Passive localization of signal source based on wireless sensor network in the air

Abstract

Passive localization of the wireless signal source attracts a considerable level of research interest for its wide applications in modern wireless communication systems. To accurately locate the signal source passively in the downtown area, sensors are carried on the unmanned aerial vehicles flying in the air, where the wireless sensor network can be established with an optimal geometry configuration conveniently. In this case, the influence of multipath fading can be avoided and the time difference of arrival measurement can be estimated precisely in Rician channel. By employing the operating center as a calibration source to refine the positions of the unmanned aerial vehicles, we present a simplified formulation of the time difference of arrival localization method according to the min-max criterion. To accurately estimate the position of the source, the nonlinear equations are relaxed using semidefinite programming to obtain an initial solution, which is utilized as the starting point of the iterative algorithm to refine the solution. In the simulation section, the validity and the robustness of the proposed methods are verified through the performance comparison with the Cramer–Rao lower bound.

Introduction

With the rapid development of information technologies and wireless communications, passive localization of the signal source has been of considerable interest in several fields such as wireless communications,^1–3 sensor networks,^4–6 microphone arrays,^7,8 sonar,^9,10 and electronic warfare (EW).^11–13 The passive localization technology only locates the position of the source through receiving the electromagnetic signals, which does not need the surveillance equipment transmitting a radio signal to the measured target. Because of the far operating distance and the good concealment advantage, passive localization becomes an important component of the electronic counter measure (ECM).^14,15 Among various classical algorithms for the signal source passive localization, the method of time difference of arrival (TDOA) based on wireless sensor network (WSN) is popular and has extensive application. The TDOA-based localization method is usually accomplished based on a two-step approach, that is, the “association” step and the “information-fusion” step; certain parameters such as the TDOA measurement at the receivers are obtained from the target source emissions. In the “information-fusion” step, the position of the target source is estimated by solving a set of nonlinear equations. ¹⁶

Over the past decades, WSN as a significant technology has attracted considerable research interest.^17,18 Due to small size, low cost, wireless network, and deployment in large numbers, WSN can provide unprecedented opportunities for information collection, monitoring, and device control.^19,20 Although the passive localization method of TDOA based on WSN has an excellent accuracy and response time, the multipath transmission channel and shadow fading worsen the localization performance greatly in the downtown area, which is largely because the sensors are bedded on the ground. However, combining the WSN and unmanned aerial vehicle (UAV) technology, these drawbacks can be overcome for the perfect environment in the free space. That means the sensors are loaded on the UAV flying in the air to capture the signal and locate the signal source, which could ensure that the transmission channel is line of sight (LOS). In practice, this scenario is already conditional on the implementation of the UAV and the WSN for the modern technologies.

The technology of the signal source passive localization using a receiver loaded on UAVs has been studied extensively. Many previous works focused on the issue of passively localizing the radar source. Mallick et al. ²¹ utilize the unscented Kalman filter (UKF) and particle filter (PF) to accurately estimate the radar’s position with UAVs capturing the signals in the air. Okello and Musicki ²² proposed the method to accurately locate the pulse signal source such as the radar source, where the leading edge of the electromagnetic pulse is detected to estimate the time information in the “association” step using an EW support sensor loaded on a single UAV. To locate the radar source more precisely, we introduced the concept of passive synthetic aperture antenna array (PSAAA) to passively localize the hostile radar’s position based on a single small-sized aircraft in Hao et al. ²³ Taking full advantage of the aircraft movement and the radar antenna circular rotation, an effective method is proposed to estimate the virtual time difference of arrival (VTDOA), while the PSAAA technology is unsuitable to passively localize the radio frequency (RF) signal source. It should be pointed out that the three methods mentioned above are exploited only about the radar source, which is unavailable for the continuous signal source. In Liang and Samn, ²⁴ block phase estimation (BPE) method is exploited to estimate the wireless channel based on two parallel antennas equipped on a small flying UAV, then the parameters of time delay are obtained based on the channel comparison. Obviously, the localization accuracy is determined by the distance between the parallel antennas and the wavelength of the RF signal. Kim et al. ²⁵ presented a regional localization scheme using UAV for the battlefield environment in the situation where the global navigation satellite system (GNSS) is locally unavailable. It is challenging to provide the exact positions of the nodes in such a scenario. For the excellent performance of the extensions of the Kalman filter algorithms, in the “information-fusion” step, Fletcher et al. ²⁶ obtained the position of the moving emitter using the extended Kalman filter (EKF) and a UKF, Plett et al. ²⁷ propose a sensor-fusion methodology based on sigma-point Kalman filtering (SPKF) techniques to localize a mobile target. Although these two methods are based on the UAV platform, the optimal geometry configuration and the positional errors of the involved sensors are out of consideration, which could generate an inevitable error in the final localization result.

To passively and accurately locate the RF signal source, a group of UAVs are utilized flying in the air to estimate the position of the source in this study. The sensors loaded on the UAV form a micro WSN in the air, and taking full advantage of the controllable UAV, we present a more precise estimation method to locate the signal source in the downtown area. The main contributions of this article are as follows:

1. Optimal geometry distribution of the UAVs in the air. The flight paths of the UAVs in the air are controlled under the instruction from the operating center, which means that the optimal geometry distribution of the sensors on the UAV can be satisfied easily under manual control. We present the optimal geometry distribution of the UAV during passive localization of the signal source, and its more accurate performance is verified in the simulation.

2. Calibration of the positional errors of the UAVs. Due to the movement of the UAV flying, the positions of the UAVs contain certain offset, which introduce a considerable estimation error in the final localization result. In this study, we utilize the operating center of the UAVs as a calibration source to refine the positions of the UAVs.

3. More accurate localization algorithm. In the “information-fusion” step, the semidefinite relaxation technique is exploited to resolve the quadratic optimization problems and obtain the initial position of the source, which is utilized as the starting point of the numerical iteration algorithm to refine the obtained position of the signal source.

The rest of the article is organized as follows: Section “System model” presents the system model of the signal source passive localization based on the optimal geometry distribution of the UAVs, which can form a WSN in the air. In section “Source localization based on TDOA,” the positional errors of the UAVs are refined and the TDOA measurement is estimated first, then the relaxation technique of semidefinite programming (SDP) and the iteration method of the Powell minimum search algorithm are given. Section “Simulations and performance analysis” illustrates the simulation results and performance comparison. The conclusions are drawn in section “Conclusion.”

System model

In three-dimensional (3D) space, at least four sensors are needed for the TDOA-based localization system to locate the signal source. Suppose the stationary signal source is fixed at $\mathit{q}=[x,y,z{]}^T$ , the position of the sensor is $p_i=[x_i,y_i,z_i{]}^T,i=1,\dots ,N$ . In practical, the localization performance is often affected seriously by the geometric distribution of the involved sensors. Theoretically, the optimal geometry distribution of the sensors involved in TDOA-based signal source localization can be obtained based on the Fisher information matrix (FIM) of the time delay information. As mentioned in Huang et al., ²⁸ the FIM of the TDOA-based signal source localization can be expressed as

$$\begin{array}{c}{F}_{\mathrm{mn}}=\frac{\partial {(\mathbf{H}\mathit{r}(\mathit{q}))}^T}{\partial u_m}{\Omega }^{-1}\frac{\partial (\mathbf{H}\mathit{r}(\mathit{q}))}{\partial u_n},m,n\in \{1,2,3\}\hfill \end{array}$$

(1)

where $F_{\mathrm{mn}}$ is the element of the FIM ( $m$ rows, $n$ columns), $u_1,u_2,u_3$ represent the coordinates of the signal source $\mathit{q}$ in the 3D space, and the covariance matrix $\Omega ={\sigma }_0^2\mathbf{H}·\mathrm{Diag}(\mathit{r}(\mathit{q}){)}^{\alpha }·{\mathbf{H}}^T$ , where $\alpha$ is the path loss parameter of the wireless signal transmission and ${\sigma }_0^2$ is the variance of the reference noise, $\mathit{r}(\mathit{q})=[r_1,r_2,\dots ,r_N{]}^T$ , $r_i=\Vert \mathit{q}-{\mathit{p}}_{\mathit{i}}\Vert ,i=1,\dots ,N$ , where $N$ is the total number of the sensors. If the first sensor is taken as the reference node, then

$$\begin{array}{c}\mathbf{H}=\left[\begin{array}{cccc}-1\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{-1}& 0\hfill & 1\hfill & 0\hfill \\ \multicolumn{1}{c}{-1}& 1\hfill & 0\hfill & 1\hfill \end{array}\right]\hfill \end{array}$$

(2)

The system model of the signal source passive localization in the downtown environment based on the WSN loaded on the UAV group is shown in Figure 1. Four homogeneous sensors $(N=4)$ are carried on the UAVs flying in the air, and their positions can be obtained using a self-positioning system on the UAV such as the global positioning system (GPS). The exact position of the operating center is ${\mathit{p}}_{\mathit{o}}=[x_o,y_o,z_o]$ . Assume that the clock of the four sensors and the operating center are rigorously synchronized. However, for the uncooperative target signal source this assumption is unnecessary, which means that the signal source can be unsynchronized with the sensors and the operating center.

Figure 1.

System model of signal source localization in the downtown environment. (The procedure of localization: source → UAVs which measure source wave → operating center which executes the localization algorithm.)

Obviously, when the sensors are loaded on the UAVs in the air, the transmission channel between the signal source and the sensors is Rician channel, which means that the received noise is distance dependent. Employing the D-optimality algorithm to maximize the determinant of the FIM, Huang et al. ²⁸ claimed that the localization system has a better accuracy when the four sensors are distributed at the vertices of any regular tetrahedron. In other words, the optimal geometry configuration of the UAVs is a regular tetrahedral distribution, which can be achieved through the instructions emitted from the operating center of the UAVs. In the following sections, we assume all the four UAVs flying parallel and keep the regular tetrahedral distribution in the air under the control of the operating center. Meanwhile, the UAVs are hovering in the same place when they receive the signal emitted from the target source.

As depicted in Figure 1, when the uncooperative signal source radiates the wireless signal into the space, the synchronized sensors loaded on the UAVs capture it immediately, and then all the received signals are sent to the operating center, where the localization algorithm is executed and the position of the signal source can be obtained. The sensors connect with each other forming a WSN in the air, which is useful to maintain synchronized time and transmit data to the operating center cooperatively. Besides execution of the localization algorithm, the operating center also sends the instructions and the calibrating signal to all the sensors continuously. Generally speaking, when we want to localize an unknown signal source in the downtown environment, it is usually an emergency circumstances, such as quickly prohibiting the harmfully illegal radio or the city anti-terrorism, where the higher the localization accuracy (e.g., less than 10 m or even), the better. The positional errors of the involved sensors and the method to accurately resolve the nonlinear equation are the major sources of the localization error, which would be mainly settled in the following section.

As for the system model depicted in Figure 1, there are some notes that should be mentioned before the signal source passive localization.

1. Even though the UAVs are assumed hovering in the air during signal collection, their positions are imprecise for the limited condition and the atmospheric fluctuation. Fortunately, we can refine the uncertain positions using the operating center as a calibration source.

2. Due to the WSN loaded on the UAV group flying over the urban area, the transmission channel between the signal source and the sensors is Rician fading. In the following sections, we do not consider the influence of multipath transmission.

3. Because the UAVs are flying with an unchanged geometrical configuration in the sky, the effect caused by the frequency difference of arrival (FDOA) is too small and can be neglected.

4. Just as indicated by Yang et al., ²⁹ although selecting an arbitrary sensor as the reference note does not change the Cramer–Rao lower bound (CRLB), localization algorithms are often biased and the accuracy of a specific localization method may change with different reference sensors. In this article, the sensor ${\mathit{p}}_1$ at the highest vertex of the regular tetrahedron is taken as the reference node. ²⁸

Source localization based on TDOA

In this section, the positions of the UAVs are refined based on the traditional time of arrival (TOA) method first, where the operating center of the UAVs is taken as the calibration source. Based on the TDOA measurement, the TDOA localization function is established as a nonconvex problem. Finally, the position of the signal source is refined using the Powell minimum search algorithm, where the initial solution is obtained by the SDP method in advance.

Refining the positions of the UAVs

Generally, UAV has many advantages, such as light weight and small size. However, many necessary devices are needed to be loaded on the UAV, such as signal receiver, control system, needful data processing unit, power supply unit, and self-positioning system. Therefore, limited by the power supply and the carrying capacity of the UAVs, the obtained UAVs’ positions are uncertain. Meanwhile, the atmospheric motion also has an influence on the self-localization of the UAVs while it is receiving the signal. It is well known that uncertain sensor positions have a significant impact on the TDOA localization result.

To accurately locate the signal source, the obtained positions of the UAVs should be refined before post-processing. Here, the operating center of the UAVs is taken as a calibration source to refine the UAVs’ positions. The operating center transmits a calibration signal and its position ${\mathit{p}}_{\mathbf{o}}$ to the UAVs when it sends out the instructions, and then the sensors refine their positions based on the TOA algorithm.

With the synchronized clock between the sensors on the UAVs and the operating center, the TOA of the calibration signal arrival at the sensor ${\mathit{p}}_{\mathit{i}}$ is

$$t_i=\frac{1}{c}\Vert {\mathit{p}}_{\mathit{i}}-{\mathit{p}}_{\mathit{o}}+\Delta {\mathit{p}}_{\mathit{i}}\Vert +e_i,i=1,\dots ,4$$

(3)

where $c$ is the signal propagation speed and $\Delta {\mathit{p}}_{\mathit{i}}$ denotes the positional error of the sensor ${\mathit{p}}_{\mathit{i}}$ . In addition, $e_i$ is the measurement noise, which is independent and identically distributed (i.i.d.) Gaussian noise with zero mean and the variance ${\sigma }^2$ . Consequently, the conditional probability density of the TOA measurement is

$$\begin{array}{c}p(t_1,t_2,t_3,t_4|{\mathit{p}}_{\mathit{0}})={\left(2\pi {\sigma }^2\right)}^2\hfill \\ \exp (-\frac{1}{2{\sigma }^2}\sum _{i=1}^4{(t_i-\frac{1}{c}\Vert {\mathit{p}}_{\mathit{i}}-{\mathit{p}}_{\mathit{0}}+\Delta {\mathit{p}}_{\mathit{i}}\Vert )}^2)\hfill \end{array}$$

(4)

Obviously, the maximum likelihood estimate (MLE) of the position error parameter $\Delta {\mathit{p}}_{\mathit{i}}$ can be calculated by

$$\begin{array}{c}\Delta {\hat{\mathit{p}}}_{\mathit{i}}=\underset{\Delta {\mathit{p}}_{\mathit{i}}}{\arg min}\sum _{i=1}^4{(t_i-\frac{1}{c}\Vert {\mathit{p}}_{\mathit{i}}-{\mathit{p}}_{\mathit{0}}+\Delta {\mathit{p}}_{\mathit{i}}\Vert )}^2\hfill \end{array}$$

(5)

Then the UAVs’ positions can be refined based on the obtained $\Delta {\hat{\mathit{p}}}_{\mathit{i}}$ . For simplicity, the symbol ${\mathit{p}}_{\mathit{i}}$ is utilized to represent the accurate positions of the UAVs in the following sections.

TDOA estimation of source localization

Time delay estimation (TDE) is a basic technique for the TDOA-based signal source passive localization. Many techniques exist for TDE, but the most popular and useful algorithms in practice are based on the generalized cross-correlation (GCC) algorithm. The delay estimate between two sensors is obtained as the time lag that maximizes the cross-correlation between the filtered versions of the received signals. This method is well studied and performs fairly well in moderately noisy and nonreverberant environments. However, if the TDE is estimated directly using the GCC algorithm, it will produce large errors in the result. To reduce the influence of noise and interference, we estimate the TDE with the inverse Fourier transform of the product of the cross-spectral density and the phase transforming in this article.

As shown in Figure 1, the signal emitted from the source $\mathit{q}$ is captured by the sensors of the WSN loaded on the UAVs. Assume that the received signal in the reference node ${\mathit{p}}_1$ is $s_1(t)=s(t-{\tau }_1)$ and in the other nodes ${\mathit{p}}_{\mathit{j}}$ is $s_j(t)=s(t-{\tau }_j),j=2,3,4$ . Obviously, the TDOA measurement is $\Delta {\tau }_j={\tau }_j-{\tau }_1$ . The generalized cross-correlation with phase transform (GCC-PHAT) algorithm is employed to perform the TDOA measurement of the received signal, which is a classical and efficient approach to estimate the time delay information for the Rician channel. In practical applications, the cross-correlation function of $s(t)$ and $s(t-{\tau }_j)$ can be calculated through the fast inverse discrete Fourier transformation of the cross power spectrum $G_{s_1,s_j}(f)$ . In order to improve the accuracy of the delay estimate $\Delta {\tau }_j$ , $s_1$ and $s_j$ should be pre-filtered, respectively, by $H_1(f)$ and $H_2(f)$ according to the GCC-PHAT algorithm, and $H_1(f)H_2^{*}(f)=1/|G_{s_1,s_j}(f)|$ .

Consequently, the GCC of $s_1$ and $s_j$ becomes

$$R_{s_1,s_j}(\tau )\stackrel{\Delta }{=}\underset{-\infty }{\overset{+\infty }{\int }}\frac{G_{s_1,s_j}(f)}{|G_{s_1,s_j}(f)|}{e}^{j2\pi f\tau }\mathrm{df}$$

(6)

Maximizing equation (6) provides an accurate estimation of the TDOA $\Delta {\tau }_j$

$$\Delta {\hat{\tau }}_j=\underset{\tau }{\arg \max }[R_{s_1,s_j}(\tau -\Delta {\tau }_j)]$$

(7)

Particularly, due to the special geometry configuration of the UAV group, it is probable to have an interesting phenomenon that all of the obtained TDOA values are $\Delta {\hat{\tau }}_j=0,j=2,3,4$ when ${\mathit{p}}_1$ is chosen as the reference node. In this case, we can confirm that the position of the signal source $\mathit{q}$ is at the subpoint of the reference sensor ${\mathit{p}}_1$ on the ground. For the other case, we should estimate its position by solving the nonlinear equations.

Establishing the TDOA localization function

Utilizing the exact positions of the sensors ${\mathit{p}}_{\mathit{i}},i=1,\dots ,4$ and the estimated TDOA measurement $\Delta {\hat{\tau }}_j,j=2,3,4$ , we can obtain a simpler model

$$\begin{array}{c}\Delta {\hat{\tau }}_j=\frac{1}{c}(\Vert {\mathit{p}}_{\mathit{j}}-\mathit{q}\Vert -\Vert {\mathit{p}}_{\mathit{1}}-\mathit{q}\Vert )+e_j\hfill \end{array}$$

(8)

where $e_j$ are i.i.d. Gaussian with zero mean and the variance ${\sigma }^2$ .

Modifying the problem formulation by rewriting equation (6) as

$$\begin{array}{c}\Delta {\hat{\tau }}_j+\frac{1}{c}\Vert {\mathit{p}}_1-\mathit{q}\Vert =\frac{1}{c}\Vert {\mathit{p}}_{\mathit{j}}-\mathit{q}\Vert +e_j\hfill \end{array}$$

(9)

we obtain

$$\begin{array}{c}{(\Delta {\hat{\tau }}_j+\frac{1}{c}\Vert {\mathit{p}}_{\mathit{1}}-\mathit{q}\Vert )}^2\hfill \\ -\frac{1}{c^2}\Vert {\mathit{p}}_{\mathit{j}}-\mathit{q}{\Vert }^2=(\frac{2}{c}\Vert {\mathit{p}}_{\mathit{j}}-\mathit{q}\Vert +e_j)e_j\hfill \end{array}$$

(10)

As for $j=2,3,4$ , without channel noise, the right-hand side of equation (10) is of course zero. Therefore, the approach to estimate the position $\mathit{q}$ of the signal source is to minimize the maximum matching error between the squares of the time delay of the sensor ${\mathit{p}}_{\mathit{j}}$ and the reference sensor ${\mathit{p}}_{\mathit{1}}$ .

To obtain a globally convergent solution, the min-max criterion will be adopted to find a simplified formulation

$$\hat{q}=\underset{q}{\arg \min }\underset{j=2,3,4}{\max }|\frac{1}{c^2}{\Vert {\mathit{p}}_{\mathit{j}}-\mathit{q}\Vert }^2-{(\frac{1}{c}\Vert {\mathit{p}}_1-\mathit{q}\Vert +\Delta {\hat{\tau }}_j)}^2|$$

(11)

It is challenging to resolve a nonconvex problem like equation (11). We introduce a simple relaxation approach to translate the issue into a convex optimization problem. To further refine the localization accuracy, the estimation will be taken as the initial starting point of the iterative algorithm.

SDP relaxation formulation of TDOA

The direct solution of the TDOA localization function in equation (11) is a nonconvex optimization problem. In Yang et al., ²⁹ a novel relaxed convex algorithm was proposed, in which all the TDOA measurements are involved. However, its reliance on independent TDOA noise assumption makes it more sensitive to the TDOA model in which the noises are correlated. In addition, the complexity can be quite high by including all the TDOA measurements.

To develop a simpler TDOA algorithm, we can designate a single sensor node as the reference node ${\mathit{p}}_1$ , consider only the TDOA between this one and the remaining sensors, and obtain the TDOA localization function as equation (11). The SDP algorithm is an effective method to resolve the nonlinear equation and has wide applications in the TDOA localization. It can avoid the local convergence problem if SDP is tight enough and thus it is possible to obtain the optimal solution. Generally speaking, the SDP algorithm has a better performance than the two-step weighted least squares (TWLS) algorithm. ³⁰ Here, we employ the SDP algorithm to resolve the TDOA localization function in equation (11) first and then refine the localization result using the iterative algorithm to cater to the requirement of high localization accuracy in the downtown.

Three new auxiliary variables are introduced into the TDOA problem, which are the inner product $q\prime ={\mathit{q}}^T\mathit{q}$ and two additional variables $d_j=\Vert {\mathit{p}}_{\mathit{j}}-\mathit{q}\Vert$ and $d\prime =d_1^2$ . Accordingly, the nonconvex optimization problem (equation 9) can be translated into

$$\begin{array}{c}\underset{\mathit{p},p\prime ,d_j,d\prime }{\mathrm{minimize}\epsilon }\hfill \\ \mathrm{s}.\mathrm{t}.-\epsilon <\frac{1}{c^2}\mathrm{Trace}\left(\left[\begin{array}{cc}\mathit{I}\hfill & \mathit{q}\hfill \\ \multicolumn{1}{c}{{\mathit{q}}^T}& q\prime \hfill \end{array}\right]\left[\begin{array}{cc}{\mathit{p}}_{\mathit{j}}^T{\mathit{p}}_{\mathit{j}}\hfill & -{\mathit{p}}_{\mathit{j}}\hfill \\ \multicolumn{1}{c}{-{\mathit{p}}_{\mathit{j}}^T}& 1\hfill \end{array}\right]\right)\hfill \\ -\frac{1}{c^2}\mathrm{Trace}\left(\left[\begin{array}{cc}1\hfill & d_1\hfill \\ \multicolumn{1}{c}{d_1}& d\prime \hfill \end{array}\right]\left[\begin{array}{cc}{s}_j^2\hfill & s_j\hfill \\ \multicolumn{1}{c}{s_j}& 1\hfill \end{array}\right]\right)<\epsilon \hfill \\ j=2,3,4,s_j=c\Delta {\hat{\tau }}_j\hfill \end{array}$$

(12)

Obviously, the two auxiliary variables $q\prime$ and $d_j^2$ are not affine in equation (12). To guarantee the whole formulation convex, the auxiliary variables $q\prime$ and $d_j^2$ are relaxed to inequalities $q\prime \succcurlyeq {\mathit{q}}^T\mathit{q}$ and $d\prime \succcurlyeq d_1^2$ . Furthermore, these inequality restrictions can be converted into linear matrix inequalities

$$\left[\begin{array}{cc}\mathit{I}\hfill & \mathit{q}\hfill \\ \multicolumn{1}{c}{{\mathit{q}}^T}& q\prime \hfill \end{array}\right]\succcurlyeq 0,\left[\begin{array}{cc}1\hfill & d_1\hfill \\ \multicolumn{1}{c}{d_1}& d\prime \hfill \end{array}\right]\succcurlyeq 0$$

(13)

Afterwards, the variable $d\prime =d_1^2$ can be expressed in terms of ${\mathit{p}}_1$ , $\mathit{q}$ , and $q\prime$ as $d\prime ={\mathit{p}}_1{\mathit{p}}_1^{\mathit{T}}-2{\mathit{p}}_1^{\mathit{T}}\mathit{q}+p\prime$ . The problem can be formulated into

$$\begin{array}{c}\begin{array}{c}\underset{\mathit{p},p\prime ,d_j,d\prime }{\mathrm{minimize}\epsilon }\hfill \\ \mathrm{s}.\mathrm{t}.-\epsilon <\mathrm{Trace}\left(\left[\begin{array}{cc}\mathit{I}\hfill & \mathit{q}\hfill \\ \multicolumn{1}{c}{{\mathit{q}}^T}& q\prime \hfill \end{array}\right]\left[\begin{array}{cc}{\mathit{p}}_j{\mathit{p}}_j^T\hfill & -{\mathit{p}}_{\mathit{j}}\hfill \\ \multicolumn{1}{c}{-{\mathit{p}}_{\mathit{j}}^T}& 1\hfill \end{array}\right]\right)\hfill \\ -\mathrm{Trace}\left(\left[\begin{array}{cc}1\hfill & d_1\hfill \\ \multicolumn{1}{c}{d_1}& d\prime \hfill \end{array}\right]\left[\begin{array}{cc}{s}_j^2\hfill & s_j\hfill \\ \multicolumn{1}{c}{s_j}& 1\hfill \end{array}\right]\right)<\epsilon \hfill \\ j=2,3,4\hfill \\ \begin{array}{c}\left[\begin{array}{cc}\mathit{I}\hfill & \mathit{q}\hfill \\ \multicolumn{1}{c}{{\mathit{q}}^T}& q\prime \hfill \end{array}\right]\succcurlyeq 0,\left[\begin{array}{cc}1\hfill & d_1\hfill \\ \multicolumn{1}{c}{d_1}& d\prime \hfill \end{array}\right]\succcurlyeq 0\hfill \\ d\prime ={\mathit{p}}_1{\mathit{p}}_1^{\mathit{T}}-2{\mathit{p}}_1^{\mathit{T}}\mathit{q}+p\prime \hfill \end{array}\hfill \end{array}\hfill \end{array}$$

(14)

The optimization problem explained in equation (14) contains both linear equality and inequality constraints, which could be resolved using the classic convex optimization methods such as the interior point. ³¹ Because without loss of information, no additional post-processing is necessary for the relaxation method presented above, which could directly yield the estimated position of the source $\mathit{q}\prime$ .

Refining the approximation solution

Generally, an approximation solution can be obtained by the SDP algorithm, which is close to the optimal solution. To further refine the obtained position $\mathit{q}\prime$ , it is necessary to take the SDP solution as the initial starting point of the traditional nonlinear search method. For the nonconvex problem as equation (11), the Powell minimum search algorithm initialized with $\mathit{q}\prime$ is employed to refine the estimated result.

The Powell minimum search algorithm is a strictly conjugate direction algorithm, which does not need the objective function differentiable and no derivatives. The method could minimize the objective function by a bidirectional search along each search vector, in turn. Then the new solution is expressed as a linear combination of the search vectors. The new displacement vector becomes a new search vector and is added to the end of the search vector list. The Powell minimum search algorithm is useful to calculate the local minimum of a continuous but complex objective function.

Because the mentioned algorithm is a classical mathematical iteration method, we present only its procedure here. For simplicity, equation (11) is replaced with $f(\mathit{q})$ in the procedure. The refined position $\hat{q}$ is the final estimation of the signal source position.

Algorithm 1.

Procedure of the Powell minimum search for the TDOA localization.

Require: The accurate position of the sensor on UAVs, ${\mathit{p}}_{\mathit{i}},i=1,2,3,4$ ; The obtained TDOA measurement, $\Delta {\hat{\tau }}_j,j=2,3,4$ ; The obtained source position according to SDP algorithm, $\mathit{q}\prime$ ; The transmission speed of the electromagnetic wave, $c$ ; ${\mathit{u}}_1=(1,0,0),{\mathit{u}}_2=(0,1,0),{\mathit{u}}_3=(0,0,1)$ ; Ensure: Refined signal source potion, $\hat{q}$ ;  1. Initialize the counter $n=0$ and set ${\mathit{x}}_{\mathit{o}}=\mathit{q}\prime$ ;  2. For $k=1,2,3$ , find the value of $\eta ={\eta }_k$ that minimizes $f({\mathit{x}}_{\mathit{k}-1}+\eta {\mathit{u}}_{\mathit{k}})$ , and set ${\mathit{x}}_{\mathit{k}}={\mathit{x}}_{\mathit{k}-1}+\eta {\mathit{u}}_{\mathit{k}}$ ;  3. Set $u_j=u_{m+1}$ for $m=1,2$ and set ${\mathit{u}}_{\mathit{n}}={\mathit{x}}_{\mathit{n}}-{\mathit{x}}_{\mathit{o}}$ ;  4. Increment the counter $i=i+1$ ;  5. Find the value of $\eta ={\eta }_{\min }$ that minimizes $f({\mathit{x}}_{\mathit{o}}+\eta {\mathit{u}}_3)$ and set ${\mathit{q}}_{\mathit{i}}={\mathit{x}}_{\mathit{o}}+{\eta }_{\min }{\mathit{u}}_3$ ;  6. Repeat steps 1 through 5 until convergence is achieved.

Simulations and performance analysis

In this section, simulations are carried out to evaluate the localization performance of the proposed method. The positional errors of the UAVs, the topological structure of the sensors and the source, and some state-of-the-art schemes and the CRLB are considered in the performance comparison. The localization accuracy of the signal source $\mathit{q}$ is computed using the root mean square error (RMSE) via

$$\begin{array}{c}\sigma (T)=\sqrt{\frac{1}{\Gamma }\sum _{\gamma =1}^{\Gamma }{({\hat{\mathit{q}}}_{\mathit{i}}^{(\gamma )}-{\mathit{q}}_{\mathit{i}})}^T({\hat{\mathit{q}}}_{\mathit{i}}^{(\gamma )}-{\mathit{q}}_{\mathit{i}})}\hfill \end{array}$$

(15)

where $\Gamma ={10}^3$ is the total time of the simulation and ${\hat{\mathit{q}}}_{\mathit{i}}^{(\gamma )}$ is the position estimation of the signal source ${\mathit{q}}_{\mathit{i}}$ in the $\gamma \mathrm{th}$ Monte Carlo simulation.

In the 3D coordinate system, the position of the UAVs’ operation center is taken as the origin of the coordinates. Four distributed sensors make up a small WSN in the sky, which are loaded on the UAVs, thus forming a regular tetrahedron. The average flight height of the UAVs is about 200 m and the distance between any pair of the sensors is 50 m. Two signal sources are fixed near the ground, which are the targets of the passive localization. The precise positions of the sensors and two sources are shown in Table 1, and their relative positions are intuitively shown in Figure 2. Because the sensors are carried in the sky, the transmission from the sources to the sensors occurs through LOS channels, even though the source is settled in a complex environment. However, the positions of the sensors contain small positional errors due to the flying of the UAVs. In the following simulations, the sensor ${\mathit{p}}_1$ at the highest vertex of the regular tetrahedron is taken as the reference node in the TDOA localization processing.

Table 1.

Positions of the sensors and sources (m).

Operation center	${\mathit{p}}_0$	0	0	0
Sensor I	${\mathit{p}}_1$	40	50	230.6
Sensor II	${\mathit{p}}_2$	25.6	25	189.8
Sensor III	${\mathit{p}}_3$	25.6	75	189.8
Sensor IV	${\mathit{p}}_4$	68.9	50	189.8
Source I	${\mathit{q}}_1$	20	60	13
Source II	${\mathit{q}}_2$	350	235	5

Figure 2.

Precise positions of the sensors and the sources.

Performance comparison when the sensors’ positions contain errors

Both the distribution of the UAVs and the sensors’ positional errors are considered in this simulation to verify the availability and robustness of the proposed method. For the signal source ${\mathit{q}}_1$ , the localization performances are compared under four different localization conditions. They are random distribution of the UAVs with positional errors, random distribution of the UAVs without positional errors, certain distribution of the UAVs with positional errors, and certain distribution of the UAVs without positional errors. Here, the positions of the randomly distributed UAVs are set as ${\mathit{p}}_1=[30.2,70.3,179.2]\mathrm{m}$ , ${\mathit{p}}_2=[75.2,53.7,190.4]\mathrm{m}$ , ${\mathit{p}}_3=[49.2,54,3,186.9]\mathrm{m}$ , and ${\mathit{p}}_4=[56.2,32,4,173.4]\mathrm{m}$ . Referring to the official statement of the GPS company, the positional errors of the UAVs are set to be less than 3 m, which means that the positional errors of the UAVs change randomly within [0,3] m. Undoubtedly, the topology of the certain distributed UAVs is regular tetrahedron. All the nonlinear localization equations established in the four different conditions are resolved using the proposed method. Then we can obtain the localization result. The simulation performance is shown in Figure 3.

Figure 3.

Performance comparison of the sensor’s distribution and positional errors.

As shown in Figure 3, the localization accuracy of the certain distributed sensors is better than that of the randomly distributed sensors, which is consistent with the conclusion of Okello and Musicki. ²² For example, when the standard deviation of the noise varies $1/\mathrm{sigm}{a}^2=30\mathrm{dB}$ , the localization accuracy of certain distributed sensors is 11.64 m, while the accuracy of the randomly distributed sensors is 17.05 m. Meanwhile, when the positions of the UAVs contain errors, the proposed sensor position refining method is effective to overcome the influence on the localization accuracy, which can demonstrate that the accuracy of the sensors’ positions with errors is almost close to that of the sensors’ positions without errors. Furthermore, certain distributed sensors has a better performance than random distributed sensors to eliminate their position error. As demonstrated in the magnified parts in Figure 3, when $1/\mathrm{sigm}{a}^2=30\mathrm{dB}$ , the performance difference between the two curves of the certain distributed sensors is $11.88-11.64=0.24(\mathrm{m})$ and that between the two curves of the randomly distributed sensors is $17.40-17.05=0.35(\mathrm{m})$ .

Performance comparison with the classical method

In this simulation, we compare the performance of the proposed algorithm with other algorithms. Recall that the state-of-the-art schemes achieve location estimation through solving a set of linear equations. By adding a redundant parameter to the solution of the classic linear algorithm, the TWLS algorithm ³⁰ improves the TDOA localization performance. The squared range-difference measurement (SRD-LS) algorithm ³² also adds a correction parameter to the LS inverse matrix. Our numerical simulation results are compared against these two algorithms. Moreover, we compare the results against the CRLB in Chan and Ho. ³⁰ To be more fair, the sensor $({\mathit{p}}_1)$ is taken as the reference node for all the mentioned algorithms. The numerical simulation results are shown in Figure 4.

Figure 4.

Performance comparison with the state-of-the-art schemes.

From Figure 4, when the distribution of the sensors has a regular tetrahedral distribution in the sky, the localization errors of the three mentioned methods are similar to each other (within 3 m), while the differences are existent. The TWLS method has a better localization performance than the SRD-LS method, and obviously the performance of the proposed method is better than the other two and closer to that of CRLB. It is mainly because the proposed method combines the excellent performance of the SDP algorithm and the iterative algorithm to resolve the nonlinear localization function.

Performance comparison for two different signal sources

In this simulation, the TWLS localization method is adopted for the performance comparison to locate the two signal sources ${\mathit{q}}_1$ and ${\mathit{q}}_2$ . The topological structure of the UAVs is a regular tetrahedron in the sky. The signal source ${\mathit{p}}_1$ is in the coverage area of the UAV group on the ground, which is the near-field condition. The signal source ${\mathit{p}}_2$ is out of the coverage area of the UAV group on the ground, which is the relatively far-field condition. The simulation performance is shown in Figure 5.

Figure 5.

Performance comparison for the two different signal sources.

As shown in Figure 5, for the near-field localization condition, the proposed method has an obvious performance advantage than the TWLS localization method, which is because the proposed method combines the SDP technology and the classical iterative method to resolve the nonlinear localization equations, thus obtaining more accurate performance while increasing the computational complexity. Furthermore, for the relatively far-field localization condition, the performance of the proposed method is worse than that of the TWLS method when $1/\sigma >30\mathrm{dB}$ . The reason is that the localization performance of the proposed method is affected by the topological structure of the sensors and the source. However, to locate the source more accurately, we can take full advantage of the movement of the UAV group. The WSN formed by the UAV group can be operated by moving in the sky to form an optimal topological structure with the signal source, which can result in more accurate localization performance.

Conclusion

In this article, we propose an effective method to passively locate the signal source in the downtown area. By loading the sensors on the UAVs to establish the WSN in the air, the challenge of the source passive localization caused by multipath fading can be overcome. Meanwhile, the operating center is taken as a calibration source to refine the positions of the UAVs, and then the accurate position of the signal source is obtained according to the classical iterative algorithm, where the starting point is generated by the SDP technology. The simulation results demonstrate the effective performance of the proposed method. In the future work, we will analyze how the distance between the sensors will affect the localization accuracy under the optimal geometry configuration.