Hindawi Publishing Corporation International Journal of Distributed Sensor Networks Volume 2015, Article ID 987526, 9 pages http://dx.doi.org/10.1155/2015/987526
Research Article CFAR Decision Fusion Approaches in the Clustered Radar Sensor Networks Using LEACH and HEED Yaoyue Hu and Jing Liang University of Electronic Science and Technology of China, No. 2006, Xiyuan Avenue, West Hi-Tech Zone, Chengdu 611731, China Correspondence should be addressed to Jing Liang;
[email protected] Received 19 November 2014; Accepted 22 December 2014 Academic Editor: Tariq S. Durrani Copyright © 2015 Y. Hu and J. Liang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We propose two constant-false-alarm-rate (CFAR) decision fusion approaches, the low-SNR and likelihood-ratio-based decision fusion in the central limit theory (LLDFCLT) and high-SNR and likelihood-ratio-based decision fusion in Kaplan-Meier estimator (HLDFKE). They are based on the clustered RSN model which combines clustering structure, target detection model, and fusion scheme. We mainly apply the clustering performances by low energy adaptive clustering hierarchy (LEACH) and hybrid energyefficient distributed clustering approach (HEED) to RSN. Their CFAR detection performances in LLDFCLT and HLDFKE are analyzed and compared. Our analyses are verified through extensive simulations in different CFARs and various numbers of initial RSs and residual RSs in RSN. Monte Carlo simulations show that LLDFCLT can provide higher probability of detection (PD) than HLDFKE; and compared to LEACH, HEED not only prolongs the lifetime of ad hoc RSN but also improves target detection performances for different CFARs.
1. Introduction and Motivation A radar sensor network (RSN) is an independent system composed of multiple radars. Due to the outstanding features like flexibility of setting and dynamic management, RSN can be applied to various fields. One of the main applications of the RSN is target detection and tracking, especially in safety and military area such as homeland security, border inspection, and defense against terrorists. Besides target detection and tracking, the lifetime is also a fundamental factor considered for the applications of RSN, especially when low-cost and energy-constrained radars are used remotely from a power source. Traditional radars, on the other hand, usually require high power for outstanding target detection performances [1, 2]. Therefore, how to improve target detection performances for RSN while keeping as energy-efficient as possible is still an open research issue [3, 4]. Investigations on improving the target detection performances of RSN are very extensive. A maximum likelihood multitarget detection algorithm [5] to estimate the number of targets present in the sensing area and a diversity scheme [6]
to reduce the interference are proposed to improve multitarget detection performances of RSN. New waveform models [1, 7, 8] are developed to alleviate blind speed problem and eliminate interference. Additionally, RSNs based on impulse radio ultrawideband (UWB) are further investigated [9–11], since UWB communications [12] are robust to clutter and interference. In particular, J. Liang and Q. Liang in [9] have exhibited an approach by applying the short time Fourier transform to the received UWB radar waveform to achieve the detection of targets in foliage environment. However, all of the above papers have not considered node topology for either detection or power loss. Considering both the detection performance and energy constraint has been rarely discussed in the existing literature about RSNs. These solutions can be divided into two groups. One is power control algorithms [13, 14] and the other is a distributed scheduling scheme [15]. Clustering topologies can be used in RSN to save energy, since [16] shows that the node clustering approaches based on the information of geographical location perform better in sensor networks than those without clustering. Current node
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clustering approaches [17–20] are robust against node failures and hold the whole network remaining connected. Also, data aggregation techniques can be used to combine several correlated data signals into a smaller set of information that maintains the effective data of the original signals [17]. Hence, much less actual data needs to be transmitted from the cluster to the base station (BS). Among the existing studies, a hybrid protocol for efficient routing and comprehensive information retrieval (APTEEN) [18] and an energy-efficient deployment and cluster formation scheme (EEDCF) [19] are typically a centralized algorithm, whereas LEACH [17] and HEED [20] are distributed cases, which are more flexible and efficient. However, none of the above papers touched target detection performances in RSN. Therefore, all the above research restrictions motivate us to analyze the target detection performance when applying the low-cost clustering topologies to RSN and to find a clustering topology with a higher PD. However, we face one key challenge. The current decision fusion rules [21–25] may not be practical for RSN. Firstly, the transmitted information has to endure both channel fading and noise/interference, whereas optimal fusion [21, 22] and the blind adaptive decision fusion [23] have been derived without regard to the communication constraints, even though this assumption may be reasonable for some applications. Secondly, the local information of radar sensors (RSs), for example, PD, may be different among each other. A maximum ratio combining (MRC) fusion [24] and an equal gain combiner (EGC) fusion [24], which have been further studied in RSN for target detection under multihop transmission [25], have been obtained in the condition of identical local sensors. Finally, a CFAR in cluster heads (CHs) and base stations (BS) can obtain predictable and consistent performance. It is crucial to exploit the CFAR decision threshold, since the above work has not taken this into consideration. In this paper, we mainly applied LEACH and HEED to RSN. Their target detection performances in LLDFCLT and HLDFKE are also evaluated. The novelty and contributions of this paper are threefold. (1) We develop the clustered RSN model combining clustering structure, target detection model, and fusion scheme. This model, as far as we are aware, is the first one which accounts for both the energy consumption in clustering and detecting and detection performance (in terms of PD and CFAR) at the same time. (2) The PD in LLDFCLT of the whole network is formulated in the condition of CFAR. It not only can approach the Monte Carlo results, but also is higher than that of HLDFKE. (3) The detection performances in different node degree with constant size of the surveillance area (CSSA) and various sizes of the area with the fixed node degree (FND) are presented for the first time to the best of our knowledge. What is more, we also studied the impact of the cluster radius of HEED and the RSs reduction because of the energy consumption on the CFAR detection performance.
The rest of the paper is organized as follows. In Section 2, LEACH and HEED are briefly discussed. Section 3 elaborates models of clustered RSN and formulates the problem. Section 4 proposes the LOLCLT and HOLKE approaches. The energy consumption model is given in Section 5. Section 6 compares and analyzes performances in LEACH and HEED and LOLCLT and HOLKE. Finally, Section 7 draws the conclusion.
2. LEACH and HEED 2.1. LEACH. LEACH is an application-specific clustering protocol, which significantly improves networks’ lifetime (e.g., compared with static clustering), latency, and application-perceived quality. Applying LEACH to RSN, we assume that each RS is reachable in a single hop and the load distribution is uniform among all RSs. LEACH assigns a fixed probability 𝑃CH with which every RS elects itself as a cluster head (CH). To balance the energy dissipation, every RS becomes a CH only once during 1/𝑃CH rounds. In each round 𝑟, each RS elects itself as a CH based on a probability model which can be expressed as 𝛿𝑛
CH ≷ 𝑇 (𝑛, 𝑟), non-cluster-head
(1)
where 𝛿𝑛 produced by the 𝑛th RS is a random number between 0 and 1. 𝑇(𝑛, 𝑟) is given by 𝑃CH { , if 𝑛 ∈ 𝑈 (2) 𝑇 (𝑛, 𝑟) = { 1 − 𝑃CH × (𝑟 mod (1/𝑃CH )) 0, otherwise, { where 𝑈 is the set of RSs that were not elected as a CH in the last 1/𝑃CH rounds. 2.2. HEED. Compared with LEACH, HEED [4] can guarantee more uniform distribution of CHs and more efficient load balancing among the network that we will show in Section 5 considering the energy depletion for detection. HEED assigns a fixed cluster radius and uses the residual energy of RSs as the primary parameter to probabilistically elect temporary CHs (𝑛) denote the residual energy of the 𝑛th RS; its (TCHs). Let 𝐸re probability of electing a TCH is CH(𝑛) prob = max {𝐶prob ×
(𝑛) 𝐸re ,𝑝 }, 𝐸ini min
(3)
where 𝐶prob is the initial probability of electing a TCH, 𝐸ini are the initial energy of RSs and same for all 𝑛’s, and 𝑝min is a certain threshold to terminate the algorithm in a constant number of iterations, for example, 0.0001. We assume that each RS is able to select the appropriate power level to communicate with its CH. Consider the case when the distance between two TCHs, named 𝑢 and V, is less than the cluster radius; 𝑢 replaces V and becomes the final CH. This case is subject to the secondary parameter, the average minimum reachability power (AMRP), which is the mean
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3 exponent. According to the pass-loss model, the power of received signal reflected from the target is
Target
𝑃𝑟𝑘 =
NCH
BS
CH
Fusion center
Figure 1: A clustered RSN model with two fusion strategies for target detection.
of the minimum power levels required by all 𝑀 RSs within the cluster range to reach 𝑢. The minimum power level for communication between 𝑢 and its 𝑖th neighbor, MinPwr𝑖 , is proportional to the square of their distance, dist𝑖 , 1 ≤ 𝑖 ≤ 𝑀. The AMRP is derived as dist2𝑖 ∑𝑀 ∑𝑀 1 MinPwr𝑖 AMRP = = 1 . 𝑀 𝑀
(4)
3. The Clustered RSN Model and Problem Formation We consider a set of RSs deployed in large numbers over a rectangular field. Some assumptions about the properties of the network are as follows. (1) The RSs in the network are quasi-stationary and location aware; (2) all RSs have similar capabilities (processing/communication) and equal initial energy; (3) all RSs are left unattended after the deployment. Figure 1 depicts a clustered RSN structure with two fusion strategies for target detection. When LEACH or HEED is applied, the non-cluster-head RSs (NCHs) are responsible for detecting the targets and transmitting the decision results to the corresponding CHs, while CHs receive and fuse messages and transmit their own decisions to the BS, which makes the second fusion and the final decision. There is a single-hop path between NCHs and the CHs and CHs and BS. Due to the two fusion strategies, the model of the clustered RSN can be referred to as a two-cross-layer design. We make the assumption that RSN is parted to 𝑐 clusters, and each cluster has 𝑁𝑖 RSs except the CH, 1 ≤ 𝑖 ≤ 𝑐. 3.1. Detection Process. In the detection process, we model the wireless propagation of RSN under the pass-loss fading, which is given by 𝑃𝑟𝑗 =
𝑃𝑡𝑖 , 𝑙𝑖𝑗𝛼
(5)
where 𝑙𝑖𝑗 is the distance between the 𝑖th transmission RS with the transmission power 𝑃𝑡𝑖 and the 𝑗th receiving RS with the receiving power 𝑃𝑟𝑗 . 𝛼 is the radio frequency attenuation
𝑃𝑡𝑘 𝐺𝛿 , 𝑑𝑘2𝛼
(6)
where 𝑑𝑘 is the distance between the 𝑘th RS and the target. 𝑃𝑡𝑘 and 𝑃𝑟𝑘 are the transmission and receiving power of the 𝑘th RS, respectively. 𝐺 is the gain of the radar antenna and 𝛿 is the radar cross section. Each RS declares either “target absent” or “target present” based on the received data. Due to the above radar detection model, the two hypotheses 𝐻0 and 𝐻1 are under test: 𝐻0 : 𝑥𝑘 = 𝐴 𝑟𝑘 + 𝑛𝑘 , 𝐻1 : 𝑥𝑘 = 𝑛𝑘 ,
(7)
where 𝑥𝑘 is the echo signal amplitude received by the 𝑘th RS in the 𝑖th cluster, 1 ≤ 𝑘 ≤ 𝑁𝑖 . 𝑛𝑘 is additive Gaussian noise with zero mean and variance 𝜎2 . 𝐴 𝑟𝑘 = √𝐺𝛿𝐴 𝑡 /𝑑𝑘𝛼 is the signal amplitude echoing to the 𝑘th RS, 𝐴 𝑡 is the transmitted signal amplitude, and 𝑑𝑘 is the distance between the target and the 𝑘th RS. Assume the 𝑘th local RS to make a binary decision 𝑢𝑘 ∈ {+1, −1} with a probability of false alarm 𝑃𝑓𝑘 and detection 𝑃𝑑𝑘 , where 𝑃[𝑢𝑘 = −1 | 𝐻0 ] = 𝑃𝑓𝑘 , 𝑃[𝑢𝑘 = +1 | 𝐻1 ] = 𝑃𝑑𝑘 . The decision threshold 𝑇𝑘 in the 𝑘th local RS can be derived by 𝑃𝑓𝑘 = ∫
𝑥2 1 exp (− 𝑘2 ) 𝑑𝑥𝑘 , √2𝜋𝜎 2𝜎
∞
𝑇
∞
𝑃𝑑𝑘 = ∫ ∫
𝑇
2
(𝑥 − 𝐴 ) 1 exp (− 𝑘 2 𝑟𝑘 ) 𝑓 (𝐴 𝑟𝑘 ) 𝑑𝑥𝑘 𝑑𝐴 𝑟𝑘 , √2𝜋𝜎 2𝜎 (8)
where 𝑓(𝐴 𝑟𝑘 ) is the probability distribution function (PDF) of 𝐴 𝑟𝑘 . Based on the Neyman-Pearson rule, 𝑃𝑑𝑘 can be obtainable when 𝑃𝑓𝑘 is assigned to a fixed number 𝑃𝑓(𝑙) . We assume that the communication range of the local RSs is the circular with radius 𝑑𝑐 , which represents the amplitude of the transmitted signals sent to the corresponding CHs. The received signal at the 𝑖th CH from the 𝑘th RS is 𝑦𝑘𝑖 = ℎ𝑘𝑖 𝑢𝑘 + 𝑛𝑘𝑖 ,
(9)
where 𝑛𝑘𝑖 ∼ 𝑁(0, 𝜎2 ). The pass-loss channel attenuation coefficient ℎ𝑘𝑖 is ℎ𝑘𝑖 = 1 −
𝑑𝑘𝑖 , 𝑑𝑐
(10)
where 𝑑𝑘𝑖 is the distance between the 𝑖th CH and the 𝑘th RS.
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3.2. Problem Formation. Using the above detection model, we can obtain the optimal likelihood-ratio-based (OL) fusion statistic of CHs. That is 𝑁𝑖
Λ𝑖 = ∏ 𝑁𝑖
=∏
𝜎𝑖𝑐2 0
| 𝐻0 )
𝑖 𝑖 𝑃𝑑𝑘 𝑃 [𝑢𝑘 = +1 | 𝐻1 ] + (1 − 𝑃𝑑𝑘 ) 𝑃 [𝑢𝑘 = −1 | 𝐻1 ]
(𝑙) 𝑘=1 𝑃𝑓 𝑃 [𝑢𝑘
=
= +1 | 𝐻1 ] + (1 − 𝑃𝑓(𝑙) ) 𝑃 [𝑢𝑘 = −1 | 𝐻1 ]
𝑖 ∏ ((𝑃𝑑𝑘 𝑘=1
exp (
− (𝑦𝑘𝑖 − ℎ𝑘𝑖 ) 2𝜎2
+ (1 −
×
(𝑃𝑓(𝑙)
exp (
− (𝑦𝑘𝑖 + ℎ𝑘𝑖 ) 2𝜎2
− (𝑦𝑘𝑖 − ℎ𝑘𝑖 ) 2𝜎2
+ (1 −
𝐻1
))
𝐻0
1 𝑖 ∑ (𝐴 (2𝑃𝑑𝑘 − 1)) 𝑁𝑖 𝑘=1 𝑘𝑖
𝜎𝑖𝑐2 1 =
𝑖 𝑖 1 (𝜎2 ∑ 𝐵𝑘𝑖 + 4 ∑ (𝐴2𝑘𝑖 𝑃𝑑𝑘 (1 − 𝑃𝑑𝑘 ))) 2 𝑁𝑖 𝑘=1 𝑘=1
𝑁
𝑁
𝐻1
− (𝑦𝑘𝑖 + ℎ𝑘𝑖 ) 2𝜎2
𝑖 𝑃𝑑𝑘 ) exp (−2𝑦𝑘𝑖 ℎ𝑘𝑖 /𝜎2 )
(𝑙) 𝑘=1 𝑃𝑓
+ (1 − 𝑃𝑓(𝑙) ) exp (−2𝑦𝑘𝑖 ℎ𝑘𝑖 /𝜎2 )
2𝑃𝑓(𝑐) − 1
𝑐
∑ 𝐶𝑖
𝑐 𝑖=1 𝑐 𝑐 1 2 2 𝜎𝑏0 = 2 (𝜎 ∑ 𝐷𝑖 + 4𝑃𝑓(𝑐) (1 − 𝑃𝑓(𝑐) ) ∑ 𝐶𝑖2 ) 𝑐 𝑖=1 𝑖=1 1 𝑐 (𝑐) 𝜇𝑏1 = ∑ (𝐶𝑖 (2𝑃𝑑𝑖 − 1)) 𝑐 𝑖=1 𝑐 𝑐 1 2 𝜎𝑏1 = 2 (𝜎2 ∑ 𝐷𝑖 + 4 ∑ (𝐶𝑖2 𝑃𝑑𝑖(𝑐) (1 − 𝑃𝑑𝑖(𝑐) ))) 𝑐 𝑖=1 𝑖=1
)
+ (1 −
−1
)) )
When the CFAR of the CHs 𝑃𝑓(𝑐) is given, the decision
threshold and the probability of detection of the 𝑖th CH, 𝑇𝑖(𝑐) (𝑐) and 𝑃𝑑𝑖 , are derived as
, (11)
where Λ 𝑖 is the fusion statistic of the 𝑖th CH. Due to the constraints among the existing research on the decision fusion rules that we mentioned in Section 1, we faced two problems: one is how to derive the alternative fusion statistics to simplify formula (11) based on the pass-loss fading channel model while taking the different PD of RSs into account; the other problem is how to obtain the CFAR according to the fusion statistics both in CHs and in BS. We shall answer these questions in the following section.
4. CFAR Decision Fusion Approaches 4.1. LLDFCLT. If 𝜎2 → ∞, the OL fusion statistic of the 𝑖th CH Λ 𝑐𝑖 can be simplified as
𝜎 →∞
𝑁
𝜇𝑖𝑐1 =
𝜇𝑏0 =
2
𝑁𝑖 𝑃𝑖 𝑑𝑘
lim log Λ 𝑐𝑖 = Λ𝐿𝑐𝑖 = 2
𝑘=1 𝑁
𝑖 𝑖 1 = 2 (𝜎2 ∑ 𝐵𝑘𝑖 + 4𝑃𝑓(𝑙) (1 − 𝑃𝑓(𝑙) ) ∑ 𝐴2𝑘𝑖 ) 𝑁𝑖 𝑘=1 𝑘=1
Table 2: Mean and variance of Λ𝐿𝑏 under 𝐻0 and 𝐻1 with 𝑁𝑖 RSs in the 𝑖th cluster.
2
𝑃𝑓(𝑙) ) exp (
𝑁𝑖
𝑁𝑖
∑ 𝐴 𝑘𝑖
) 2
𝑖 𝑃𝑑𝑘 ) exp (
2𝑃𝑓(𝑙) − 1
𝑁
2
𝑁𝑖
=∏
𝜇𝑖𝑐0 =
𝐻0
𝑓 (𝑦𝑘𝑖 | 𝐻1 )
𝑖 𝑘=1 𝑓 (𝑦𝑘
Table 1: Mean and variance of Λ𝐿𝑐𝑖 under 𝐻0 and 𝐻1 with 𝑁𝑖 RSs in the 𝑖th cluster.
𝑁
1 𝑖 ∑ (𝑃 − 𝑃𝑓𝑘 ) ℎ𝑘𝑖 𝑦𝑘𝑖 . 𝑁𝑖 𝑘=1 𝑑𝑘
(12)
Obviously, Λ𝐿𝑐𝑖 is the sums of independent and identically distributed (i.i.d.) random variables whose PDF can be approximately obtained through the central limit theorem (CLT). In order to use the CLT, the first and second statistics of Λ𝐿𝑐𝑖 , are derived and summarized in Table 1, where 𝐴 𝑘𝑖 = (ℎ𝑘𝑖 )2 (𝑃𝑑𝑘 − 𝑃𝑓(𝑙) ) and 𝐵𝑘𝑖 = (ℎ𝑘𝑖 (𝑃𝑑𝑘 − 𝑃𝑓(𝑙) ))2 .
𝑇𝑖(𝑐) = 𝜎𝑖𝑐0 𝑄−1 (𝑃𝑓(𝑐) ) + 𝜇𝑖𝑐0 , (𝑐) = 𝑄( 𝑃𝑑𝑖
𝑇𝑖(𝑐) − 𝜇𝑖𝑐1 𝜎𝑖𝑐1
(13) ).
The 𝑖th CH also makes a binary decision 𝑢𝑖 ∈ {+1, −1}. Assuming that the communication range of CHs is the circular with radius 𝑑𝑏 , which represents the amplitude of the transmitted signals sent to the BS, the fusion statistic of BS is 1 𝑐 (𝑐) Λ𝐿𝑏 = ∑ (𝑃𝑑𝑖 − 𝑃𝑓(𝑐) ) ℎ𝑖𝑐 𝑦𝑖𝑐 , 𝑐 𝑖=1
(14)
where 𝑦𝑖𝑐 is the received signal from the 𝑖th CH and ℎ𝑖𝑐 = 1 − 𝑑𝑖𝑐 /𝑑𝑏 and 𝑑𝑖𝑐 is the distance between the 𝑖th CH and BS. The first and second statistics of Λ𝐿𝑏 are derived and summarized (𝑐) (𝑐) − 𝑃𝑓(𝑐) ) and 𝐷𝑖 = (ℎ𝑖𝑐 (𝑃𝑑𝑖 − in Table 2, where 𝐶𝑖 = (ℎ𝑖𝑐 )2 (𝑃𝑑𝑖 𝑃𝑓(𝑐) ))2 .
Similarly, when the CFAR of the RSN 𝑃𝑓(𝑏) is given, the decision threshold of the BS and the probability of detection of the RSN, namely, 𝑇(𝑏) and 𝑃𝑑(𝑏) , are derived as 𝑇(𝑏) = 𝜎𝑏0 𝑄−1 (𝑃𝑓(𝑏) ) + 𝜇𝑏0 , 𝑃𝑑(𝑏) = 𝑄 (
𝑇(𝑏) − 𝜇𝑏1 𝜎𝑏1
(15) ).
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4.2. HLDFKE. If 𝜎2 → 0, the OL fusion statistic of the 𝑖th CH Λ 𝑐𝑖 can be simplified as Λ𝐻𝑐𝑖 = lim log Λ 𝑐𝑖 2 𝜎 →0
=
∑
log
sign(𝑦𝑘𝑖 )=1
+
𝑃𝑑𝑘
𝑃𝑓(𝑙)
(16)
1 − 𝑃𝑑𝑘 log . ∑ 1 − 𝑃𝑓(𝑙) sign(𝑦𝑘𝑖 )=−1
𝐹 (𝑡) = 𝑃 {𝑆 > 𝑡} = 𝑃 {𝑆 > 𝑡, 𝑆 > 𝑡𝑢 } = 𝑃 {𝑆 > 𝑡 | 𝑆 > 𝑡𝑢 } 𝑃 {𝑆 > 𝑡𝑢 }
= 𝑃 {𝑆 > 𝑡 | 𝑆 > 𝑡𝑢 } 𝑃 {𝑆 > 𝑡𝑢 | 𝑆 > 𝑡𝑢−1 } ⋅ ⋅ ⋅ 𝑃 {𝑆 > 𝑡1 | 𝑆 > 𝑡0 } 𝑃 {𝑆 > 𝑡0 } 𝑢
≈ ∏𝑃 {𝑆 > 𝑡𝑗 | 𝑆 > 𝑡𝑗−1 }
𝑗=1
𝑑𝑗
(17) where 𝑑𝑗 denotes the number of 𝑆 samples more than 𝑡𝑗−1 ; 𝑥𝑗 denotes the number of 𝑆 samples more than 𝑡𝑗−1 and not less than 𝑡𝑗 . The required threshold 𝑇𝑐 is 𝐹 (𝑇𝑐 ) = 𝑃𝑓(𝑐) .
(18)
Suppose that 𝑇𝑐𝑖 is the decision threshold of the 𝑖th CH; its PD is ∞
𝑃𝑑𝑖 = ∫ 𝑓 (Λ𝐻𝑐𝑖 | 𝐻1 ) 𝑑Λ𝐻𝑐𝑖 . 𝑇𝑐𝑖
(19)
Then the fusion statistic of BS is Λ𝐻𝑏 =
∑ sign(𝑦𝑖𝑐 )=1
log
(21)
where 𝑑𝑜 = √𝜖𝑓𝑠 /𝜖𝑚𝑝 , and to receive this message, the radio expends 𝐸𝑅𝑥 (𝑙) = 𝐸𝑅𝑥-elec (𝑙) = 𝑙𝐸elec .
(22)
Presumably the distance to the CH is small, so the energy consumption follows the Friss free-space model (power loss). Then the energy consumed by a NCH during one detection 𝐸NCH is (23)
where 𝐸𝑇 det and 𝐸𝑅 det are energy for transmitting and 2 receiving detection signals, 𝐸𝑇 det ∝ 𝐸𝑅 det /𝑑to Ta , and 𝑑to Ta and 𝑑to CH are the distance from the RS to the target and the CH, respectively. The energy consumed by a CH within a cluster of 𝑁 NCHs is (24)
where 𝐸DA is the energy for data aggregation and 𝐸𝑇𝑥 depends on the distance between the CHs and the BS.
,
(𝑐)
𝑙𝐸elec + 𝑙𝜖𝑓𝑠 𝑑2 , 𝑑 < 𝑑𝑜 𝑙𝐸elec + 𝑙𝜖𝑚𝑝 𝑑4 , 𝑑 ≥ 𝑑𝑜 ,
𝐸CH = 𝑁 (𝑙𝐸elec + 𝑙𝐸DA ) + 𝐸𝑇𝑥 ,
𝑗=1
≈∏
={
2 𝐸NCH = 𝐸𝑇 det + 𝐸𝑅 det + 𝑙𝐸elec + 𝑙𝜖𝑓𝑠 𝑑to CH ,
= 𝑃 {𝑆 > 𝑡 | 𝑆 > 𝑡𝑢 } 𝑃 {𝑆 > 𝑡𝑢 | 𝑆 > 𝑡𝑢−1 } 𝑃 {𝑆 > 𝑡𝑢−1 }
𝑑𝑗 − 𝑥𝑗
The energy is primarily consumed for detection and data transmission. We implement the free space and the multipath fading channel models. The transmitter dissipates energy to run the radio electronics and the power amplifier, and the receiver dissipates energy to run the radio electronics. To transmit an l-bit detection or fusion message to a distance 𝑑, the radio expends 𝐸𝑇𝑥 (𝑙, 𝑑) = 𝐸𝑇𝑥-elec (𝑙) + 𝐸𝑇𝑥-amp (𝑙, 𝑑)
In order to obtain the CFAR for the whole network, the decision threshold based on the empirical cumulative distribution function (ECDF) of Λ𝐻𝑐𝑖 under 𝐻0 is estimated by Kaplan-Meier estimator (KE) from the fusion statistic data samples for 𝐻0 during every clustering process. Here, we briefly summarize the KE. Suppose 𝑡0 < 𝑡1 < ⋅ ⋅ ⋅ < 𝑡V are preestablished threshold values, and 𝑡0 is designed to satisfy the equation 𝑃{𝑆 > 𝑡0 } = 1, where 𝑆 is the fusion statistic under 𝐻0 . For 𝑡𝑢−1 ≤ 𝑡 ≤ 𝑡𝑢 , the ECDF of 𝑆 can be derived by
𝑢
5. Radio Energy Consumption Model
(𝑐) 𝑃𝑑𝑖 𝑃𝑓(𝑏)
+
∑ sign(𝑦𝑖𝑐 )=−1
log
(𝑐) 1 − 𝑃𝑑𝑘 . 1 − 𝑃𝑓(𝑏)
(20)
We could easily acquire the decision threshold by secondly using KE.
6. Performances Evaluation In this section, we present and analyze the detection performances of the three groups, LEACH and HEED, LLDFCLT and HLDFKE, and CSSA and FND, respectively. Specific simulation setup is listed as follows. (1) Simulation parameters (SPs) shown in Table 3 are for the first and second groups, and some SPs of radio energy consumption model are similar to those in [17]. Here, we briefly describe them. We assume that 100 RSs with the same initial energy, 0.5 J/battery, are uniformly dispersed into a square field with dimensions 100 m × 100 m; the CFARs of NCHs and CHs, given as 𝑃𝑓(𝑙) and 𝑃𝑓(𝑐) , are set as 0.05 and 0.01, respectively; the CFAR of BS 𝑃𝑓(𝑏) is fixed as 10−3 except Figures 4 and 7(b).
(2) As for the last group, we only changed the number of initial RSs and network grid in Figure 8 and the cluster radius of HEED (from 12 m to 40 m) in Figure 9. For example, if the number of initial RSs is
6
International Journal of Distributed Sensor Networks 25
Table 3: Simulation parameters.
Network
HEED
Energy model
Parameter Network grid BS Initial energy Initial number of RSs Cluster radius 𝑝min 𝐸elec 𝜖fs 𝜖mp 𝐸DA 𝐸Tdet Data packet size Broadcast packet size NCHs: 𝑃𝑓(𝑙)
Value From (0, 0) to (100, 100) At (50, 175) 0.5 J/battery 100 25 m 10−4 50 nJ/bit 10 pJ/bit/m2 0.0013 pJ/bit/m4 5 J/bit/signal 12.64 uJ/signal 10 bytes 5 bytes 0.05
BS: 𝑃𝑓(𝑏)
10−3
CHs: 𝑃𝑓(𝑐)
CFAR
0.01
10
5
0 −2
−1
0
1
2 3 SNR (dB)
4
5
6
LEACH HEED
simulation and by numerical approximation using LLDFCLT. While some discrepancy exists, approximations using LLDFCLT match relatively well to the corresponding simulation results. Application of CLT also allows a more intuitive explanation and analysis. From Stein’s lemma [26], the relative entropy (Kullback-Leibler distance) between the two distributions under test is directly related to the detection performance in an asymptotic regime. The relative entropy between two Gaussian distributions can be presented as
1 0.9 0.8 0.7 0.6 PD
15
Figure 3: Kullback-Leibler distance (relative entropy) between the two hypotheses for both LEACH and HEED in LLDFCLT.
106 times
Monte Carlo simulations
20
Deflection
Type
0.5 0.4 0.3
2
2 2 𝜎1 (𝜎0 − 𝜎1 ) + (𝜇0 − 𝜇1 ) + . 𝐷 (𝑃1 ‖ 𝑃2 ) = log 𝜎0 2𝜎12
0.2 0.1 0 −2
−1
0
1
2 3 SNR (dB)
Monte Carlo for LEACH Monte Carlo for HEED
4
5
6
Approximation for LEACH Approximation for HEED
Figure 2: The PD of LEACH and HEED in LLDFCLT.
200, the dimensions are 100√2 m × 100√2 m in FND but keep constant 100 m × 100 m in CSSA. Note that the distribution of RSs is still uniform and that other parameters’ values remain the same with Table 3. (3) All of the ROC curves except Figure 5 are generated using 106 Monte Carlo runs. First of all, we will analyze CFAR detection performances of LEACH and HEED in LLDFCLT. 6.1. CFAR Detection Performance Analysis of LEACH and HEED in LLDFCLT. Figure 2 presents the receiver operating characteristic (ROC) curves obtained both by Monte Carlo
(25)
We can therefore get the asymptotic relative entropy as a function of SNR for LLDFCLT of both LEACH and HEED by plugging in the corresponding mean and variance from Table 2. Figure 3 shows the results for both LEACH and HEED for the same parameter setting. Both Figures 2 and 3 illustrate that HEED has a better CFAR detection performance than LEACH. To better understand the performance differences as various CFARs of the system, we change the value of previous 𝑃𝑓(𝑏) shown in Table 3 and plot the PD of RSN in LEACH and HEED versus different CFARs in Figure 4. Figure 4 shows that, at a fixed SNR, the PD of both LEACH and HEED is increasing with the rise of CFAR, whereas HEED has a better improved PD than that of LEACH. The lifetime of RSN in LEACH and HEED is shown in Figure 5. The first RS dies in LEACH ahead of 50 intervals. The last RS dies during the 2860th interval in HEED, which extends 14.86 percent of lifetime. Therefore, HEED can reduce more energy dissipation and prolong the lifetime of RSN compared with LEACH. Based on Figure 5, we also want to know the PD of RSN in LEACH and HEED when the number of RSs is decreasing
International Journal of Distributed Sensor Networks
7 1
1
0.9
0.9
0.8 0.7
0.7
0.6
0.6
0.5
PD
PD
0.8
0.4
0.5
0.3
0.4
0.2
0.3
0.1
0.2 10−5
10−4
10−2
10−3
10−1
100
0 −2
0
Pf LEACH (SNR = 2 dB) HEED (SNR = 2 dB)
LEACH (SNR = 4 dB) HEED (SNR = 4 dB)
Figure 4: The PD of LEACH and HEED in LLDFCLT with constant SNRs versus different CFARs. 100
4 SNR (dB)
NRR = 90–75 (LEACH) NRR = 90–75 (HEED)
6
8
10
NRR = 60–45 (LEACH) NRR = 60–45 (HEED)
Figure 6: The PD of RSN in LEACH and HEED when NRR is in the range between 90 and 75 and between 60 and 45.
fading while LLDFCLT holds almost complete knowledge;
90 The number of resident RSs (NRR)
2
80
(2) different with LLDFCLT, the PD of HEED is higher than that of LEACH when SNR is more than 2 dB.
70 60 50 40 30 20 10 0
0
500
1000 1500 2000 Maximum intervals
2500
3000
LEACH HEED
Figure 5: The lifetime of RSN in LEACH and HEED.
because of the energy consumption, which is presented in Figure 6. Figure 6 shows that the PD of RSN both in LEACH and in HEED is declined when the number of residual RSs (NRR) is diminishing, but when SNR is more than 4 dB, HEED with the number of residual RSs between 60 and 45 can have an approximate PD to LEACH with the number of residual RSs between 90 and 75. 6.2. The Comparison between LLDFCLT and HLDFKE. Figure 7 gives the CFAR detection performance of LEACH and HEED in LLDFCLT and HLDFKE. Figure 7 shows that (1) for both LEACH and HEED, HOLKE offers lower PD than LLDFCLT, since HLDFKE ignores the channel
6.3. Detection Performances in CSSA and FND versus Different Cluster Radius of HEED. Since LLDFCLT has a better CFAR detection performance than HLDFKE, we studied the impact of the cluster radius of HEED, CSSA, and FND on PD of the whole network, as observed in Figures 8 and 9, respectively. From Figure 9, we can observe that the PD of HEED degrades as the cluster radius increases. The conclusion from Figure 8 is as follows. (1) Whether in CSSA or in FND, the PD of both LEACH and HEED can be enhanced by the increasing number of initial RSs but will remain constant from a certain number of initial RSs. (2) CSSA makes a distinct improvement of PD compared with FND.
7. Conclusion In this paper, we propose two CFAR detection approaches based on the clustered RSN model, namely, LLDFCLT and HLDFKE, which combine clustering structure, target detection model, and fusion schemes. We compare the clustering performances of LEACH and HEED, and the target detection performances in both LLDFCLT and HLDFKE are also analyzed and compared. We demonstrate that (1) LLDFCLT outperforms HLDFKE; (2) compared with LEACH, HEED not only prolongs the lifetime of the network but provides better target detection performances for different CFARs and
International Journal of Distributed Sensor Networks 1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6 PD
PD
8
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0 −2
−1
0
1
2 3 SNR (dB)
HLDFKE + LEACH HLDFKE + HEED
4
5
6
0
0
1
LLDFCLT + LEACH LLDFCLT + HEED
2
5
6
LLDFCLT + LEACH LLDFCLT + HEED
HLDFKE + LEACH HLDFKE + HEED
(a)
4
3 SNR (dB)
(b)
Figure 7: The PD of LEACH and HEED in LLDFCLT and HLDFKE. (a) 𝑃𝑓(𝑏) = 10−3 and (b) 𝑃𝑓(𝑏) = 10−4 .
1
1
0.9
0.8
0.8
0.7
0.7
PD
PD
0.9
0.6
0.6
0.5
0.5
0.4 80
100 120 140 160 180 200 220 240 260 280 300 The number of initial RSs FND for LEACH FND for HEED
CSSA for LEACH CSSA for HEED
Figure 8: The PD of RSN in LEACH and HEED and CSSA and FND.
0.4
12
16
20
24
28
32
36
40
The cluster radius of HEED (m) SNR = 1 dB SNR = 2 dB SNR = 3 dB
Figure 9: The PD of RSN in HEED versus different cluster radius.
entire SNR values in LLDFCLT and for moderate-tohigh-SNR values in HLDFKE; (3) the detection performance can be improved by the increasing number of initial RSs but will remain constant from a certain number of initial RSs, while the less the number of residual RSs in RSN or the larger the cluster radius of HEED, the worse the detection performance at the same SNR. Accordingly, the HEED in LLDFCLT approach has the robust CFAR detection performances. We can apply it to RSN to improve the PD and also keep energy efficiency. In future
work, we may investigate multitarget detection performance of clustered RSN and multihop clustering methods for RSN.
Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments This work was supported by the National Natural Science Foundation of China Project no. 61102140, Doctoral Fund of
International Journal of Distributed Sensor Networks Ministry of Education of China Project no. 20110185120003, and the Fundamental Research Funds for the Central Universities Project no. ZYGX2012J015.
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