SIGNAL DETECTION IN NON-GAUSSIAN NOISE BY A KURTOSIS-BASED PROBABILITY DENSITY FUNCTION MODEL

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1 SIGNAL DETECTION IN NON-GAUSSIAN NOISE BY A KURTOSIS-BASED PROBABILITY DENSITY FUNCTION MODEL A. Tesei, and C.S. Regazzoni Department of Biophysical and Electronic Engineering (DIBE), University of Genoa Via all'opera Pia A Genova - ITALY phone: , fax: tale@dibe.unige.it carlo@dibe.unige.it 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 (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.. 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) [] 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. 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) []: 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 993), in the context of MAST-I SNECOW project [4].. DESCRIPTION OF THE APPROACH 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 presents the two main classes of approaches used

2 (conventiaonal and HOS-based techniques) and the tests derived by their application. time sequence conventional HOS stationarity test Gaussianity test sources with weak decreasing power are the traffic ships. Figures (a) and (b) present the scheme of the experiment area centred in the oceanographic ship in a certain time sample: in Fig. (a) the approximately equal distribution of ships is shown; in the scheme (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. Block diagram of the main steps for statistical noise 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 x i have an iperbolic distribution of power, according with the following expression of the noise variable n: N xi n =. () i = i 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 sensor ship (a) 4 sensor ship (b) Fig. (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 β, e.g., the ratio between the fourth and the square of the second moments [], 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.8 β 4., under the above 6 8

3 hypothesis about components distribution. For β =3, the pdf approximately has a Gaussian shape, so that the model is feasible even for Gaussian noise. The Champernowne pdf expression follows: F πα I senh HG 3 α KJ f ( n) = α F πα I F n I cosh cosh HG KJ + π 3 HG α 3 α KJ () where α = 5β (3) and m4 β af n = m n b afg. (4) Among many known established binary statistical testing criteria (e.g. LOD, Neyman Pearson, etc.) [], the LOD approach [] has been selected, as it is particularly suitable for the critical case of weak signals, and its non-linearity g lo 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 λ lo and the statistical threshold T a, given the significance level α: M T λ α = > H lo = glo ( i ) s( i ) < T i RST α = > H 0 = (5) where f n glo ( n ) ' ( ) =, (6) f ( n) z + PFA = α = pλ H λ lo H dλ lo / ( / ) 0 0 (7) Tα and {s(i), i=,..,m} is the signal to detect. According to the theory of statistical binary hypothesis testing, the signal is detected when H is decided, otherwise the null hypothesis H0 (i.e., signal absence) is selected. A scheme of LOD tests is shown in Figure 3. noise samples noise samples current data Champernowne model Gaussian model locally optimum detector locally optimum detector λ C lo λ G lo H T' α Tα H Fig.3 Scheme of the two LOD tests implemented In order to evaluate the performances of the proposed method, it is compared with the LOD under the conventional ideal Gaussian hypothesis. It is to notice that the additional information introduced by the proposed approach is contained in a single 4th-order parameter (β ), being very simple and quick to estimate; moreover, no constraint is requested about the statistics of the signal to detect. On the other hand the main limits of the LOD consist in the following characteristics:. it needs a value of P FA, fixed by the user;. it needs complete a-priori knowledge about the signal when it is acquired (in terms of time shape); this aspect is particularly critical if distortion phenomena occurs during the propagation: in these cases, not only complete a-prriori knowledge on the transmitted signal needs, but also a realistic channel model has to be inserted; 3. only for few pdf models the test threshold T α can be computed analytically on the basis of the LOD λ lo expression (7); otherwise, the threshold has to be computed by means of numerical procedures. > > < < H0 H0 3. EXPERIMENTAL RESULTS

4 An extensive test phase has been carried out by applying the LOD detector with the Gaussian and Champernowne noise models on a large set of known deterministic signals, in the presence of real underwater acoustic shipping noise. Noise was acquired in a coastal shallow-water sea area. The presence of high ship traffic and phenomena of reflection and refraction from the sea bottom and surface [6] makes the problem complex. More details about noise and signal shape selection will be presented in the full paper. The results on the LOD performances are summarized in Figure 4 in terms of Probability of Detection, P det, for different values of SNR, under the Champernowne and Gaussian assumptions. Detection Probability Detection Probability SNR (db) (a) SNR (db) (b) Fig. 4 Results of the LOD under the Gaussian (a) Champernowne (b) hypothesis about noise pdf. The tests have been carried out under the following working boundary conditions: fixed Probability of False Alarm P FA =α=5%, non Gaussian real underwater acoustic ship-traffic noise (Kurtosis approximately equal to.84), fixed window amplitude (equal for signal and receiver) M=000 samples, fixed sample period (noise data sampling at KHz and detection process each 00 samples). 4. CONCLUSIONS AND FUTURE WORK In the context of digital signal processing addressed to communications, this paper has focused attention on the problem of optimizing signal detection in presence of additive independent stationary non-gaussian noise under the conditions of weak signals. In order to optimize the Probability of Detection P det for low SNR values, the selected binary statistical testing approach consists in a LO detector, whose test stochastic variable λ lo has been computed on the basis of a kurtosis-based model of the background-noise pdf. A wide set of experimental tests have proved significant performance improvement of the LOD test applied with the selected model, in comparison with the same test applied under the ideal Gaussian-noise assumption. As future trends, various innovative analytical models expressed in terms of different HOS statistical parameters (of the third and fourth orders) are going to be investigated and developed. In particular, different symmetric (depending on the 4 th -order kurtosis parameter) and asymmetric (in terms of the 3 th -order skewness parameter) functions are studied, in order to describe realistically the non-gaussian noise and, so, to improve the detector performances. 5. REFERENCES [] C. Nikias, J. Mendel, "Signal Processing with Higher- Order Spectra", IEEE SP Mag., pp. 0-37, 993. [] S.A. Kassam, Signal Detection in Non-Gaussian Noise, Springer Verlag, Berlin, 988. [3] R.J. Webster, "Ambient Noise Statistics", IEEE Trans. SP, Vol. 4 (6), pp , 993. [4] DIBE, "SNECOW-MAST 009-C(A) Final Report-Task 5: Ship traffic noise statistical evaluation," Dec [5] C.S. Regazzoni, A. Tesei, G. Tacconi, "A comparison between spectral and bispectral for ship detection from acoustical time series," in Proc. of ICASSP94, pp. 89-9, April 994.

5 [6] M.J. Buckingham, "Ocean-acoustic propagation models", J. Acoustique, 3, pp. 3-87, June 99.

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