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3464 6. Discussion We have studied the usefulness of a specific nonlinear preprocessor, a parallel array of two-state quantizers, to design simple suboptimal detectors in non-Gaussian noise. To do so, we have considered a standard two hypotheses detection problem and we have assessed the performance of two kinds of detectors: (i) Standard linear detectors, optimal in Gaussian noise, composed of a correlation receiver followed by a binary test. (ii) Nonlinear detectors, composed of the nonlinear preprocessor under study followed by a structure identical to the one of the linear detectors. The performance of these linear and nonlinear detectors have been compared in a Bayesian and in a Neyman–Pearson detection strategy when the signal to be detected and the native non-Gaussian noise are known a priori. This comparison is meaningful since the linear detectors are often used even when the noise is a priori known to be non- Gaussian. This suboptimal choice is specially made for fast real-time hard processing, to maintain a simple test statistic on which to base the detection. The nonlinear detector considered here is almost as fast to compute as the linear detector. In addition of being compatible with a fast real-time hard implementation, we have shown that this nonlinear detector can achieve better performance compared to the linear detector for various non-Gaussian noise of practical interest. As illustrated in this article, this occurs for both Bayesian and Neyman–Pearson detection strategy with a large range of input signal sðtÞ amplitude with various types including constant signals (in Figs. 4 and 5), periodic signals (in Fig. 6) and aperiodic signals (in Fig. 8). These results have been obtained with a nonstandard configuration of the parallel array quantizers. Instead of adopting a classical distributedthreshold configuration, we have used a simpler common-threshold configuration in which some noise have purposely been injected in order to shape the input–output characteristic of the array seen as a single device. The resulting input–output characteristic of the array depends on the choice of the injected noises and quantizer characteristics. In this article, we have chosen to study an array of two-state quantizers associated with noises in the array that were uniform or Gaussian. Many other ARTICLE IN PRESS D. Rousseau et al. / Signal Processing 86 (2006) 3456–3465 associations could be investigated with variations concerning the type of noises injected in the array or the characteristic of the quantizers. It is remarkable to note that the simple associations tested in this study can be useful to design suboptimal detectors for the variety of non-Gaussian noise pdf that we have explored here. Such simple associations of standard noises and existing electronic devices which can easily be replicated, at a micro- or nano-scale, to constitute large arrays, could also exist for other detection problem in non-Gaussian noises of practical interest (like speckle noise in coherent imaging). These research could benefit from the theoretical framework of this study since it allows to consider any kind of static nonlinearity. Another possible interesting extension would be to draw the design of adaptive procedures, which would extend those introduced for conventional stochastic resonance [21] and which would let the array automatically increase the array noises above zero until an optimal efficacy is reached. At a general level, this report contributes to extend our understanding of noise enhanced signal processing. The idea that an addition of a certain controlled amount of noise can be useful in the processing of information carrying signals by static quantizers is not new in itself. All occurrences of this phenomenon in static quantizers are often thought to illustrate the dithering effect [22]. In this well-known dithering effect, a noise induces threshold crossing and linearizes on average the quantizer characteristic. This is clearly not the case here since when the input noise has a non-Gaussian pdf, averaging the individual independent response of the two-state quantizers leads to a collective detection performance that outperforms the linear detector applied to the original analog data. This proves that the noise-enhanced detection performance reported here cannot be interpreted as a noise-induced linearization of a nonlinear system. Many counter-intuitive contributions of noise in nonlinear signal processing may still be uncovered and ought to be analyzed in detail. References [1] D.L. Duttweiler, T. Kailath, RKHS approach to detection and estimation problems—part IV: non-Gaussian detection, IEEE Trans. Inform. Theory 19 (1973) 19–28. [2] S. Kay, Fundamentals of Statistical Signal Processing: Detection Theory, Prentice-Hall, Englewood Cliffs, NJ, 1998. 93/197
[3] N.H. Lu, B.A. Einstein, Detection of weak signals in non- Gaussian noise, IEEE Trans. Inform. Theory IT-27 (1981) 755–771. [4] S. Kassam, Optimum quantization for signal detection, IEEE Trans. Comm. 25 (1977) 479–484. [5] H.V. Poor, J.B. Thomas, Applications of Ali–Silvey distance measures in the design of generalized quantizers for binary decision systems, IEEE Trans. Comm. 25 (1977) 893–900. [6] B. Aazhang, H.V. Poor, On optimum and nearly optimum data quantization for signal detection, IEEE Trans. Comm. 32 (1984) 745–751. [7] B. Picinbono, P. Duvaut, Optimum quantization for detection, IEEE Trans. Comm. 36 (1988) 1254–1258. [8] C. Duchene, P.-O. Amblard, S. Zozor, Quantizer-based suboptimal detectors: noise-enhanced performance and robustness, International Symposium on Fluctuations and Noise (FN’05), Proceedings of the SPIE, vol. 5847, Austin, TX, USA, 23–26 May 2005, pp. 100–110. [9] N.G. Stocks, Suprathreshold stochastic resonance in multilevel threshold systems, Phys. Rev. Lett. 84 (2000) 2310–2313. [10] N.G. Stocks, Information transmission in parallel threshold arrays: suprathreshold stochastic resonance, Phys. Rev. E 63 (2001) 041114,1–9. [11] M.D. McDonnell, D. Abbott, C.E.M. Pearce, A characterization of suprathreshold stochastic resonance in an array of comparators by correlation coefficient, Fluctuation Noise Lett. 2 (2002) L205–L220. [12] D. Rousseau, F. Duan, F. Chapeau-Blondeau, Suprathreshold stochastic resonance and noise-enhanced Fisher information in arrays of threshold devices, Phys. Rev. E 68 (2003) 031107,1–10. ARTICLE IN PRESS D. Rousseau et al. / Signal Processing 86 (2006) 3456–3465 3465 [13] D. Rousseau, F. Chapeau-Blondeau, Suprathreshold stochastic resonance and signal-to-noise ratio improvement in arrays of comparators, Phys. Lett. A 321 (2004) 280–290. [14] D. Rousseau, F. Chapeau-Blondeau, Constructive role of noise in signal detection from parallel arrays of quantizers, Signal Processing 85 (2005) 571–580. [15] X. Godivier, F. Chapeau-Blondeau, Noise-assisted signal transmission in a nonlinear electronic comparator: experiment and theory, Signal Processing 56 (1997) 293–303. [16] L. Gammaitoni, P. Ha¨nggi, P. Jung, F. Marchesoni, Stochastic resonance, Rev. Mod. Phys. 70 (1998) 223–287. [17] E. Pantazelou, F. Moss, D. Chialvo, Noise sampled signal transmission in an array of Schmitt triggers, in: P.H. Handel, A.L. Chung (Eds.), Noise in Physical Systems and 1=f Fluctuations, AIP Conference Proceedings, vol. 285, New York, 1993, pp. 549–552. [18] J.J. Collins, C.C. Chow, T.T. Imhoff, Stochastic resonance without tuning, Nature 376 (1995) 236–238. [19] F. Chapeau-Blondeau, Nonlinear test statistic to improve signal detection in non-Gaussian noise, IEEE Signal Process. Lett. 7 (2000) 205–207. [20] F.W. Machell, C.S. Penrod, G.E. Ellis, Statistical characteristics of ocean acoustic noise process, in: E.J. Wegman, S.C. Schwartz, J.B. Thomas (Eds.), Topics in Non-Gaussian Signal Processing, Springer, Berlin, 1989, pp. 29–57. [21] B. Kosko, S. Mitaim, Robust stochastic resonance: signal detection and adaptation in impulsive noise, Phys. Rev. E 64 (2001) 051110,1–11. [22] L. Gammaitoni, Stochastic resonance and the dithering effect in threshold physical systems, Phys. Rev. E 52 (1995) 4691–4698. 94/197
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