Collaborative Beamforming for Multi‐Target Localization Using Passive Anchors

Introduction

Inspired by the radio-frequency (RF) wireless power transfer technology and backscatter communication, passive sensor network (PSN) demonstrates significant advantages in prolonging lifetime, low power consumption, and cost-effective deployment and maintenance [1]. In this case, PSN can serve as a localization system, and these passive sensor nodes are the passive anchors which provide ranging measurements to the targets.

For PSN, the localization availability and accuracy are two main crucial issues. Considering the wireless power transfer constraints, several passive anchors may not be powered up when the targets send the requests. Then, the localization availability means that at least three anchors should be activated to locate a target. One scheme is to let each target send the request independently to the anchors and attain the ranging measurements for its own localization [2]. In this case, the localization is achieved with at least three anchors. However, it is energy consuming if each target periodically sends the WPT based localization request. In addition, sending the requests together by all targets is an efficient way to power up more anchors, and improve the localization effectively [3]. In [4], we proposed a beamforming scheme for hybrid active and passive cooperative localization. However, the proposed scheme only considers the ideal scenario that all the passive sensors can be powered up without any power constraints.

In this letter, we mainly analyse the localization performance using PSN. Such localization system consists passive sensors as anchors and multiple active devices as targets. The targets jointly generate the WPT waves to power up the anchors and attain the ranging measurements for self-localization. We derive the Fisher information matrix (FIM) of this localization scheme and the related squared position error bound (SPEB) which is the trace of inverse form of FIM. Then, we propose a collaborative beamforming strategy for the targets to achieve the optimal localization accuracy. Our strategy considers the power constraints and the localization availability requirements, and then employ a two-stage semi-definite programming method to attain the optimal beamforming solution. Compared with independent requesting sending scheme and the joint power allocation strategy, the simulation results demonstrate that our proposed strategy effectively increases the localization accuracy.

System Model

Passive Sensor Network

The architecture of PSN based localization system is depicted in Figure 1, which consists of the active targets and the passive anchors. The active targets generate WPT waveforms as localization request to power up the passive anchors. The passive anchors broadcast ranging data with the collected energy from targets for self-localization. The wireless power signals should cover at least a part of anchors to power them up.

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Localization on Request Schemes

We assume there are $$K$$ targets and $$N$$ passive anchors. Each target or anchor contains single antenna. Let $$\mathbf {G} \triangleq [\mathbf {g}_1, \mathbf {g}_2, \ldots, \mathbf {g}_{N}]^T$$ be the channel matrix between the targets and anchors, where $$g_{kn} \triangleq [\mathbf {G}]_{kn}$$ is the channel coefficient between the $$k$$th target and the $$n$$th anchor. Then, the microwave signals are transmitted through $$\mathbf {G}$$ to power up the anchors. We define the $$n$$th anchor's 2D position as $$\mathbf {p}_n = [p^X_n, \ p^Y_n]^T$$ and the $$k$$th target's position as $$\mathbf {p}_k = [p^X_k,\ p^Y_k]^T$$. Then, the range between the $$n$$th anchor and the $$k$$th target is $$d_{n, k} = ||\mathbf {p}_n - \mathbf {p}_k||$$, where $$||\cdot ||$$ indicates the Euclidean distance between two vectors. Considering that only parts of the anchors can be powered up by the wireless power, we use $$N_{\text{A}} \le N$$ to indicate the total number of all the passive anchors that are active.

The targets jointly form a request complex signal beam $$\mathbf {x} = [x_1, x_2, \ldots, x_K]^H$$, and the passive anchors are activated by $$\mathbf {x}$$. The received signal for $$n$$th anchor is: 1 $$$\begin{equation} y_n = \mathbf {g}_n \mathbf {x} + v^n_0 = \sum ^K_{k=1} g_{kn} x_k + v^n_0 \end{equation}$$$where $$v^n_0$$ is the additive noise which follows zero-mean Gaussian distribution. Here, we assume that the received energy is mainly from the WPT beam, and $$v^n_0$$ is too small to power up the device, which cannot be collected as the energy source. Then, the received power $$r^n_y = \mathbb {E}_{y_n}(||y_n||^2)$$ of the $$n$$th anchor is: 2 $$$\begin{equation} r^n_y = \mathbb {E}_{\mathbf {x}}(\mathbf {x}^H \mathbf {g}^H_n \mathbf {g}_n \mathbf {x}) \end{equation}$$$If $$r^n_y$$ exceeds a typical threshold $$P_{\text{th}}$$, the $$n$$th anchor becomes active to broadcast ranging signal. Then, the received ranging signal from the $$n$$-th anchor to the $$k$$th target is: 3 $$$\begin{equation} {z_{nk} = h_{n, k}(d_{n, k}) y_n + v^{nk}_e} \end{equation}$$$where $$n \ne k$$, $$h_{n,k}(\cdot)$$ is the channel response of $$y_n$$ which considers the path loss according to the range $$d_{n, k} = ||\mathbf {p}_n - \mathbf {p}_k||$$ between $$n$$th anchor and $$k$$th target. Take a flat fading propagation channel for instance, we have $$h_{nk}(d_{n, k}) = (\frac{f_c}{4 \pi d_{n, k}})^{\beta } \sqrt {\xi _{n, k}}$$ where $$f_{\text{c}}$$ is the central frequency of the signal, $$\beta$$ is the fading factor, and $$\xi _{n, k}$$ is the channel gain.

FIM and SPEB

We employ FIM to derive the fundamental performance. Here, we define $$\mathbf {\theta }$$ as the vector of the unknown target position states: 4 $$$\begin{equation} \mathbf {\theta } = [\mathbf {p}^T_1 \ \mathbf {p}^T_2 \ \ldots \ \mathbf {p}^T_{K}]^T \end{equation}$$$Then, we define $$\mathbf {z}$$ as the observation vector of all the received signals on the target side: 5 $$$\begin{equation} {\mathbf {z} = [{\mathbf {z}_1}^T \ {\mathbf {z}_2}^T \ \ldots \ {\mathbf {z}_k}^T \ \ldots \ {\mathbf {z}_{K}}^T]^T} \end{equation}$$$where $$\mathbf {z}_i$$ indicates the received signal vectors of target $$k$$ from anchors: 6 $$$\begin{equation} {\mathbf {z}_k = [z_{1k} \ z_{2k} \ \ldots \ z_{nk} \ \ldots \ z_{N_A k}]^T} \end{equation}$$$

Next, we define $$\widetilde{\mathbf {\theta }}$$ as the unbiased estimator of $$\mathbf {\theta }$$ based on $$\mathbf {z}$$. The CRLB indicates that the covariance of $$\widetilde{\mathbf {\theta }}$$ should satisfy the information inequality: 7 $$$\begin{equation} \mathbb {E}_{\mathbf {z}, \mathbf {\theta }}\lbrace (\widetilde{\mathbf {\theta }} - \mathbf {\theta })(\widetilde{\mathbf {\theta }} - \mathbf {\theta })^T\rbrace \succeq \mathbf {J}^{-1}_{\mathbf {\theta }} \end{equation}$$$where $$\mathbf {J}_{\mathbf {\theta }}$$ is the FIM for $$\mathbf {\theta }$$ given by: 8 $$$\begin{equation} \mathbf {J}_{\mathbf {\theta }} = \mathbb {E}_{\mathbf {z}, \mathbf {\theta }} \lbrace \nabla _{\mathbf {\theta }} \ln f(\mathbf {\theta }, \mathbf {z}) [\nabla _{\mathbf {\theta }} \ln f(\mathbf {\theta }, \mathbf {z})]^T\rbrace \end{equation}$$$where $$\nabla _{\mathbf {\theta }}$$ is the first order partial derivatives operator of $$\mathbf {\theta }$$. According to [5], the FIM of each target is given as follows: 9 $$$\begin{equation} {\mathbf {J}_A(\mathbf {p}_k) = \sum ^{N_A}_{n = 1} \lambda _{n, k} \mathbf {J}(\phi _{n, k})} \end{equation}$$$where $$\mathbf {J}(\phi _{n, k}) = \mathbf {q}_{n, k} \mathbf {q}^T_{n, k}$$ is the angle-of-arrival (AOA) matrix, $$\mathbf {q}_{n, k} = [\cos \phi _{n, k} \ \sin \phi _{n, k}]^T$$ with $$\phi _{n, k} = \arctan \frac{p^X_k - p^X_n}{p^Y_k - p^Y_n}$$ denoting the AOA from the $$n$$th anchor to the $$k$$th target; and $$\lambda _{n, k}$$ is the equivalent ranging coefficient (ERC): 10 $$$\begin{equation} {\lambda _{n, k} = r^{n}_y \frac{\xi _{n, k} \beta ^2 {\left(\frac{f_c}{4 \pi }\right)}^{2\beta }}{d^{2 \beta + 2}_{n, k} N_0}} \end{equation}$$$Then, for multiple target localization, the FIM is: 11 $$$\begin{equation} \mathbf {J}_{\mathbf {\theta }} = \mathrm{diag}[\mathbf {J}_A(\mathbf {p}_1), \ldots, \mathbf {J}_A(\mathbf {p}_K)] \end{equation}$$$In this case, SPEB which is the trace of $$\mathbf {J}^{-1}_{\mathbf {\theta }}$$ is employed as the optimal performance metric [6]. For a single target node, we define SPEB as: 12 $$$\begin{equation} {\mathcal {P}(\mathbf {p}_k) \triangleq \mathrm{tr} \lbrace \mathbf {J}^{-1}_A(\mathbf {p}_k)\rbrace } \end{equation}$$$Then, for multiple target nodes, the SPEB is expressed as follows: 13 $$$\begin{equation} {\mathcal {P}(\mathbf {\theta }) \triangleq \sum ^{K}_{k = 1} \mathrm{tr} \lbrace \mathbf {J}^{-1}_A(\mathbf {p}_k)\rbrace = \mathrm{tr} \lbrace \mathbf {J}^{-1}(\mathbf {\theta })\rbrace } \end{equation}$$$

Problem Formulation

Here, we employ the derived SPEB as the main metric. Considering the sum of transmitter power is constrained, the minimum SPEB is to decide how to design the wireless power wave forms to achieve the minimum location estimation error: 14 $$$\begin{equation} \begin{split} (\mathbb {P}_1): &\min \limits _{\mathbf {x}} \mathcal {P}(\mathbf {\theta })\\ &\ \text{s.t.}\ \; \mathbb {E}_{\mathbf {x}}(\mathbf {x}^H \mathbf {x}) \le P_c\\ &\ \ \ \ \ \ \ r^n_y \ge P_{\text{th}}\; \ n \in [1, 2, \ldots, N_{\text{A}}]\\ &\ \ \ \ \ \ \ 3 \le N_{\text{A}} \le N\\ \end{split} \end{equation}$$$where $$P_{\text{c}}$$ indicates the total transmitted power constraint from all the active targets, $$P_{\text{th}}$$ is the received power threshold that make a single passive anchor active. A passive anchor can only be powered up when its received power exceeds $$P_{\text{th}}$$. Note that, there should be at least three powered up anchors to make the localization available. Thus, the available anchors form a subset from the total anchor set, whose total number is $$N_{\text{A}}$$. And the number $$N_{\text{A}}$$ of them should be no smaller than 3. It is clearly observed that $$\mathbb {P}_1$$ is a nonconvex and nonincreasing function of $$\mathbf {J}(\mathbf {\theta })$$. Note that, our objective is the localization error minimization problem with Tx power constraint. For energy efficiency issue, we can reduce $$P_{\text{c}}$$ to save the power consumption. In addition, $$\mathbb {P}_1$$ can also be converted into the energy efficiency problem by constructing the dual functions. However, such issue is out of scope in this paper.

SDP Based Collaborative Beamforming

Relaxation

To relax $$\mathbb {P}_1$$ to a convex optimization problem, we eigen-decompose $$\mathbf {J}(\mathbf {\theta })$$ as follows: 15 $$$\begin{equation} \mathbf {J}(\mathbf {\theta }) = \mathbf {U}_\psi {\left( \def\eqcellsep{&}\begin{array}{cccc}\tau _1 & & & \\[3pt] & \tau _2 & & \\[3pt] & & \ddots & \\[3pt] & & & \tau _{2 N_T} \end{array} \right)} \mathbf {U}^T_\psi \end{equation}$$$where $$\mathbf {U}_\psi$$ is the eigenvector matrix and $$\tau _1, \ldots, \tau _{2K}$$ are the corresponding eigenvalues. Then, $$\mathrm{tr}\lbrace \mathbf {J}(\mathbf {\theta })\rbrace = \sum ^{2 K}_{i = 1} \tau _i$$ and $$\mathcal {P}(\mathbf {\theta }) = \sum ^{2 K}_{i = 1} \frac{1}{\tau _i}$$. Since each eigenvalue is positive in FIM, the traces of FIM and corresponding CRLB have the inequality relationship which is consistent with the AM–HM relation. Please refer to our previous work in [4] for detailed proof: 16 $$$\begin{equation} {\frac{2 K}{\sum ^{2 K}_{i = 1} \frac{1}{\tau _i}} \le \frac{1}{2 K} \sum ^{2 K}_{i = 1} \tau _i} \end{equation}$$$Then, we obtain $$\mathcal {P}(\mathbf {\theta }) \ge 4 K^2 \frac{1}{\mathrm{tr}\lbrace \mathbf {J}(\mathbf {\theta })\rbrace }$$, and $$\mathbb {P}_1$$ is relaxed as: 17 $$$\begin{equation} \begin{split} (\mathbb {P}^*_1): &\max \limits _{\mathbf {x}} \mathrm{tr}\lbrace \mathbf {J}(\mathbf {\theta })\rbrace \\ &\ \text{s.t.}\ \; \mathbb {E}_{\mathbf {x}}(\mathbf {x}^H \mathbf {x}) \le P_c\\ &\ \ \ \ \ \ \ r^n_y \ge P_{th}\; \ n \in [1, 2, \ldots, N_A]\\ &\ \ \ \ \ \ \ 3 \le N_A \le N\\ \end{split} \end{equation}$$$Note that, $$\mathrm{tr}\lbrace \mathbf {J}(\mathbf {\theta })\rbrace = \sum ^{K}_{k = 1} \sum ^{N_A}_{n = 1} \lambda _{n, k}$$. Substitute Equation (10) into $$\mathrm{tr}\lbrace \mathbf {J}(\mathbf {\theta })\rbrace$$, we attain: 18 $$$\begin{equation} {\mathrm{tr}\lbrace \mathbf {J}(\mathbf {\theta })\rbrace = \mathbf {x}^T \mathbf {G}^T \sum ^{K}_{k = 1} \mathbf {\Lambda }_k \mathbf {G} \mathbf {x}} \end{equation}$$$where $$\mathbf {\Lambda }_k = \mathrm{diag}(\frac{\xi _{1, k} \beta ^2 (\frac{f_c}{4 \pi })^{2\beta }}{d^{2 \beta + 2}_{1, k} N_0}, \ldots, \frac{\xi _{N, k} \beta ^2 (\frac{f_c}{4 \pi })^{2\beta }}{d^{2 \beta + 2}_{N, k} N_0})$$.

Two-Stage SDP Solution

However, $$\mathbb {P}^*_1$$ is still not convex, since different beamforming signal waves result in different received power on the passive anchor side. Thus, which anchors are powered up is not decided yet. Then, we divide $$\mathbb {P}^*_1$$ into two sub-problems to solve it. Firstly, we assume all the anchors can be activated without considering the power threshold. In this case, $$\mathbb {P}^*_1$$ can be solved using SDP. Then, we apply the beamforming result to calculate which anchors gain enough power which is over the threshold $$P_{\text{th}}$$, and form the sub-set of anchors. Then, we execute the SDP again to attain the final beamforming solution. The technique details are as follows:

Stage I

According to the above procedure, we employ SDP to attain the initial beamforming solution. Firstly, let $$\mathbf {X} = \mathbf {x} \mathbf {x}^H$$ and it is obvious that $$\mathrm{rank}(\mathbf {X}) = 1$$. Consider $$\mathbf {x} \mathbf {x}^H = \mathrm{tr}(\mathbf {X})$$ and $$\mathbf {x}^H \mathbf {\Omega } \mathbf {x} = \mathrm{tr}(\mathbf {Q} \mathbf {X})$$, where $$\mathbf {\Omega }$$ indicate any symmetric matrix, we solve the following sub-problem: 19 $$$\begin{equation} \begin{split} (\mathbb {P}^*_2): &{\max \limits _{\mathbf {X}} \mathrm{tr}{\left\lbrace \mathbf {G}^T \sum ^{K}_{k = 1} \mathbf {\Lambda }_k \mathbf {G} \mathbf {X}\right\rbrace} }\\ &\ \text{s.t.}\ \; \mathrm{tr}(\mathbf {X}) \le P_c\\ &\ \ \ \ \ \ \ \mathbf {X} \ge 0\\ \end{split} \end{equation}$$$which is a typical SDP problem. We employ CVX tool to solve the above SDP problem. The CVX is a powerful Matlab solver of convex optimization. Here, SDPT3 in CVX which uses the interior point method is applied. When $$\mathbf {X}$$ is attained, we have $$\mathbf {X} = \lambda _0 \mathbf {q}_0 \mathbf {q}^T_0$$, where $$\lambda _0$$ is the eigenvalue of rank one matrix and $$\mathbf {q}_0$$ is the corresponding eigenvector. Then, the feasible solution of $$\mathbf {x}$$ is $$\mathbf {x}^{\ast } = \sqrt {\lambda _0} \mathbf {q}_0$$.

Stage II

We apply $$\mathbf {x}^{\ast }$$ as the beamforming vector to evaluate each received power $$r^n_y$$. Then, we collect all the available anchors from which the received power exceed the threshold $$P_{\text{th}}$$ and execute the SDP again. Such two stages are executed iteratively until the distance between $$\mathbf {x}^{\ast }$$ of each stage is below a threshold. Consequently, the final $$\mathbf {x}^{\ast }$$ is attained. The whole scheme is presented in Algorithm 1.

1

ALGORITHM

BEAM: Two-stage SDP beamforming.

1:	Initialization: Broadcast the Localization Request;
2:	Receive the ranging feedbacks from anchors and maintain the anchor list;
3:	while $$||\mathbf {x}^{\ast }_1 - \mathbf {x}^{\ast }_2||^2 \le \varepsilon$$ do
4:	Stage I:
5:	Solve $$\mathbb {P}^*_2$$ using SDP, attain $$\mathbf {x}^{\ast }_1$$
6:	Stage II
7:	Calculate $$r^n_y$$ for each anchor
8:	Select anchors whose feedback power exceed $$P_{th}$$
9:	Re-solve $$\mathbb {P}^*_2$$ using SDP, attain $$\mathbf {x}^{\ast }_2$$
10:	end while
11:	Final beamforming vector $$\mathbf {x}^{\ast }$$;

Simulation

Following the similar simulation settings [7], all anchors and targets are randomly deployed in a $$500 \times 500\ \mathrm{m^2}$$ playing field, which is a typical scenario for IoT applications. The signal carrier frequency is 2.4 GHz which is the open frequency for IoT. The maximum allowed total transmitted power $$P_{\text{c}}$$ of the active targets is 30 dBm (1 W), otherwise it will generate harmful radiations for the environment. According to the statistical parameters, the received power threshold $$P_{\text{th}}$$ of each passive anchor for communications and harvesting energy is −90 dBm, and the power of the background noise is −130 dBm [2]. We employ the averaged SPEB as the multi-target localization performance metric. We compare our proposed scheme which is named BEAM with other two schemes. The first one is the independent request (IR), in which each target sends the request independently. The second one is the power allocation (PA), in which the targets send orthogonal signals and ignore channel correlations [8]. We set 1000 Monte-Carlo simulations, in which the anchors and targets are randomly deployed in each simulation.

Sequential Evaluation

We evaluate the schemes in a sequential way, in which the FIM calculation fuses the previous FIM to derive the current SPEB. The averaged SPEBs of all the compared schemes are depicted in Figure 2. In the initial time step, the averaged SPEB of BEAM is lower than IR and PA. However, with the increased time step and enough previous FIM participating, the averaged SPEBs are converged closely. This means that using a sequential localization tool, for example, Kalman filter, the localization performance can overpass the drawback of the IR and PA. However, without previous information, our strategy is promising to achieve high accurate localization.

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Passive Anchor Evaluation

Then, we evaluate the impact of passive anchors. In addition to the algorithm comparison, we also consider different anchor placement schemes, which are uniform placement and random placement. As illustrated in Figure 3, the SPEB of IR and PA decrease with more anchors participating in the localization, although it still has some fluctuations. BEAM outperforms other schemes even with only 10 anchors. Note that, BEAM in both placement schemes have similar SPEB, which indicates that BEAM needs no special optimal placement method.

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Target Evaluation

We also increase the target number from 10 to 35, and fix the number of anchors as 20. The results are illustrated in Figure 4, and the tendency is quite similar to Figure 3. Note that the SPEB of PA quickly drops down with more targets, because more targets are shared with the ranging information and average down the SPEBs. Still, the BEAM can achieve a quite low value of SPEB and outperforms other schemes.

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Conclusion

In this letter, we derive the FIM and SPEB for the passive sensor based localization system. The ranging information from the anchors are activated by the wireless power request signals from the targets. Then, we propose a collaborative beamforming scheme to optimize the localization accuracy. The simulations demonstrate that our scheme outperforms the power allocation and the independent request scheme for high accurate multi-target localization.

Author Contributions

Xingjun Lai: data curation, resources. Hongyuan Liu: investigation. Xiaofan Li: methodology. Yubin Zhao: writing – original draft, writing – review and editing.

Acknowledgement

This work was partially supported by Open fund of Key Laboratory of Short Range Radio Equipment Testing and Evaluation, Ministry of Industry and Information Technology (No. SRTC2025-SZ-HY-017).

Conflicts of Interest

The authors declare no conflicts of interest.

Data Availability Statement

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

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