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IEEE COMMUNICATIONS LETTERS, VOL. 16, NO. 2, FEBRUARY 2012
Distributed Detection and Fusion of Weak Signals in Fading Channels with Non-Gaussian Noises Jintae Park, Student Member, IEEE, Georgy Shevlyakov, Member, IEEE, and Kiseon Kim, Senior Member, IEEE
Abstract—Distributed detection and information fusion have received recent research interest due to the success of emerging wireless sensor network (WSN) technologies. For the problem of distributed detection in WSNs under energy constraints, a weak signal model in the canonical parallel fusion scheme with additive non-Gaussian noises and fading channels is considered. To solve this problem in the Neyman-Pearson setting, a unified asymptotic fusion rule generalizing the maximum ratio combiner (MRC) fusion rule is proposed. Explicit formulas for the threshold and detection probability applicable for wide classes of fading channels and noise distributions are written out. Both asymptotic analysis and Monte Carlo modeling are used to examine the performance of the proposed detection fusion rule. Index Terms—Distributed detection, decision fusion, weak signal, non-Gaussian noise, wireless sensor networks.
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I. I NTRODUCTION
HE problem of distributed detection using multiple sensors has been extensively studied over the past few decades. The recent success of emerging wireless sensor network (WSN) technology has encouraged researchers to develop distributed algorithms in this field [1]. Many applications in WSNs can be regarded as a two-hypotheses detection problem corresponding to the target-present or target-absent hypotheses. In practice, it is frequently observed that one hypothesis, target-absent, is more likely. The energy constraint in WSNs is typically connected with densely deployed sensor nodes having limited capabilities such as lifetime, processing power, etc., due to the cost. Thus, in order to save energy, each sensor node only transmits signals when its decision corresponds to the target-present hypothesis, and it is designed to use the minimum amount of transmitting power. A research effort to the energy efficient detection for the send/no-send transmission scenario can be found in [2]. With advances in wireless technologies, the wireless channel layer has become an important bottleneck in the design of the detection framework in WSNs [1]. Recently, decision fusion problems have been studied basing on the parallel fusion model incorporated with fading and Gaussian noise channels [3], [4], where several suboptimal fusion rules are obtained by the signal-to-noise ratio (SNR) approximation
Manuscript received September 5, 2011. The associate editor coordinating the review of this letter and approving it for publication was A. Banchs. This research was supported by the World-Class University Program funded by the Ministry of Education, Science, and Technology through the National Research Foundation of Korea (R31-10026). J. Park and K. Kim are with the School of Information and Communication, Department of Nanobio Materials and Electronics, World-Class University (WCU), Gwangju Institute of Science & Technology (GIST), Gwangju 500712, South Korea (e-mail:
[email protected]). G. Shevlyakov is with the Department of Applied Mathematics, St. Petersburg State Polytechnic University, St. Petersburg, 195251, Russian Federation. Digital Object Identifier 10.1109/LCOMM.2011.121311.111870
method. In particular, the low SNR approximation is used in [3], where lim𝜎2 →∞ is considered with the Gaussian noise variance 𝜎 2 and the likelihood ratio Λ. In this paper, we extend those results to non-Gaussian noise cases to consider more practical channel noises known to have impulsive characteristics [5]-[7]. Thus, our aim is to find a unified fusion rule that can be directly applicable for any symmetric noise probability density function (pdf). In order to propose a unified decision fusion rule, we first concurrently integrate non-Gaussian noise channels into the fusion model using the asymptotic weak signal approach as in [5], [8]. Note that, for the Gaussian noise, both the low SNR approximation [3] and the weak signal approach give the maximum ratio combiner (MRC) fusion rule [3]. In this paper by replacing the sample size with the number of sensor nodes, the weak signal is used to model each sensor output for decision of target-present. This idea involving a large number of sensors in fusion has been studied for both asymptotic and non-asymptotic cases with a binary symmetric channel assumption [9]. In detection of a weak signal, one necessarily uses a relatively large sample size to get a reasonable value for the probability of detection [5]. This can be achieved by fusing decisions with the large number of participating sensor nodes. Under standard assumptions of regularity imposed on channel noise pdfs, we can thereby derive an asymptotic fusion rule of a general form that is directly applicable for wide classes of fading channels and non-Gaussian noise pdfs. The remainder of this paper is organized as follows. In Section II, we formulate the parallel fusion problem using the weak signal approach and propose a unified asymptotic fusion rule. In Section III, the asymptotic performance analysis is conducted for the Neyman-Pearson setting. In Section IV, the asymptotic optimality of the proposed fusion rule in case of a known channel noise pdf is numerically justified basing both on theoretical and experimental results. Finally, we conclude in Section V. II. A SYMPTOTIC F USION RULE Fig. 1 depicts a parallel fusion model used in [3], where 𝑢𝑘 ∈ {0, 𝜃} is the 𝑘th sensor output having the false alarm and detection probabilities of 𝑃𝑓 𝑘 = 𝑃 [𝑢𝑘 = 𝜃∣𝐻0 ] and 𝑃𝑑𝑘 = 𝑃 [𝑢𝑘 = 𝜃∣𝐻1 ], respectively. After passing through the communication channel, the received decision 𝑦𝑘 = ℎ𝑘 𝑢𝑘 + 𝑛𝑘 , where ℎ𝑘 > 0 is the attenuation of fading (assumed known) and 𝑛𝑘 is the additive channel noise with a symmetric pdf 𝑓 . A similar constant signal model can be found in [8]. Here, we assume the phase coherent reception similar to [3]. In this model, the sensor decision 𝜃 is a weak signal in√the sense that its amplitude decreases with 𝐾 as 𝜃 = 𝜃𝐾 = 𝜈/ 𝐾 with the finite constant 𝜈 > 0.
c 2012 IEEE 1089-7798/12$31.00 ⃝
PARK et al.: DISTRIBUTED DETECTION AND FUSION OF WEAK SIGNALS IN FADING CHANNELS WITH NON-GAUSSIAN NOISES
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Fig. 1. The parallel fusion model consisting of 𝐾 sensor nodes and the fusion center.
The optimal fusion rule for this fusion model is given by the likelihood ratio (LR) [3]. Given the conditional independence assumption of local observations, the LR can be written as Λ𝐾 (¯ 𝑦) =
𝐾 ∏ 𝑃𝑑𝑘 𝑓 (𝑦𝑘 − ℎ𝑘 𝜃) + (1 − 𝑃𝑑𝑘 )𝑓 (𝑦𝑘 ) , 𝑃𝑓 𝑘 𝑓 (𝑦𝑘 − ℎ𝑘 𝜃) + (1 − 𝑃𝑓 𝑘 )𝑓 (𝑦𝑘 )
(1)
𝑘=1
where 𝑦¯ = [𝑦1 , . . . , 𝑦𝐾 ]𝑇 is the observation vector. The fusion rule utilizing this LR statistic is optimal for a noise pdf 𝑓 satisfying regularity conditions. From (1) it directly follows equation (2-32) in [5] providing that with the asymptotic weak signal approach (𝜃 → 0 as 𝐾 → ∞) the locally optimum detector fusion statistic is given by 𝐾 1 ∑ 𝑀𝐿 (¯ 𝑦) = √ (𝑃𝑑𝑘 − 𝑃𝑓 𝑘 )ℎ𝑘 𝜓𝑀𝐿 (𝑦𝑘 ) , 𝑇𝐾 𝐾 𝑘=1
(2)
where 𝜓𝑀𝐿 (𝑥) = −𝑓 ′ (𝑥)/𝑓 (𝑥) is the maximum likelihood (ML) score function. In the Gaussian noise case, the fusion statistic (2) is linear with respect to observations and equiva∑𝐾 𝑀𝐿 ∝ 𝑘=1 (𝑃𝑑𝑘 − 𝑃𝑓 𝑘 )ℎ𝑘 𝑦𝑘 [3]. lent to the MRC statistic: 𝑇𝐾 Generalizing the MRC fusion rule, we consider the fusion rule based on an arbitrary score function 𝜓 satisfying regularity conditions specified in Section III. 𝐾 1 ∑ 𝑦) = √ (𝑃𝑑𝑘 − 𝑃𝑓 𝑘 )ℎ𝑘 𝜓(𝑦𝑘 ) 𝑇𝐾 (¯ 𝐾 𝑘=1
𝐻1
≷
𝐻0
𝜆𝛼 ,
(3)
where 𝜆𝛼 is the threshold value providing the required rate of the false alarm probability 𝑃𝐹 = 𝛼. Here we consider the score functions 𝜓 introduced in [10] as a generalization of the maximum likelihood score function 𝜓𝑀𝐿 implicitly defining an 𝑀 -estimate which in its turn is a generalization of the maximum likelihood estimate (MLE). Thus, the proposed generalization allows for further development of suboptimal and robust fusion rules. III. A SYMPTOTIC P ERFORMANCE A NALYSIS In what follows, we use decision rule (3) for which the related performance evaluation characteristics are computed. In the Neyman-Pearson setting, the false alarm probability is defined as (4) 𝑃𝐹 = 𝑃 {𝑇𝐾 > 𝜆∣𝐻0 } ≤ 𝛼. All further results hold under general regularity conditions imposed on score functions 𝜓 and noise pdfs 𝑓 (see, for
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example, [11], pp. 125-127). Note that since 𝑓 is a symmetric pdf, the mean value of 𝜓 is zero (see Appendix B). Now we briefly comment on them. In the literature, the conditions imposed on pdfs and score functions take different forms depending on the pursued goals: in general, one may strengthen the conditions on pdfs and weaken those on scores, and vice versa (various suggestions can be found in [11], [12] ). In this paper, we use a balanced set of conditions following [11]: they define smooth pdfs and allow for a finite number of points of discontinuity for scores and their derivatives; moreover, they require the existence of Fisher information and other integrals to prove an analog of the central limit theorem (CLT). To derive the asymptotic formulas for the false alarm and detection probabilities, we need to have the conditional means 𝐸[𝑇𝐾 ∣𝐻𝑖 ] and variances Var[𝑇𝐾 ∣𝐻𝑖 ] (𝑖 = 0, 1) computed under the hypotheses 𝐻0 and 𝐻1 . Those results are given by Lemma 1 and Lemma 2. Lemma 1: The conditional means of statistic 𝑇𝐾 under hypotheses 𝐻0 and 𝐻1 are as follows 𝐸[𝑇𝐾 ∣𝐻0 ]= 𝜈𝜓 ′ 𝐻𝑓 𝐾 and 𝐸[𝑇𝐾 ∣𝐻1 ] = 𝜈𝜓 ′ 𝐻𝑑𝐾 , (5) ∫∞ ∑𝐾 1 where 𝜓 ′ = −∞ 𝜓 ′ (𝑥)𝑓 (𝑥)𝑑𝑥, 𝐻𝑓 𝐾 = 𝐾 𝑘=1 𝑃𝑓 𝑘 (𝑃𝑑𝑘 − ∑ 𝐾 1 2 𝑃𝑓 𝑘 )ℎ2𝑘 + 𝑜(1/𝐾) and 𝐻𝑑𝐾 = 𝐾 𝑃 (𝑃 𝑑𝑘 − 𝑃𝑓 𝑘 )ℎ𝑘 + 𝑘=1 𝑑𝑘 𝑜(1/𝐾). Proof: See Appendix A. Lemma 2: The conditional variances of statistic 𝑇𝐾 under hypotheses 𝐻0 and 𝐻1 are given by (6) Var[𝑇𝐾 ∣𝐻0 ]= Var[𝑇𝐾 ∣𝐻1 ] = 𝜓 2 𝐻𝐾 , ∫∞ 2 ∑ 𝐾 1 where 𝜓 2 = −∞ 𝜓 (𝑥)𝑓 (𝑥)𝑑𝑥 and 𝐻𝐾 = 𝐾 𝑘=1 (𝑃𝑑𝑘 − 2 2 𝑃𝑓 𝑘 ) ℎ𝑘 + 𝑜(1/𝐾). Proof: See Appendix B. Assume that the attenuations ℎ𝑘 (𝑘 = 1, . . . , 𝐾) are i.i.d. random variables with a common density 𝑝(ℎ) with the bounded first four moments. Then the quantities 𝐻𝑑𝐾 , 𝐻𝑓 𝐾 and 𝐻𝐾 converge in probability to 1 ∑𝐾 the finite values 𝐻𝑑 = ℎ2 lim𝐾→∞ 𝐾 𝑘=1 𝑃𝑑𝑘 (𝑃𝑑𝑘 − ∑ 𝐾 1 𝑃𝑓 𝑘 ), 𝐻𝑓 = ℎ2 lim𝐾→∞ 𝐾 𝑃 (𝑃 − 𝑃𝑓 𝑘 ), and 𝑓 𝑘 𝑑𝑘 𝑘=1 1 ∑𝐾 2 2 lim 𝐻 = ℎ (𝑃 − 𝑃 ) , where ℎ2 = 𝐾→∞ 𝐾 𝑑𝑘 𝑓𝑘 𝑘=1 ∫∞ 2 ℎ 𝑝(ℎ)𝑑ℎ < ∞. −∞ The sketch of proof for convergence of 𝐻∑ 𝐾 is as fol𝐾 1 lows. Set 𝐻 = ℎ2 𝐴, where 𝐴 = lim𝐾→∞ 𝐾 𝑘=1 𝑎𝑘 and 0 < 𝑎𝑘 = (𝑃𝑑𝑘 − 𝑃𝑓 𝑘 )2 ≤ 1. The existence of the limit 𝐴 is justified by the Cauchy convergence test. Then by the Chebyshev inequality, we get that 𝑃 {∣𝐻𝐾 − 𝐻 𝐾 ∣ ≥ 𝜀} ≤
Var[𝐻𝐾 ] Var[ℎ2 ](𝑎𝑚𝑎𝑥 )2 , ≤ 2 𝜀 𝜀2 𝐾
where 𝑎𝑚𝑎𝑥 = max 𝑎𝑘 and Var[ℎ2 ] < ∞. Since 𝐻 𝐾 = ℎ2 𝐴𝐾 ∑𝑘𝐾 1 with 𝐴𝐾 = 𝐾 𝑘=1 𝑎𝑘 , 𝐻𝐾 converges to 𝐻 in probability. The the asymptotic expressions for the false alarm and detection probabilities are given by Theorem 1. Theorem 1: The false alarm probability has the following asymptotic form ( ) 𝜆𝛼 − 𝜈𝜓 ′ 𝐻𝑓 √ 𝑃𝐹 = lim 𝑃 {𝑇𝐾 > 𝜆𝛼 ∣𝐻0 } = 1 − Φ ,(7) 𝐾→∞ 𝜓2 𝐻
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IEEE COMMUNICATIONS LETTERS, VOL. 16, NO. 2, FEBRUARY 2012
𝑃𝐷 = lim 𝑃 {𝑇𝐾 > 𝜆𝛼 ∣𝐻1 } 𝐾→∞ ( ) /√ = 1 − Φ Φ−1 (1 − 𝛼) − 𝜈𝜓 ′ (𝐻𝑑 − 𝐻𝑓 ) 𝜓 2 𝐻 .(9) /√ Proof: By the CLT, the ratio (𝑇𝐾 − 𝐸[𝑇𝐾 ]) Var[𝑇𝐾 ] converges weakly to the standard normal random variable so that the false alarm probability can be written as { } 𝑇𝐾 − 𝐸[𝑇𝐾 ∣𝐻0 ] 𝜆𝛼 − 𝐸[𝑇𝐾 ∣𝐻0 ] > 𝑃𝐹 = lim 𝑃 . 𝐾→∞ Var[𝑇𝐾 ∣𝐻0 ] Var[𝑇𝐾 ∣𝐻0 ] Similarly to the aforementioned reasoning, the asymptotic expression for the detection probability is obtained by using the CLT with 𝜆𝛼 from (8), and the conditional mean and variance under hypothesis 𝐻1 given by Lemma 1 and Lemma 2, respectively. The obtained results are connected with the well-known Huber’s formula for the asymptotic/ variance of 𝑀 -estimates of location [6], [10]: 𝑉 (𝜓, 𝑓 ) = 𝜓 2 (𝜓 ′ )2 . Thus, the detection probability can be rewritten as ( ) /√ 𝑃𝐷 = 1 − Φ Φ−1 (1 − 𝛼) − 𝜈(𝐻𝑑 − 𝐻𝑓 ) 𝑉 (𝜓, 𝑓 )𝐻 .(10) In the particular case of the maximum likelihood score function 𝜓 = 𝜓𝑀𝐿 = −𝑓 ′ /𝑓 , formulas (9) and (10) take the following form ( √ /√ ) 𝑃𝐷 = 1 − Φ Φ−1 (1 − 𝛼) − 𝜈 𝐼(𝑓 )(𝐻𝑑 − 𝐻𝑓 ) 𝐻 , ∫ where 𝐼(𝑓 ) = (𝑓 ′ (𝑛)/𝑓 (𝑛))2 𝑓 (𝑛)𝑑𝑛 is Fisher information for location, hence providing the maximum value of detection probability when the channel noise pdf 𝑓 is known. IV. N UMERICAL R ESULTS In this section, we focus on the asymptotic optimality of the proposed fusion rule in case of a known noise pdf, and compare both the analytical and experimental results under Gaussian, Laplace, and Cauchy noises. Throughout this paper, we assume that the attenuations ℎ𝑘 , 𝑘 = 1, . . . , 𝐾 have the Rayleigh distribution of unit power with the pdf: 𝑝𝑅 (ℎ) = 2 2 ℎ𝑎−2 𝑒−ℎ /2𝑎 , where 𝑎2 = 1/2. Also identical sensor nodes are assumed: 𝑃𝑑𝑘 = 0.5 and 𝑃𝑓 𝑘 = 0.05 for all 𝑘. = √For the Gaussian noise with pdf 𝑓𝐺 (𝑥) ( 2𝜋𝜎)−1 exp(−𝑥2 /2𝜎 2 ) and with the maximum likelihood 2 , the detection probability score function 𝜓𝑀𝐿 (𝑥) = ( 𝑥/𝜎 ) is −1 given by 𝑃𝐷𝐺 = 1 − Φ Φ (1 − 𝛼) − (𝑃𝑑 − 𝑃𝑓 )𝜈/𝜎 . For the Laplace noise with pdf 𝑓𝐿 (𝑥) = (2𝑠)−1 exp(−∣𝑥∣/𝑠) and the detection probability) is given 𝜓𝑀𝐿 (𝑥) = 𝑠−1 sign(𝑥), ( by 𝑃𝐷𝐿 = 1 − Φ Φ−1 (1 − 𝛼) − (𝑃𝑑 − 𝑃𝑓 )𝜈/𝑠 . Finally, the detection probability for the Cauchy noise with pdf 𝑓𝐶 (𝑥) = 𝛾/(𝜋(𝑥2 + 𝛾 2()) and 𝜓𝑀𝐿 (𝑥) = 2𝑥/(𝑥2 + 𝛾√2 ) )is given by 𝑃𝐷𝐶 = 1 − Φ Φ−1 (1 − 𝛼) − (𝑃𝑑 − 𝑃𝑓 )𝜈/(𝛾 2) .
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The asymptotic expression for the detection probability is given by
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Fig. 2. ROC curves for Gaussian noise at SNR = 0, 10, 15, and 20 dB with 𝐾 = 100. 1 Detection Probability
∫𝑧 where Φ(𝑧) = (2𝜋)−1/2 −∞ exp(−𝑡2 /2)𝑑𝑡 is the Gaussian cumulative. The threshold 𝜆𝛼 is obtained from (7) equating 𝑃𝐹 to 𝛼 √ (8) 𝜆𝛼 = 𝜈𝜓 ′ 𝐻𝑓 + 𝜓 2 𝐻Φ−1 (1 − 𝛼).
0.9 0.8 0.7 Asymptotic formulas Simulations − LR Simularions − proposed detector
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Fig. 3. Detection probability as a function of the number 𝐾 of sensor nodes for Gaussian noise at 20 dB SNR for the false alarm probability 𝑃𝐹 = 0.01.
Next, we define SNR in a common way as the ratio of the network energy and the noise variance: 𝑆𝑁 𝑅 = ℰ/𝜎 2 (𝑓 ). So, we set the variance of the Gaussian equal to unit, i.e., 𝜎 2 = 1. However, since the Cauchy distribution does not have moments, we use a geometric SNR (GSNR) [7]: 𝐺𝑆𝑁 𝑅 = 2𝐶𝑔 𝐴2 /𝑆02 where 𝐶𝑔 = 𝑒𝐶𝑒 ≈ 1.78 is the exponential of the Euler constant, 𝐴 is the amplitude of a modulated signal, and 𝑆0 is the geometric power given by 𝑆0 = 𝑆0 (𝑋) = 𝑒𝐸 log ∣𝑋∣ with a logarithmic-order random variable 𝑋. For the Cauchy = 𝛾. To obtain density, geometric power is given by 𝑆0 √ 𝐺𝑆𝑁 𝑅 = ℰ, thus, we set 𝐴2 = ℰ and 𝛾 = 2𝐶𝑔 . Fig. 2 shows the receiver operating characteristic (ROC) curves for the Gaussian noise case with 𝐾 = 100; the results of the asymptotic analysis can be directly obtained using 𝑃𝐷𝐺 . Since we apply the weak signal approach, it is quite natural that the smaller the SNR, the better the match between theoretical and experimental results. Fig. 3 shows the asymptotic optimality for the Gaussian noise at 20 dB SNR and at 𝑃𝐹 = 0.01. Naturally, as the number of sensor nodes increases, the detection probability tends to the optimal ML rate. Note that the analytic result (asymptotic formulas) provides upper bound on asymptotic detection performance. The numerical results for the Cauchy noise shown in Fig. 4 are similar to the Gaussian example from the viewpoint of asymptotic optimality.
PARK et al.: DISTRIBUTED DETECTION AND FUSION OF WEAK SIGNALS IN FADING CHANNELS WITH NON-GAUSSIAN NOISES
1
Therefore, under 𝐻0 , { with 1 − 𝑃𝑓 𝑘 𝜓(𝑛𝑘 ) √ 𝜓(𝑦𝑘 ) = ′ 𝜓(𝑛𝑘 ) + ℎ𝑘 𝜃𝜓 (𝑛𝑘 ) + 𝑜(1/ 𝐾) with 𝑃𝑓 𝑘 , √ 𝜈𝜓 ′ and 𝐸[𝜓(𝑦𝑘 )∣𝐻0 ] = 𝜓 + 𝑃𝑓 𝑘 ℎ𝑘 √ + 𝑜(1/ 𝐾) where 𝜓 = ∫ ∫ ′𝐾 ′ 𝜓(𝑛)𝑓 (𝑛)𝑑𝑛 = 0 and 𝜓 = 𝜓 (𝑛)𝑓 (𝑛)𝑑𝑛. Hence, since 𝑛𝑘 are i.i.d. random variables, the conditional mean takes the form 𝐸[𝑇𝐾 ∣𝐻0 ] = 𝜈𝜓 ′ 𝐻𝑓 𝐾 , where 𝐻𝑓 𝐾 = 1 ∑𝐾 2 𝑘=1 𝑃𝑓 𝑘 (𝑃𝑑𝑘 − 𝑃𝑓 𝑘 )ℎ𝑘 + 𝑜(1/𝐾). 𝐾 Similarly,∑under the 𝐻1 , 𝐸[𝑇𝐾 ∣𝐻1 ] = 𝜈𝜓 ′ 𝐻𝑑𝐾 where 𝐾 1 2 𝐻𝑑𝐾 = 𝐾 𝑘=1 𝑃𝑑𝑘 (𝑃𝑑𝑘 − 𝑃𝑓 𝑘 )ℎ𝑘 + 𝑜(1/𝐾).
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A PPENDIX B P ROOF OF L EMMA 2
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10
Fig. 4. ROC curves for the Cauchy noise at GSNR = 0, 10, 20, and 30 dB with 𝐾 = 100.
The numerical experiments are also conducted for the Laplace noise. However, results are not included due to limited space. In this case the asymptotic optimality is also justified at 0 dB SNR, but the discordance at 10 dB SNR is larger than other cases because of the less accuracy of the CLT approximations for pdfs that are not bell-shaped [13]. On the whole, Monte Carlo modeling shows that analytical and experimental results match quite well, in particular, they practically coincide on samples 𝑁 > 60. V. C ONCLUDING R EMARKS The decision fusion problem in WSNs having constraints in energy efficiency is studied in this paper. By using a weak signal model and non-Gaussian noise channels into a parallel fusion scheme, a unified asymptotic decision fusion rule based on an arbitrary score function was subsequently proposed. In the case of a known noise pdf, this rule is optimal with the maximum likelihood score function. In the Neyman-Pearson setting, the performance characteristics of the proposed fusion rule such as the false alarm and detection probabilities together with the threshold value have been analytically obtained in the closed form. These results naturally incorporate the available information about channel noises and fading channel attenuation. The asymptotic optimality of the proposed rule is confirmed basing on numerical theoretical and experimental results. A PPENDIX A P ROOF OF L EMMA 1 Given ℎ𝑘 , from (3) it follows that 𝐾 1 ∑ 𝐸[𝑇𝐾 ∣𝐻0 ] = √ ℎ𝑘 (𝑃𝑑𝑘 − 𝑃𝑓 𝑘 )𝐸[𝜓(𝑦𝑘 )∣𝐻0 ]. 𝐾 𝑘=1 √ Since 𝜃 = 𝜈 𝐾 → 0 as 𝐾 → ∞, the√following expansion holds with the remainder of order 𝑜(1/ 𝐾) √ 𝜓(𝑛𝑘 + ℎ𝑘 𝜃) = 𝜓(𝑛𝑘 ) + ℎ𝑘 𝜃𝜓 ′ (𝑛𝑘 ) + 𝑜(1/ 𝐾).
The conditional variance of statistic 𝑇𝐾 under hypothesis 2 𝐻0 is given by Var[𝑇𝐾 ∣𝐻0 ] = 𝐸[𝑇𝐾 ∣𝐻0 ] − (𝐸[𝑇𝐾 ∣𝐻0 ])2 . Then, by using asymptotic expansions based on elementary but tedious transformations, it can be shown that 2 𝐸[𝑇𝐾 ∣𝐻0 ]=
where 𝜓 2 = is given by
∫
𝐾 1 ∑ (𝑃𝑑𝑘 − 𝑃𝑓 𝑘 )2 ℎ2𝑘 𝜓 2 + 𝜈 2 (𝜓 ′ )2 𝐻𝑓2𝐾 , 𝐾 𝑘=1
2
𝜓 (𝑛)𝑓 (𝑛)𝑑𝑛. Thus, the variance Var[𝑇𝐾 ∣𝐻0 ] Var[𝑇𝐾 ∣𝐻0 ] = 𝜓 2 𝐻𝐾 ,
∑𝐾
(11)
1 2 2 where 𝐻𝐾 = 𝐾 𝑘=1 (𝑃𝑑𝑘 −𝑃𝑓 𝑘 ) ℎ𝑘 +𝑜(1/𝐾). By a similar procedure it can be shown that the conditional variance under hypothesis 𝐻1 is the same as in (11): Var[𝑇𝐾 ∣𝐻1 ] = 𝜓 2 𝐻𝐾 .
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