistical threshold Ta, given the significance level α:
S ∑ glo ( i ) ⋅ s ( i ) R T
H1 => H0
(5)
where
glo ( n ) = − PFA = α =
f ' (n) , f (n)
+∞
zp
Λ lo / H0
( λ lo / H 0 ) dλ
H0
G
λ lo
Tα
H1
2
λ lo =
locally optimum
af. bm af n g
