Optimal local sensor decision rule for target detection with channel fading statistics in multi-sensor networks

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Journal of the Franklin Institute 354 (2017) 530–555 www.elsevier.com/locate/jfranklin

Optimal local sensor decision rule for target detection with channel fading statistics in multi-sensor networks Yongsheng Yana,n, Haiyan Wanga, Teng Xuea, Xuan Wangb,1 a

School of Marine Science and Technology, Northwestern Polytechnical University, Xi'an, 710072, China b Electrical and Computer Engineering, University of Wisconsin, Madison, WI 53706, USA Received 4 December 2014; received in revised form 21 October 2015; accepted 11 October 2016 Available online 25 October 2016

Abstract In this paper, we investigate the optimal local sensor decision rule based on non-ideal transmission channels between local sensors and the fusion center for distributed target detection system. The optimality of a likelihood-ratio test (LRT)-based local decision rule at local sensor, which requires only the knowledge of channel statistics instead of instantaneous channel state information (CSI), is established. The coupled local decision rule at each sensor is derived in a closed-form for coherent BPSK and OOK and non-coherent OOK. The iterative person-by-person optimization (PBPO) algorithm is employed to solve the coupled local thresholds. Simulation analysis reveals that the derived thresholds according to the local decision rule are consistent to the exhaustive searching. Furthermore, the detection performance of the system with the proposed optimal local decision rule for different reception modes and modulations is analyzed and compared. & 2016 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

1. Introduction Multi-sensor networks (MSNs) have been extensively used in varieties of application including civilian monitoring and military surveillance. The rapid deployment, wide range of n

Corresponding author. E-mail addresses: [email protected] (Y. Yan), [email protected] (H. Wang), [email protected] (T. Xue), [email protected] (X. Wang). 1 Xuan Wang co-authored this work when she was with School of Electronics and Information, Northwestern Polytechnical University, Xi'an, 710072, China. http://dx.doi.org/10.1016/j.jfranklin.2016.10.025 0016-0032/& 2016 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

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surveillance, reliable performance and low-cost sensors make MSNs to be a pervasive tool to implement widespread monitoring task, such as disaster forecasting [1], biomedical applications [2], target detection, localization and tracking in battlefield [3–5] and intrusion detection [6]. We are particularly interested in the distributed target detection for MSNs, which has been attracting researchers' significant attention due to the development of wireless sensor networks (WSNs). The distributed target detection can be dated back to the 1980s for detecting airplanes with distributed radars, where the costly data transmission among radars makes data compression a reality [7]. An MSN consists of geographically dispersed sensors and a fusion center, where local sensors obtain observations of the region of interest, make preliminary decisions according to the decision rules of local sensors and then transmit the corresponding results to the fusion center over cable or wireless channel. The fusion center combines all the data and makes a final decision about the state of targets according to the fusion rule. In this paper, we focus on the distributed target detection based on MSNs, more specifically, the process of optimal decision fusions of local sensors is investigated to give the further results in the presence of fading and noisy transmission channels with fading statistics. In a distributed detection system with MSNs, many researchers concentrate on the optimal local decision rules at sensors as well as the optimal fusion rule at the fusion center under Neyman-Pearson or Bayesian detection criteria. Under the assumption of conditionally independent observations given H0 (target absence) and H1 (target presence) hypotheses, the optimality of likelihood ratio tests (LRTs) both at the fusion center and local sensors was established for binary local sensor outputs [8,9]. This result is also compatible when the assumption is relaxed to multilevel quantization outputs of local sensors. Also, the optimality holds regardless of the ideal or non-ideal channel between local sensors and the fusion center [10,11]. The optimality of LRT was summarized in [12]. This study dramatically reduces the search space for the optimal rules at the fusion center as well as the local sensors. The optimal detection rule is an LRT, nevertheless, different models, assumptions and the knowledge of transmission channels result in different formulations. The optimization of the detection performance can be achieved by considering the following three different levels: (1) the fusion rule at the fusion center is optimized while the local sensor decision rules at local sensors are fixed; (2) both the fusion rule and the local sensor decision rules are optimized; (3) the sensor decision rules are optimized while the fusion rule is fixed. Previous study mainly concentrated on the level 1), i.e., the optimization of the fusion rule at the fusion center for fixed local sensor decision rules [13–20]. In [13], the optimal ChairVarshney fusion rule was proposed based on the assumption that the fusion center obtains the local detection performance indices, i.e., the detection probabilities and the false alarm probabilities. However, the prior information is not available at the fusion center. To solve this problem, a Bayesian approach by averaging probability density under H1 hypothesis over the prior distribution of the targets location was developed in [14]. Also, a generalized LRT was described in [15], where the target location is estimated by the maximum likelihood (ML). The Bayesian approach and the generalized LRT are computationally expensive for resourceconstrained detection systems with MSNs. An intuitive heuristic counting fusion rule that uses the total number of “1”s of the local decisions was proposed in [16,17]. In addition to these fusion rules, an alternative fusion rule that is insensitive to the probability distributions of sensors' observations was presented based on multi-times and multilevel quantization of local sensors outputs in [18]. Traditionally, the above fusion rules are based on the assumption that the transmission channels between local sensors and the fusion center are error-free. However, the transmission channels are always subject to channel fading, environmental noise and

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interferences. The necessity of integrating the channel-aware design with the fusion algorithms was given in [12]. For coherent reception and binary phase keying (BPSK) modulation, researchers in [19–21] obtained different LRT-based fusion rules according to availability of channel state information (CSI) at the fusion center. For non-coherent reception and on-off keying (OOK) modulation, an LRT-based fusion rule in [22] was proposed under Rayleigh, Ricean, and Nakagami fading channel models, respectively. For the optimization of both the fusion rule and local decision rules, the optimal rules reduce to the threshold searching since the statistic of the optimal LRT is easily derived for given priori knowledge, such as probability distribution under Hi, i¼ 0,1 hypothesis, CSI. An exhaustive searching method can be employed to derive the coupled thresholds [23]. However, the computational complexity of the searching method exponentially increases with the number of local sensors. A popular method to design optimal distributed target detection system with the coupled thresholds is to employ a person-by-person optimization (PBPO) technique [8]. For such a technique, one of the thresholds of the local decision rules is optimized at a time while keeping other thresholds fixed. The overall performance of the distributed detection system is guaranteed to improve with every iteration [24]. For the optimization of local decision rules, the main task is also to search local sensor thresholds for the given fusion rule of the fusion center. The authors in [25] provided a multiobjective optimization method to search the local sensor thresholds. Such a method considers maximizing the detection performance while minimizing transmission costs by employing OOK modulation based on the error-free transmission channels assumption. To analyze the performance loss caused by non-ideal channels, the optimal thresholds of local sensors were derived for fixed fusion rule at the fusion center in [10,11]. The non-ideal channels are assumed to be binary symmetric channel for a decode-then-fuse target detection system, where the fusion center decodes the local sensor results before data fusion. However, for a direct fusion system without communication data decoding, the design of the optimal local decision rule with nonideal transmission channels characterized by channel fading statistics under different reception modes and modulations in MSNs is an unaddressed problem. In this paper, we focus on the local decision rules based on the channel fading statistics, where different modulations and reception modes are considered. Our goal here is to address the following questions: (1) what are the optimal thresholds of decision rules at local sensors to minimize the system average error probability for the given fusion rule at the fusion center? (2) how are the optimal thresholds of local decision rules with channel fading statistics related to the reception mode and modulation? (3) for the optimal thresholds of the local decision rules, how does the detection performance of the integrated target detection system based on channel fading statistics vary for different modulations and reception modes? To answer these questions, the closed-form solution of the optimal thresholds is derived for the given linear fusion rule at the fusion center. The local thresholds are coupled with each other and are also linked to the fusion rule of the fusion center. Also, numerous design examples for coherent reception and noncoherent reception, respectively, are given. For coherent reception, the transmitter at each sensor sends BPSK or OOK modulated decision to the fusion center. For non-coherent reception, the transmitter sends OOK modulated decision to the fusion center. As such, we can understand the system detection performance limitations imposed by the loss of modulation at local sensors and the reception modes at the fusion center. Our main contributions of this paper are summarized as follows.

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Fig. 1. Parallel fusion model with three hierarchical structures for testing two hypotheses. Fig. (a) illustrates the case of coherent reception. Fig. (b) gives the case of non-coherent reception.

(1) An LRT-based local decision rule at local sensors is derived. The result turns out to be equivalent to the previous conclusion. (2) To derive the coupled optimal local decision rule at sensors in a closed-form, the optimal fusion rule at the fusion center for coherent OOK is proposed according to the channel Rayleigh fading statistics instead of CSI and a linear approximation in low SNR channel regime is also developed. (3) The channel fading statistics-based coupled local decision rule at each sensor is derived in a closed-form for coherent BPSK and OOK and non-coherent OOK. The organization of this paper is as follows. Next section introduces the system model and some assumptions. Section 3 gives the optimal local decision rules. Section 4 describes several design examples of different modulation schemes and different reception modes. In Section 5 several simulations and corresponding results are provided, followed by conclusions in Section 6.

2. Problem statements Consider a target detection system of testing two hypotheses (H0 and H1) illustrated in Fig. 1 based on a parallel MSN with N local sensors and a fusion center. Each sensor collects observations Xi of H 1 =H 0 phenomenon. We assume that the observations are conditionally

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independent, that is, N

pðX 1 ; X 2 ; …; X N ∣H k Þ ¼ ∏ pðX i ∣H k Þ;

k ¼ 0; 1:

ð1Þ

i¼1

In general, the conditionally independent assumption across sensors is reasonable if the noisy observations at the sensors result in inaccuracies [24]. To meet the severe resource and limited bandwidth constraints of MSN-based target detection system, a binary decision ui is made according to the ith local sensor decision rule ui ¼ γðX i Þ:

ð2Þ

Then, each local decision result ui is transmitted over parallel fading channel and yi is the output. We assume that the fading channels between the local sensors and the fusion center are independent of each other, i.e., N

pðy∣uÞ ¼ ∏ pðyi ∣ui Þ:

ð3Þ

i¼1

For the fading channels between sensors and the fusion center, it is realistic for small-scale sensor networks to employ orthogonal transmission in which each sensor transmits data to the fusion center through an independent channel [12]. We investigate the combining process for the following two cases: coherent reception and non-coherent reception. For coherent reception, the phase of the transmission channel is obtained by transmitting a training sequence before effective data transmission for stationary channels, or, by differential encoding for fast fading channels [20]. The output of fading channel with phase coherent reception shown in Fig. 1(a), can be expressed as yi ¼ hi ui þ n i

ð4Þ

where hi is the channel envelop and ni is the additive Gaussian noise with mean zero and variance s2i . In a scattering environment without a dominant path, the channel envelop can be modeled as Rayleigh distribution. We assume a Rayleigh fading channel with unit power, and the pdf of hi is f ðhi Þ ¼ 2hi e  hi : 2

ð5Þ

The coherent reception is based on the assumption that the receiver (fusion center) has obtained both the channel phase and propagation delay information. This extra parameter estimation is costly especially for the time-selective fading channels when sensors and the fusion center have relative motion such as the fluctuation of sensors in an underwater acoustic MSN, the motion of sensors (e.g., underwater unmanned vehicle) in a mobile MSN. In this case, non-coherent phase reception is suitable for data transmission between local sensors and the fusion center shown in Fig. 1(b). The channel output of sensor i is yi ¼ hi ejφi ui þ ni

ð6Þ

where hi also has Rayleigh distribution with unit power, φi is the phase of the ith fading channel with uniform distribution, i.e., φi  ½  π; þπÞ and ni  Nð0; s2i Þ. Each local decision ui is transmitted to the fusion center over fading channels with ui ¼ 1 or ui ¼  1 for BPSK modulation and ui ¼ 1 or ui ¼ 0 for OOK modulation. The fusion center employs coherent or non-coherent reception to obtain the data from local sensors and distinguishes between two hypothesises H1 (target-presence) and H0 (target-absence) according

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to the fusion rule u0 ¼ γ 0 ðyÞA fH 0 ; H 1 g:

ð7Þ

The signal transmission process forms a Markov chain H 1 =H 0 -x-u-y-u0 , where x ¼ ½X 1 ; X 2 ; …; X N T and u ¼ ½u1 ; u2 ; …; uN T . The optimal fusion rule under the Bayesian criterion is u0 ¼ 1

pðy∣H 1 Þ Z π 0 pðy∣H 0 Þ o π 1

ð8Þ

u0 ¼ 0

where u0 ¼ 0 means target-absence and u0 ¼ 1 means target-presence, π 1 ¼ PðH 1 Þ and π 0 ¼ PðH 0 Þ ¼ 1 π 1 are prior probabilities of H1 and H0, respectively. Note that the optimal fusion rule is an LRT involving the comparison of the corresponding fusion statistic with a constant (threshold). 3. The optimal decision rule of local sensors This section derives the optimal decision rule of local sensors for the given fusion rule of the fusion center under the Bayesian criterion. The result turns out to be consistent with the previous conclusion in [11] for the LRT optimality. It is worth noting that the fusion rule with channel statistics at the fusion center is first investigated, since it determines the optimal local sensor decision rules. We examine the linear fusion rule, which has less computational complexity than that of the optimal LRT fusion rule [26,27]. Also, the linear fusion rule is a low channel SNR approximation of the optimal LRT fusion rule. Numerous two-sensor design examples for coherent reception with BPSK and OOK modulations, and non-coherent reception with OOK modulation are given. The system average error probability at the fusion center can be expressed as Pe ¼ π 0 Pðu0 ¼ 1∣H 0 Þ þ π 1 Pðu0 ¼ 0∣H 1 Þ # Z Z " X X ¼ π 0 Pðu0 ¼ 1; x; y; u∣H 0 Þ þ π 1 Pðu0 ¼ 0; x; y; u∣H 1 Þ dxdy y

x

y

x

u u Z Z " X ¼ π 0 Pðu0 ¼ 1∣yÞpðy∣uÞPðu∣xÞpðx∣H 0 Þ u # X þπ 1 Pðu0 ¼ 0∣yÞpðy∣uÞPðu∣xÞpðx∣H 1 Þ dxdy

ð9Þ

u

where Pð∣Þ means the probability density function of discrete random variables and pð∣Þ means the probability density function of continuous random variables. Under the Bayesian criterion, the goal is to minimize the system average error probability. To obtain the optimal decision rules of local sensors, the term Pðui ¼ 1∣xi Þ is required. Equation (9) can be further expressed as Z Z X Pe ¼ Pðu0 ¼ 1∣yÞ pðy∣uÞPðujxÞ½π 0 pðx∣H 0 Þ π 1 pðx∣H 1 Þdxdy þ π 1 y x Z Z Zu XX ¼ Pðu0 ¼ 1∣yÞ pðy∣uÞPðui ∣xi ÞPðui ∣xi Þ y

x i x i ui

ui

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   π 0 pðxi ∣H 0 Þpðxi ∣H 0 Þ π 1 pðxi ∣H 1 Þpðxi ∣H 1 Þ dxi dxi dy þ π 1

ð10Þ

where xi ¼ ½x1 ; …; xi  1 ; xiþ1 ; …; xN T and similarly ui ¼ ½u1 ; …; ui  1 ; uiþ1 ; …; uN T . Note that in Equation (10), the first equation is written as such since Pðu0 ¼ 0jyÞ ¼ 1  Pðu0 ¼ 1jyÞ. The can be derived due to x ¼ ½xi ; xi T and u ¼ ½ui ; ui T . Using the fact Rsecondi equation i i i i xi Pðu ∣x Þpðx ∣H k Þdx ¼ Pðu ∣H k Þ, k ¼ 0,1, we have ( Z Z X  Pe ¼ Pðui ¼ 1∣xi Þ Pðu0 ¼ 1∣yÞ pðy∣ui ; ui ¼ 1Þ  π 0 pðxi ∣H 0 ÞPðui ∣H 0 Þ y

xi i



i



ui

 π 1 pðxi ∣H 1 ÞPðu ∣H 1 Þ Z X  þPðui ¼ 0∣xi Þ Pðu0 ¼ 1∣yÞ pðy∣ui ; ui ¼ 0Þ  π 0 pðxi ∣H 0 ÞPðui ∣H 0 Þ y

ui

 π pðx ∣H ÞPðu ∣H 1 Þ gdydxi þ π 1 Z 1 i 1 9 Pðui ¼ 1∣xi Þ½π 0 pðxi ∣H 0 ÞAi  π 1 pðxi ∣H 1 ÞBi dxi þ C

ð11Þ

xi

where Pðui ¼ 0∣xi Þ ¼ 1  Pðui ¼ 1∣xi Þ is used to derive the equation. In the last step, we define ( ) Z X   i i i Ai ¼ Pðu0 ¼ 1∣yÞ Pðu ∣H 0 Þ pðy∣u ; ui ¼ 1Þ pðy∣u ; ui ¼ 0Þ dy ð12Þ y

Z Bi ¼ and

y

ui

(

) X   Pðu0 ¼ 1∣yÞ Pðui ∣H 1 Þ pðy∣ui ; ui ¼ 1Þ pðy∣ui ; ui ¼ 0Þ dy

ð13Þ

ui

Z Z C¼ xi y

Pðu0 ¼ 1∣yÞ

X

 pðy∣ui ; ui ¼ 0Þ  π 0 pðxi ∣H 0 Þpðui ∣H 0 Þ

ui

  π 1 pðxi ∣H 1 Þpðui ∣H 1 Þ dydxi þ π 1 Z X   ¼ Pðu0 ¼ 1∣yÞ pðy∣ui ; ui ¼ 0Þ  π 0 pðui ∣H 0 Þ  π 1 pðui ∣H 1 Þ dy þ π 1 : y

ð14Þ

ui

It is worth noting that, the authors in [11] also derived the optimal decision rules of local sensors which have almost the same expression except for the coefficient Bi and the constant C. In fact, Bi in Equation (11) is equivalent to the threshold coefficient in [11] since we have the following equation: Z X   Bi ¼ ½1  Pðu0 ¼ 0∣yÞ Pðui ∣H 1 Þ pðy∣ui ; ui ¼ 1Þ pðy∣ui ; ui ¼ 0Þ dy Z ¼

y

ui

½pðy∣ui ¼ 1; H 1 Þ  pðy∣ui ¼ 0; H 1 Þ Z X    Pðu0 ¼ 0∣yÞ Pðui ∣H 1 Þ pðy∣ui ; ui ¼ 1Þ pðy∣ui ; ui ¼ 0Þ dy y

Z ¼ y

y

ui

X   Pðu0 ¼ 0∣yÞ Pðui ∣H 1 Þ pðy∣ui ; ui ¼ 0Þ pðy∣ui ; ui ¼ 1Þ dy ui

ð15Þ

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R where the fact y ½pðy∣ui ¼ 1; H 1 Þ  pðy∣ui ¼ 0; H 1 Þ ¼ 0 is used. The similar derivation is also applied to the constant C, which is omitted here. From [11], the coefficient Ai in Equation (20) can be further written as Z   Pðu0 ¼ 1∣yi ; ui ¼ 1Þ Pðu0 ¼ 1∣yi ; ui ¼ 0Þ pðyi ∣H 0 Þdyi : ð16Þ Ai ¼ yi

Similarly, for the threshold coefficient Bi, we have Z   Bi ¼ Pðu0 ¼ 1∣yi ; ui ¼ 1Þ Pðu0 ¼ 1∣yi ; ui ¼ 0Þ pðyi ∣H 1 Þdyi :

ð17Þ

yi

From Equation (11), we can see that C is a constant, which is independent of the decision rule of sensor i Pðui ¼ 1∣xi Þ in the presence of non-ideal channel. The necessary condition of the optimal local sensor decision rule to minimize the average error probability can be written as ( 1 π 0 pðxi ∣H 0 ÞAi  π 1 pðxi ∣H 1 ÞBi r 0 Pðui ¼ 1∣xi Þ ¼ ð18Þ 0 π 0 pðxi ∣H 0 ÞAi  π 1 pðxi ∣H 1 ÞBi 40 From Eqs. (16) and (17), we can see that if the term I i 9 Pðu0 ¼ 1∣yi ; ui ¼ 1Þ Pðu0 ¼ 1∣yi ; ui ¼ 0Þ

ð19Þ

satisfies the inequality I i Z 0, we have Ai Z 0 and Bi Z 0. Thus, the optimal decision rule of sensor i amounts to an LRT, shown as ui ¼ 1

pðxi ∣H 1 Þ Z π 0 Ai : pðxi ∣H 0 Þ o π 1 Bi

ð20Þ

ui ¼ 0

If the term Ii ¼ 0, i.e., Ai ¼ Bi ¼ 0, we choose Pðui ¼ 1∣xi Þ ¼ 0 by defining 00 ¼ 1. For a monotone fusion rule of the fusion center, it is easy to satisfy the condition I i Z 0 [11,8]. The optimal threshold of sensor i depends on the fusion rule Pðu0 ¼ 1∣yÞ of the fusion center, and other local decision rules under both H1 and H0 hypotheses, i.e., Pðui ∣H 0 Þ and Pðui ∣H 1 Þ. We can see that, the optimal fusion rule (8) and the local sensor decision rule (20) are all LRTs under the Bayesian criterion. Also, we can see that the optimal threshold of the fusion rule is a constant while the optimal threshold of sensor i is ππ 01 ABii . The proposed optimal local decision rule can be treated as a feedback process. Each sensor derives local decision based on its local measurement and transmits the local decision to the fusion center, where a final detection result about the presence/absence of a target is given. For a feedback-based sensor network, the final decision is transmitted back to the local sensor, where the finial decision is combined to formulate the local decision rule. Such a feedback can improve the detection performance compared with the multi-sensor detection system without feedback. This is because that the optimal local decision rule with feedback is derived based on the global optimization of the average error probability, while the multi-sensor distributed detection system only with its local measurements is based on the local optimization. That is, the local decision rule is derived based on its Bayesian criterion for the simple decision rule, however, the main goal is to minimize the average error probability of the multi-sensor distributed detection system under the Bayesian criterion.

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4. Several design examples with different reception modes and modulations In this section, we investigate a two-sensor target detection system to illustrate the optimal local sensor decision rule in the presence of non-ideal communication channel with fading statistics instead of the instantaneous CSI. The two cases: coherent reception and non-coherent reception are considered. For the coherent case, the BPSK modulation with ui ¼ þ 1 (or ui ¼  1) and the OOK modulation with ui ¼ 1 (or ui ¼ 0) are employed. The OOK modulation amounts to a censoring signal transmission at local sensors where each sensor sends local decision result to the fusion center only if the optimal LRT statistic exceeds the corresponding threshold. For the non-coherent case, the OOK modulation is considered, where the phase information of the transmission channel is no longer required. 4.1. The coherent reception case The optimal local decision rule depends on the optimal fusion rule at the fusion center. Thus, we first give the optimal fusion rule and then derive the optimal local sensor decision rule. For coherent reception with OOK modulation, the optimal log-LRT of equation (8) based on channel fading statistics is   pðy∣H 1 Þ ΛCO ¼ log CS pðy∣H 0 Þ " # N X pðyi ∣ui ¼ 1ÞPid þ pðyi ∣ui ¼ 0Þð1  Pid Þ ¼ log ð21Þ pðyi ∣ui ¼ 1ÞPif þ pðyi ∣ui ¼ 0Þð1  Pif Þ i¼1 where Pid ¼ Pðui ¼ 1jH 1 Þ denotes the detection probability of sensor i, Pif ¼ Pðui ¼ 1jH 0 Þ denotes the false alarm probability of sensor i. In Eq. (21), the assumption of the conditional independence of local decisions is used. The notation “CS” denotes channel fading statistics and “CO” denotes coherent reception and OOK modulation. pðyi ∣ui ¼ 1Þ in Eq. (21) is given as [20]  y2  pffiffiffiffiffi  i2     ðai yi Þ2 2si 2s e i 1 þ 2π ai yi e 2 Q  ai yi p yi ∣ui ¼ 1 ¼ pffiffiffiffiffi ð22Þ 2π ð1 þ 2s2i Þ 1 ffi. Similarly, we have where ai ¼ pffiffiffiffiffiffiffiffiffi 2 si



1þ2si



y2

 i2 1 p yi ∣ui ¼ 0 ¼ pffiffiffiffiffi e 2si : 2π si

ð23Þ

pffiffiffiffiffi   ðai yi Þ2 With the above results, by defining 2π ai yi e 2 Q  ai yi ¼ bi , the log-LRT can be written as 2 3 y2 y2  i2   1  2si2 2s i pffiffiffiffiffi 2si i e i ð1 þ bi Þ þ 1 Pd pffiffiffiffi e i 7 6 Pd N 2π si X 6 7 2π ð1 þ 2s2i Þ CO 6 7 ΛCS ¼ log 6 2 2 y y 7

1  i2  i2 5 4 i 2si i¼1 e 2si ð1 þ bi Þ þ 1  Pif pffiffiffiffiffi e 2si Pf pffiffiffiffiffi 2π si 2π ð1 þ 2s2i Þ 2 3  u ¼1 Pid  2  N 61 þ 1 þ 2s2 2si bi  1 7 0Z X π0 6 7 i ¼ log 6 log ð24Þ 7 i o 4 5 π1   P f i¼1 2b  1 u0 ¼ 0 1þ 2s i i 1 þ 2s2i

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π0 π1

where log

is the log-threshold of the optimal fusion rule at the fusion center. As we can see,

the optimal fusion rule based on channel fading statistics is related to the channel SNR defined as SNR ¼

E½h2i  ¼ s12 for Rayleigh fading channel s2i i (Pid and Pif) and the received data yi.

with unit power, local detection performance

indices For the optimal fusion rule at the fusion center, it is difficult to derive the closed-form solutions of the optimal local sensor decision rule according to (20), since the local thresholds are coupled with the nonlinear fusion statistic in (24). Hence, an alterative linear fusion rule can be employed to optimize the local sensor decision rules, where the comparable detection performance can be achieved compared with the optimal LRT fusion rule [27]. Further, the linear fusion rule at the fusion center is the low-channel SNR approximation of the optimal fusion rule. Next, we first determine the linear fusion rule by considering the low-SNR approximation of the optimal LRT fusion rule based on channel fading statistics before determining the optimal local sensor decision rule. Lemma 1. When the channel noise variance s2i -1, the SNR of the Rayleigh fading channel with unit power is SNR ¼ s12 -0. The optimal fusion rule based on channel fading statistics for i coherent reception with OOK modulation ΛCO CS reduces to the following fusion statistic Λ1, i.e., lim

s2i -1

ΛCO CS

u ¼1 

0Z pffiffiffi  i 1 π0 i ¼ Λ1 ¼  1 P  P log : π y i d f 2 o π1 1 þ 2s i i¼1 N X

ð25Þ

u0 ¼ 0

If local sensors have identical detection performance indices and channel SNRs, i.e., Pid ¼ Pd , Pif ¼ Pf , and a2i ¼ a2 for all i, and Pd 4Pf , then ΛCO CS further reduces to a rule in the form of an equal gain combiner Λ2, given as    1 1 þ 2s2 π0 pffiffiffi N þ yi log Λ2 ¼ : o π Pd  Pf π1 i¼1 N X

u0 ¼ 1

Z

ð26Þ

u0 ¼ 0

1 ffi -0. Qð ai yi Þ can be approximated by the first Proof. When s2i -1, the term ai ¼ pffiffiffiffiffiffiffiffiffi si 1þ2s2i order Taylor series expansion, i.e.,   1 ai y Q  ai yi  þ pffiffiffiffiffii : ð27Þ 2 2π

Similarly, the term e ðai yi Þ2 2

ðai yi Þ2 2

a2i y2i

can be also approximated by

: ð28Þ 2 The high order terms ðai yi Þ2 , ðai yi Þ3 and ðai yi Þ4 are ignored. Thus, with the above results, the fusion statistic in (24) can be written as 8  i9 1  2 pffiffiffiffiffi > > > 2π a 1 þ s y  1 Pd > i = < N i X 1 þ 2s2i i CO ð29Þ lim ΛCS ¼ log  i >: 1  2 pffiffiffiffiffi > s2 -1 > > i¼1 ; :1 þ 2π a s y  1 P i i f 1 þ 2s2i i e

1þ

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pffiffiffiffiffi 1 2 2π ai yi ¼ Since s2i -1, it is straightforward to derive that 1þ2s 2 -0 and si i With these results and the fact limx-0 log ð1 þ xÞ ¼ x, we derive Λ1 

N X i¼1

u ¼1 

 0Z pffiffiffi  i 1 π0 i π y  1 P  P log : i d f 2 o π1 1 þ 2si

pffiffiffiffi 2 pffiffiffi sp 2π yi ffi ffiffiffiffiffiffiffiffiffi - π yi . s 1þ2s2

ð30Þ

u0 ¼ 0

When Pid ¼ Pd , Pif ¼ Pf and s2i ¼ s2 for all i, the fusion rule reduces to Λ2 shown in (26).□ We can see that the optimal fusion rule reduces to an affine function of receiving data yi for i ¼ 1; 2; …; N for low channel SNR. With the above fusion rule and its low channel SNR approximation, we can determine the coefficients Ai and Bi from (16) and (17). Without loss of generality, we illustrate the optimal local decision rule for a two-sensor target detection system by the following theorem. Theorem 1. For a two-sensor distributed target detection system assuming only the knowledge of channel Rayleigh fading statistics, given the optimal linear fusion rule at the fusion center, the optimal local decision rule of sensor i for coherent reception and OOK modulation amounts to the following necessary condition: ui ¼ 1

pðxi ∣H 1 Þ Z i τ pðxi ∣H 0 Þ o OOK

ð31Þ

ui ¼ 0

where the two thresholds are     T 1 ðy2 2 Þ  1 ffiQ  a1 T 1 ðy2 Þ e 2s1 þ1 p y2 ∣H 0 dy2 π 0  1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2s1 þ 1 τ1OOK ¼ 2 2Þ      T2s1 ðy2 þ1 R þ1 1 1 ffiQ  a1 T 1 ðy2 Þ e π 1  1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p y2 ∣H 1 dy2 2 2s1 þ 1

ð32Þ

   T 1 ðy2 1 Þ   1 ffiQ  a2 T 1 ðy1 Þ e 2s2 þ1 p y1 ∣H 0 dy1 π 0  1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2s2 þ 1 τ2OOK ¼ : 2 1Þ      T2s1 ðy2 þ1 R þ1 1 2 ffiQ  a2 T 1 ðy1 Þ e π 1  1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p y1 ∣H 1 dy1 2s22 þ 1

ð33Þ

R þ1

2

and R þ1

We define   1 T 1 yi ¼ pffiffiffi þ π

2

 log





pffiffiffi  π0 π yi  1 Pid  Pif 1 þ 2s2ð3  iÞ  a2i s2i π1

pffiffiffi ð3  iÞ  iÞ π Pd  Pð3 f

ð34Þ

for i ¼ 1,2. QðÞis the  complementary distribution function of the standard Gaussian defined as R y2 1 2 1 QðxÞ ¼ 2π dy. x e

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541

□

Proof. See Appendix A.

The above results give the closed-form solution of the coupled thresholds based on channel fading statistics for coherent reception and OOK modulation. Next, the BPSK modulation case is considered. The optimal fusion rule based on the channel fading statistics for coherent reception and BPSK modulation is derived in [20], given as   pðy∣H 1 Þ CP ΛCS ¼ log pðy∣H 0 Þ " # N X pðyi ∣ui ¼ 1ÞPid þ pðyi ∣ui ¼  1Þð1  Pid Þ log ¼ pðyi ∣ui ¼ 1ÞPif þ pðyi ∣ui ¼  1Þð1  Pif Þ i¼1 9 8  i pffiffiffiffiffi ðai yi Þ2 > > < N X 1 þ Pd  Qðai yi Þ 2π ai yi e 2 = ð35Þ ¼ log h ipffiffiffiffiffi ða y Þ2 > ; :1 þ Pif  Qðai yi Þ 2π ai yi e i 2i > i¼1 where the notation “CP” denotes coherent reception and BPSK modulation. It is obvious that the fusion statistic ΛCP CS is a nonlinear function of received data yi, which leads to difficult derivation of the optimal local sensor decision rule. We rely on the analysis to find the low channel SNR approximation [20], which is given by Λ3 ¼ lim ΛCP CS ¼ s2i -1

2

pffiffiffiffiffi X 2π ai yi : Pid  Pif

ð36Þ

i¼1

Based on this linear fusion statistic, we give the following theorem. Theorem 2. For a two-sensor distributed target detection system with channel Rayleigh fading statistics, given the linear fusion rule at the fusion center, the optimal local decision rule of sensor i for coherent reception and BPSK modulation is pðxi ∣H 1 Þ pðxi ∣H 0 Þ

ui ¼ þ1

Z

o

τiBPSK

ð37Þ

ui ¼  1

where the two thresholds are τ1BPSK

¼

π0 π1

R þ1



T 2 ðy2 Þ2 2s2 þ1 1



T 2 ðy2 Þ2 2s2 þ1 1

R þ1



T 2 ðy1 Þ2 2s2 þ1 2

R þ1



T 2 ðy1 Þ2 2s2 þ1 2

1

R þ1 1

e e

  p y2 ∣H 0 dy2 

ð38Þ



p y2 ∣H 1 dy2

and τ2BPSK

¼

π0 π1

1 e 1

e

  p y1 ∣H 0 dy1 



p y1 ∣H 1 dy1

:

ð39Þ

542

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T 2 ðyi Þ is defined as 

pffiffiffiffiffi π0 log  2π Pid  Pif ai yi   π

T 2 yi ¼ pffiffiffiffiffi 1  iÞ ð3  iÞ 2π Pð3  P að3  iÞ d f

ð40Þ

for i¼ 1,2.

Proof. See Appendix B.

□

For coherent reception with BPSK and OOK modulations, the optimal local decision rules are derived according to the optimality of LRT. As we can see from Theorems 1 and 2, the thresholds are coupled with each other, and are also related to the signal modulation and channel fading statistics. For target detection system with more than two sensors, the optimal thresholds are determined by the multiple integral on yi , which can be derived by numerical integration. 4.2. The non-coherent reception case This subsection gives the optimal local decision rule at each local sensor for non-coherent reception, which is suitable for the scenario where the channel phase information is intractable to obtain when there is the relative motion between the transmitter and the receiver in the data transmission [22]. The non-coherent reception allows phase information-free and only relies on the power of the channel output. Let zi denote the signal power of the ith channel output, i.e., zi ¼ jyi j2 . Given the Rayleigh fading channel statistics (see Equation (5)) and the channel model (see Eq. (6)), the authors in [22] gave the conditional pdfs of zi with OOK modulation, which are given as 8 zi < 1 e  2s2i ; z Z 0 i pðzi jui ¼ 0Þ ¼ 2s2i ð41Þ : 0; zi o0 and

8 <

z

 i 2 1 e 1þ2si ; 2 pðzi jui ¼ 1Þ ¼ 1 þ 2si : 0;

zi Z 0

ð42Þ

zi o0

With the above results, one can easily derived the log-LRT fusion rule and its low channel SNR approximation, which have been addressed in [22]. The optimal log-LRT for noncoherent reception at the fusion center with Rayleigh channel fading statistics and OOK modulation is 2 z  i 2   1  2s12 3 u ¼ 1 1 1þ2s i i i þ 1P  p e e i 0Z N d d X 6 1 þ 2s2i 7 π0 2s2i NO 6 7 ΛCS ¼ log 4 log ð43Þ zi

1 5   o 1 1 π1 1þ2s2 2s2 i i¼1 i þ i u ¼ 0 pif e 1  P e 0 f 1 þ 2s2i 2s2i

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543

where the notation “NO” denotes non-coherent reception. And, the low channel SNR approximation of ΛNO CS is u0 ¼ 1  Z zi π0   Λ4 ¼ log : 2 2 o π1 i ¼ 1 2si 1 þ 2si N X

ð44Þ

u0 ¼ 0

Based on this linear fusion rule, the following theorem is developed in order to derive the optimal local sensor decision rule, where we also take a two-sensor target detection system as an example. Theorem 3. For a two - sensor distributed target detection system with channel Rayleigh fading statistics, given the linear fusion rule at the fusion center, the optimal local decision rule of sensor i for non - coherent reception and OOK modulation is ui ¼ 1

pðxi ∣H 1 Þ Z i τ pðxi ∣H 0 Þ o OOK

ð45Þ

ui ¼ 0

where the thresholds are  F1 ðz2 Þ F ðz Þ   1 22 R t2  2s 2 þ1 π 0 0 e 1  e 2s1 pðz2 ∣H 0 Þdz2 1 τOOK ¼  F1 ðz2 Þ F ðz Þ   1 22 R t2  2s 2 þ1 π 1 0 e 1  e 2s1 pðz2 ∣H 1 Þdz2 and π0 τ2OOK ¼ π1

R t1 0

R t1 0





e 



e

F 1 ðz1 Þ 2s2 þ1 2 F 1 ðz1 Þ 2s2 þ1 2

e e





F 1 ðz1 Þ 2s2 2



F 1 ðz1 Þ  2s2 2

ð46Þ

pðz1 ∣H 0 Þdz1 :

ð47Þ

pðz1 ∣H 1 Þdz1

We define

and

  

π0 zi 2 2 F 1 ðzi Þ ¼ log 1 þ 2s  2 2s ð3  iÞ ð3  iÞ π1 2si ð1 þ 2s2i Þ

ð48Þ

   1 π0 2s2i 1 þ 2s2i i t i ¼ log π1 Pd  Pif

ð49Þ

for i¼ 1,2.

Proof. See Appendix C.

□

As we can see from Theorems 1–3, given the linear fusion rule at the fusion center, the optimal local decision rule is an LRT involving the comparison of the corresponding local decision statistic with a threshold. Several features about the optimal local sensor decision rule with channel Rayleigh fading statistics are stated as follows:

544

  

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The threshold of the optimal local sensor decision rule is coupled with each other, and is also related to the fusion rule of the fusion center. Different physical layer specifications, including the reception mode at the receiver (fusion center), the modulation at the transmitters (local sensors) and the channel fading statistics, result in different thresholds of the optimal local decision rule. Compared with BSPK modulation, the OOK modulation can dramatically reduces the energy consumption of the transmitters (local sensors) by censoring signal transmission scheme, where the sensors send local results to the fusion center only when the optimal LRT statistic exceeds the corresponding threshold.

5. Simulations and results In this section, we evaluate the detection performance of the distributed target detection system with the proposed optimal local sensor decision rule for the given linear fusion rule only based on the knowledge of channel fading statistics. For the thresholds solving, we employ the exhaustive searching and the PBPO thresholds solving method, respectively, to compare the detection performance under the Bayesian criterion. We consider the case where the known signal S is detected in additive Gaussian noise with independent and identically distributed (i.i.d), i.e., H0 : Xi ¼ Ni H1 : Xi ¼ S þ Ni

ð50Þ

where Ni is Gaussian random variable with zero-mean and variance s . In the following examples, we assume S¼ 1, s2 ¼ 0:01. Each sensor make a local decision according to the optimal local decision rule given by (20). Then, the local result is transmitted to the fusion center via Rayleigh fading channel. The fusion center makes last decision about the phenomenon H 1 =H 0 according to the linear fusion rule given by Eq. (8). Here, by taking a two-sensor distributed target detection system as an example, the thresholds solving procedure of the optimal local decision rule using PBPO can be summarized as follows [11]: 2

(1) Initialize the two local thresholds τ1 and τ2. (2) Compute the local detection probability Pid ¼ Pðui ¼ 1jH 1 Þ and the local false alarm probability Pif ¼ Pðui ¼ 0jH 0 Þ of sensor i according to the initialized τ1 and τ2 in step 1. (3) Obtain the linear fusion rule of the fusion center according to Pid and Pif. (4) For given fusion rule (25), (36) and (44) and the threshold τ2, the threshold τ1 is calculated by using Eqs. (32), (38) and (46), respectively. (5) For the given fusion rule and τ1, calculate τ2 using Eqs. (33), (39) and (47), respectively. (6) If the derived τ1 and τ2 from the above steps are identical to that from the previous iteration, then stop the iteration. Otherwise, go to step 2. Next, for different reception modes and modulations, we evaluate the performance of the proposed optimal local decision rule by solving the thresholds using PBPO. Also, the derived thresholds are compared with the exhaustive searching method.

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5.1. The case with coherent reception and OOK modulation For coherent reception and OOK modulation, the channel output is yi ¼ hi ui þ ni , ui A f1; 0g for i ¼ 1,2. The average channel SNR is defined as SNR ¼ s12 . The prior probability of H0 is i assumed to be π0 ¼ 0.6. Fig. 2 illustrates the average error probability as the function of the thresholds ðτ1 ; τ2 Þ with SNR ¼ 0 dB. From the results, the optimal thresholds that minimize the average error probability can be found by searching the whole range of τ1 and τ2, i.e., the exhaustive searching method. The thresholds solving precision depends on the searching grid size. That is, the smaller the grid size is, the more accurate the exhaustive searching is. The optimal thresholds obtained by the exhaustive searching method with size 0.005 are (0.5017, 0.5017) and the minimum average error probability is 0.265041526373685. Contrastively, the results (0.501176, 0.501176) are exactly derived according to the proposed

Fig. 2. The average error probability for the two-sensors distributed detection system with channel Rayleigh fading statistics, coherent reception and OOK modulation. The average channel SNR is 0 dB.

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local decision thresholds in a closed-form that is solved by the iterative PBPO method, and the achieved minimum average error probability is 0.265041526320205. As we can see from the above results, the two methods converge to almost the same point. The proposed optimal local decision threshold in a closed-form, which is solved by the iterative PBPO, exhibits slightly better performance than the exhaustive searching. The solving accuracy of the exhaustive searching method depends on the searching grid size. Intuitively, one can get arbitrarily close to the closed-form solution of the optimal local thresholds by reducing the searching grid size. Accordingly, the computational complexity is increasing with the number of the grid. We call this phenomenon “cell effect”.

Fig. 3. The average error probability for the two-sensors distributed detection system with channel Rayleigh fading statistics, coherent reception and BPSK modulation. The average channel SNR is 0 dB.

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547

5.2. The case with coherent reception and BPSK modulation For the case with coherent reception and BPSK modulation, the channel output is yi ¼ hi ui þ ni for ui Afþ1;  1g. The channel SNR is also assumed to be 0 dB and the prior probability of H0 is π 0 ¼ 0:6. Fig. 3 shows the average error probability Pe versus ðτ1 ; τ2 Þ based on channel fading statistics for coherent reception and BPSK modulation. We find that the optimal thresholds is reached at the point (0.5025, 0.5025) for ðτ1 ; τ2 Þ by the exhaustive searching method with the grid size 0.005. Accordingly, the minimum achievable average error probability is 0.123597813055458. By comparison, the closed-form solution of the optimal local thresholds, which is solved by the iterative PBPO method always converge iteratively to the point (0.502241, 0.502241) with average error probability 0.123597813039197. We can see that achieved average error probability of the closed-form solution of the optimal local thresholds is slightly smaller than that of the exhaustive searching. That is, the proposed method in a closedform provides slightly better performance than the exhaustive searching.

Fig. 4. The average error probability for the two-sensors distributed detection system with channel Rayleigh fading statistics, non-coherent reception and OOK modulation. The average channel SNR is 0 dB.

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Y. Yan et al. / Journal of the Franklin Institute 354 (2017) 530–555

5.3. The case with non-coherent reception and OOK modulation For the case with non-coherent reception and OOK modulation, the channel output is yi ¼ hi ejφi ui þ ni for ui A f1; 0g. The power of channel output is developed for the fusion rule at the fusion center. Fig. 4 gives the average error probability Pe versus ðτ1 ; τ2 Þ by using zi ¼ jyi j2 instead of yi. The thresholds are (0.5100, 0.5100) with the minimum average error probability 0.319695975459464 using exhaustive searching method with the grid size of 0.005. By comparison, the PBPO converges iteratively to the point (0.5070, 0.5070) with the minimum error probability 0.319695975238531. Not surprisingly, the proposed method in a closed-form provides slightly better performance than the exhaustive searching method. However, the performance of the exhaustive searching method can be improved by employing smaller gird size. Accordingly, the computational complexity will increase exponentially.

Table 1 Comparison of the local sensor decision rule with different reception modes and modulations. Reception and modulation

Exhaustive searching

PBPO

CO

(0.4000,04000) 0.341332801491134

(0.4022,04022) 0.341332777932102

CP

(0.4000,04000) 0.238477475975058

(0.4026,04026) 0.238477388400797

NO

(0.4050,04050) 0.381027727500608

(0.4052,04052) 0.381027725460478

Fig. 5. The comparison of the average error probability versus channel SNR with different reception modes and modulations. The prior probability of H0 is π 0 ¼ 0:6.

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549

5.4. Comparison of the different cases To better understand the robust performance of the proposed local sensor decision rule based on channel fading statistics, we give anther simulation results with the parameter setting SNR ¼  5 dB, S ¼ 0.8. The other parameter setting is identical to the above example. Table 1 illustrates both the coherent reception case and the non-coherent reception case. The thresholds and the achieved minimum system average error probability are given. From this table, it is clear that the optimal local threshold in a closed-form iteratively solved by PBPO converges to approximately consistent result with the exhaustive searching for each case. Furthermore, the proposed method with the optimal local threshold in a closed form provides slightly better performance than the exhaustive searching method with gird size of 0.005. Compared with different cases, the optimal threshold of local sensor decision rule with different reception modes and modulation schemes is different with each other even for the identical detection model.

Fig. 6. Local threshold convergence of the proposed local decision rule based on the channel fading statistics by using PBPO.

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Y. Yan et al. / Journal of the Franklin Institute 354 (2017) 530–555

Fig. 5 gives the comparison of the average error probability versus channel SNR with different reception modes and modulations. It is worth noting that, when we plot the average error probability versus channel SNR, every point on the curve corresponds to a coupled thresholds solving using PBPO. Comparing the average error probability for the above three cases, we observe that the detection performance of the distributed detection system with coherent reception and BPSK modulation is maximized. This is attributed to the less information loss of the BPSK modulation, however, it is at the expense of higher energy consumption since each symbol (including þ1 and  1) is encoded and transmitted. For OOK modulation, the case with coherent reception presents better detection performance than that with non-coherent reception. This is due to the smaller average bit error rate of the communication for the coherent reception. Furthermore, Fig. 5 shows that the average error probability is capped at 0.4, since the system average error probability is no worse than that for ignoring the local decision and deciding H0 or H1, i.e., Pe r maxfπ 0 ¼ 0:6; π 1 ¼ 0:4g. From Fig. 5, we can see that when the channel SNR changes from 5 dB to 10 dB, the average error probability of the multi-sensor detection system for non-coherent OOK does not significantly change. That is, there is a saturation effect. When the channel SNR increases to a value, the average error probability will be capped at a value that depends on reception mode, modulation and channel fading statistics. This is because that the linear fusion rule is a linear approximation of the optimal fusion rule at the fusion center under low channel SNR ranges. Furthermore, the channel error does not have a significant impact on the average error probability for high channel SNR ranges. 5.5. Convergence analysis For the proposed coupled local decision rule based on the channel fading statistics, the local threshold in a closed-form is iteratively derived by using PBPO. In order to analyze the convergence, we give anther simulation results with different initial thresholds. The channel SNR is set to be SNR ¼  5 dB and S¼ 0.8. The prior probability of H0 is π 0 ¼ 0:6. Fig. 6 gives the two thresholds versus iteration with 10 different initial τ1 and τ2. From Fig. 6, we can see that the proposed local decision rule in a closed-form solved by using iterative PBPO algorithm converges in one iteration. Its convergence rate is fast. When the two solved thresholds converge, each iteration has the identical result to the previous iteration. Furthermore, different initial thresholds does not impact on the convergence. From Figs. 2–4, we can see that there is only one global minimum. Thus, the exhaustive searching algorithm always converge to the optimal thresholds. However, the convergence rate depends on the searching grid size and the searching range. As the grid size becomes smaller and the searching range becomes larger, the convergence rate is significantly slow. 6. Conclusion Considering a distributed target detection system that is tasked with a binary hypothesis testing, we investigated the optimal local sensor decision rule based on the channel fading statistics instead of instantaneous CSI, and mainly considered the effect of wireless channel uncertainty on the detection performance of the system. Specifically, we provided an optimal local sensor decision rule, as well as the specific formulations for different reception modes and modulations by taking a two-sensor distributed detection system as an example. The optimal decision rule at each local sensor amounts to an LRT, consisting of a statistic and the

Y. Yan et al. / Journal of the Franklin Institute 354 (2017) 530–555

551

corresponding threshold. The results show that the threshold of the optimal local sensor decision rule is coupled with each other, and is also related to the fusion rule of the fusion center. Several simulations with channel Rayleigh fading statistics for different reception modes and modulations were given. Several results show that the proposed method with the optimal local threshold in a closed form derives consistent result with the exhaustive searching method, and even provides slightly better performance than the exhaustive searching method in some cases. Furthermore, the proposed local decision rule in a closed-form solved by using iterative PBPO algorithm converges in one iteration while the computational complexity of the exhaustive searching is increasing with the number of gird size. The detection performance of the distributed system with coherent reception and BPSK modulation is better than that of the other cases. Appendix A. Proof of Theorem 1 For a two-sensor distributed detection system with coherent reception and OOK modulation, given linear fusion rule at the fusion center, the term Pðu0 ¼ 1∣yi ; ui ¼ 1Þ with i¼ 1 in (16) can be expressed as   P u0 ¼ 1∣y2 ; u1 ¼ 1 !  2

X pffiffiffi  i 1 π0 i π yi  1 Pd  Pf 4log jy2 ; u1 ¼ 1 ¼P π1 1 þ 2s2i i¼1  9 P y1 4T 1 ðy2 Þju1 ¼ 1Þ ðA:1Þ where T 1 ðy2 Þ is given in (34). And, similarly, we have Pðu0 ¼ 1∣y2 ; u1 ¼ 0Þ  9 P y1 4T 1 ðy2 Þju1 ¼ 0Þ:

ðA:2Þ

We can see that the above expressions are the function of random variable y2. And using the conditional pdf pðy1 ju1 ¼ 1Þ and pðy1 ju1 ¼ 0Þ, Eqs. (A.1) can be written as  P y1 4T 1 ðy2 Þju1 ¼ 1Þ  y2  Z þ1 pffiffiffiffiffi  12   ða1 y1 Þ2 2s1 2s 2 1 1 þ pffiffiffiffiffi ¼ e 2π a y e Q  a y 1 1 1 1 dy1 2π ð1 þ 2s21 Þ T 1 ðy2 Þ  2 2Þ    T2s1 ðy2 þ1 T 1 ðy2 Þ 1 ffi Q  a1 T 1 ðy2 Þ e 1 ¼Q ðA:3Þ þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s1 2s21 þ 1 where QðÞ is the complementary distribution function of the standard Gaussian. Similarly, equation (A.2) is  P y1 4T 1 ðy2 Þju1 ¼ 0Þ y2 Z þ1  12 1 2s p ffiffiffiffiffi e 1 dy1 ¼ 2π s1 T 1 ðy2 Þ  T 1 ðy2 Þ ¼Q : ðA:4Þ s1 With the above results, it is straightforward to derive the threshold π 0 A1 τ1OOK ¼ π 1 B1

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¼

π0

R þ1 

1 Pðu0 R þ1  π 1  1 Pðu0 Z þ1

 ¼ 1jy2 ; u1 ¼ 1Þ Pðu0 ¼ 1jy2 ; u1 ¼ 0Þ pðy2 jH 0 Þdy2  ¼ 1jy2 ; u1 ¼ 1Þ Pðu0 ¼ 1jy2 ; u1 ¼ 0Þ pðy2 jH 1 Þdy2

   T 1 ðy2 2 Þ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQ  a1 T 1 ðy2 Þ e 2s1 þ1 2s21 þ 1 1 pðy2 jH 0 Þdy2 2    T 1 ðy2 2 Þ  R þ1 1 ffiQ  a1 T 1 ðy2 Þ e 2s1 þ1 p y2 jH 1 Þdy2 : π 1  1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2s21 þ 1 2

¼ π0

ðA:5Þ

Similarly, we have     T 1 ðy2 1 Þ  1 ffiQ  a2 T 1 ðy1 Þ e 2s2 þ1 p y1 ∣H 0 dy1 π 0  1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2s2 þ 1 τ2OOK ¼ : 2 1Þ      T2s1 ðy2 þ1 R þ1 1 2 ffiQ  a2 T 1 ðy1 Þ e π 1  1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p y1 ∣H 1 dy1 2s22 þ 1 R þ1

2

ðA:6Þ

Appendix B. Proof of Theorem 2 For a two-sensor distributed target detection system with channel fading statistics, given the linear fusion rule at the fusion center, the term Pðu0 ¼ 1jy2 ; u1 ¼ þ 1Þ with i ¼ 1 is    P u0 ¼ 1 y 2 ; u1 ¼ þ 1 !  2 pffiffiffiffiffi X π0 ¼P jy2 ; u1 ¼ þ 1 2π ai yi 4log π1 i¼1     9 P y1 4T 2 y2 y2 ; u1 ¼ þ 1  2 2Þ    T2s2 ðy2 þ1 T 2 ðy2 Þ 1 1 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ¼Q ðB:1Þ Q  a1 T 2 ðy2 Þ e þ s1 2s21 þ 1 where T 2 ðy2 Þ is defined as (40). Similarly, we have    P u0 ¼ 1 y 2 ; u1 ¼  1 !  2 pffiffiffiffiffi X π0 2π ai yi 4log ¼P jy ; u1 ¼  1 π1 2 i¼1     9 P y1 4T 2 y2 y2 ; u1 ¼  1  y2  Z þ1 pffiffiffiffiffi  12 ða1 y1 Þ2   2s1 2s 2 1 1 pffiffiffiffiffi ¼ e 2π a y e Q a y 1 1 1 1 dy1 2π ð1 þ s21 Þ T 2 ðy2 Þ  2 2Þ    T2s2 ðy2 þ1 T 2 ðy2 Þ 1 ffi Q a1 T 2 ðy2 Þ e 1 : ¼Q  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s1 2s21 þ 1 Thus, the threshold of sensor 1 is π 0 A1 τ1BPSK ¼ π 1 B1

ðB:2Þ

Y. Yan et al. / Journal of the Franklin Institute 354 (2017) 530–555

¼

¼

π0

R þ1 

1 R þ1 π1  1

π0 π1



553

 Pðu0 ¼ 1jy2 ; u1 ¼ þ 1Þ Pðu0 ¼ 1jy2 ; u1 ¼  1Þ pðy2 jH 0 Þdy2  Pðu0 ¼ 1jy2 ; u1 ¼ þ 1Þ Pðu0 ¼ 1jy2 ; u1 ¼  1Þ pðy2 jH 1 Þdy2

R þ1



T 2 ðy2 Þ2 2s2 þ1 1

R þ1



T 2 ðy2 Þ2 2s2 þ1 1

1 e

  p y2 jH 0 dy2 

ðB:3Þ



e p y2 jH 1 dy2     where the fact Q  a1 T 2 ðy2 Þ ¼ 1 Q a1 T 2 ðy2 Þ is used. Similarly, we have π 0 A2 τ2BPSK ¼ π 1 B2 2  R þ1  T 2 ðy2 1 Þ  π 0  1 e 2s2 þ1 p y1 jH 0 dy1 ¼ : 2 1Þ   R þ1  T2s2 ðy2 þ1 π 1  1 e 2 p y1 jH 1 dy1 1

ðB:4Þ

Appendix C. Proof of Theorem 3 For a two-sensor distributed target detection system with known Rayleigh channel fading statistics and non-coherent reception at the fusion center, the term Pðu0 ¼ 1jz2 ; u1 ¼ 1Þ with i¼ 1 is Pðu0 ¼ 1jz2 ; u1 ¼ 1Þ !  2 X zi π0   4log ¼P jz2 ; u1 ¼ 1 2 2 π1 i ¼ 1 2si 1 þ 2si 9 Pðz1 4F 1 ðz2 Þjz2 ; u1 ¼ 1Þ ðC:1Þ h i

zi 2 2 where F 1 ðz2 Þ is F 1 ðzi Þ ¼ log ππ01  2s2 ð1þ2s for i¼ 2. From (41) and (42), 2 Þ 2sð3  iÞ 1 þ 2sð3  iÞ i

i

we can see that the conditional pdf is a piecewise function, thus, equation could be  the above  considered piecewise. For F 1 ðz2 ÞZ 0, i.e., z2 r t 2 ¼ log ππ01 2s22 1 þ 2s22 P2 1 P2 , we have d

Pðu0 ¼ 1jz2 ; u1 ¼ 1Þ Z þ1 z  12 1 1þ2s 1 dz ¼ e 1 2 F 1 ðz2 Þ 1 þ 2s1 ¼e



F 1 ðz2 Þ 1þ2s2 1

:

  For F 1 ðz2 Þo0, i.e., z2 4t 2 ¼ log ππ01 2s22 1 þ 2s22 P2 1 P2 , we have Pðu0 ¼ 1jz2 ; u1 ¼ 1Þ Z 0 Z ¼ 0dz1 þ F 1 ðz2 Þ

¼ 1:

0

d

þ1

f

ðC:2Þ

f

z

 12 1 1þ2s 1 dz e 1 1 þ 2s21

Similarly, we have Pðu0 ¼ 1jz2 ; u1 ¼ 0Þ !  2 X zi π0   4log ¼P jz2 ; u1 ¼ 0 2 2 π1 i ¼ 1 2si 1 þ 2si

ðC:3Þ

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¼

8 <



e

: 1;

F 1 ðz2 Þ 2s2 1

; z2 r t 2 z2 4t 2

With the above results, the threshold of sensor 1 can be written as π 0 A1 τ1OOK ¼ π 1 B1 R þ1 π 0 0 ½Pðu0 ¼ 1jz2 ; u1 ¼ 1Þ Pðu0 ¼ 1jz2 ; u1 ¼ 0Þpðz2 jH 0 Þdz2 ¼ R þ1 π 1 0 ½Pðu0 ¼ 1jz2 ; u1 ¼ 1Þ Pðu0 ¼ 1jz2 ; u1 ¼ 0Þpðz2 jH 1 Þdz2  F1 ðz2 Þ F ðz Þ   1 22 R t2  2s 2 þ1 π 0 0 e 1  e 2s1 pðz2 ∣H 0 Þdz2 ¼ :  F1 ðz2 Þ F ðz Þ   1 22 R t2  2s 2 þ1 2s 1 π1 0 e 1  e pðz2 ∣H 1 Þdz2 Similarly, the threshold of sensor 2 is π 0 A2 τ2OOK ¼ π 1 B2 R þ1 π 0 0 ½Pðu0 ¼ 1jz1 ; u2 ¼ 1Þ Pðu0 ¼ 1jz1 ; u2 ¼ 0Þpðz1 jH 0 Þdz1 ¼ R þ1 π 1 0 ½Pðu0 ¼ 1jz1 ; u2 ¼ 1Þ Pðu0 ¼ 1jz1 ; u2 ¼ 0Þpðz1 jH 1 Þdz1  F1 ðz1 Þ F ðz Þ   2  1 21 Rt π 0 01 e 2s2 þ1  e 2s2 pðz1 ∣H 0 Þdz1 ¼ :  F1 ðz1 Þ F ðz Þ   1 21 R t1  2s 2 þ1 2s 2 π1 0 e 2  e pðz1 ∣H 1 Þdz1

ðC:4Þ

ðC:5Þ

ðC:6Þ

References [1] K. Bouabdellah, H. Noureddine, S. Larbi, Using wireless sensor networks for reliable forest fires detection, Procedia Comput. Sci. 19 (2013) 794–801. [2] H. Alemdar, C. Ersoy, Wireless sensor networks for healthcare: a survey, Comput. Netw. 54 (15) (2010) 2688–2710. [3] R. Daher, R. Adve, A notion of diversity order in distributed radar networks, IEEE Trans. Aerosp. Electron. Syst. 46 (2) (2010) 818–831. [4] X. Sheng, Y.-H. Hu, Maximum likelihood multiple-source localization using acoustic energy measurements with wireless sensor networks, IEEE Trans. Signal Process. 53 (1) (2005) 44–53. [5] A.M. Aziz, Fuzzy track-to-track association and track fusion approach in distributed multisensor-multitarget multiple-attribute environment, Signal Process. 87 (6) (2007) 1474–1492. [6] Y. Wang, W. Fu, D.P. Agrawal, Gaussian versus uniform distribution for intrusion detection in wireless sensor networks, IEEE Trans. Parallel Distrib. Syst. 24 (2) (2013) 342–355. [7] R.R. Tenney, N.R. Sandell Jr, Detection with distributed sensors, IEEE Trans. Aerosp. Electron. Syst. 17 (1981) 501–510. [8] P.K. Varshney, Distributed Detection and Data Fusion, Springer-Verlag New York, Inc., 1996. [9] I. Bahceci, G. Al-Regib, Y. Altunbasak, Parallel distributed detection for wireless sensor networks: performance analysis and design, in: Proceedings of the Global Telecommunications Conference, 2005. GLOBECOM'05. IEEE, Vol. 4, IEEE, 2005, pp. 2420–2424. [10] H. Chen, B. Chen, P.K. Varshney, Further results on the optimality of the likelihood-ratio test for local sensor decision rules in the presence of nonideal channels, IEEE Trans. Inf. Theory 55 (2) (2009) 828–832. [11] B. Chen, P.K. Willett, On the optimality of the likelihood-ratio test for local sensor decision rules in the presence of non-ideal channels, IEEE Trans. Inf. Theory 51 (2) (2005) 693–699.

Y. Yan et al. / Journal of the Franklin Institute 354 (2017) 530–555

555

[12] B. Chen, L. Tong, P.K. Varshney, Channel aware distributed detection in wireless sensor networks, in: IEEE Signal Processing Mag, Citeseer, 2006. [13] Z. Chair, P. Varshney, Optimal data fusion in multiple sensor detection systems, IEEE Trans. Aerosp. Electron. Syst. 1 (1986) 98–101. [14] M. Guerriero, L. Svensson, P. Willett, Bayesian data fusion for distributed target detection in sensor networks, IEEE Trans. Signal Process. 58 (6) (2010) 3417–3421. [15] R. Niu, P.K. Varshney, Joint detection and localization in sensor networks based on local decisions, in: Proceedings of the IEEE Fortieth Asilomar Conference on Signals, Systems and Computers, 2006. ACSSC'06, 2006, pp. 525– 529. [16] R. Niu, P.K. Varshney, Q. Cheng, Distributed detection in a large wireless sensor network, Inf. Fusion 7 (4) (2006) 380–394. [17] R. Niu, P.K. Varshney, Performance analysis of distributed detection in a random sensor field, IEEE Trans. Signal Process. 56 (1) (2008) 339–349. [18] A.M. Aziz, A new multiple decisions fusion rule for targets detection in multiple sensors distributed detection systems with data fusion, Inf. Fusion 18 (2014) 175–186. [19] B. Chen, R. Jiang, T. Kasetkasem, P.K. Varshney, Channel aware decision fusion in wireless sensor networks, IEEE Trans. Signal Process. 52 (12) (2004) 3454–3458. [20] R. Niu, B. Chen, P.K. Varshney, Fusion of decisions transmitted over rayleigh fading channels in wireless sensor networks, IEEE Trans. Signal Process. 54 (3) (2006) 1018–1027. [21] J. Park, G. Shevlyakov, K. Kim, Distributed detection and fusion of weak signals in fading channels with nongaussian noises, Commun. Lett., IEEE 16 (2) (2012) 220–223. [22] R. Jiang, B. Chen, Fusion of censored decisions in wireless sensor networks, IEEE Trans. Wirel. Commun. 4 (6) (2005) 2668–2673. [23] S.C. Thomopoulos, R. Viswanathan, D.K. Bougoulias, Globally optimal computable distributed decision fusion, in: Conference Proceedings, 1987, pp. 1846–1847. [24] V.V. Veeravalli, J.-F. Chamberland, Detection in Sensor Networks, John Wiley & Sons, 2007. [25] E. Masazade, R. Rajagopalan, P.K. Varshney, C.K. Mohan, G.K. Sendur, M. Keskinoz, A multiobjective optimization approach to obtain decision thresholds for distributed detection in wireless sensor networks, IEEE Trans. Syst., Man, Cybern., Part B: Cybern. 40 (2) (2010) 444–457. [26] Z. Quan, S. Cui, A.H. Sayed, Optimal linear cooperation for spectrum sensing in cognitive radio networks, IEEE J. Sel. Top. Signal Process. 2 (1) (2008) 28–40. [27] Z. Quan, W.-K. Ma, S. Cui, A.H. Sayed, Optimal linear fusion for distributed detection via semidefinite programming, IEEE Trans. Signal Process. 58 (4) (2010) 2431–2436.