CFAR Decision Fusion Approaches in the Clustered Radar Sensor Networks Using LEACH and HEED

Abstract

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 energy-efficient 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 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 $P_{\text{CH}}$ with which every RS elects itself as a cluster head (CH). To balance the energy dissipation, every RS becomes a CH only once during $\mathrm{1}/P_{\text{CH}}$ rounds. In each round r, each RS elects itself as a CH based on a probability model which can be expressed as

$$\begin{array}{cc}& {\delta }_n\begin{array}{c}\text{CH}\\ \gtrless \\ \text{non-cluster-head}\end{array}T\left(n,r\right),\end{array}$$

(1)

where ${\delta }_n$ produced by the nth RS is a random number between 0 and 1. $T(n,r)$ is given by

$$\begin{array}{cc}& T(n,r)=\left\{\begin{array}{cc}\frac{P_{\text{CH}}}{\mathrm{1}-P_{\text{CH}}\times (r\mathrm{mod}\left(\mathrm{1}/P_{\text{CH}}\right))},\hfill & \text{if}\hspace{0.17em}\hspace{0.17em}n\in U\hfill \\ \mathrm{0},\hfill & \text{otherwise},\hfill \end{array}\right.\end{array}$$

(2)

where U is the set of RSs that were not elected as a CH in the last $\mathrm{1}/P_{\text{CH}}$ 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 (TCHs). Let $E_{\text{re}}^{(n)}$ denote the residual energy of the nth RS; its probability of electing a TCH is

$$\begin{array}{cc}& {\text{CH}}_{\text{prob}}^{\left(n\right)}=\max \left\{C_{\text{prob}}\times \frac{E_{\text{re}}^{\left(n\right)}}{E_{\text{ini}}},p_{\min }\right\},\end{array}$$

(3)

where $C_{\text{prob}}$ is the initial probability of electing a TCH, $E_{\text{ini}}$ are the initial energy of RSs and same for all n's, and $p_{\min }$ 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 ( $\text{AMRP}$ ), which is the mean 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, $\text{MinPw}{\text{r}}_i$ , is proportional to the square of their distance, ${\mathrm{dist}}_i$ , $\mathrm{1}\le i\le M$ . The $\text{AMRP}$ is derived as

$$\begin{array}{cc}& \text{AMRP}=\frac{{\sum }_{\mathrm{1}}^M‍\text{MinPw}{\text{r}}_i}{M}=\frac{{\sum }_{\mathrm{1}}^M‍{\mathrm{dist}}_i^2}{M}.\end{array}$$

(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.

Figure 1

A clustered RSN model with two fusion strategies for target detection.

We make the assumption that RSN is parted to c clusters, and each cluster has $N_i$ RSs except the CH, $\mathrm{1}\le i\le c$ .

3.1. Detection Process

In the detection process, we model the wireless propagation of RSN under the pass-loss fading, which is given by

$$\begin{array}{cc}& P_{rj}=\frac{P_{ti}}{l_{ij}^{\alpha }},\end{array}$$

(5)

where $l_{ij}$ is the distance between the ith transmission RS with the transmission power $P_{ti}$ and the jth receiving RS with the receiving power $P_{rj}$ . α is the radio frequency attenuation exponent. According to the pass-loss model, the power of received signal reflected from the target is

$$\begin{array}{cc}& P_{rk}=\frac{P_{tk}G\delta }{d_k^{\mathrm{2}\alpha }},\end{array}$$

(6)

where $d_k$ is the distance between the kth RS and the target. $P_{tk}$ and $P_{rk}$ 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 $H_{\mathrm{0}}$ and $H_{\mathrm{1}}$ are under test:

$$\begin{array}{cc}& H_{\mathrm{0}}:x_k=A_{rk}+n_k,\\ H_{\mathrm{1}}:x_k=n_k,\end{array}$$

(7)

where $x_k$ is the echo signal amplitude received by the kth RS in the ith cluster, $\mathrm{1}\le k\le N_i$ . $n_k$ is additive Gaussian noise with zero mean and variance ${\sigma }^{\mathrm{2}}$ . $A_{rk}=\sqrt{G\delta }{A}_t/d_k^{\alpha }$ is the signal amplitude echoing to the kth RS, $A_t$ is the transmitted signal amplitude, and $d_k$ is the distance between the target and the kth RS.

Assume the kth local RS to make a binary decision $u_k\in \{+\mathrm{1},-\mathrm{1}\}$ with a probability of false alarm $P_{fk}$ and detection $P_{dk}$ , where $P\left[u_k=-\mathrm{1}\hspace{0.17em}|\hspace{0.17em}{H}_{\mathrm{0}}\right]=P_{fk}$ , $P\left[u_k=+\mathrm{1}\hspace{0.17em}|\hspace{0.17em}{H}_{\mathrm{1}}\right]=P_{dk}$ . The decision threshold $T_k$ in the kth local RS can be derived by

$$\begin{array}{cc}& P_{fk}={{\displaystyle \int }}_T^{\infty }‍\frac{\mathrm{1}}{\sqrt{\mathrm{2}\pi }\sigma }\exp \left(-\frac{x_k^{\mathrm{2}}}{\mathrm{2}{\sigma }^{\mathrm{2}}}\right)d_{x_k},\\ P_{dk}={\displaystyle \int }‍{{\displaystyle \int }}_T^{\infty }‍\frac{\mathrm{1}}{\sqrt{\mathrm{2}\pi }\sigma }\exp \left(-\frac{{\left(x_k-A_{rk}\right)}^{\mathrm{2}}}{\mathrm{2}{\sigma }^{\mathrm{2}}}\right)f\left(A_{rk}\right)d_{x_k}{d}_{A_{rk}},\end{array}$$

(8)

where $f(A_{rk})$ is the probability distribution function (PDF) of $A_{rk}$ . Based on the Neyman-Pearson rule, $P_{dk}$ can be obtainable when $P_{fk}$ is assigned to a fixed number $P_f^{(l)}$ .

We assume that the communication range of the local RSs is the circular with radius $d_c$ , 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

$$\begin{array}{cc}& y_k^i=h_k^i{u}_k+n_k^i,\end{array}$$

(9)

where $n_k^i~N(\mathrm{0},{\sigma }^{\mathrm{2}})$ . The pass-loss channel attenuation coefficient $h_k^i$ is

$$\begin{array}{cc}& h_k^i=\mathrm{1}-\frac{d_k^i}{d_c},\end{array}$$

(10)

where $d_k^i$ is the distance between the ith CH and the kth RS.

3.2. Problem Formation

Using the above detection model, we can obtain the optimal likelihood-ratio-based (OL) fusion statistic of CHs. That is

$$\begin{array}{l}{\Lambda }_i={\displaystyle \prod }_{k=\mathrm{1}}^{N_i}‍\frac{f\left(y_k^i\hspace{0.17em}|\hspace{0.17em}{H}_{\mathrm{1}}\right)}{f\left(y_k^i\hspace{0.17em}|\hspace{0.17em}{H}_{\mathrm{0}}\right)}\\ ={\displaystyle \prod }_{k=\mathrm{1}}^{N_i}‍\frac{P_{dk}^{i}P\left[u_k=+\mathrm{1}\hspace{0.17em}|\hspace{0.17em}{H}_{\mathrm{1}}\right]+\left(\mathrm{1}-P_{dk}^i\right)P\left[u_k=-\mathrm{1}\hspace{0.17em}|\hspace{0.17em}{H}_{\mathrm{1}}\right]}{P_f^{\left(l\right)}P\left[u_k=+\mathrm{1}\hspace{0.17em}|\hspace{0.17em}{H}_{\mathrm{1}}\right]+\left(\mathrm{1}-P_f^{\left(l\right)}\right)P\left[u_k=-\mathrm{1}\hspace{0.17em}|\hspace{0.17em}{H}_{\mathrm{1}}\right]}\\ ={\displaystyle \prod }_{k=\mathrm{1}}^{N_i}‍\left(\left(P_{dk}^i\exp \left(\frac{-{\left(y_k^i-h_k^i\right)}^{\mathrm{2}}}{\mathrm{2}{\sigma }^{\mathrm{2}}}\right)\right.\phantom{{\left.+\left(\mathrm{1}-P_f^{\left(l\right)}\right)\exp \left(\frac{-{\left(y_k^i+h_k^i\right)}^{\mathrm{2}}}{\mathrm{2}{\sigma }^{\mathrm{2}}}\right)\right)}^{-1}}\right.\\ \phantom{ssssdsssssss}\left.+(\mathrm{1}-P_{dk}^i)\exp \left(\frac{-{\left(y_k^i+h_k^i\right)}^{\mathrm{2}}}{\mathrm{2}{\sigma }^{\mathrm{2}}}\right)\right)\\ \phantom{sssssdsss}\times \left(P_f^{\left(l\right)}\exp \left(\frac{-{\left(y_k^i-h_k^i\right)}^{\mathrm{2}}}{\mathrm{2}{\sigma }^{\mathrm{2}}}\right)\right.\\ \phantom{ssssssssssssss}\left.{\left.+\left(\mathrm{1}-P_f^{\left(l\right)}\right)\exp \left(\frac{-{\left(y_k^i+h_k^i\right)}^{\mathrm{2}}}{\mathrm{2}{\sigma }^{\mathrm{2}}}\right)\right)}^{-1}\right)\\ ={\displaystyle \prod }_{k=\mathrm{1}}^{N_i}‍\frac{P_{dk}^i+\left(\mathrm{1}-P_{dk}^i\right)\exp \left(-\mathrm{2}{y}_k^i{h}_k^i/{\sigma }^{\mathrm{2}}\right)}{P_f^{\left(l\right)}+\left(\mathrm{1}-P_f^{\left(l\right)}\right)\exp \left(-\mathrm{2}{y}_k^i{h}_k^i/{\sigma }^{\mathrm{2}}\right)},\end{array}$$

(11)

where ${\Lambda }_i$ 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 ${\sigma }^{\mathrm{2}}\to \infty$ , the OL fusion statistic of the ith CH ${\Lambda }_{ci}$ can be simplified as

$$\begin{array}{cc}& \underset{{\sigma }^{\mathrm{2}}\to \infty }{\lim }\log {\Lambda }_{ci}={\Lambda }_{ci}^L=\frac{\mathrm{1}}{N_i}{\displaystyle \sum }_{k=\mathrm{1}}^{N_i}‍\left(P_{dk}-P_{fk}\right)h_k^i{y}_k^i.\end{array}$$

(12)

Obviously, ${\Lambda }_{ci}^L$ 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 ${\Lambda }_{ci}^L$ , are derived and summarized in Table 1, where $A_{ki}=(h_k^i{)}^{\mathrm{2}}(P_{dk}-P_f^{(l)})$ and $B_{ki}=(h_k^i(P_{dk}-P_f^{(l)}){)}^{\mathrm{2}}$ .

Table 1

Mean and variance of $${\mathrm{\Lambda }}_{ci}^L$$ under $$H_{\mathrm{0}}$$ and $$H_{\mathrm{1}}$$ with $$N_i$$ RSs in the ith cluster.

$$H_{\mathrm{0}}$$	$${\mathrm{\mu }}_{i{c}_{\mathrm{0}}}=\frac{\mathrm{2}{P}_f^{(l)}-\mathrm{1}}{N_i}{{\displaystyle \sum }}_{k=\mathrm{1}}^{N_i}\mathrm{‍}{A}_{ki}$$
	$${\mathrm{\sigma }}_{i{c}_{\mathrm{0}}}^{\mathrm{2}}=\frac{\mathrm{1}}{{N_i}^{\mathrm{2}}}\left({\mathrm{\sigma }}^{\mathrm{2}}{{\displaystyle \sum }}_{k=\mathrm{1}}^{N_i}\mathrm{‍}{B}_{ki}+\mathrm{4}{P}_f^{\left(l\right)}\left(\mathrm{1}-P_f^{\left(l\right)}\right){{\displaystyle \sum }}_{k=\mathrm{1}}^{N_i}\mathrm{‍}{A}_{ki}^{\mathrm{2}}\right)$$
$$H_{\mathrm{1}}$$	$${\mathrm{\mu }}_{i{c}_{\mathrm{1}}}=\frac{\mathrm{1}}{N_i}{{\displaystyle \sum }}_{k=\mathrm{1}}^{N_i}\mathrm{‍}\left(A_{ki}\left(\mathrm{2}{P}_{dk}-\mathrm{1}\right)\right)$$
	$${\mathrm{\sigma }}_{i{c}_{\mathrm{1}}}^{\mathrm{2}}=\frac{\mathrm{1}}{{N_i}^{\mathrm{2}}}\left({\mathrm{\sigma }}^{\mathrm{2}}{{\displaystyle \sum }}_{k=\mathrm{1}}^{N_i}\mathrm{‍}{B}_{ki}+\mathrm{4}{{\displaystyle \sum }}_{k=\mathrm{1}}^{N_i}\mathrm{‍}\left(A_{ki}^{\mathrm{2}}{P}_{dk}\left(\mathrm{1}-P_{dk}\right)\right)\right)$$

When the CFAR of the CHs $P_f^{(c)}$ is given, the decision threshold and the probability of detection of the ith CH, $T_i^{(c)}$ and $P_{di}^{(c)}$ , are derived as

$$\begin{array}{cc}& T_i^{\left(c\right)}={\sigma }_{i{c}_{\mathrm{0}}}{Q}^{-\mathrm{1}}\left(P_f^{\left(c\right)}\right)+{\mu }_{i{c}_{\mathrm{0}}},\\ P_{di}^{\left(c\right)}=Q\left(\frac{T_i^{\left(c\right)}-{\mu }_{i{c}_{\mathrm{1}}}}{{\sigma }_{i{c}_{\mathrm{1}}}}\right).\end{array}$$

(13)

The ith CH also makes a binary decision $u_i\in \{+\mathrm{1},-\mathrm{1}\}$ . Assuming that the communication range of CHs is the circular with radius $d_b$ , which represents the amplitude of the transmitted signals sent to the BS, the fusion statistic of BS is

$$\begin{array}{cc}& {\Lambda }_b^L=\frac{\mathrm{1}}{c}{\displaystyle \sum }_{i=\mathrm{1}}^c‍\left(P_{di}^{\left(c\right)}-P_f^{\left(c\right)}\right)h_i^c{y}_i^c,\end{array}$$

(14)

where $y_i^c$ is the received signal from the ith CH and $h_i^c=\mathrm{1}-d_i^c/d_b$ and $d_i^c$ is the distance between the ith CH and BS. The first and second statistics of ${\Lambda }_b^L$ are derived and summarized in Table 2, where $C_i=(h_i^c{)}^{\mathrm{2}}(P_{di}^{(c)}-P_f^{(c)})$ and $D_i=(h_i^c(P_{di}^{(c)}-P_f^{(c)}){)}^{\mathrm{2}}$ .

Table 2

Mean and variance of $${\mathrm{\Lambda }}_b^L$$ under $$H_{\mathrm{0}}$$ and $$H_{\mathrm{1}}$$ with $$N_i$$ RSs in the ith cluster.

$$H_{\mathrm{0}}$$	$${\mathrm{\mu }}_{b_{\mathrm{0}}}=\frac{\mathrm{2}{P}_f^{\left(c\right)}-\mathrm{1}}{c}{{\displaystyle \sum }}_{i=\mathrm{1}}^c\mathrm{‍}{C}_i$$
	$${\mathrm{\sigma }}_{b_{\mathrm{0}}}^{\mathrm{2}}=\frac{\mathrm{1}}{c^{\mathrm{2}}}\left({\mathrm{\sigma }}^{\mathrm{2}}{{\displaystyle \sum }}_{i=\mathrm{1}}^c\mathrm{‍}{D}_i+\mathrm{4}{P}_f^{\left(c\right)}\left(\mathrm{1}-P_f^{\left(c\right)}\right){{\displaystyle \sum }}_{i=\mathrm{1}}^c\mathrm{‍}{C}_i^{\mathrm{2}}\right)$$
$$H_{\mathrm{1}}$$	$${\mathrm{\mu }}_{b_{\mathrm{1}}}=\frac{\mathrm{1}}{c}{{\displaystyle \sum }}_{i=\mathrm{1}}^c\mathrm{‍}\left(C_i\left(\mathrm{2}{P}_{di}^{\left(c\right)}-\mathrm{1}\right)\right)$$
	$${\mathrm{\sigma }}_{b_{\mathrm{1}}}^{\mathrm{2}}=\frac{\mathrm{1}}{c^{\mathrm{2}}}\left({\mathrm{\sigma }}^{\mathrm{2}}{{\displaystyle \sum }}_{i=\mathrm{1}}^c\mathrm{‍}{D}_i+\mathrm{4}{{\displaystyle \sum }}_{i=\mathrm{1}}^c\mathrm{‍}\left(C_i^{\mathrm{2}}{P}_{di}^{\left(c\right)}\left(\mathrm{1}-P_{di}^{\left(c\right)}\right)\right)\right)$$

Similarly, when the CFAR of the RSN $P_f^{(b)}$ is given, the decision threshold of the BS and the probability of detection of the RSN, namely, $T^{(b)}$ and $P_d^{(b)}$ , are derived as

$$\begin{array}{cc}& T^{\left(b\right)}={\sigma }_{b_{\mathrm{0}}}{Q}^{-\mathrm{1}}\left(P_f^{\left(b\right)}\right)+{\mu }_{b_{\mathrm{0}}},\\ P_d^{\left(b\right)}=Q\left(\frac{T^{\left(b\right)}-{\mu }_{b_{\mathrm{1}}}}{{\sigma }_{b_{\mathrm{1}}}}\right).\end{array}$$

(15)

4.2. HLDFKE

If ${\sigma }^{\mathrm{2}}\to \mathrm{0}$ , the OL fusion statistic of the ith CH ${\Lambda }_{ci}$ can be simplified as

$$\begin{array}{l}{\Lambda }_{ci}^H=\underset{{\sigma }^{\mathrm{2}}\to \mathrm{0}}{\lim }\log {\Lambda }_{ci}\\ ={\displaystyle \sum }_{\mathrm{sign}\left(y_k^i\right)=\mathrm{1}}‍\log \frac{P_{dk}}{P_f^{\left(l\right)}}\\ +{\displaystyle \sum }_{\mathrm{sign}\left(y_k^i\right)=-\mathrm{1}}‍\log \frac{\mathrm{1}-P_{dk}}{\mathrm{1}-P_f^{\left(l\right)}}.\end{array}$$

(16)

In order to obtain the CFAR for the whole network, the decision threshold based on the empirical cumulative distribution function (ECDF) of ${\Lambda }_{ci}^H$ under $H_{\mathrm{0}}$ is estimated by Kaplan-Meier estimator (KE) from the fusion statistic data samples for $H_{\mathrm{0}}$ during every clustering process. Here, we briefly summarize the KE.

Suppose $t_{\mathrm{0}}<t_{\mathrm{1}}<\cdots <t_v$ are preestablished threshold values, and $t_{\mathrm{0}}$ is designed to satisfy the equation $P\{S>t_{\mathrm{0}}\}=\mathrm{1}$ , where S is the fusion statistic under $H_{\mathrm{0}}$ . For $t_{u-\mathrm{1}}\le t\le t_u$ , the ECDF of S can be derived by

$$\begin{array}{l}F\left(t\right)=P\{S>t\}=P\{S>t,\hspace{0.17em}\hspace{0.17em}S>t_u\}\\ =P\left\{S>t\hspace{0.17em}|\hspace{0.17em}S>t_u\right\}P\left\{S>t_u\right\}\\ =P\{S>t\hspace{0.17em}|\hspace{0.17em}S>t_u\}P\{S>t_u\hspace{0.17em}|\hspace{0.17em}S>t_{u-\mathrm{1}}\}P\left\{S>t_{u-\mathrm{1}}\right\}\\ =P\left\{S>t\hspace{0.17em}|\hspace{0.17em}S>t_u\right\}P\left\{S>t_u\hspace{0.17em}|\hspace{0.17em}S>t_{u-\mathrm{1}}\right\}\\ \cdots P\left\{S>t_{\mathrm{1}}\hspace{0.17em}|\hspace{0.17em}S>t_{\mathrm{0}}\right\}P\left\{S>t_{\mathrm{0}}\right\}\\ \approx {\displaystyle \prod }_{j=\mathrm{1}}^u‍P\left\{S>t_j\hspace{0.17em}|\hspace{0.17em}S>t_{j-\mathrm{1}}\right\}\\ \approx {\displaystyle \prod }_{j=\mathrm{1}}^u‍\frac{d_j-x_j}{d_j},\end{array}$$

(17)

where $d_j$ denotes the number of S samples more than $t_{j-\mathrm{1}}$ ; $x_j$ denotes the number of S samples more than $t_{j-\mathrm{1}}$ and not less than $t_j$ . The required threshold ${\stackrel{-}{T}}_c$ is

$$\begin{array}{cc}& F\left({\stackrel{-}{T}}_c\right)=P_f^{\left(c\right)}.\end{array}$$

(18)

Suppose that ${\stackrel{-}{T}}_{ci}$ is the decision threshold of the ith CH; its PD is

$$\begin{array}{cc}& {\stackrel{-}{P}}_{di}^{\left(c\right)}={{\displaystyle \int }}_{{\stackrel{-}{T}}_{ci}}^{\infty }‍f\left({\Lambda }_{ci}^H\hspace{0.17em}|\hspace{0.17em}{H}_{\mathrm{1}}\right)d_{{\Lambda }_{ci}^H}.\end{array}$$

(19)

Then the fusion statistic of BS is

$$\begin{array}{cc}& {\Lambda }_b^H={\displaystyle \sum }_{\mathrm{sign}\left(y_i^c\right)=\mathrm{1}}‍\log \frac{P_{di}^{\left(c\right)}}{P_f^{\left(b\right)}}+{\displaystyle \sum }_{\mathrm{sign}\left(y_i^c\right)=-\mathrm{1}}‍\log \frac{\mathrm{1}-P_{dk}^{\left(c\right)}}{\mathrm{1}-P_f^{\left(b\right)}}.\end{array}$$

(20)

We could easily acquire the decision threshold by secondly using KE.

5. Radio Energy Consumption Model

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

$$\begin{array}{l}{E}_{Tx}\left(l,d\right)=E_{Tx\text{-elec}}\left(l\right)+E_{Tx\text{-amp}}(l,d)\\ =\left\{\begin{array}{ll}l{E}_{\text{elec}}+l{ϵ}_{fs}{d}^{\mathrm{2}},\hfill & d<d_o\hfill \\ l{E}_{\text{elec}}+l{ϵ}_{mp}{d}^{\mathrm{4}},\hfill & d\ge d_o,\hfill \end{array}\right.\end{array}$$

(21)

where $d_o=\sqrt{{ϵ}_{fs}/{ϵ}_{mp}}$ , and to receive this message, the radio expends

$$\begin{array}{cc}& E_{Rx}\left(l\right)=E_{Rx\text{-elec}}\left(l\right)=l{E}_{\text{elec}}.\end{array}$$

(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 $E_{\text{NCH}}$ is

$$\begin{array}{cc}& E_{\text{NCH}}=E_{T\det }+E_{R\det }+l{E}_{\text{elec}}+l{ϵ}_{fs}{d}_{\text{to}\hspace{0.17em}\text{CH}}^{\mathrm{2}},\end{array}$$

(23)

where $E_{T\det }$ and $E_{R\det }$ are energy for transmitting and receiving detection signals, $E_{T\det }\propto E_{R\det }/d_{\text{to}\hspace{0.17em}\text{Ta}}^2$ , and $d_{\text{to}\hspace{0.17em}\text{Ta}}$ and $d_{\text{to}\hspace{0.17em}\text{CH}}$ 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

$$\begin{array}{cc}& E_{\text{CH}}=N\left(l{E}_{\text{elec}}+l{E}_{\text{DA}}\right)+E_{Tx},\end{array}$$

(24)

where $E_{\text{DA}}$ is the energy for data aggregation and $E_{Tx}$ depends on the distance between the CHs and the BS.

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 $P_f^{(l)}$ and $P_f^{(c)}$ , are set as 0.05 and 0.01, respectively; the CFAR of BS $P_f^{(b)}$ is fixed as $\mathrm{1}{\mathrm{0}}^{-\mathrm{3}}$ 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 200, the dimensions are 100 $\sqrt{\mathrm{2}}$  m × 100 $\sqrt{\mathrm{2}}$  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 $\mathrm{1}{\mathrm{0}}^{\mathrm{6}}$ Monte Carlo runs.

Table 3

Simulation parameters.

Type	Parameter	Value
Network	Network grid	From (0, 0) to (100, 100)
	BS	At (50, 175)
	Initial energy	0.5 J/battery
	Initial number of RSs	100
HEED	Cluster radius	25 m
	$$p_{\min }$$	$$\mathrm{1}{\mathrm{0}}^{-\mathrm{4}}$$
Energy model	$$E_{\text{elec}}$$	50 nJ/bit
	$${\mathrm{ϵ}}_{\text{fs}}$$	10 pJ/bit/m2
	$${\mathrm{ϵ}}_{\text{mp}}$$	0.0013 pJ/bit/m4
	$$E_{\text{DA}}$$	5 J/bit/signal
	$$E_{\text{Tdet}}$$	12.64 uJ/signal
	Data packet size	10 bytes
	Broadcast packet size	5 bytes
CFAR	NCHs: $$P_f^{(l)}$$	0.05
	CHs: $$P_f^{(c)}$$	0.01
	BS: $$P_f^{(b)}$$	$$\mathrm{1}{\mathrm{0}}^{-\mathrm{3}}$$
	Monte Carlo simulations	$$\mathrm{1}{\mathrm{0}}^{\mathrm{6}}$$ times

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 simulation and by numerical approximation using LLDFCLT. While some discrepancy exists, approximations using LLDFCLT match relatively well to the corresponding simulation results.

Figure 2

The PD of LEACH and HEED in LLDFCLT.

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

$$\begin{array}{cc}& D\left(P_{\mathrm{1}}\parallel P_{\mathrm{2}}\right)=\log \frac{{\sigma }_{\mathrm{1}}}{{\sigma }_{\mathrm{0}}}+\frac{\left({\sigma }_{\mathrm{0}}^{\mathrm{2}}-{\sigma }_{\mathrm{1}}^{\mathrm{2}}\right)+{\left({\mu }_{\mathrm{0}}-{\mu }_{\mathrm{1}}\right)}^{\mathrm{2}}}{\mathrm{2}{\sigma }_{\mathrm{1}}^{\mathrm{2}}}.\end{array}$$

(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.

Figure 3

Kullback-Leibler distance (relative entropy) between the two hypotheses for both LEACH and HEED in LLDFCLT.

Figure 4

The PD of LEACH and HEED in LLDFCLT with constant SNRs versus different CFARs.

Figure 5

The lifetime of RSN in LEACH and HEED.

To better understand the performance differences as various CFARs of the system, we change the value of previous $P_f^{(b)}$ 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 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.

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.

Figure 7

The PD of LEACH and HEED in LLDFCLT and HLDFKE. (a) $P_f^{(b)}=\mathrm{1}{\mathrm{0}}^{-\mathrm{3}}$ and (b) $P_f^{(b)}=\mathrm{1}{\mathrm{0}}^{-\mathrm{4}}$ .

Figure 8

The PD of RSN in LEACH and HEED and CSSA and FND.

Figure 9

The PD of RSN in HEED versus different cluster radius.

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 fading while LLDFCLT holds almost complete knowledge;

(2)

different with LLDFCLT, the PD of HEED is higher than that of LEACH when SNR is more than 2 dB.

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 entire SNR values in LLDFCLT and for moderate-to-high-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.