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Graphical deployment strategies in radar sensor networks (RSN) for target detection
EURASIP Journal on Wireless Communications and Networkingvolume 2013, Article number: 55 (2013)
Abstract
Nodes deployment is a major challenge to a successful implementation of radar sensor network (RSN). The goal of the deployment is to ensure that the target can achieve expected detection performance with highenergy efficiency. In this article, two deployment strategies, named Hexagonal Deployment Strategy (HDS) and Diamond Deployment Strategy (DDS), are proposed to solve this problem. Each Radar Sensor (RS), separately, obtains probability of target detection depending on the position of RS according to the deployment strategies. To appraise these two deployment strategies, two decision fusion rules are derived over passloss fading channel in multihop RSN. We combine these two decision fusion rules with unfixed local detection performance. Simulations results show that given a finite number of RSs, our proposed strategies are far superior to Random Deployment Strategy in terms of detection probability and energy consumption to satisfy detection and false alarm requirements. The DDS achieves higher probability of detection and consumption fewer energy than HDS, no matter in decision fusion rules with Binary Transmission (BT) or without Binary Transmission (NBT). The BT fusion rules performs better than NBT and the number of RS needed for expected detection performance is not the more the better.
1 Introduction
Radar Sensor Networks (RSN) have attracted growing interest in various applications. They can monitor a large area and observe targets from many different angles. These networks will be included in the tactical combat systems that are deployed on airborne, surface, and subsurface unmanned vehicles in order to protect critical infrastructure from terrorist activities [1–3]. In a resourceconstrained RSN, a deployment strategy is used to ensure that targets in the sensor field can be detected properly and the energy consumed should be as little as possible.
For target detection problem in RSN, Ly and Liang [4] exhibited a diversity scheme to improve detection performance of RSN in the presence of strong interference, especially clutters and noise, and then proposed a maximum likelihood multitarget detection algorithm to estimate the possible number of targets in a surveillance area in RSNs [5]. Deng [6] introduced a distributed RSNs systems for effective target detection and the detection performance of this system improves dramatically with the increase of the number of radar sensors (RSs) in the system. Shu and Liang [7] studied the decision fusion rules of multiple fluctuating targets in multiradar sensor networks under multihop transmission. They all discussed the detection performance and considered the information integration in RSN. However, none of the above papers touched deployment strategy in RSN, which is practical and useful in RSN applications.
RSN is highly related to Wireless Sensor Networks (WSN) and many deployment strategies have been proposed in recent years in WSN. Yu et al. [8] and Luo et al. [9] applied grid to help deploy the sensor nodes according to detection features of the sensors. Xu et al. [10] and Mageid and Ramadan [11] investigated overall field coverage problems to deploy WSNs. Nevertheless all the above deployment strategies or placement algorithms are not considering the information integration, which is necessary in an RSN. Lin et al. [12] researched on the decision fusion rules for a WSN, but did not consider the deployment strategy of sensors. Aitsaadi et al. [13] derived a new WSN deployment strategy named multiobjective deployment algorithm based on evolutionary and neighborhood search algorithms. Kapnadak and Coyle [14] and Gogu et al. [15] determined the optimal spatial node densities for target detection. Xu et al. [16] and Ababnah and Natarajan [17] modeled the deployment based on the quantitative analysis of connectivity and network lifetime based on the collaborative detection model, respectively. However, most of these investigations have same assumption that the information sent from the sensors is perfectly recovered at the fusion center. This is not realistic for RSN as the transmitted information suffers channel distortions such as pathloss, lognormal, and multipath fading. Thus, these deployment strategies are not suitable for RSN and new schemes are necessary.
In this study, we propose two graphical deployment strategies, namely, Hexagonal Deployment Strategy (HDS) and Diamond Deployment Strategy (DDS), to realize target detection with satisfying probability of false alarm and probability of target detection in RSN. In RSN, RSs send signals out and get echoes back for targets detection. When detecting targets and transmitting information, the RSs receive signals based on random channel. The target detection probability of each RS is obtained according to the position of the RS, independently. Based on the passloss fading channel environment, two fusion rules with Binary Transmission (BT) and without Binary Transmission (NBT) are derived to evaluate the performance of the two graphical deployment strategies in a multihop RSN. For better evaluating the two deployment strategies, the RS number and energy consumption for expected detection performance are researched. These two deployment strategies are applied to resourceconstrained RSN for reducing resource consumption and improving RSN performance in target detection at the same time. They are suitable for monitoring area of rectangular or area having the approximate shape of a rectangular.
The remainder of the article is as follows. Section 2 introduces the principles of deployment we use in this RSN and proposes the fading channel model and detection rules. Section 3 depicts deployment methods and processes with a finite number of RSs. Section 4 elaborates two decision fusion rules. Based on the simulation result, Section 5 analyzes performance evaluation of deployment methods. Finally, Section 6 draws the conclusion.
2 Design principles and channel model
Assume that there is a rectangular area S(L×W) under surveillance, where L and W are the length and width, respectively. N RSs will be deployed to detect targets within this monitoring area. N1 RSs individually radiates signals, receives echoes, and makes local decision whether there are targets in the monitoring area or not. These N1 RSs also transmit their independent decision to the fusion center (the remaining one node) through a number of relay nodes (selected from these N1 sensors). The fusion center is placed to collect data and make a final decision. Our purpose is to investigate how to deploy these RSs with proper fusion rules so that satisfying detection performance and energy efficiency can be achieved.
In RSN, radiofrequency (RF) signals have unique attenuation characteristic, passloss model. Assume that the transmission power is P_{ t } and the receiving power is P_{ r }, the model on the ground is given by
where l is the distance between transmission node and receiving node and α is the RF attenuation exponent. Due to multipath and other interference effects, α is typically in the range of 2 to 5 [18].
Each RS transmits a known waveform and receives the echoes from targets, independently. According to the passloss model, the power of received echo signal is given by
where d is the distance between RS and target, G is the gain of radar antenna, and δ is radar cross section.
There are two hypotheses under test for each RS that either having a target (H_{1}) or having no target (H_{0}). Due to the above radar detection model, the two hypotheses H_{0} and H_{1} under test can be given by
where x_{ k }is the echo signal amplitude received by the k th sensor, k = 1,2,…,N  1, n_{ k }is additive Gaussian noise with zero mean and variance σ^{2} and ${A}_{\mathit{\text{rk}}}=\frac{\sqrt{\mathrm{G\delta}}{A}_{t}}{{d}_{k}^{\alpha}}$ is the signal amplitude echoing to the k th sensor, A_{ t }is transmitted signal amplitude, d_{ k }is the distance between target and the k th sensor.
Assume that the targets emerge in the rectangular area homogeneously that the targets obey uniform distribution of the rectangular. According to the relationship between d and A_{ r k }, we know our A_{ r k }is a random variable unlike traditional test. For a fixed position RS, the probability density function of A_{ r k }can be written as f(A_{ r k }) which is related to the position of the RS. Assume that the k th RS makes a binary decision u_{ k }∈{+1,1}, ${P}_{\mathit{\text{dk}}}=P\phantom{\rule{0.25em}{0ex}}[{u}_{k}^{0}=+1{H}_{1}]$ represent probability of detection, ${P}_{\mathit{\text{fk}}}=P\phantom{\rule{0.25em}{0ex}}[{u}_{k}^{0}=+1{H}_{0}]$ represent probability of false alarm. When there is just one target in our monitoring area, false alarm probability P_{ f k }and detection probability P_{ d k }can be derived by
where T can be get according to the relationship between P_{ f k }and T when the given P_{ f k }, T_{1k}, and T_{2k}are the minimum and maximum signal amplitude echoing to the k th sensor. Obviously, the P_{ d k }is related to the position of the RS. When there are M targets in the monitoring rectangular area, considering the influence among targets, we define the k th local false alarm and detection probabilities as ${P}_{f{k}_{M}}=1{(1{P}_{\mathit{\text{fk}}})}^{\frac{M}{4}+\frac{3}{4}}$ and ${P}_{d{k}_{M}}={\left({P}_{\mathit{\text{dk}}}\right)}^{\frac{M}{4}+\frac{3}{4}}$.
3 Graphical deployment strategies
In practice, we can usually regard the monitoring area as a rectangle for target detection. In this section, the two deployment strategies for detecting a rectangular region are proposed as follows.
3.1 HDS
The HDS is a strategy that place finite RSs to form mutually mosaic hexagons in the monitoring plane area. The following processes are taken.

1.
Take the vertex K of bottom left margin of this rectangle as a starting point and R as the length of each segment, dividing equally the rectangle hemline. Starting with these equal division points, make rays with 60° and 120° until they intersect with boundary of this rectangle.

2.
Take K as a starting node and $\frac{\sqrt{3}}{2}R$ as the length of each segment, dividing equally the rectangle left boundary. Starting with these equal division nodes, make rays with 60° and rays paralleling to the hemline of this rectangle until they intersect with boundary of this rectangle.

3.
Now we need to select the proper points to place RSs there to form hexagons. First, pick out the points in the rectangle from these crossing points obtained according to above process. Second, in each odd row, respectively, taking the first point as the starting point, pick out every third point and then taking the third point as the starting point, pick out every third point. Third, in each even row, respectively, taking the first point as the starting point, pick out every third point and then taking the second point as the starting point, pick out every third point.

4.
Finally, select the node which is closest to the geometric center of this rectangle to place a fusion center of RSN from nodes composing these hexagons and nodes in the center of these hexagons.
The number of sensors in the RSNs monitoring rectangular area for HDS is
where N_{1} is a approximate number of sensors rather than a accurate one, [∗] is an integer of not less than ∗.
After the above placing process, a whole nodes’ placement diagram is completed. Here, we use a 400×300 m^{2} area (notice that the square is a special case of rectangle) as an example to exhibit the HDS. Figure 1 shows the graphical deployment of N_{1}=30 nodes in a 400×300 m^{2} area using HDS.
3.2 DDS
The DDS is a strategy that place finite RSs to form mutually mosaic diamonds in the monitoring plane area. The following processes are used to deploy sensors in DDS.

1.
Take the vertex K of bottom left margin of this rectangle as a starting point and R as the length of each segment, dividing equally the rectangle bottom boundary. Starting with these equal division points, make rays with 45° and 135° until they intersect with boundary of this rectangle.

2.
Take K as a starting point and $\frac{\sqrt{3}}{2}R$ as the length of each segment, dividing equally the rectangle left boundary. Starting with these equal division points, make rays with 45° and rays paralleling to the hemline of this rectangle until they intersect with boundary of this rectangle.

3.
Select the nodes in the rectangle from these crossing nodes to place RSs there to form diamonds.

4.
Finally, select the node which is closest to the geometric center of this rectangle to place a fusion center of RSN from nodes composing these diamonds and nodes in the center of these diamonds.
The number of sensors in the RSNs monitoring rectangular area for DDS is
where the N_{2} is not a accurate one, but a approximate number of sensors instead.
After the above placing process, a whole array nodes diagram is completed. Here, we use a 400 × 300 m^{2} area as example to exhibit the DDS strategy. Figure 2 shows the graphical deployment of N_{2} = 30 nodes in a 300 × 400 m^{2} area using DDS.
4 Decision fusion rules and energy analysis in multihop RSN
N1 RSs transmit a known signal and collect data generated according to the echo. They make local decisions (their independent decisions) according to these data and then transmit these decisions to a fusion center through several relay nodes over fading and noisy channels. The relay nodes are selected following a certain routing protocol. Each relay node sends the decision from its source node to the next node until it reaches the fusion center. The fusion center tries to decide whether or not to have a target in the monitoring area based on the received information. For the significantly improving of detection performance, the fusion center uses likelihood ratio to fuse each sensor’s local decision. In this section, we will describe two fusion rules based on different relay rules.
4.1 Decision fusion rules with BT
In this decision fusion rule, each relay node tries to retrieve the decision sent from its source node in spite of fading and noise distortion. These relay nodes make a binary decision when receiving signals. Assume that all the channels are independent of each other and each of them can be modeled as a pathloss channel. Noise in all channels are Gaussian with zero mean and variance σ^{2} and are independent of each other.
The signal amplitude that every RS sends for detection is A_{ a }and for relay is A_{ b }. Assume that M_{ k }denotes the number of relay nodes between the k th local RS and the fusion center, with k = 1,2,…,N1. The ${h}_{k}^{i}$ is the corresponding channel gain and i = 0,1,…,M_{ k }is the hop index. The process of target detection in BT is described below.

1.
Every RS sends signal with amplitude A _{ a }out for detection and receives echo from target, independently.

2.
According to the echoes, each RS individually makes a binary decision (local decision) : ${u}_{k}^{0}=+1$ is made if H _{1} is decided, and ${u}_{k}^{0}=1$ is made otherwise. They each sends signal ${v}_{k}^{0}={A}_{b}{u}_{k}^{0}$ out for relay.

3.
Local decisions made at N1 RSs are transmitted over passloss fading channels to the fusion center through several relay nodes. Every relay node makes a binary decision which ${u}_{k}^{i}$ is either +1 or 1 and sends signal ${v}_{k}^{i}$ out. There
$$\begin{array}{l}{u}_{k}^{i}\phantom{\rule{2pt}{0ex}}=\phantom{\rule{2pt}{0ex}}\mathtt{\text{sign}}({v}_{k}^{i1}{h}_{k}^{i1}+{n}_{k}^{i1})\phantom{\rule{2em}{0ex}}\end{array}$$(8)$$\begin{array}{l}{v}_{k}^{i}\phantom{\rule{1.5pt}{0ex}}=\phantom{\rule{1.5pt}{0ex}}{A}_{b}{u}_{k}^{i}\phantom{\rule{2em}{0ex}}\end{array}$$(9) 
4.
The decisions are sent to the fusion center, finally. Let y _{ k }denotes the input signal of the fusion center from the k th RS, thus,
$${y}_{k}={A}_{b}{u}_{k}^{{M}_{k}}{h}_{k}^{{M}_{k}}+{n}_{k}^{{M}_{k}}$$(10)
So, when ${u}_{k}^{{M}_{k}}$ is determined, y_{ k }obey the Gaussian distribution with mean ${A}_{b}{u}_{k}^{{M}_{k}}{h}_{k}^{{M}_{k}}$ and variance σ^{2}. Based on the received data y_{ k }for all RSs, the fusion center decides whether having a target or not.
Define ${P}_{\mathit{\text{dk}}}^{\left(c\right)}$ and ${P}_{\mathit{\text{fk}}}^{\left(c\right)}$ as the probability of detection and probability of false alarm, respectively, at the last relay.
They are different from the local performance indices P_{ d k }and P_{ f k }.
The optimal LRbased fusion statistics for the multihop systems with BT is denoted by Λ_{1}. Given ${P}_{\mathit{\text{dk}}}^{\left(c\right)}$ and ${P}_{\mathit{\text{fk}}}^{\left(c\right)}$, the LR with the fusion statistic can be written as
4.2 Decision fusion rules NBT
In this fusion rule, we assume that relay nodes do not make binary decision when transmitting data. They simply forward the information from source nodes to the fusion center. Other conditions are the same as BT case. The process of target detection in NBT is described as follows.

1.
Every RS sends signal with amplitude A _{ a }out for detection and receives echo from target, independently.

2.
According to the echoes, each RS individually makes a binary decision (local decision) : ${u}_{k}^{0}=+1$ is made if H _{1} is decided, and ${u}_{k}^{0}=1$ is made otherwise. They each sends signal ${v}_{2k}^{0}={A}_{b}{u}_{k}^{0}$ for relay.

3.
Local decisions made at N1 RSs are transmitted over passloss fading channels to the fusion center through several relay nodes. Every relay node simply forward the information ${v}_{2k}^{i}$ from source sensor.
$${v}_{2k}^{i}={v}_{2k}^{i1}{h}_{k}^{i1}+{n}_{k}^{i1}$$(14) 
4.
The decisions are sent to the fusion center, finally. The input signal of the fusion center from the k th RS is
$$\begin{array}{l}{y}_{2k}={h}_{k}^{{M}_{k}}{v}_{2k}^{{M}_{k}}+{n}_{k}^{{M}_{k}}\\ \phantom{\rule{2em}{0ex}}={h}_{k}^{{M}_{k}}\left[\dots {h}_{k}^{2}\right({h}_{k}^{1}({h}_{k}^{0}{v}_{2k}^{0}+{n}_{k}^{0})+{n}_{k}^{1})+{n}_{k}^{2}\dots ]+{n}_{k}^{{M}_{k}}\\ \phantom{\rule{2em}{0ex}}={h}_{k}^{{M}_{k}}\dots {h}_{k}^{2}{h}_{k}^{1}{h}_{k}^{0}{v}_{2k}^{0}+{h}_{k}^{{M}_{k}}\dots {h}_{k}^{2}{h}_{k}^{1}{n}_{k}^{0}+{h}_{k}^{{M}_{k}}\dots {h}_{k}^{2}{n}_{k}^{1}\\ \phantom{\rule{2.8em}{0ex}}+{h}_{k}^{{M}_{k}}\dots {h}_{k}^{3}{n}_{k}^{2}+\cdots +{h}_{k}^{{M}_{k}}{n}_{k}^{{M}_{k}1}+{n}_{k}^{{M}_{k}}\end{array}$$(15)
On account of that ${n}_{k}^{i}$ is additive Gaussian noise, when the ${u}_{k}^{0}$ is fixed, y_{2k}obeys the Gaussian distribution. We set ${\mu}_{k}={h}_{k}^{{M}_{k}}\dots {h}_{k}^{2}{h}_{k}^{1}{h}_{k}^{0}{A}_{b}$ and ${\sigma}_{k}^{2}=[{\left({h}_{k}^{{M}_{k}}\dots {h}_{k}^{2}{h}_{k}^{1}\right)}^{2}+{\left({h}_{k}^{{M}_{k}}\dots {h}_{k}^{2}\right)}^{2}+\cdots +{\left({h}_{k}^{{M}_{k}}\right)}^{2}+{\left(1\right)}^{2}]{\sigma}^{2}$. When ${u}_{k}^{0}=1$ at the k th local sensor, y_{2k} obeys the Gaussian distribution with mean μ_{ k } and variance ${\sigma}_{k}^{2}$ and when ${u}_{k}^{0}=0$ at the k th local sensor, y_{2k} obeys the Gaussian distribution with mean μ_{ k } and variance ${\sigma}_{k}^{2}$.
Due to 15, f(y_{2k}H_{1}) and f(y_{2k}H_{0}) are decomposed into
Therefore, in NBT situation, the LR can be written as
4.3 Energy analysis
The total transmitting power P can be divided into two main components: the power consumption of detecting target P_{det} and the power consumption of forwarding decisions P_{for}. There P = P_{det} + P_{for}. The P_{det} is the sum of power sent at each RS for target detection. The P_{for} is the summation of power consumed at every relay node for forwarding the information from the N1 RSs.
Therefore, the total energy consumption per bit for a fixedrate system can be obtained as
where R_{b} is the bit rate.
In view of these N1 RSs in our area of interest, the average energy consumption of every RS for transmitting is derived by
5 Simulation results
In this section, we analyze the performances of HDS, DDS, and RDS deployment in terms of probability of detection and energy efficiency with BT and NBT decision fusion rules. We place finite RSs in a 400 × 300 m^{2} rectangular surveillance area as an example. The local detection probabilities P_{ d k }are generated mainly based on distances between sensors and target. As an illustration, set false alarm probability P_{fa} = 0.001 and local false alarm probability P_{ f k }=0.05. The SNR is the average of SNR for detection and SNR for forwarding based on passloss fading channel.
Figure 3 compares the probability of detection versus channel SNR for three deployment strategies with different conditions. Besides HDS and DDS, we include random deployment strategy (RDS) for better performance comparison. Each point are obtained using 10^{5} Monte Carlo runs. The simulation result shows the following facts:

1.
It can be seen that the performance for BT fusion rules is better than that for NBT fusion rules.

2.
Both HDS and DDS are better than RDS in terms of detection probability, and DDS is superior to HDS, no matter in BT or NBT. It means that arraying nodes regularly can get higher detection probability than deploying them randomly. Both HDS and DDS are efficient in target detection.
To better understand the performance versus channel SNR, Figures 4 and 5 compare the detection probability with different RS numbers. We can see from Figure 4 that, no matter which deployment strategy is used, the detection probability increases with the reduction of the RS number under same channel SNR for NBT. While Figure 5a, b shows a totally different result that the best performance is achieved when RS number is 30 for both HDS and DDS with BT at moderate to high channel SNR. Figure 5c presents that the detection probability grows along with the RS number increment in RDS for BT at moderate to high SNR. Hence, for the highest detection probability, there is an optimum choice of the RS number. It is not the more the better.
Figure 6 presents the average energy E_{ave} of every RS versus RS number N. Figure 6a compares the average energy E_{ave} for three deployment strategies in two fusion rules with system detection probability P_{ d } = 0.9, while Figure 6b applies P_{ d } = 0.95. Figures 6a,b shows that the BT fusion rules consume fewer energy than the NBT fusion rules. In BT, the energy consumption decreases as the increase of the RS number and DDS is more energy efficient than HDS. RDS consumes the most energy. While in NBT, when the RS number is smaller than 35 at P_{ d } = 0.9 and smaller than 25 at P_{ d } = 0.95, DDS is more energy efficient than HDS. However, as the RS number grows, HDS out performs DDS. RDS is still the wrest in energy efficiency. Thus, we may conclude that among DDS/HDS/RDS and BT/NBT, DDS with BT fusion rule is the best for target detection in RSN due to the high probability of detection, high energy efficiency, and the optimum RS.
6 Conclusion
In this article, we propose two deployment strategies, namely, HDS and DDS to deploy finite RSs to achieve a higher expected detection probability with low energy consumption to satisfy the target detection performance in RSN. Simulation results show that under same channel SNR, the DDS achieves highest probability of detection, the HDS gets a lower one and the RDS is lowest no matter in BT and NBT. To achieve the highest detection performance, there is an optimum choice of the sensor number. It is not the more the better. When the number of sensors is 30, RSN gets highest probability of detection in DDS and BT. The DDS consumes less energy than HDS in multihop RSN with passloss fading channel. The two deployment strategies both are more energy efficient than RDS. Due to the high detection probability, high energy efficiency and the optimum RS, DDS with BT fusion rule is the best among DDS/HDS/RDS and BT/NBT for target detection in RSN.
References
 1.
Liang J, Liang Q: Design and analysis of distributed radar sensor networks. IEEE Trans. Parallel Distrib. Syst 2011, 22(11):19261933.
 2.
Haykin S: Cognitive radar networks. In 1st IEEE International Workshop on Computational Advances in MultiSensor Adaptive Processing. Nashville April; 2005:13.
 3.
Dutta PK, Arora AK, Bibyk SB: Towards radarenabled sensor networks. In Fifth International Conference on Information Processing in Sensor Networks. Nashville, TN April; 2006:467474.
 4.
Ly HD, Liang Q: Spatialtemporalfrequency diversity in radar sensor networks. In 2006 IEEE Military Communications Conference. Washington, DC, October; 2006:17.
 5.
Ly HD, Liang Q: Collaborative multitarget detection in radar sensor networks. In 2007 IEEE Military Communications Conference. Orlando, FL, October; 2007:17.
 6.
Deng H: Target detection with distributed radar sensor networking systems (DRASENS). In 2010 IEEE 10th International Conference on Signal Processing (ICSP). Beijing, October; 2010:19511954.
 7.
Shu H, Liang Q: Data fusion in a multitarget radar sensor network. In 2007 IEEE Radio and Wireless Symposium. Long Beach, CA, January; 2007:129132.
 8.
Yu S, Wang R, Xu H, Wan W, Gao Y, Jin Y: WSN nodes deployment based on artificial fish school algorithm for traffic monitoring system. In IET International Conference on Smart and Sustainable City. Shanghai, July; 2011:15.
 9.
Luo H, Liu Z, Xue F: A deployment strategy for target surveillance sensor networks based on acoustic energy measurements. In 2nd International Conference on Future Computer and Communication. Wuhan, May; 2010:686690.
 10.
Xu K, Wang Q, Hassanein H, Takahara G: Optimal wireless sensor networks (WSNs) deployment: minimum cost with lifetime constraint. IEEE International Conference on Wireless And Mobile Computing, Networking And Communications August, 2005, 2224.
 11.
Mageid SA, Ramadan RA: Efficient deployment algorithms for mobile sensor networks. In International Conference on Autonomous and Intelligent Systems. Povoa de Varzim, June; 2010:16.
 12.
Lin Y, Chen B, Varshney PK: Decision fusion rules in multihop wireless sensor networks. IEEE Trans. Aerospace Electron. Syst 2005, 41(2):475488. 10.1109/TAES.2005.1468742
 13.
Aitsaadi N, Achir N, Boussetta K, Pujolle G: Multiobjective WSN deployment: quality of monitoring, connectivity and lifetime. In 2010 IEEE International Conference on Communications (ICC). Cape Town, May; 2010:16.
 14.
Kapnadak V, Coyle EJ: Optimal nonuniform deployment of sensors for detection in singlehop wireless sensor networks. In 8th Annual IEEE Communications Society Conference on Sensor, Mesh and Ad Hoc Communications and Networks (SECON). Salt Lake City, UT, June; 2011:8997.
 15.
Gogu A, Nace D, Challal Y: A note on joint optimal transmission range assignment and sensor deployment for wireless sensor networks. In 14th International conference on Telecommunications Network Strategy and Planning Symposium (NETWORKS). Warsaw, September; 2010:16.
 16.
Xu K, Hassanein H, Takahara G, Wang Q: Relay node deployment strategies in heterogeneous wireless sensor networks. IEEE Trans. Mob. Comput 2011, 9(2):145159.
 17.
Ababnah A, Natarajan B: Optimal sensor deployment for valuefusion based detection. In 2009 IEEE Global Telecommunications Conference. Honolulu, HI, November–December; 2009:16.
 18.
Zhao F, Guibas L: Wireless Sensor Networks: An Information Processing Approach. New York: Morgan Kaufmann; 2004:77.
Acknowledgements
This study was supported by the National Natural Science Foundation of China under Young Scientists Fund 61102140.
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Keywords
 Graphical deployment strategies
 Decision fusion rules
 Radar sensor networks
 Target detection
 Energy efficiency