Apparatus and method for RSS/AoA target 3-D localization in wireless networks

Abstract

An apparatus and a method for RSS/AoA target 3-D localization in wireless networks and wireless sensor networks (WSNs), utilizing combined measurements of received signal strength (RSS) and angle of arrival (AoA) are disclosed herein. By using the spherical coordinate conversion and available AoA observations to establish new relationships between the measurements and the unknown target location, a simple closed-form solution is developed. The method disclosed herein has a straightforward adaptation to the case where the target's transmit power is also not known. A representative set of simulations and experiments verify the potential performance improvement realized with embodiments of the method for RSS/AoA target 3-D localization in wireless networks.

Claims

1. An apparatus for target localization in wireless networks comprising: M targets and N anchor receivers, wherein each anchor receiver includes at least one directive or antenna array to receive a signal sent from a target; a processing unit configured to receive the RSS (received signal strength) information form said receivers and to process the RSS information and to compute a distance measurement between said targets and said receivers; a processing unit configured to receive AoA (angle of arrival) information from said receivers, wherein the AOA information includes angles of azimuth and elevation measurements of the incoming signal transmitted by each one of the targets present in the wireless network; a processing unit configured to compute a conversion from Cartesian coordinates of the RSS and AOA information to spherical coordinates in order to merge the RSS and AoA information; and an estimator configured to estimate a location of each target based on the merged RSS and AoA information and WLS (weighted least squares) criterion, wherein the WLS criterion determines weights based on a ML (maximum likelihood) estimate of the distance measurement.

2. A method for target localization in wireless networks with M targets and N anchors receivers, wherein each anchor receiver includes at least one directive or antenna array to receive a signal sent from a target comprising the following steps: receive RSS (received signal strength) information form said receivers and process the RSS information to compute a distance measurement between said targets and said receivers; receive AoA (angle of arrival) information from said receivers, wherein the AOA information includes angles of azimuth and elevation measurements of the incoming signal transmitted by each one of the targets present in the wireless network; compute a conversion from Cartesian coordinates of the RSS and AoA information to spherical coordinates in order to merge the RSS and AoA information; and estimate a location of each target based on the merged RSS and AoA information and WLS (weighted least squares) criterion, wherein the WLS criterion determines weights based on a ML (maximum likelihood) estimate of the distance measurement.

3. The method of claim 2, when transmitted power information P.sub.T of each target is known, comprising the following steps: a. define and compute the following relations:
.sub.ixa.sub.id.sub.0 for i=1, . . . ,N,(7)
c.sub.i.sup.T(xa.sub.i)0, for i=1, . . . ,N,(8)
k.sup.T(xa.sub.i)xa.sub.icos(.sub.i), for i=1, . . . ,N,(9) where ${}_i={10}^{\frac{P_i}{10}},={10}^{\frac{P_0}{10}},$ c.sub.i=[sin(.sub.i),cos(.sub.i),0].sup.T and k[0,0,1].sup.T, x=[x.sub.x,x.sub.y,x.sub.z].sup.T and a.sub.i=[a.sub.ix,a.sub.iy,a.sub.iz].sup.T represent the coordinates of the target and the i-th anchor, respectively, while d.sub.i, .sub.i, and .sub.i denote respectively the distance, azimuth angle and elevation angle between the target and the i-th anchor, d.sub.0 (d.sub.0d.sub.i) is a reference distance, is the path loss exponent (PLE), and d.sub.i is the distance between the target and the i-th anchor, and convert the Cartesian to spherical coordinates write xa.sub.i=r.sub.iu.sub.i, where the unit vector u.sub.i=[cos(.sub.i)sin(.sub.i),sin(.sub.i)sin(.sub.i),cos(.sub.i)].sup.T can be obtained from the AoA measurements; b. next apply the conversion from Cartesian to spherical coordinates to (7) and (9), and multiply by 1 (formed as u.sub.i.sup.Tu.sub.i), to respectively get:
.sub.iu.sub.i.sup.Tu.sub.id.sub.0.sub.iu.sub.i.sup.T(xa.sub.i)d.sub.0,(10)
and
k.sup.Tr.sub.iu.sub.iu.sub.i.sup.Tr.sub.iu.sub.i cos(.sub.i)(cos(.sub.i)u.sub.ik).sup.T(xa.sub.i)0,(11) c. give more importance to nearby links by introducing weights, w=[{square root over (w.sub.i)}], where each w.sub.i is defined as $\begin{array}{cc}{w}_i=1-\frac{{\hat{d}}_i}{\underset{i=1}{\overset{N}{.Math.}}{\hat{d}}_i},\mathrm{with}{\hat{d}}_i=d_0{10}^{\frac{P_0-P_i}{10}}& \left(12\right)\end{array}$ being the ML (maximum likelihood) estimate of the distance obtained from $P_i=P_0-10{\log }_{10}\frac{d_i}{d_0}+n_i,\mathrm{for}i=1,\mathrm{.Math.},N;$ d. next according to the WLS (weighted least squares) criterion and (10), (8), (11) and (12), obtain the following estimator: $\begin{array}{cc}\hat{x}=\arg \underset{x}{\min }\underset{i=1}{\overset{N}{.Math.}}{w_i\left({}_i{u}_i^T\left(x-a_i\right)-d_0\right)}^2+\underset{i=1}{\overset{N}{.Math.}}{w_i\left(c_i^T\left(x-a_i\right)\right)}^2+\underset{i=1}{\overset{N}{.Math.}}{w_i\left({\left(\cos \left({}_i\right)u_i-k\right)}^T\left(x-a_i\right)\right)}^2,& \left(13\right)\end{array}$ which can be writen as $\begin{array}{cc}\underset{x}{\mathrm{minimize}}{.Math.W\left(\mathrm{Ax}-b\right).Math.}^2,& \left(14\right)\end{array}$ where W=I.sub.3.Math.diag(w), with .Math. denoting the Kronecker product, and $A=\left[\begin{array}{c}\mathrm{.Math.}\\ {}_i{u}_i^T\\ \mathrm{.Math.}\\ c_i^T\\ \mathrm{.Math.}\\ {\left(\cos \left({}_i\right)u_i-k\right)}^T\\ \mathrm{.Math.}\end{array}\right],b=\left[\begin{array}{c}\mathrm{.Math.}\\ {}_i{u}_i^T+d_0\\ \mathrm{.Math.}\\ c_i^T{a}_i\\ \mathrm{.Math.}\\ {\left(\cos \left({}_i\right)u_i-k\right)}^T{a}_i\\ \mathrm{.Math.}\end{array}\right],$ with the solution given by
{circumflex over (x)}=(A.sup.TW.sup.TWA).sup.1(A.sup.TW.sup.Tb).(15)

4. The method of claim 2, when the transmitted power information P.sub.T of each target is unknown, comprising the following steps: a. first, define and compute the following relations:
.sub.ixa.sub.id.sub.0 for i=1, . . . ,N,(7)
c.sub.i.sup.T(xa.sub.i)0m for i=1, . . . ,N,(8)
k.sup.T(xa.sub.i)xa.sub.icos(.sub.i), for i=1, . . . ,N,(9) where ${}_i={10}^{\frac{P_i}{10}},={10}^{\frac{P_0}{10}},$ c.sub.i=[sin(.sub.i),cos(.sub.i),0].sup.T and k=[0,0,1].sup.T, x=[x.sub.x,x.sub.y,x.sub.z].sup.T and a.sub.i=[a.sub.ix,a.sub.iy,a.sub.iz].sup.T represent the coordinates of the target and the i-th anchor, respectively, while d.sub.i, .sub.i, and .sub.i denote respectively the distance, azimuth angle and elevation angle between the target and the i-th anchor, is the path loss exponent (PLE), d.sub.i is the distance between the target and the i-th anchor, d.sub.0 (d.sub.0d.sub.i) is a reference distance and, convert from Cartesian to spherical coordinates to write xa.sub.i=r.sub.iu.sub.i, where the unit vector u.sub.i=[cos(.sub.i)sin(.sub.i),sin(.sub.i)sin(.sub.i),cos(.sub.i)].sup.T can be obtained from the AoA measurements; b. apply the described conversion in (7) and (9), and multiply with 1 (formed as u.sub.i.sup.Tu.sub.i), to respectively get
.sub.iu.sub.i.sup.Tu.sub.id.sub.0.sub.iu.sub.i.sup.T(xa.sub.i)d.sub.0,(10)
and
k.sup.Tr.sub.iu.sub.iu.sub.i.sup.Tr.sub.iu.sub.i cos(.sub.i)(cos(.sub.i)u.sub.ik).sup.T(xa.sub.i)0,(11) c. give more importance to nearby links by introducing weights, {tilde over (w)}=[{square root over ({tilde over (w)}.sub.i)}], such that $\begin{array}{cc}{\tilde{w}}_i=1-\frac{P_i}{\underset{i=1}{\overset{N}{.Math.}}{P}_i};& \left(16\right)\end{array}$ d. apply the WLS (weighted least squares) and (10), (8), (11) and (16), to get: $\begin{array}{cc}\left(\hat{x},\hat{}\right)=\underset{x,}{\arg \min }\underset{i=1}{\overset{N}{.Math.}}{{\tilde{w}}_i\left({}_i{u}_i^T\left(x-a_i\right)-d_0\right)}^2+\underset{i=1i}{\overset{N}{.Math.}}{{\tilde{w}}_i\left(c_i^T\left(x-a_i\right)\right)}^2+\underset{i=1}{\overset{N}{.Math.}}{{\tilde{w}}_i\left({\left(\cos \left({}_i\right)u_i-k\right)}^T\left(x-a_i\right)\right)}^2,& \left(17\right)\end{array}$ which can be rewritten as: $\begin{array}{cc}\underset{y={\left[x^T,\right]}^T}{\mathrm{minimize}}{.Math.\tilde{W}\left(\tilde{A}y-\tilde{b}\right).Math.}^2,& \left(18\right)\end{array}$ where {tilde over (W)}=I.sub.3.Math.diag({tilde over (w)}), with .Math. denoting the Kronecker product and $\tilde{A}=\left[\begin{array}{cc}\mathrm{.Math.}& \mathrm{.Math.}\\ {}_i{u}_i^T& -d_0\\ \mathrm{.Math.}& \mathrm{.Math.}\\ c_i^T& 0\\ \mathrm{.Math.}& \mathrm{.Math.}\\ {\left(\cos \left({}_i\right)u_i-k\right)}^T& 0\\ \mathrm{.Math.}& \mathrm{.Math.}\end{array}\right],\tilde{b}=\left[\begin{array}{c}\mathrm{.Math.}\\ {}_i{u}_i^T{a}_i\\ \mathrm{.Math.}\\ c_i^T{a}_i\\ \mathrm{.Math.}\\ {\left(\cos \left({}_i\right)u_i-k\right)}^T{a}_i\\ \mathrm{.Math.}\end{array}\right],$ whose solution is given by:
=(.sup.T{tilde over (W)}.sup.T{tilde over (W)}).sup.1(.sup.T{tilde over (W)}.sup.T{tilde over (b)}).(19)

Description

BRIEF DESCRIPTION OF DRAWINGS

(1) The various aspects of embodiments disclosed here, including features and advantages of the present invention outlined above are described more fully below in the detailed description in conjunction with the drawings where like reference numerals refer to like elements throughout, in which:

(2) FIG. 1 is a block diagram of the geolocalization apparatus, where block 101 represents the i-th anchor receiver that receives the signal sent from the target. The receiver gives as output the received signal strength indicator (RSS). Block 102 is the i-th anchor receiver that receives the signal sent from the target and calculates the angles of azimuth and elevation, giving these values in the output. Note that anchor nodes can have multiple antennas or directional antennas to obtain AoA measurements, both azimuth angle and elevation. Block 103 is the central node where all data processing is executed (fusion center), using the readings of RSS and AoA that anchors conveyed to this node. The block 104 represents the estimator implemented by the estimation process.

(3) FIG. 2 shows anchor and target locations in a 3-D space where x=[x.sub.x,x.sub.y,x.sub.z].sup.T and a.sub.i=[a.sub.ix,a.sub.iy,a.sub.iz].sup.T represent the coordinates of the target and the i-th anchor, respectively, while d.sub.i, .sub.i and .sub.i denote respectively the distance, azimuth angle and elevation angle between the target and the i-th anchor;

(4) FIG. 3 shows the azimuth angle measurement error for different distances;

(5) FIG. 4 shows the RMSE versus N comparison, when .sub.n.sub.i=6 dB, .sub.m.sub.i=10 deg, .sub.v.sub.i=10 deg. PLE is fixed to =2.5. However, to account for a realistic measurement model mismatch and test the robustness of the new algorithms to imperfect knowledge of the PLE, the true PLE was drawn from a uniform distribution on an interval [2.2; 2.8], i.e., .sub.iU[2.2,2.8], for i=1, . . . , N., =2.5, B=15 m, P.sub.0=10 dBm, d.sub.0=1 M.sub.c=50000;

(6) FIG. 5 shows the RMSE versus .sub.n.sub.i (dB) comparison, when N=4, .sub.m.sub.i=10 deg, .sub.v.sub.i=10 deg, .sub.iU[2.2,2.8], =2.5, B=15 m, P.sub.0=10 dBm, d.sub.0=1 m, M.sub.c=50000;

(7) FIG. 6 shows the RMSE versus .sub.m.sub.i (deg) comparison, when N=4, .sub.n.sub.i=6 dB, .sub.v.sub.i=10 deg, .sub.iU[2.2,2.8], =2.5, B=15 m, P.sub.0=10 dBm, d.sub.0=1 m, M.sub.c=50000.

(8) FIG. 7 shows the RMSE versus .sub.v.sub.i (deg) comparison, when N=4, .sub.n.sub.i=6 dB, .sub.m.sub.i=10 deg, .sub.iU[2.2,2.8], =2.5, B=15 m, P.sub.0=10 dBm, d.sub.0=1 m, M.sub.c=50000.

DETAILED DESCRIPTION OF THE INVENTION

(9) The present application describes the apparatus and a method for RSS/AoA target 3-D localization in wireless networks. Let x.sup.3 be the unknown location of the target and a.sub.i.sup.3, for i=1, . . . , N, be the known location of the i-th anchor. In order to determine the target's location, a hybrid system that combines range and angle measurements is employed. As shown in FIG. 2 x=[x.sub.x,x.sub.y,x.sub.z].sup.T and a.sub.i=[a.sub.ix,a.sub.iy,a.sub.iz].sup.T represent the coordinates of the target and the i-th anchor, respectively, while d.sub.i, .sub.i and .sub.i denote respectively the distance, azimuth angle and elevation angle between the target and the i-th anchor.

(10) The determination of the locations is done using a hybrid system that combines the distance and angle measurements obtained at the blocks 101 and 102 of FIG. 1, respectively. The combination of the two radio signal measurements provides more information for the user and is capable of improving estimation accuracy.

(11) It is assumed that the distance is drawn in 101 from the RSS information exclusively, since ranging based on RSS does not require additional hardware [9]. The noise-free RSS between the target and the i-th anchor is defined as [29, Ch.3]

(12) $\begin{array}{cc}{P}_i\left(W\right)={P_T\left(\frac{d_0}{d_i}\right)}^{}{10}^{\frac{L_0}{10}},\mathrm{for}i=1,\mathrm{.Math.},N,& \left(1\right)\end{array}$
where P.sub.T is the transmit power of the target, L.sub.0 is the path loss value measured at a short reference distance d.sub.0 (d.sub.0d.sub.i), is the path loss exponent (PLE), and d.sub.i is the distance between the target and the i-th anchor. The RSS model in (1) can be rewritten in a logarithmic form as

(13) $\begin{array}{cc}{P}_i=P_0-10{\log }_{10}\frac{d_i}{d_0}+n_i,\mathrm{for}i=1,\mathrm{.Math.},N,& \left(2\right)\end{array}$
where P.sub.0 is the received power (dBm) at d.sub.0, and n.sub.iN(0,.sub.n.sub.i.sup.2) is the log-normal shadowing term modeled as zero-mean Gaussian random variable with variance .sub.n.sub.i.sup.2. Note that P.sub.0 is dependent on P.sub.T. It is assumed that the sensors are static and there is no node and link failure during the computation period, and all sensors can convey their measurements to a central processor represented by block 103 of FIG. 1.

(14) The AoA measurements performed in 102 can be obtained by installing directional antenna or antenna array [15], or video cameras [30]) at anchors. Thus, by applying simple geometry in 102, azimuth and elevation angle measurements are modeled respectively as [15]:

(15) $\begin{array}{cc}{}_i={\tan }^{-1}\left(\frac{x_y-a_{\mathrm{iy}}}{x_x-a_{\mathrm{ix}}}\right)+m_i,\mathrm{for}i=1,\mathrm{.Math.},N,& \left(3\right)\\ {}_i={\cos }^{-1}\left(\frac{x_z-a_{\mathrm{iz}}}{.Math.x-a_i.Math.}\right)+v_i,\mathrm{for}i=1,\mathrm{.Math.},N,& \left(4\right)\end{array}$
where m.sub.iN(0,.sub.m.sub.i.sup.2) and v.sub.iN(0,.sub.v.sub.i.sup.2) are the measurement errors of azimuth and elevation angles, respectively. Given the observation vector =[P.sup.T,.sup.T,.sup.T] (.sup.3N), where P=[P.sub.i], =[.sub.i], =[.sub.i], the conditional probability density function (PDF) is given as:

(16) 0$\begin{array}{cc}p\left(.Math.x\right)=\underset{i=1}{\overset{3N}{.Math.}}\frac{1}{\sqrt{2{}_I^2}}\exp \left\{-\frac{{\left({}_i-f_i\left(x\right)\right)}^2}{2{}_i^2}\right\},\mathrm{where}f\left(x\right)={\left[P_0-10{\log }_{10}\frac{d_i}{d_0},{\tan }^{-1}\left(\frac{x_y-a_{\mathrm{iy}}}{x_x-a_{\mathrm{ix}}}\right),{\cos }^{-1}\left(\frac{x_z-a_{\mathrm{iz}}}{.Math.x-a_i.Math.}\right)\right]}^T\mathrm{and}={\left[{}_{n_i},{}_{m_i},{}_{v_i}\right]}^T.& \left(5\right)\end{array}$

(17) The ML estimate, {circumflex over (x)}, of the unknown location is obtained by maximizing the log of the likelihood function (5) with respect to x [31, Ch. 7], as:

(18) $\begin{array}{cc}\hat{x}=\underset{x}{\arg \min }\underset{i=1}{\overset{3N}{.Math.}}{\frac{1}{{}_i^2}\left[{}_i-f_i\left(x\right)\right]}^2.& \left(6\right)\end{array}$

(19) The above ML estimator (6) is non-convex and does not have a closed-form solution. The 3-D localization method in wireless networks disclosed in this application is implemented in block 103 and aproximates (6) by another estimator whose solution is given in a closed-form, and it is composed by the following steps:
.sub.ixa.sub.id.sub.0 for i=1,. . . ,N,(7)
c.sub.i.sup.T(xa.sub.i)0, for i=1,. . . ,N,(8)
k.sup.T(xa.sub.i)xa.sub.i|cos(.sub.i), for i=1, . . . ,N,(9) where

(20) ${}_i={10}^{\frac{P_1}{10}},={10}^{\frac{P_0}{10}},$ c.sub.i=[sin(.sub.i),cos(.sub.i),0].sup.T and k=[0,0,1].sup.T, and apply Cartesian to spherical coordinates conversion to write xa.sub.i=r.sub.iu.sub.i, where the unit vector u.sub.i=[cos(.sub.i)sin(.sub.i),sin(.sub.i)sin(.sub.i),cos(.sub.i)].sup.T can be obtained from the available AoA measurements; Next apply the described conversion in (7) and (9), and multiply with 1 (formed as u.sub.i.sup.Tu.sub.i), to respectively get:
.sub.iu.sub.i.sup.Tr.sub.iu.sub.id.sub.0.sub.iu.sub.i.sup.T(xa.sub.i)d.sub.0,(10)
and
k.sup.Tr.sub.iu.sub.iu.sub.i.sup.Tr.sub.iu.sub.i cos(.sub.i)(cos(.sub.i)u.sub.ik).sup.T(xa.sub.i)0.(11) To give more importance to nearby links, introduce weights, w=[{square root over (w.sub.i)}], where each w.sub.i is defined as

(21) $\begin{array}{cc}{w}_i=1-\frac{{\hat{d}}_i}{\underset{i=1}{\overset{N}{.Math.}}{\hat{d}}_i},\mathrm{with}{\hat{d}}_i=d_0{10}^{\frac{P_0-P_i}{10}}& \left(12\right)\end{array}$ being the ML estimate of the distance obtained from (2);

(22) The reason for defining the weights in this manner is because both RSS and AoA short-range measurements are trusted more than long ones. The RSS measurements have constant multiplicative factor with range [9], which results in a greater error for remote links in comparison with the nearby ones. In FIG. 2, .sub.i and {circumflex over ()}.sub.i represent respectively the true and the measured azimuth angle between an anchor and two targets, x.sub.1 and x.sub.2, located along the same line, but unequally distant from the anchor. The goal is to determine the locations of the two targets based on the available information. Consequently, the location estimates of the two targets are at points {circumflex over (x)}.sub.1 and {circumflex over (x)}.sub.2. However, from FIG. 2, it can be seen that the estimated location of the target physically closer to the anchor ({circumflex over (x)}.sub.1) is much closer to its true location than the one further away. Next according to the WLS criterion and (10), (8), (11) and (12), obtain the following estimator:

(23) $\begin{array}{cc}\hat{x}=\arg \underset{x}{\min }\underset{i=1}{\overset{N}{.Math.}}{w}_i{\left({}_i{u}_i^T\left(x-a_i\right)-d_0\right)}^2+\underset{i=1}{\overset{N}{.Math.}}{w_i\left(c_i^T\left(x-a_i\right)\right)}^2+\underset{i=1}{\overset{N}{.Math.}}{w_i\left({\left(\cos \left({}_i\right)u_i-k\right)}^T\left(x-a_i\right)\right)}^2,& \left(13\right)\end{array}$ which can be writen as

(24) $\begin{array}{cc}\underset{x}{\mathrm{minimize}}{.Math.W\left(\mathrm{Ax}-b\right).Math.}^2,& \left(14\right)\end{array}$ where W=I.sub.3.Math.diag(w), with .Math. denoting the Kronecker product, and

(25) $A=\left[\begin{array}{c}\mathrm{.Math.}\\ {}_i{u}_i^T\\ \mathrm{.Math.}\\ c_i^T\\ \mathrm{.Math.}\\ {\left(\cos \left({}_i\right)u_i-k\right)}^T\\ \mathrm{.Math.}\end{array}\right],b=\left[\begin{array}{c}\mathrm{.Math.}\\ {}_i{u}_i^T+d_0\\ \mathrm{.Math.}\\ c_i^T{a}_i\\ \mathrm{.Math.}\\ {\left(\cos \left({}_i\right)u_i-k\right)}^T{a}_i\\ \mathrm{.Math.}\end{array}\right],$ with the closed form solution given by
{circumflex over (x)}=(A.sup.TW.sup.TWA).sup.1(A.sup.TW.sup.Tb).(15) We label (15) as WLS1 in the remaining text.

(26) When the transmitted power information from block 101 is unavailable, which corresponds to not knowing P.sub.0 in (2), the estimation is perfomed by the estimator of block 104 that introduces weights {tilde over (w)}=[{square root over ({tilde over (w)}.sub.i)}], such that

(27) $\begin{array}{cc}{\tilde{w}}_i=1-\frac{P_i}{\underset{i=1}{\overset{N}{.Math.}}{P}_i}.& \left(16\right)\end{array}$

(28) From the WLS principle and (10), (8), (11) and (16), we get:

(29) $\begin{array}{cc}\left(\hat{x},\hat{}\right)=\underset{x,}{\arg \min }\underset{i=1}{\overset{N}{.Math.}}{{\tilde{w}}_i\left({}_i{u}_i^T\left(x-a_i\right)-d_0\right)}^2+\underset{i=1i}{\overset{N}{.Math.}}{{\tilde{w}}_i\left(c_i^T\left(x-a_i\right)\right)}^2+\underset{i=1}{\overset{N}{.Math.}}{{\tilde{w}}_i\left({\left(\cos \left({}_i\right)u_i-k\right)}^T\left(x-a_i\right)\right)}^2,& \left(17\right)\end{array}$ which can be rewritten as:

(30) $\begin{array}{cc}\underset{y={\left[x^T,\right]}^T}{\mathrm{minimize}}{.Math.\tilde{W}\left(\tilde{A}y-\tilde{b}\right).Math.}^2,& \left(18\right)\end{array}$ where {tilde over (W)}=I.sub.3.Math.diag({tilde over (w)}), and

(31) 0$\tilde{A}=\left[\begin{array}{cc}\mathrm{.Math.}& \mathrm{.Math.}\\ {}_i{u}_i^T& -d_0\\ \mathrm{.Math.}& \mathrm{.Math.}\\ c_i^T& 0\\ \mathrm{.Math.}& \mathrm{.Math.}\\ {\left(\cos \left({}_i\right)u_i-k\right)}^T& 0\\ \mathrm{.Math.}& \mathrm{.Math.}\end{array}\right],\tilde{b}=\left[\begin{array}{c}\mathrm{.Math.}\\ {}_i{u}_i^T{a}_i\\ \mathrm{.Math.}\\ c_i^T{a}_i\\ \mathrm{.Math.}\\ {\left(\cos \left({}_i\right)u_i-k\right)}^T{a}_i\\ \mathrm{.Math.}\end{array}\right],$ whose solution is given by:
=(.sup.T{tilde over (W)}.sup.T{tilde over (W)}).sup.1(.sup.T{tilde over (W)}.sup.T{tilde over (b)}).(19) We will refer to (19) as WLS2 in the further text.

(32) Assuming that K is the maximum number of steps in the bisection procedure used in [21], Table 1 provides an overview of the considered algorithms together with their worst case computational complexities.

(33) TABLE-US-00001 TABLE 1 Summary of the Considered Algorithms Algorihm Description Complexity WLS1 The proposed WLS for known P.sub.T O(N) WLS2 The proposed WLS for unknown P.sub.T O(N) SOCP The SOCP method in [10] for known P.sub.T O(N.sup.3.5) SR-WLS The bisection method in [11] for known P.sub.T O(KN) LS The LS method in [5] for known P.sub.T O(N)

(34) Table 1 shows that the computational complexity of the considered methods depends mainly on the network size, i.e., the total number of anchors in the network. This property is a characteristic of methods operating in a centralized manner [21], where all information is conveyed to a central processor. From Table 1, we can see that the computational complexity of the proposed methods is linear.

(35) Performance of the proposed algorithm was verified through computer simulations. It was assumed that radio measurements were generated by using (2), (3) and (4). All sensors were deployed randomly inside a box with an edge length B=15 m in each Monte Carlo M.sub.c) run. The reference distance is set to d.sub.0=1 m, the reference path loss to P.sub.0=10 dBm, and the PLE was fixed to =2.5. However, to account for a realistic measurement model mismatch and test the robustness of the new algorithms to imperfect knowledge of the PLE, the true PLE was drawn from a uniform distribution on an interval [2.2,2.8], i.e., .sub.iU[2.2,2.8] for i=1, . . . , N. For SR-WLS method in [11], K=30 is used. The performance metric used here is the root mean square error (RMSE), defined as

(36) $\mathrm{RMSE}=\sqrt{\underset{i=1}{\overset{M_c}{.Math.}}\frac{{.Math.x_i-{\hat{x}}_i.Math.}^2}{M_c}},$
where {circumflex over (x)}.sub.i denotes the estimate of the true target location, x.sub.i, in the i-th M.sub.c run.

(37) FIG. 3 shows that all methods benefit from additional information introduced by increasing N. Although computationally extremely light, the new method exhibits superior performance for all N, as well as the robustness to not knowing P.sub.T. It is important to note that in FIG. 3 the noise powers were set to a relatively high value, and that for such a setting our method behaves exceptionally well.

(38) FIGS. 4, 5 and 6 show the quality of different types of measurements on the performance of the considered approaches. More precisely, FIGS. 4, 5 and 6 illustrate the RMSE versus .sub.n.sub.i (dB), .sub.m.sub.i (deg) and .sub.v.sub.i (deg) comparison, respectively, for N=4. From these figures, one can observe that the performance of all methods impairs as the quality of a certain measurement drops, as expected. However, not all measurements have equal impact on the performance of the considered methods. For example, the quality of the RSS measurement has very little influence on the performance of the proposed method, while the quality of AoA measurements have greater impact on its performance. This is due to the fact that the new method relies more on the quality of angle measurements than range ones in its derivation. Nevertheless, the performance deterioration is moderate for such a wide span of noise power.

(39) The above description of illustrated embodiments is not intended to be exhaustive or limited by the disclosure. While specific embodiments of, and examples are described herein for illustrative purposes, various equivalent modifications are possible, as those skilled in the relevant art will recognize.