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1438 IEEE SENSORS JOURNAL, VOL. 10, NO. 9, SEPTEMBER 2010

Large-Signal Robustness of the Chair-Varshney Fusion RuleUnder Generalized-Gaussian Noises

Jintae Park, Student Member, IEEE, Eunchan Kim, Student Member, IEEE, and Kiseon Kim, Senior Member, IEEE

AbstractTheChair-Varshney rule (CVR) has been used to pro-vide a large signal-to-noise ratio (SNR) approximation of the op-timal fusion rule under Gaussian noise. For more practical use insensor networks, this paper extends CVR to Generalized-Gaussiannoise channels, along with verification of the suboptimality and ro-bustness of CVR under the Generalized-Gaussian channel noisethrough the use of Monte Carlo simulations.

Index TermsDecision fusion, generalized Gaussian, wirelesssensor networks.

I. INTRODUCTION

DATA fusion for wireless sensor networks (WSNs) has in-creasingly been the focus of research interest due to its

wide range of potential applications, including environmental

event monitoring and distributed disaster protection scenarios[1], where there may be a sufficient power supply but the net-work needs to satisfy such requirements as near-optimal per-formance, robustness to a harsh communication environment,and a low communication bandwidth. In [2], the Chair-Varshneyrule (CVR) was derived, which can determine the probabilitiesof false alarms and detection of local decisions. In [3], CVRis regarded as the admissible fusion rule (FR) for the gener-alized Gaussian noise of local sensors observations. Interest-ingly, CVR was shown to be a robust suboptimal FR underfading and Gaussian noise channels at large signal-to-noise ra-tios (SNRs), as an approximation of the optimal log-likelihoodratio (LLR)-based FR [4]. It should be noted that CVR doesnot depend on the Gaussian channel parametersi.e., it is ro-bust against diverse Gaussian channel conditions; subsequently,it has potential for use in the development of a robust FR undergeneralized noises covering non-Gaussian channels.

II. PRELIMINARIES

Fig. 1 depicts a parallel fusion model, where the th sensoroutputs , with false alarm and detection proba-bilities of and ,respectively. The received decision is represented as

, where is the attenuation of fading andis the additive channel noise having a symmetric probability

density function (pdf) of about zero. The final decisionis made at the fusion center based on the FR.Under the NeymanPearson (NP) sense, the FR is given by

Manuscript received November 02, 2009; revised January 18, 2010; acceptedFebruary 23, 2010. Date of publication June 07, 2010; date of current versionJuly 14, 2010. This work was supported by a Basic Research Program of theAgency for Defense Development (ADD), Korea, under Grant ADD080601.The associate editor coordinating the review of this paper and approving it forpublication was Prof. Eduardo Nakamura.

The authors are with the School of Information and Mechatronics, GwangjuInstitute of Science and Technology, Gwangju, 500-712, Korea (e-mail: jtpark@gist.ac.kr; tokec@gist.ac.kr; kskim@gist.ac.kr).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/JSEN.2010.2045157

Fig. 1. Parallel fusion model incorporating fading and noise channels.

. Here, is a fusion statistic (FS)

and is a threshold chosen to satisfy the false alarm proba-bility of the system, . Then, the optimal FSgiven by the likelihood ratio (LR) for conditionally independentlocal observations can be written as

(1)

A. No Noise Case

As the trivial case, when there is no channel noise, i.e.,and , the optimal LLR statistic for the NP

test, namely the Chair-Varshney FS, is given as [2]

(2)

B. Gaussian Noise Case

A popular wireless communication channel noise is theGaussian model, widely accepted as means of modelingreal-life phenomena [6]. CVR is derived as a large SNR alter-native to the optimal FS under Gaussian noise with the knownfading channel [4]. This result still hold for .

III. FUSIONRULE AT ALARGESNR

The generalized Gaussian (GG) distribution, , a broadclass of non-Gaussian noises used to describe various electro-magnetic radio noises showing impulsive characteristics, is awell-known model. When the concentration of values aroundthe mean and the tail behavior are of particular interest [5]

(3)

where , , is the scale parameter, the power , ,

reflects the pdf tail behavior, and is the gamma-function.Equation(3) yieldsa Gaussian pdfwith and a Laplacepdf

with , whereas in the limiting case as tends to , the

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converges to a uniform distribution. In other words, thelower the value of the power parameter , the more impulsivethe distribution.

The optimal LR FS under GG pdfs can be obtained from thedefinition of and (3), such that

(4)

where Notethat becomes infinite, i.e.,

when

when

Then, considering that is a positive number and , toensure , it is necessary that

. Furthermore, is a symmetric andconvex function and the condition

is equivalent to , regardless of . Subsequently, similarto [4], for a large SNR approximation, i.e., , using thedefinitions and , theoptimal LR FS can be simplified as

where the exponential terms in (4) become infinite for the firstproduct group and the corresponding coefficients terms remain;the corresponding coefficients disappear due to the exponen-tial terms becoming zero. By taking further logarithms on bothsides, we obtain the equivalence of FSs, i.e.,

. Again, note that this FS does not depend on the char-

acteristics of the GG channels.

IV. PERFORMANCE EVALUATION

To verify the usefulness of CVR under non-Gaussian noise,we compared its performance to the optimal LR test in termsof suboptimality, and then verified its robustness under contam-inated noise with respect to other well-known schemes, suchas equal gain combining (EGC) and maximum ratio combining(MRC) in [4]. The simulations were conducted with the sameconditions in the Gaussian case [4]; a Rayleigh fading with unitpower, and .

A. Evaluation of Suboptimality

To show the suboptimality of CVR, we performed extensiveMonte Carlo simulations for , , and ; how-ever, only the Laplace case results are presented here due to lim-ited space. Fig. 2 gives the detection probability for the Laplacedensity, and it shows that CVR has suboptimality at a large SNRas for the Gaussian case, in [4]. CVR outperforms bothEGC and MRC at large SNRs. Furthermore, for the other ex-treme case of , the simulation results show a similarsuboptimality for CVR.

B. Evaluation of Robustness

To verify robustness, we consider the well-known -contam-

inated distribution introduced by Huber in [7]; specifically, theLaplace contaminated Gaussian density given by

Fig. 2. Probability of detection versus channel SNR for Laplace noise witheight sensors.

Fig. 3. Probability of detection versus channel SNR for Laplace contaminatedGaussian noise with eight sensors.

. The results in Fig. 3 were obtained bysetting for and for . With , the

performance of the LR rule starts decreasing at SNRs of higherthan 25 dB, even though the SNR may be increased. In otherwords, CVR still provides near optimal performance for SNRslarger than around 18 dB.

V. CONCLUSION

In this paper, we showed that CVR is robust to generalizedGaussian channel noise behavior at large SNRs. Furthermore,to consider more a realistic channel environment, the -contam-inated noise density was evaluated to show the robustness of theCVR.

REFERENCES

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[2] Z. Chair and P. K. Varshney, Optimal data fusion in multiple sensordetection systems,IEEE Trans. Aerosp. Electron. Syst., vol. AES-22,pp. 98101, Jan. 1986.

[3] W. Shi, T. W. Sun, and R. D. Wesel, Quasi-convexity and optimalbinary fusion for distributed detection with identical sensors in gener-alized Gaussian noise,IEEE Trans. Inf. Theory, vol. 47, pp. 446450,Jan. 2001.

[4] B. Chen, R. Jiang, T. Kasetkasem, and P. K. Varshney, Channelaware decision fusion in wireless sensor networks, IEEE Trans.Signal Process., vol. 52, no. 12, pp. 34543458, Dec. 2004.

[5] S. A. Kassam, Signal Detection in Non-Gaussian Noise. Berlin, Ger-many: Springer-Verlag, 1988.

[6] K. Kim and G. Shevlyakov, Why Gaussianity? (an attempt to explain

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