Received signal strength–based target localization under spatially correlated shadowing via convex optimization relaxation

Abstract

Received signal strength–based target localization methods normally employ radio propagation path loss model, in which the log-normal shadowing noise is generally assumed to follow a zero-mean Gaussian distribution and is uncorrelated. In this article, however, we represent the simplified additive noise by the spatially correlated log-normal shadowing noise. We propose a new convex localization estimator in wireless sensor networks by using received signal strength measurements under spatially correlated shadowing environment. First, we derive a new non-convex estimator based on weighted least squares criterion. Second, by using the equivalence of norm, the derived estimator can be reformulated as its equivalent form which has no logarithm in the objective function. Then, the new estimator is relaxed by applying efficient convex relaxation that is based on second-order cone programming and semi-definite programming technique. Finally, the convex optimization problem can be efficiently solved by a standard interior-point method, thus to obtain the globally optimal solution. Simulation results show that the proposed estimator solves the localization problem efficiently and is close to Cramer–Rao lower bound compared with the state-of-the-art approach under correlated shadowing environment.

Introduction

Wireless sensor networks (WSNs) are composed of a number of sensor nodes which can be further classified into anchor nodes and target node.^1,2 In general, the location of the anchor nodes are known, whereas the location of target node is unknown, which needs to be determined. Due to the merit of WSNs for its tiny, low device costs, and multi-functional sensors, it has gained wide applications in various areas such as target tracking, navigation, emergency services, friends finding, and intelligent transportation.^3,4 Most of the current positioning approaches for WSNs commonly rely on range measurements between the anchor nodes and target node.^5,6 Depending on the available hardware, range measurements can be extracted from different characteristics of the radio signal, ⁷ such as time of arrival (ToA),^8,9 time difference of arrival (TDoA),^10,11 angle of arrival (AoA),^12,13 and received signal strength (RSS)^14–20 or their combination.^21–23 Recently, RSS-based localization has received tremendous attention due to its low-cost, low-complexity, and easy-implementation solution. ²⁴ Maximum likelihood (ML)^25,26 and least squares (LS)^17,18 are two typical estimators based on RSS measurements. Although the ML estimator is very important due to its asymptotic normality performance, the ML estimator is non-convex and has multiple local optimal solutions. Moreover, it depends highly on the initial point, and a poor initialization commonly leads to bad effects in WSNs, making it difficult to find the globally optimal solution. To overcome those problems, the LS estimator is put forward. Though the LS estimator is much easier and has explicit solution, its accuracy is not as good as the ML estimator, especially when the variance of the measurement noise is large.

Recently, some other methods have been proposed, of which there are the semi-define programming (SDP) relaxation ¹⁶ and second-order cone programming (SOCP) relaxation.^17,18 The basic idea of these methods is to transform the objective function into a convex problem via SDP/SOCP relaxation, then the globally optimal solution of wireless localization problem can be efficiently solved.^27,28 To circumvent the non-convexity of ML estimator, Ouyang et al. ¹⁶ reformulate the problem by eliminating the logarithm and relaxing it as an SDP optimization problem. Tomic et al.^17,18 propose methods to solve the localization problem by converting the original ML non-convex problem into an SOCP convex optimization.

As we know, most of localization algorithms are based on the radio propagation path loss model with the noise generally assumed identically independent distributed (i.i.d.) and they follow a zero-mean Gaussian distribution. However, the noise is not always uncorrelated, especially in complex environment such as indoor environment or forest environment.^29–31 Thus the algorithm based on traditional uncorrelated noise would result in bed effects in WSNs.

In this article, we design an SOCP/SDP estimator for the RSS-based localization problem while the noise is spatially correlated. It shows that when the correlation is taken into consideration among the RSS measurements, the accuracy of localization is improved. ³² First, we present the corresponding ML estimator for the RSS measurements model. Then, we derive a non-convex objective function based on weighted least squares (WLS) criterion but has no logarithm in the objective function. It demonstrates that the developed objective function can be transformed into a convex estimator by applying SOCP and SDP relaxation technique, which can be solved exactly by using the MATLAB package CVX, ³³ where the solver is SeDuMi ³⁴ or SDPT3. ³⁵ Simulation results show that the proposed convex estimator efficiently solves the localization problem and confirms the effectiveness compared with the state-of-the-art approach.

Notation

The following notations are adopted throughout the article. Bold-face lower case letters and bold-face upper case letters denote the vectors and matrices, respectively. ${\mathbb{R}}^n$ denotes the set of $n$ -dimensional real column vectors. ${\mathit{I}}_N$ denotes the $N\times N$ identity matrix. $Q_{\mathrm{ij}}$ denotes the $(i,j)\mathrm{th}$ entry of the matrix $\mathit{Q}$ . $r_i$ denotes the $i\mathrm{th}$ entry of the vector $\mathit{r}$ . In addition, for any symmetric matrix $\mathit{A}$ , $\mathit{A}\ge 0$ means that $\mathit{A}$ is positive semi-definite. $||·|{|}_1$ , $||·|{|}_2$ , and $||·|{|}_{\infty }$ , denote the ${\ell }_1$ norm, ${\ell }_2$ norm, and ${\ell }_{\infty }$ norm, respectively.

System model and problem formulation

Log-normal shadowing path loss model

We consider WSNs of $N$ anchor nodes and one target node in a two-dimensional $(2D)$ localization scenario, where the locations of anchor nodes are known and denoted by ${\mathit{s}}_1,\dots ,{\mathit{s}}_N$ , the location of target node is unknown and denoted by $\mathit{x}$ . For the sake of simplicity and without loss of generality, we assume that the anchor nodes are equipped with omnidirectional antennas. Figure 1 shows the $i\mathrm{th}$ sensoring link in the WSNs model.

Figure 1.

The $i\mathrm{th}$ sensoring link in the wireless sensor networks (WSNs) model.

For $i=1,2,\dots ,N$ , the RSS path loss between the $i\mathrm{th}$ anchor node and the target node that can be denoted in the following radio propagation log-normal shadowing path loss model (in dB) ³⁰

$$L_i=L_0+10\gamma \underset{10}{\log }\frac{\parallel \mathit{x}-{\mathit{s}}_i{\parallel }_2}{d_0}+n_i,i=1,\dots ,N$$

(1)

where $L_i$ is path loss from target node and received by $i\mathrm{th}$ anchor node, $L_0$ is the reference value at the reference distance $d_0$ $(\parallel \mathit{x}-{\mathit{s}}_i{\parallel }_2\ge d_0)$ , $\gamma$ is the path loss exponent (PLE), and in this article, we assume that the propagation with respect to all anchor nodes can be accurately modeled using the same PLE $\gamma$ which usually varies between $2$ and $4$ according to different propagation environment, ¹⁹ let $\mathit{n}=(n_1,n_2,\dots ,n_N{)}^T$ be the vector of log-normal shadowing components which are zero-mean Gaussian random variables with a covariance matrix $\mathit{Q}$ . In most literatures, the log-normal shadowing are generally modeled as i.i.d. zero-mean Gaussian random variables with covariance matrix $\mathit{Q}={\sigma }^2{\mathit{I}}_N$ .^14–16 In this article, we assume that log-normal shadowing are spatially correlated and model the covariance matrix as ¹⁹

$${\mathit{Q}}_{\mathrm{ij}}=\{\begin{array}{cc}{\sigma }^2,\hfill & i=j\hfill \\ \multicolumn{1}{c}{\rho {\sigma }^2,}& i\ne j\hfill \end{array}$$

(2)

where $\sigma$ is the standard deviation (SD) of the log-normal shadowing, $0\le \rho <1$ is the correlation coefficient between any two different links $i$ and $j$ to the target node for $i,j=1,2,\dots ,N$ . This correlation between a pair of sensor nodes mainly depends on the relative angles and distances. ³⁶ The results of empirical studies suggest that the value of the correlation coefficient typically varies from $0.2$ to $0.8$ . ³⁷

Problem formulation

The resulting joint ML estimation of target location $\mathit{x}$ can be formulated as

$$\underset{\mathbf{x}}{min}(\mathit{L}-\mathit{f}(\mathit{x}){)}^T{\mathit{Q}}^{-1}(\mathit{L}-\mathit{f}(\mathit{x}))$$

(3)

where $\mathit{L}=(L_1,L_2,\dots ,L_N{)}^T$ , which denotes the vector of observations given by equation (1), $\mathit{f}(\mathit{x}{)}_i=L_0+10\gamma \underset{10}{\log }(\parallel \mathit{x}-{\mathit{s}}_i{\parallel }_2/d_0)$ .

It is clear that the ML estimator equation (3) is non-convex and difficult to find the global minimum solution. In this article, we propose a new estimator to solve this problem.

Localization via convex relaxation

In order to develop a convex estimator, we first introduce the following Lemma.

Lemma 1

If $\mathit{x}\in {\mathbb{R}}^d$ , then

1. $\parallel \mathit{x}{\parallel }_{\infty }\le \parallel \mathit{x}{\parallel }_2\le \sqrt{d}\parallel \mathit{x}{\parallel }_{\infty }$ ;

2. $\parallel \mathit{x}{\parallel }_{\infty }\le \parallel \mathit{x}{\parallel }_1\le d\parallel \mathit{x}{\parallel }_{\infty }$ .

Lemma 2

The norm $\parallel \mathit{x}{\parallel }_a$ and $\parallel \mathit{x}{\parallel }_b$ are equivalent, if there exists positive number $c_1,c_2$ , such that $c_1\parallel \mathit{x}{\parallel }_b\le \parallel \mathit{x}{\parallel }_a\le c_2\parallel \mathit{x}{\parallel }_b$ for any $\mathit{x}\in {\mathbb{R}}^d$ .

To develop efficient method for tackling the problem (3), we formulate the target localization problem as the following WLS problem

$$\underset{\mathit{x}}{min}\sum _{i=1}^N\frac{{(L_i-L_0-10\gamma \underset{10}{\log }\frac{\parallel \mathit{x}-{\mathit{s}}_i{\parallel }_2}{d_0})}^2}{{\sigma }_i^2}$$

(4)

Because of

$$\begin{array}{c}\multicolumn{1}{c}{L_i-L_0-10\gamma \underset{10}{\log }\frac{\parallel \mathit{x}-{\mathit{s}}_i{\parallel }_2}{d_0}=L_i-L_0}\\ \multicolumn{1}{c}{-5\gamma \underset{10}{\log }{\left(\frac{\parallel \mathit{x}-{\mathit{s}}_i{\parallel }_2}{d_0}\right)}^2}\\ \multicolumn{1}{c}{=-5\gamma (\underset{10}{\log }{\left(\frac{\parallel \mathit{x}-{\mathit{s}}_i{\parallel }_2}{d_0}\right)}^2-\frac{L_i-L_0}{5\gamma })}\\ \multicolumn{1}{c}{=-5\gamma (\underset{10}{\log }{\left(\frac{\parallel \mathit{x}-{\mathit{s}}_i{\parallel }_2}{d_0}\right)}^2-\underset{10}{\log }{10}^{\frac{L_i-L_0}{5\gamma }})}\\ \multicolumn{1}{c}{=-5\gamma \underset{10}{\log }\frac{\parallel \mathit{x}-{\mathit{s}}_i{\parallel }_2^2}{d_0^2{10}^{\frac{L_i-L0}{5\gamma }}}}\end{array}$$

(5)

Then, from equations (4) and (5), we get

$$\underset{\mathbf{x}}{min}\sum _{i=1}^N\frac{{(-5\gamma \underset{10}{\log }\frac{\parallel \mathit{x}-{\mathit{s}}_i{\parallel }_2^2}{d_0^2{10}^{\frac{L_i-L0}{5\gamma }}})}^2}{{\sigma }_i^2}$$

(6)

According to Lemma $1$ and Lemma $2$ , by replacing the ${\ell }_2$ norm with the ${\ell }_{\infty }$ , that is, the Chebyshev norm, we have

$$\sum _{i=1}^N\frac{{(-5\gamma \underset{10}{\log }\frac{\parallel \mathit{x}-{\mathit{s}}_i{\parallel }_2^2}{d_0^2{10}^{\frac{L_i-L0}{5\gamma }}})}^2}{{\sigma }_i^2}\iff \underset{i}{max}\left|\frac{1}{{\sigma }_i}\underset{10}{\log }\frac{\parallel \mathit{x}-{\mathit{s}}_i{\parallel }_2^2}{{\omega }_i^2}\right|$$

(7)

where

$${\omega }_i^2=d_0^2{10}^{\frac{L_i-L_0}{5\gamma }}$$

Since $\underset{10}{\log }x$ is strictly monotonically increasing function over its domain $(0,\infty )$ , equation (7) can be written in an equivalent form as

$$\underset{i}{max}\left|\frac{1}{{\sigma }_i}\underset{10}{\log }\frac{\parallel \mathit{x}-{\mathit{s}}_i{\parallel }_2^2}{{\omega }_i^2}\right|\iff \underset{i}{max}(\frac{\parallel \mathit{x}-{\mathit{s}}_i{\parallel }_2^2}{{\sigma }_i{\omega }_i^2},\frac{{\omega }_i^2}{{\sigma }_i\parallel \mathit{x}-{\mathit{s}}_i{\parallel }_2^2})$$

(8)

Then the primal optimization problem (4) can be expressed as

$$\underset{\mathit{x},\{{\tau }_i\}}{min}\sum _{i=1}^N{\tau }_i$$

(9a)

$$s.t.\frac{\parallel \mathit{x}-{\mathit{s}}_i{\parallel }_2^2}{{\sigma }_i{\omega }_i^2}\le {\tau }_i,i=1,\dots ,N$$

(9b)

$$\frac{{\omega }_i^2}{{\sigma }_i\parallel \mathit{x}-{\mathit{s}}_i{\parallel }_2^2}\le {\tau }_i,i=1,\dots ,N$$

(9c)

By introducing auxiliary variables $r_i=\parallel \mathit{x}-{\mathit{s}}_i{\parallel }_2$ and $\mathit{X}={\mathit{x}}^T\mathit{x}$ , we have the following optimization problem

$$\underset{\mathit{x},\mathit{X},\{r_i\},\{{\tau }_i\}}{min}\sum _{i=1}^N{\tau }_i$$

(10a)

$$s.t.\frac{r_i^2}{{\sigma }_i{\omega }_i^2}\le {\tau }_i,i=1,\dots ,N$$

(10b)

$$\frac{{\omega }_i^2}{{\sigma }_i(\mathrm{tr}(X)-2{\mathit{s}}_i^T\mathit{x}+\parallel {\mathit{s}}_i{\parallel }_2^2)}\le {\tau }_i,i=1,\dots ,N$$

(10c)

$$\parallel \mathit{x}-{\mathit{s}}_i{\parallel }_2=r_i,i=1,\dots ,N$$

(10d)

$$\mathbf{X}={\mathit{x}}^T\mathit{x}$$

(10e)

Clearly, the minimization problem (10) is still non-convex and difficult to solve, since the constraints (10b), (10c), (10d), and (10e) remain non-convex. However, we can use SOCP and SDP relaxation technique to relax $\parallel \mathit{x}-{\mathit{s}}_i{\parallel }_2=r_i$ into $\parallel \mathit{x}-{\mathit{s}}_i{\parallel }_2\le r_i$ and relax (10e) into an inequality constraint $\mathit{X}\ge {\mathit{x}}^T\mathit{x}$ , then equation (10) can be relaxed into an SOCP/SDP estimator, which can be expressed as

$$\underset{\mathit{x},\mathit{X},\{r_i\},\{{\tau }_i\}}{min}\sum _{i=1}^N{\tau }_i$$

(11a)

$$s.t.\parallel [2r_i;{\omega }_i^2+{\sigma }_i{\tau }_i]{\parallel }_2\le {\omega }_i^2-{\sigma }_i{\tau }_i,i=1,\dots ,N$$

(11b)

$$\left[\begin{array}{cc}\mathrm{tr}(\mathit{X})-2{\mathit{s}}_i^T\mathit{x}+\parallel {\mathit{s}}_i{\parallel }_2^2\hfill & \frac{{\omega }_i}{\sqrt{{\sigma }_i}}\hfill \\ \multicolumn{1}{c}{\frac{{\omega }_i}{\sqrt{{\sigma }_i}}}& {\tau }_i\hfill \end{array}\right]\ge 0,i=1,\dots ,N$$

(11c)

$$\parallel \mathit{x}-{\mathit{s}}_i{\parallel }_2\le r_i,i=1,\dots ,N$$

(11d)

$$\left[\begin{array}{cc}\mathit{X}\hfill & \mathit{x}\hfill \\ \multicolumn{1}{c}{{\mathit{x}}^T}& 1\hfill \end{array}\right]\ge 0$$

(11e)

Problem (11) is an SOCP/SDP convex problem and can be efficiently solved by a standard interior-point method. ³⁸

Complexity analysis

There exists inherent trade-off between the estimation accuracy and the implementation complexity among all proposed estimators. The estimator for the worst-case complexity of the mixed SD/SOCP ³⁸ is used to analyze the complexities of the proposed estimator and other considered estimators in this article. The formula of computing complexities is given here

$$\mathcal{O}(\sqrt{U}(m\sum _{i=1}^{N_{\mathrm{sd}}}{n}_i^{s{d}^3}+m^2\sum _{i=1}^{N_{\mathrm{sd}}}{n}_i^{s{d}^2}+m^2\sum _{i=1}^{N_{\mathrm{soc}}}{n}_i^{\mathrm{soc}}+\sum _{i=1}^{N_{\mathrm{soc}}}{n}_i^{\mathrm{so}{c}^2}+m^3))$$

(12)

where $U$ is the number of iterations of the estimator, $m$ is the number of equality constrains, $N_{\mathrm{sd}},N_{\mathrm{soc}}$ are respectively the number of the semi-definite cone (SDC) and second-order constrains (SOC), and $n_i^{\mathrm{sd}},n_i^{\mathrm{soc}}$ are the number of dimensions of the $i\mathrm{th}$ SDC and $i\mathrm{th}$ SOC, respectively. We assume that $K_{\max }$ is the maximum number of steps in the bisection procedure used by Wang and Yang. ¹⁴ The formula (12) and Table 1 show that the proposed “SOCP/SDP-NEW” estimator has the same complexity as that of the “SDP” estimator and “SDP-correlated” estimator, and a higher complexity than that of the “UT-WLS” estimator and “SOCP2” estimator.Table 2 shows the running times of the discussed estimators, which also verified the analysis results.

Table 1.

Summary of the considered methods.

Estimator	Description	Complexity
UT-WLS	WLS method based on UT 14	$\mathcal{O}(K_{\max }N)$
SOCP2	SOCP method based on LS 17	$2·\mathcal{O}(N^{3.5})$
SDP	SDP relaxation method by Ouyang et al. 16	$\mathcal{O}(N^{4.5})$
SDP-correlated	SDP consider correlation by Vaghefi and Buehrer 19	$\mathcal{O}(N^{4.5})$
SOCP/SDP-NEW	The proposed convex method (11)	$\mathcal{O}(N^{4.5})$

WLS: weighted least squares; SOCP: second-order cone programming; SDP: semi-define programming; LS: least squares.

Table 2.

Average running times of various methods (s).

Method	UT-WLS	SOCP2	SDP
Time (s)	0.0022	0.4464	0.6924
Method	SDP-correlated	SOCP/SDP-NEW
Time (s)	0.4738	0.6248

WLS: weighted least squares; SOCP: second-order cone programming; SDP: semi-define programming.

Simulation results

In this section, computer simulation results are provided to compare with the performance among the discussed estimators including “UT-WLS” by Wang and Yang, ¹⁴ “SOCP2” by Tomic et al., ¹⁷ “SDP” by Ouyang et al., ¹⁶ “SDP-correlated” by Vaghefi and Buehrer ¹⁹ and the proposed estimator denoted as “SOCP/SDP-NEW.” Cramer–Rao lower bound (CRLB) is given as performance comparison benchmarks.^39,40 For convenience, the discussed estimators are listed in Table 3. It is better mentioning that for the sake of fairness, the propagation model (1) is used to generate the range measurements. The anchor nodes are assumed uniformly to locate at a circle with radius $20\mathrm{m}$ , while the target node is uniformly randomly chosen from a square region: $\{(x,y)|-10\le x\le 10,-10\le y\le 10\}$ in each Monte Carlo (Mc) runs. Unless stated otherwise, the reference distance is set to $d_0=1\mathrm{m}$ , ${\sigma }_i=\sigma$ for $i=1,\dots ,N$ , the reference path loss is $L_0=40\mathrm{dB}$ , and the PLE is fixed to $\gamma =3$ . Typical values of PLE in building have been found experimentally to approximately $3$ for obstructed paths. ³⁷ From Liberti and Rappaport, ³⁷ we know that the correlation coefficient highly dependent on the specific topography of the anchors geometry. Moreover, the empirical studies results also suggest that the value of the correlation coefficient typically varies from $0.2$ to $0.8$ . Without loss of generality, for the rest of the article, we assume $\rho =0.8$ . The covariance matrix of log-normal shadowing, $\mathit{Q}$ , is a symmetric and positive-definite matrix, which can be Cholesky decomposed as $\mathit{Q}=\mathit{B}{\mathit{B}}^T$ , where $\mathit{B}$ is a lower triangular matrix. Then, the log-normal shadowing term in equation (1) can be generated as ¹⁹

$$\mathit{n}=\mathit{Bw}$$

(13)

where $\mathit{w}$ is a vector of zero-mean i.i.d. Gaussian variable with unit variance. $3000\mathrm{Mc}$ runs show the main performance metric which uses the root-mean-square error (RMSE), defined as $\mathrm{RMSE}=\sqrt{\sum _{i=1}^{\mathrm{Mc}}\frac{\parallel \mathit{x}-\hat{\mathit{x}}{\parallel }_2^2}{\mathrm{Mc}}}$ , where $\hat{\mathit{x}}$ is the estimated location of the $i\mathrm{th}$ Mc runs.

Table 3.

Summary of the compared methods.

Method	Description
UT-WLS	WLS method based on UT by Wang and Yang 14
SOCP2	SOCP relaxation method by Tomic et al. 17
SDP	SDP relaxation method by Ouyang et al. 16
SDP-correlated	SDP-correlated method by Vaghefi and Buehrer 19
SOCP/SDP-NEW	The proposed method equation (11)

WLS: weighted least squares; SOCP: second-order cone programming; SDP: semi-define programming.

Figure 2 illustrates the cumulative distribution function (CDF) of localization comparison of the proposed estimator when uncorrelated case, that is, $\rho =0$ . The log-normal shadowing term are generated using equation (13) and its SD, $\sigma =4\mathrm{dB}$ . The figure shows that the proposed “SOCP/SDP-NEW” estimator is slightly better than the other discussed estimators when shadowing is uncorrelated.

Figure 2.

The cumulative distribution function (CDF) of localization error for compared estimators when standard deviation $\sigma =4\mathrm{dB}$ , the number of anchor nodes $N=6$ , the correlation coefficient $\rho =0$ (uncorrelated case).

In Figure 3, we plot the CDF of localization comparison of the proposed estimator when shadowing highly correlated. Just like the aforementioned case, the log-normal shadowing term are generated using equation (13), its SD and correlation coefficient were $\sigma =4\mathrm{dB}$ and $\rho =0.8$ , respectively. Comparing with Figure 2, it can be seen that the proposed “SOCP/SDP-NEW” estimator exhibits the best performance in the localization accuracy and has an improvement at about $30\mathrm{\%}$ in correlated shadowing. The other discussed estimators do not perform the same way in this case, since “UT-WLS,”“SOCP2,” and “SDP” do not consider the correlation of shadowing components. From Figure 3, it can be seen that the proposed “SOCP/SDP-NEW” shows the better performance under correlated shadowing environment than the other discussed estimators.

Figure 3.

The CDF of localization error for compared estimators when the noise standard deviation $\sigma =4\mathrm{dB}$ , the number of anchor nodes $N=6$ , the correlation coefficient $\rho =0.8$ (correlated case).

Figure 4 illustrates the RMSE versus the noise SD $\sigma (\mathrm{dB})$ comparison when the number of anchor nodes is fixed as $N=6$ . The figure shows that the performance of the discussed estimators are worsening when the noise SD $\sigma$ grows. Furthermore, it also clearly shows the superiority of the proposed “SOCP/SDP-NEW” estimator over the other estimators for the whole range of the noise SD $\sigma$ , and its performance is very close to the CRLB. In order to demonstrate this, the cases when $\sigma =3\mathrm{dB}$ and $\sigma =6\mathrm{dB}$ are taken into consideration, respectively. For the former case, it is clear that “SOCP/SDP-NEW” shows a gain of approximately over $0.6\mathrm{m}$ when compared with the other existing estimators. For the latter case, “SOCP/SDP-NEW” estimator outperforms “SOCP2,”“SDP-correlated,”“UT-WLS,” and “SDP” estimators by $1.2\mathrm{m}$ , $2.3\mathrm{m}$ , $4.8\mathrm{m}$ , and $7.6\mathrm{m}$ , respectively. It is clear that the proposed “SOCP/SDP-NEW” estimator outperforms the existing estimators and its performance falls on top of the CRLB. Therefore, considering the correlation among RSS measurements model (1) is essential.

Figure 4.

Root-mean-square error (RMSE) versus the noise standard deviation $\sigma$ when the number of anchor nodes $N=6$ , the correlation coefficient $\rho =0.8$ .

For the sake of completeness, Figure 5 is provided to compare the RMSE versus the number of anchor nodes. In this scenario, the noise SD is fixed as $\sigma =4\mathrm{dB}$ , the correlation coefficient $\rho =0.8$ , and the number of anchor nodes vary from $4$ to $16$ . This figure shows that the performance can be significantly improved by increasing the number of anchor nodes. Furthermore, the performance of the proposed estimator is superior to the other estimators in this case and the performance gap increases as N grows. To demonstrate this, we consider the cases when $N=6$ and $N=12$ , respectively. In the former case, the “SOCP/SDP-NEW” estimator shows a gain of roughly $0.9\mathrm{m}$ when compared with the other discussed estimators, whereas in the latter case, “SOCP/SDP-NEW” estimator outperforms “SDP-correlated,”“SOCP2,”“UT-WLS,” and “SDP” approaches by $0.3\mathrm{m}$ , $2.1\mathrm{m}$ , $3\mathrm{m}$ , and $5.6\mathrm{m}$ , respectively. Furthermore, its performance falls on top of the CRLB again. This shows that the proposed estimator has the best estimate accuracy compared with the other discussed estimators when taking the correlation into account.

Figure 5.

RMSE versus the number of anchor nodes N when the noise standard deviation $\sigma =4\mathrm{dB}$ , the correlation coefficient $\rho =0.8$ .

To show the effect of spatially correlated shadowing on localization accuracy, Figure 6 compares the RMSE versus the correlation coefficient $\rho$ when the noise SD is fixed as $\sigma =4\mathrm{dB}$ , the number of anchor nodes is fixed as $N=8$ , the correlation coefficient vary from $0.2$ to $0.8$ . The figure shows that the RMSE decreases with the increasing of correlation coefficient for the discussed estimators, and “SOCP/SDP-NEW” estimator provides better performance that is superior to the other discussed estimators. “SOCP2” and “SDP-correlated” also show good performance, although they are worse than CRLB and “SOCP/SDP-NEW.” While the performance of the “SDP” and “UT-WLS” become poor, as the latter two methods do not consider the correlation.

Figure 6.

RMSE versus the correlation coefficient $\rho$ when the noise standard deviation $\sigma =4\mathrm{dB}$ , the number of anchor nodes $N=8$ .

In summary, simulation and theoretical results verify that the proposed “SOCP/SDP-NEW” estimator provide the best performance versus the noise SD, the number of anchor nodes, and the correlation coefficient, respectively, under correlated shadowing environment.

Conclusion

In this article, we consider the log-normal shadowing noise in RSS measurements model and examined its effects under correlated shadowing environment. Based on this model, we investigated the localization problem in WSNs and put forward a new convex estimator. In order to avoid the ML estimator convergence problem, we first developed a non-convex objective function from the RSS measurements model under correlated shadowing environment. Then, we show that the developed objective function can be transformed into a convex estimator by applying SOCP and SDP relaxation technique. Computer simulations were conducted to compare the proposed SOCP/SDP estimator with the other previously discussed estimators. Simulation results show that the proposed convex estimator is better than the other discussed estimators and its performance is very close to the CRLB. In short, the proposed estimator exhibited an excellent performance for RSS-based wireless localization under correlated shadowing environment.