Multi-User Tracking in Reconfigurable Intelligent Surface Aided Near-Field Wireless Communications System

Abstract

An uplink multi-user tracking problem aided by multiple passive reconfigurable intelligent surfaces (RISs) is addressed in this work. Under a near-field circumstance, a multi-antenna base station (BS) localizes multiple moving single-antenna users by processing the received signals transmitted by users and reflected by RISs. Considering the users’ mobility and the potential obstruction of line-of-sight paths, a multi-user tracking system based on the extended Kalman filter (EKF) which fully exploits the temporal correlations between each user’s coordinate changes is designed. Then, the Bayesian Cramér–Rao bound (BCRB) of tracking errors is derived in a pattern consistent with the EKF process. Subsequently, an optimization scheme for passive phase shift design at the RISs is devised by minimizing the derived BCRB and is solved using the Gradient Descent method. Numerical results indicate that the accuracy of our tracking algorithm can approach the BCRB. With abundant RISs deployed and optimized, high-precision multi-user tracking via a single BS can be realized even in harsh localization environments.

1. Introduction

High-precision positioning, as an essential concern in sensing, holds great promise for future sixth-generation (6G) mobile networks, encompassing various applications such as augmented reality, Internet of Vehicles, virtual reality, and unmanned aerial vehicle communications [1,2,3]. As carrier frequencies move toward millimetre waves (mmWaves) and higher sub-THz spectrum, large antenna arrays enabling high sensing resolution can be widely used, making radio-based high-precision positioning from signals possible [1,2,3,4]. Typical radio localizing methods consist of measuring received signal strength (RSS), angle of arrival (AOA), angle of departure (AOD), time of arrival (TOA), and time difference of arrival (TDOA) of incident signals from multiple base stations (BSs) [5,6,7,8,9].

According to the electromagnetic (EM) propagation theory, increasing the radio frequency to meet the demands of ultra-high-speed data transmission in 6G networks can significantly weaken the effects of diffraction [1,2]. Therefore, there is heavy reliance on line-of-sight (LoS) paths to guarantee satisfactory RSS and carry location information of targets. The existing methods may suffer from severe performance degradation in harsh EM environments, especially in mobile scenarios, where complex conditions such as multipath, clutter, and non-line-of-sight (NLoS) propagation frequently occur [10].

To solve the aforementioned positioning problem, reconfigurable intelligent surfaces (RISs) are introduced as energy-saving and low-cost devices [11,12,13,14]. RISs are essentially large EM meta-surfaces; each element is capable of independently adjusting the amplitude and phase of an incident radio wavefront passively, thereby enabling precise control over the reflection of the waves [15]. Once installed in wireless environments, RISs can precisely direct signals towards the desired direction, thus significantly enhancing the virtual line-of-sight (VLoS) links [11,16]. From a positioning perspective, RISs can serve as an extra reference point, offering supplementary measurements that help to avoid dependence on the LoS path [17].

Introducing RISs into positioning recently sparked widespread interest due to its promising benefits. For instance, Björnson et al. [18] analyzed a downlink orthogonal frequency division multiplexing (OFDM) scenario aided by RIS from both the communication and positioning perspectives. Similarly, Wymeersch et al. [19] derived the Cramér–Rao lower bound (CRLB) for positioning error in a downlink setting, emphasizing how deploying RIS in the far field can outperform conventional methods relying solely on naturally occurring scatterers for better positioning accuracy. Elzanaty et al. [20] explored the upper limits of positioning performance of an uplink multiple-input multiple-output (MIMO) system, proposed a near-field propagation model, and demonstrated the significant promotion of positioning accuracy brought by a single RIS. Zhang et al. [21] applied a RIS-assisted user positioning scheme based on RSS in an indoor scenario, where a high-resolution signal intensity map was built to localize the target through iterative optimization of RIS phase shift. Lin et al. [22] introduced a beam searching method based on hierarchical codebook design to achieve accurate positioning indirectly by utilizing RIS-assisted channel parameters. Similarly, the user position was estimated from the channel information of RIS, and channel angle parameters were obtained through the maximum likelihood method [23]. For MIMO-OFDM systems, He et al. [24] implemented a twin-RIS structure and applied advanced signal processing techniques to extract relevant information from the received signals. Dardari et al. [25] considered a single-anchor positioning problem under NLoS conditions characterized by frequent applications. More practical RIS-assisted localization schemes can be found in [26,27].

The design of the RIS is the most critical part of assisting either positioning or communication. Literature concerning the optimization of RIS-assisted communications under various channel state information (CSI) and a priori location knowledge is abundant [28,29,30]. However, existing optimization solutions for RIS aimed at improving communication are not necessarily practical or optimal in improving positioning. RIS optimization focusing on positioning is less frequently investigated. Lin et al. [31] proposed a new RIS conformal structure that can be used to estimate the AoD and AoA of RISs and solve scatterer and user localization problems based on it. Fascista et al. [32] studied joint positioning and synchronization problems and optimized the BS precoding and RIS phase shift based on their proposed novel beamforming codebook. Rahal et al. [33] and Gao et al. [34] optimized a RIS by minimizing positioning error bounds (PEB), where the blockage of LoS and hardware limitations were considered. Feng et al. [35] proposed an optimization scheme of RIS based on CRLB for positioning and transmit power minimization and adopted a semi-positive semi-definite relaxation method for the power minimization part.

Most existing works involving RIS-assisted localization consider snapshot localization, and only a few works address the user tracking problem. Teng et al. [36] proposed a Bayesian tracking scheme which is realized indirectly by estimating AoAs. If the user’s initial location information is known, better positioning accuracy can be achieved through directional beamforming on the BS side [32]. Furthermore, works in [37,38,39,40] resolve the problem in a double-scale way, significantly reducing the RIS reconfiguration rate. Fewer works consider multi-user tracking problems. Recently, Yu et al. [41] studied a multi-user localization problem and introduced two semi-passive RISs to estimate the AoA on the RIS; the consequent hardware overhead would be inconsistent with the low-cost intention of introducing RIS. Teng et al.proposed a multi-user tracking algorithm based on Bayesian theory, carrying out positioning by estimating AoA and optimizing passive beamforming of RIS [42]. However, the model adopted is only valid for the far-field scenario, and a two-step approach starting from intermediate parameters is suboptimal to direct means, as it demands an appropriate characterization of the measurement statistics.

Methods and types of the most relevant RIS-assisted positioning literature are listed in

Table 1. Methods and types of the most relevant RIS-assisted positioning literature.

	Uplink	LoS-Blocked	Near-Field	Multi-User	Direct Approach 1	Dynamic Tracking 2	Passive RIS 3	CRLB Utilized 4
Elzanaty et al. [20]	✓	✗	✓	✗	✓	✗	✓	✓
Zhang et al. [21]	✗	✗	✗	✓	✗	✗	✓	✗
Lin et al. [22]	✓	✓	✗	✗	✗	✗	✓	✗
Wang et al. [23]	✗	✓	✗	✓	✗	✗	✓	✓
He et al. [24]	✗	✓	✗	✗	✗	✗	✓	✗
Dardari et al. [25]	✗	✗	✓	✗	✓	✗	✓	✓
Lin et al. [31]	✓	✓	✗	✗	✗	✗	✓	✗
Fascista et al. [32]	✗	✗	✗	✗	✗	✗	✓	✓
Teng et al. [36]	✗	✓	✗	✗	✗	✓	✓	✓
Palmucci et al. [37]	✗	✗	✓	✗	✓	✓	✓	✗
Yu et al. [41]	✓	✓	✗	✓	✗	✗	✗	✗
Teng et al. [42]	✓	✓	✗	✓	✗	✓	✓	✓

¹ The direct approach denotes that the target position is estimated directly from the measured data, and the corresponding method is the two-stage method, which estimates the target position by estimating intermediate parameters. ² Dynamic tracking denotes that the localizing is conducted in a dynamic scenario and the other corresponding method is snapshot localization without exploiting the temporal correlations of user locations. ³ RIS is the abbreviation of reconfigurable intelligent surface. ⁴ CRLB is the abbreviation of Cramér–Rao lower bound. The passive beamforming of RIS is designed by utilizing (Bayesian) CRLB, for instance, by minimizing (B) CRLB.

. With carrier frequencies moving toward mmWaves and higher sub-THz spectrum, large antenna arrays are widely used in tracking scenarios, where the targets often move in near-field regions. However, the signal model adopted in existing works is only valid for the far-field scenario. Among existing works involving RIS-assisted localization, most consider snapshot localization without exploiting the temporal correlations of user locations. In dynamic scenarios, the localization performance of these methods can be severely limited. Further, few works investigate the multi-user tracking problems in RIS-assisted systems.

Enlightened by the correlated works and considering their limitations, we study an uplink multi-user tracking problem aided by multiple passive RISs under the near-field circumstance, where the BS and users are equipped with multiple antennas and a single antenna, respectively. As indicated in Figure 1, the LoS paths between the BS and users are presumed to be blocked during the tracking process. The core insights offered by this research include:

• Considering the temporal correlations between each user’s coordinates changes, a multi-user tracking system is designed based on the extended Kalman filter (EKF), where the non-linear relationships between the signals received at the BS antennas and the coordinates of the moving users are directly utilized.
• Based on the near-field model accounting for the incident spherical wavefront, its geometric properties are further exploited. Additional measurement dimensions can be provided by the RISs to suffice high-precision multi-user tracking with a single BS, thus avoiding the reliance on the LoS paths.
• The Bayesian Cramér–Rao bound (BCRB) is derived for the multi-user tracking system in a pattern consistent with the EKF process, providing a theoretical lower bound of mean square error (MSE) for the tracking problem.
• An optimizing scheme of passive phase shift design at the RISs is devised. The issue is formatted as a problem of minimizing the derived BCRB of tracking errors and solved by leveraging the Gradient Descent (GD) method. Numerical results indicate that the accuracy of the proposed tracking scheme can approach the derived BCRB.

The remainder of the study unfolds as follows. The tracking scenario, geometry, and signal model of the proposed system are described in Section 2. The basic navigation framework, including the transition model, observation model, and proposed tracking algorithm, is elaborated in Section 3. The BCRB of the proposed RIS-aided tracking scheme and the optimization of RISs’ phase shift are presented in Section 4. A series of simulations are conducted, and the validity and accuracy of the proposed tracking system are demonstrated in Section 5. The conclusions are stated in Section 6. Additionally, notations and operators used in this work are enumerated in

Table 2. Notations and operators.

Symbol	Meaning
x	scalar
$\mathbf{x}$	vector
$\mathbf{X}$	matrix
${\left(·\right)}^{\mathrm{T}}$	transpose
${\left(·\right)}^{*}$	conjugate
${\left(·\right)}^{-1}$	inverse
∇	partial derivative
$tr\left(\mathbf{X}\right)$	trace of $\mathbf{X}$
$\mathrm{diag}\left(·\right)$	diagonal matrix
$\mathbb{E}\left\{·\right\}$	expectation
$\mathbb{C}$	complex domain
$\mathbb{R}$	real domain
$A\times B$	dimension of matrices
$\mathfrak{R}\left(·\right)$	real part of complex value
$\mathfrak{I}\left(·\right)$	imaginary part of complex value
$\left|·\right|$	module of complex value
$∥·∥$	$l_2$-norm
$j=\sqrt{-1}$	the imaginary unit
$\mathcal{B}$, $\mathcal{K}$, $\mathcal{P}$, $\mathcal{M}$	set
$\mathbf{I}$	identity matrix
$\mathbf{0}$	zero matrix

.

2. System Model

2.1. Tracking Scenario and Geometry

We set up a multi-RIS-aided localization scenario in a Cartesian coordinate system, where a multi-antenna BS localizes M moving single-antenna users by processing the received signals transmitted by users and reflected by K RISs consisting of multiple elements.

According to Figure 2, we denote ${\mathbf{p}}_{\mathrm{B}}\in {\mathbb{R}}^3$ as the centre of BS, ${\mathbf{p}}_b\in {\mathbb{R}}^3$ as the coordinate of the b-th antenna on the BS with $b\in \mathcal{B}\triangleq \{0,1,\dots ,N_B-1\}$, ${\mathbf{p}}_k\in {\mathbb{R}}^3$ as the centre of k-th RIS, ${\mathbf{p}}_{k,p}\in {\mathbb{R}}^3$ as the coordinate of the p-th element on the k-th RIS with $k\in \mathcal{K}\triangleq \{1,\dots ,K\}$ and $p\in \mathcal{P}\triangleq \{0,1,\dots ,N_R-1\}$, ${\mathbf{p}}_m\in {\mathbb{R}}^3$ as the coordinate of the m-th user with $m\in \mathcal{M}\triangleq \{1,\dots ,M\}$. The distances between elements can be represented as $d_{Bk}=∥{\mathbf{p}}_{\mathrm{B}}-{\mathbf{p}}_k∥$, $d_{km}=∥{\mathbf{p}}_k-{\mathbf{p}}_m∥$, $d_{k;b,p}=∥{\mathbf{p}}_b-{\mathbf{p}}_{k,p}∥$ and $d_{k;m,p}=∥{\mathbf{p}}_m-{\mathbf{p}}_{k,p}∥$. The elevation and azimuth angles between two elements are $\theta$ and $\varphi$, respectively. The local reflection coefficient of an element of RIS is $r_{k,m;b,p}=G_c{e}^{j{\psi }_{k,p}}\sqrt{F\left({\mathsf{\Phi }}_{k;b,p}\right)F\left({\mathsf{\Phi }}_{k,m;p}\right)}$, where $G_c$ is the gain of the unit element and $e^{j{\psi }_{k,p}}$ is the unit-norm load coefficient [25,28]. The variable set $\mathsf{\Phi }$ includes $\{\theta ,\varphi \}$, and $F\left(\mathsf{\Phi }\right)$ represents the normalized power radiation pattern per element, which remains constant across the frequencies of interest and is characterized in an exponential-Lambertian radiation pattern with parameter q as

$$F\left(\mathsf{\Phi }\right)=\left\{\begin{array}{cc}{sin}^q\left(\theta \right),\hfill & \theta \in \left[\begin{array}{c}0,\pi /2\end{array}\right],\varphi \in \left[\begin{array}{c}0,2\pi \end{array}\right]\hfill \\ 0,\hfill & otherwise.\hfill \end{array}\right.$$

(1)

2.2. Signal Model for Incident Spherical Wavefronts

In the uplink, M users operate on orthogonal frequencies but share the same configurations of RISs. Given the carrier frequency $f_c$, subcarrier spacing $\Delta f$, the frequency of the m-th user can be expressed with $f_m=f_c+m\Delta f-B/2$, and the signal bandwidth is $B=M\Delta f$. Each user transmits a single data symbol $p\left[n\right]=1$, which meets the criteria of $\mathbb{E}\left\{p\left[n\right]p^{*}\left[n\right]\right\}=1$. The LoS paths between the BS and users are presumed to be blocked, while VLoS paths reflected by RISs always exist. The received signal from the m-th user at the BS, ${\mathbf{y}}_m\in {\mathbb{C}}^{N_B\times 1}$, is

$${\mathbf{y}}_m=\sqrt{P}\sum _{k=1}^K{\mathbf{H}}_k{\mathsf{\Omega }}_k{\mathbf{h}}_{k,m}+{\mathbf{n}}_m,$$

(2)

where P denotes the transmit signal power, ${\mathbf{H}}_k=\left\{h_{k;b,p}\right\}\in {\mathbb{C}}^{N_B\times N_R}$ and ${\mathbf{h}}_{k,m}=\left\{h_{k,m;p}\right\}\in {\mathbb{C}}^{N_R\times 1}$ are the channel matrices of BS-(k-th)RIS and (k-th)RIS-(m-th)user links, respectively. Here, we put the normalized power radiation pattern and gain of each RIS element into the channel matrices. Thus, ${\mathsf{\Omega }}_k=\mathrm{diag}(c_{k,0},\dots ,c_{k,p},\dots ,c_{k,N_R-1})\in {\mathbb{C}}^{N_R\times N_R}$ only contains the unit-norm load coefficient of each RIS element, i.e., $c_{k,p}$ denotes the unit-norm load coefficient of the p-th element of the k-th RIS and $c_{k,p}=e^{j{\psi }_{k,p}}$. ${\mathbf{n}}_m\in {\mathbb{C}}^{N_B\times 1}$ denotes the additive white Gaussian noise (AWGN) sequences with variance ${\sigma }_{n_m}^2$. BS estimates the users’ positions by exploiting the spherical waveform model. The components of the channel matrix are modelled as

$$h_{k;b,p}={\rho }_{k;b,p}{e}^{-j\frac{2\pi }{{\lambda }_m}{d}_{k;b,p}},$$

(3)

$$h_{k,m;p}=\sqrt{\frac{{\kappa }_h}{{\kappa }_h+1}}{\rho }_{k,m;p}{e}^{-j\frac{2\pi }{{\lambda }_m}{d}_{k,m;p}}+\sqrt{\frac{1}{{\kappa }_h+1}}{\delta }_{k,m;p},$$

(4)

where ${\rho }_{k;b,p}\triangleq \frac{{\lambda }_m}{4\pi }\frac{\sqrt{G_b{G}_{p}F\left({\mathsf{\Phi }}_{k;b,p}\right)}}{d_{Bk}}$, ${\rho }_{k,m;p}\triangleq \frac{{\lambda }_m}{4\pi }\frac{\sqrt{G_p{G}_{m}F\left({\mathsf{\Phi }}_{k,m;p}\right)}}{d_{km}}$ are the attenuation due to propagation of the LoS components, and ${\lambda }_m$ satisfies the relation of ${\lambda }_m{f}_m=c$, where c is the speed of light. We adopt a simple Rice channel model to describe the links between RISs and users, and ${\kappa }_h\ge 0$ are the Ricean factors for (k-th)RIS-(m-th)user links. We assume that the LoS paths always exist, as the probability of the LoS condition is close to 1 in the near-field region [37]. ${\delta }_{k,m;p}\sim \mathcal{C}\mathcal{N}(0,{\rho }_{k,m;p}^2)$ denotes the random complex fading coefficients of the NLoS components. The received signal at the b-th antenna on the BS from the m-th user results in

$$\begin{array}{cc}\hfill y_{m;b}& =\sqrt{\frac{{\kappa }_{h}P}{{\kappa }_h+1}}\sum _{k=0}^{K-1}\sum _{p=0}^{N_R-1}{\rho }_{k;b,p}{\rho }_{k,m;p}{c}_{k,p}{e}^{-j\frac{2\pi }{{\lambda }_m}(d_{k;b,p}+d_{k,m;p})}+w_{m;b},\hfill \end{array}$$

(5)

where $w_{m;b}\sim \mathcal{C}\mathcal{N}\left(0,{\sigma }_{m;b}^2\right)$ is the interference-plus-noise term containing the AWGN noise and the multipath components and is given by

$$w_{m;b}=n_{m;b}+\sqrt{\frac{P}{{\kappa }_h+1}}\sum _{k=0}^{K-1}\sum _{p=0}^{N_R-1}{\rho }_{k;b,p}{\delta }_{k,m;p}{c}_{k,p}{e}^{-j\frac{2\pi }{{\lambda }_m}{d}_{k;b,p}}.$$

(6)

3. Navigation Framework

M users move with uniform velocity in the tracking scenario. To describe the tracking system, we choose EKF among the Bayesian estimators. The state-space model comprises a transition model, which details how the state evolves over successive time intervals, and an observation model, which defines the relationship between the measurements taken and the current state of the users.

The state vector is denoted as

$$\mathbf{s}\left(t\right)\triangleq {\left[\begin{array}{c}{\mathbf{s}}_1\left(t\right),\dots ,{\mathbf{s}}_m\left(t\right),\dots ,{\mathbf{s}}_M\left(t\right)\end{array}\right]}^{\mathsf{T}},$$

(7)

where

$${\mathbf{s}}_m\left(t\right)\triangleq {\left[\begin{array}{c}{\mathbf{p}}_m{\left(t\right)}^{\mathsf{T}},{\dot{\mathbf{p}}}_m{\left(t\right)}^{\mathsf{T}}\end{array}\right]}^{\mathsf{T}}.$$

(8)

${\mathbf{p}}_m\left(t\right)={\left[\begin{array}{c}{x}_m\left(t\right),y_m\left(t\right),z_m\left(t\right)\end{array}\right]}^{\mathsf{T}}$ and ${\dot{\mathbf{p}}}_m\left(t\right)={\left[\begin{array}{c}{v}_{x_m}\left(t\right),v_{y_m}\left(t\right),v_{z_m}\left(t\right)\end{array}\right]}^{\mathsf{T}}$ are 3-D position and velocity parameters of the m-th user, respectively.

3.1. Transition Model

The position of the m-th user in the $t+1$ slot is ${\mathbf{p}}_m(t+1)={\mathbf{p}}_m\left(t\right)+{\dot{\mathbf{p}}}_m\left(t\right)T$, where T denotes the sampling interval. Denote the random disturbance received during the target movement as ${\mathbf{u}}_m\left(t\right)={\left[\begin{array}{c}{u}_{x_m}\left(t\right),u_{y_m}\left(t\right),u_{z_m}\left(t\right)\end{array}\right]}^{\mathsf{T}}$. For the m-th user, the transition dynamics can be modelled as

$${\mathbf{s}}_m(t+1)={\mathbf{F}}_m{\mathbf{s}}_m\left(t\right)+{\mathsf{\Gamma }}_m{\mathbf{u}}_m\left(t\right),$$

(9)

where

$${\mathbf{F}}_m=\left[\begin{array}{cc}{\mathbf{I}}_{3\times 3}& T{\mathbf{I}}_{3\times 3}\\ {\mathbf{0}}_{3\times 3}& {\mathbf{I}}_{3\times 3}\end{array}\right],$$

(10)

$${\mathsf{\Gamma }}_m={\left[\begin{array}{cc}\frac{1}{2}{T}^2{\mathbf{I}}_{3\times 3}& T{\mathbf{I}}_{3\times 3}\end{array}\right]}^{\mathsf{T}}.$$

(11)

The random disturbance ${\mathbf{u}}_m\left(t\right)$ can be characterized by the covariance matrix

$${\mathbf{U}}_m=\mathrm{diag}({\sigma }_{x_m}^2,{\sigma }_{y_m}^2,{\sigma }_{z_m}^2),$$

(12)

where ${\sigma }_{x_m}^2,{\sigma }_{y_m}^2,{\sigma }_{z_m}^2$ are adjustable parameters that denote process noise in the moving process. Thus,

$${\mathbf{Q}}_m={\mathsf{\Gamma }}_m{\mathbf{U}}_m{\mathsf{\Gamma }}_m^{\mathsf{T}}.$$

(13)

Accordingly, $\mathbf{F}=\mathrm{diag}({\mathbf{F}}_1,\dots ,{\mathbf{F}}_m,\dots ,{\mathbf{F}}_M)$ and $\mathbf{Q}=\mathrm{diag}({\mathbf{Q}}_1,\dots ,{\mathbf{Q}}_m,\dots ,{\mathbf{Q}}_M)$.

3.2. Observation Model

Approaches starting from intermediate parameters demand proper and accurate characterization of the measurements. We consider estimating the users’ positions in a direct way, where the non-linear relationships between the signals received at the BS antennas and the coordinates of the moving users are directly utilized. Thus, the measurement vector is defined as

$${\mathbf{z}}_t\triangleq g\left({\mathbf{s}}_t\right)+{\mathbf{v}}_t={\left[\begin{array}{c}{\widehat{\mathbf{z}}}_1^{\mathsf{T}},\dots ,{\widehat{\mathbf{z}}}_m^{\mathsf{T}},\dots ,{\widehat{\mathbf{z}}}_M^{\mathsf{T}}\end{array}\right]}^{\mathsf{T}},\forall m\in \mathcal{M},$$

(14)

where ${\widehat{\mathbf{z}}}_m={\left[\begin{array}{c}\mathfrak{R}{\left\{{\widehat{\mathbf{y}}}_m\right\}}^{\mathsf{T}},\mathfrak{I}{\left\{{\widehat{\mathbf{y}}}_m\right\}}^{\mathsf{T}}\end{array}\right]}^{\mathsf{T}}$ is the measurement vector of the m-th user, and

$${\widehat{\mathbf{y}}}_m={\left[\begin{array}{c}{\widehat{y}}_{m;0},\dots ,{\widehat{y}}_{m;b},\dots ,{\widehat{y}}_{m;N_B-1}\end{array}\right]}^{\mathsf{T}},\forall b\in \mathcal{B},$$

(15)

and ${\mathbf{v}}_t$ denotes a zero-mean white noise sequence unavoidably caused by measurement, the covariance matrix of which is defined as

$$\mathbf{R}=\mathrm{diag}{({\sigma }^2,\dots ,{\sigma }^2)}_{2N_{B}M\times 2N_{B}M}.$$

(16)

The detailed Jacobian calculation is provided in Appendix A, and the EKF procedure is sketched in Algorithm 1.

Algorithm 1 EKF-based Multi-user Tracking with RISs

Initialization Define initial distribution ${\mathbf{s}}_0\sim \mathcal{N}({\widehat{\mathbf{s}}}_{0|0},{\mathbf{P}}_{\mathbf{0}|\mathbf{0}})$.  Prediction Step      Given ${\mathbf{s}}_{t+1}=\mathbf{F}{\mathbf{s}}_t+{\mathbf{w}}_t$, where ${\mathbf{w}}_t\sim \mathcal{N}(0,{\mathbf{Q}}_t)$. $${\widehat{\mathbf{s}}}_{t+1|t}=\mathbf{F}{\widehat{\mathbf{s}}}_{t|t}.$$ (17) $${\mathbf{P}}_{t+1|t}=\mathbf{F}{\mathbf{P}}_{t|t}{\mathbf{F}}^{\mathsf{T}}+{\mathbf{Q}}_t.$$ (18)  Update Step      Given ${\mathbf{z}}_{t+1}=g\left({\widehat{\mathbf{s}}}_{t+1|t}\right)+{\mathbf{v}}_{t+1}$, where ${\mathbf{v}}_{t+1}\sim \mathcal{N}(0,{\mathbf{R}}_{t+1})$. $${\mathbf{G}}_{t+1}=\frac{\partial g}{\partial \mathbf{s}}\left({\widehat{\mathbf{s}}}_{t+1|t}\right).$$ (19) $${\mathbf{K}}_{t+1}={\mathbf{P}}_{t+1|t}{\mathbf{G}}_{t+1}^{\mathsf{T}}{({\mathbf{G}}_{t+1}{\mathbf{P}}_{t+1|t}{\mathbf{G}}_{t+1}^{\mathsf{T}}+{\mathbf{R}}_{t+1})}^{-1}.$$ (20) $${\widehat{\mathbf{s}}}_{t+1|t+1}={\widehat{\mathbf{s}}}_{t+1|t}+{\mathbf{K}}_{t+1}({\mathbf{z}}_{t+1}-g\left({\widehat{\mathbf{s}}}_{t+1|t}\right)).$$ (21) $${\mathbf{P}}_{t+1|t+1}={\mathbf{P}}_{t+1|t}-{\mathbf{K}}_{t+1}{\mathbf{G}}_{t+1}{\mathbf{P}}_{t+1|t}.$$ (22)

4. RIS-Aided Tracking Error Bound and Optimization

4.1. Derivation of BCRB

The inaccuracy in modelling users’ dynamic patterns and the limitations of the adopted measurement method may bring uncertainty to the users’ state estimates. Denote the users’ estimated state as ${\widehat{\mathbf{s}}}_{t+1}$; the MSE of the users’ state estimation satisfies

$$\begin{array}{cc}\hfill {\mathbb{E}}_{{\mathbf{s}}_{t+1},{\mathbf{z}}_{t+1}}& \left\{({\widehat{\mathbf{s}}}_{t+1}\left({\mathbf{z}}_{t+1}\right)-{\mathbf{s}}_{t+1}){({\widehat{\mathbf{s}}}_{t+1}\left({\mathbf{z}}_{t+1}\right)-{\mathbf{s}}_{t+1})}^{\mathsf{T}}\right\}\hfill \\ \hfill & \ge {\mathbf{J}}^{-1}\left({\mathbf{s}}_{t+1}\right),\hfill \end{array}$$

(23)

where ${\mathbb{E}}_{{\widehat{\mathbf{s}}}_t,{\mathbf{z}}_t}\left\{·\right\}$ denotes mathematical expectations for users’ states and observations; $\mathbf{J}\left({\mathbf{s}}_{t+1}\right)$ is the Bayesian Fisher information matrix (BFIM) of users’ states ${\mathbf{s}}_t$, which can be expressed as [43]

$$\mathbf{J}\left({\mathbf{s}}_{t+1}\right)={\mathbf{J}}_{\mathsf{P}}\left({\mathbf{s}}_{t+1}\right)+{\mathbf{J}}_{\mathsf{D}}\left({\mathbf{s}}_{t+1}\right),$$

(24)

where ${\mathbf{J}}_{\mathsf{P}}\left({\mathbf{s}}_{t+1}\right)$ and ${\mathbf{J}}_{\mathsf{D}}\left({\mathbf{s}}_{t+1}\right)$ are the BFIM of the users’ a priori information [44] and the data of observations[45], respectively. Specific expressions of them are

$$\begin{array}{cc}\hfill & {\mathbf{J}}_{\mathsf{P}}\left({\mathbf{s}}_{t+1}\right)={[{\mathbf{Q}}_t+{\mathbf{F}}_{t+1}{\mathbf{J}}^{-1}\left({\mathbf{s}}_t\right){\mathbf{F}}_{t+1}^{\mathsf{T}}]}^{-1}\hfill \\ \hfill & {\mathbf{J}}_{\mathsf{D}}\left({\mathbf{s}}_{t+1}\right)={\mathbf{G}}_{t+1}^{\mathsf{T}}{\mathbf{R}}_{t+1}^{-1}{\mathbf{G}}_{t+1}.\hfill \end{array}$$

(25)

Given initialized $\mathbf{J}\left({\mathbf{s}}_0\right)$, the Fisher information matrix at time $t+1$ can be recursively calculated, and the BCRB of users’ state estimation error can be attained by

$$\begin{array}{cc}\hfill \mathbf{C}\left({\mathbf{s}}_{t+1}\right)& ={\mathbf{J}}^{-1}\left({\mathbf{s}}_{t+1}\right)\hfill \\ \hfill & ={\left[{\left[{\mathbf{Q}}_t+{\mathbf{F}}_{t+1}{\mathbf{J}}^{-1}\left({\mathbf{s}}_t\right){\mathbf{F}}_{t+1}^{\mathsf{T}}\right]}^{-1}+{\mathbf{G}}_{t+1}^{\mathsf{T}}{\mathbf{R}}_{t+1}^{-1}{\mathbf{G}}_{t+1}\right]}^{-1}.\hfill \end{array}$$

(26)

Note that $\mathbf{J}\left({\mathbf{s}}_t\right)$ should be parameterized by $\mathbf{s}\left(t\right)$ theoretically but may not always be available. Instead, we rely on the predictions made during the previous time interval. Since the diagonal elements of the BCRB include the lower bound of the MSE of both users’ position and velocity estimates, (26) cannot be used directly as the metric of tracking accuracy. Here, we denote

$${\mathbf{T}}_m=\left[\begin{array}{cc}{\mathbf{I}}_{3\times 3}& {\mathbf{0}}_{3\times 3}\\ {\mathbf{0}}_{3\times 3}& {\mathbf{0}}_{3\times 3}\end{array}\right],$$

(27)

and $\mathbf{T}=\mathrm{diag}({\mathbf{T}}_1,\dots ,{\mathbf{T}}_m,\dots ,{\mathbf{T}}_M)$. Thus, the metric of tracking accuracy can be prescribed as

$$\begin{array}{cc}\hfill L\left({\mathbf{s}}_t\right)& =tr\left(\mathbf{C}({\mathbf{p}}_1,\dots ,{\mathbf{p}}_m,\dots ,{\mathbf{p}}_M)\right)\hfill \\ \hfill & =tr\left(\mathbf{T}\mathbf{C}\right).\hfill \end{array}$$

(28)

The derived BCRB serves as a benchmark for the proposed tracking scheme, based on which an optimization scheme for the phase shift of RISs is proposed in the subsequent section.

4.2. GD-Based RIS Phase Shift Optimization

To achieve high-precision multi-user tracking, we design the phase shift induced at the RISs to minimize the BCRB mentioned above, i.e.,

$$\begin{array}{cc}\hfill \underset{\mathsf{\Omega }}{ min}& L\left(\mathsf{\Omega }\right)\hfill \\ \hfill s.t.& \left|c_{k,p}\right|=1,\forall k\in \mathcal{K},p\in \mathcal{P}.\hfill \end{array}$$

(29)

where $\mathsf{\Omega }$ denotes the set of RISs’ reflection coefficient matrices ${\left\{{\mathsf{\Omega }}_k\right\}}_{k\in \mathcal{K}}$. To tackle the optimization problem, we utilize the classical GD-based method. By applying the second-order Taylor expansion at the given point ${\mathsf{\Omega }}^r$, the approximate function is given by [46]

$$\begin{array}{c}\hfill \widehat{L}\left(\mathsf{\Omega }\right|{\mathsf{\Omega }}^r)=L\left({\mathsf{\Omega }}^r\right)+{\nabla }_{\mathsf{\Omega }}L{\left({\mathsf{\Omega }}^r\right)}^{\mathbf{T}}(\mathsf{\Omega }-{\mathsf{\Omega }}^r)+\frac{{\alpha }_r}{2}{\parallel \mathsf{\Omega }-{\mathsf{\Omega }}^r\parallel }^2.\end{array}$$

(30)

As long as the GD updating step ${\alpha }_r$ is chosen properly, which can be determined by the Armijo rule [47], the approximation function $\widehat{L}\left(\mathsf{\Omega }\right|{\mathsf{\Omega }}^r)$ satisfies the following three properties[48,49]

$$\begin{array}{cc}\hfill \widehat{L}\left(\mathsf{\Omega }\right|{\mathsf{\Omega }}^r)& \ge L\left(\mathsf{\Omega }\right),\hfill \\ \hfill \widehat{L}\left({\mathsf{\Omega }}^r\right|{\mathsf{\Omega }}^r)& =L\left({\mathsf{\Omega }}^r\right),\hfill \\ \hfill {\nabla }_{\mathsf{\Omega }}\widehat{L}\left({\mathsf{\Omega }}^r\right|{\mathsf{\Omega }}^r)& ={\nabla }_{\mathsf{\Omega }}L\left({\mathsf{\Omega }}^r\right).\hfill \end{array}$$

(31)

Then, the optimal solution of $\mathsf{\Omega }$ at the $(r+1)$-th iteration is given by

$$\begin{array}{c}\hfill {\mathsf{\Omega }}^{r+1}=arg\underset{\mathsf{\Omega }}{min}\widehat{L}\left(\mathsf{\Omega }\right|{\mathsf{\Omega }}^r).\end{array}$$

(32)

It is proved in [46] that the optimal ${\mathsf{\Omega }}^{r+1}$ follows the update rule of the GD method, which is given by

$$\begin{array}{c}\hfill {\mathsf{\Omega }}^{r+1}={\mathsf{\Omega }}^r-\frac{1}{{\alpha }_r}{\nabla }_{\mathsf{\Omega }}L\left({\mathsf{\Omega }}^r\right).\end{array}$$

(33)

Notably, the calculation of ${\nabla }_{\mathsf{\Omega }}L\left({\mathsf{\Omega }}^r\right)$ can be converted into the gradient calculation of the phase shift ${\left\{{\psi }_{k,p}\right\}}_{k\in \mathcal{K},p\in \mathcal{P}}$ of each element, which is given by [50]

$$\begin{array}{c}\hfill {\nabla }_{{\psi }_{k,p}}L\left({\psi }_{k,p}^r\right)=-\frac{1}{{\sigma }^2}tr\left\{{\mathbf{J}}^{-1}\left({\mathbf{s}}_{t+1}\right)\mathbf{T}{\mathbf{J}}^{-1}\left({\mathbf{s}}_{t+1}\right)\left\{{\left[\frac{\partial {\mathbf{G}}_{t+1}}{\partial {\psi }_{k,p}}\right]}^{\mathsf{T}}{\mathbf{G}}_{t+1}+{\mathbf{G}}_{t+1}^{\mathsf{T}}\frac{\partial {\mathbf{G}}_{t+1}}{\partial {\psi }_{k,p}}\right\}\right\},\end{array}$$

(34)

the detailed derivation of which is shown in Appendix B. The corresponding update process of ${\left\{{\psi }_{k,p}\right\}}_{k\in \mathcal{K},p\in \mathcal{P}}$ is given by

$$\begin{array}{c}\hfill {\psi }_{k,p}^{r+1}={\psi }_{k,p}^r-\frac{1}{{\beta }_r}{\nabla }_{{\psi }_{k,p}}L\left({\psi }_{k,p}^r\right).\end{array}$$

(35)

This procedure will repeat until it satisfies the stopping criteria. The whole procedure entailing the RISs’ phase shift design is sketched in Algorithm 2.

Algorithm 2 Optimization for RISs’ phase shift

Initialization ${\mathsf{\Omega }}^t$. Set iteration index $r=0$.  repeat       Compute ${\mathbf{G}}_{t+1}$ and $L\left({\mathbf{s}}_{t+1}\right)$ according to (19) and (28), respectively.       For each element in ${\mathsf{\Omega }}_k$, calculate ${\nabla }_{{\psi }_{k,p}}L\left({\psi }_{k,p}^r\right)$ by (34).       For each element in ${\mathsf{\Omega }}_k$, update ${\psi }_{k,p}^{r+1}$ by (35).       Set $r=r+1$.  until the fractional decrement of the target value is below a certain threshold.  Output: ${\mathsf{\Omega }}^{t+1}$

5. Results

A series of simulations are conducted in this section to assess the validity of the proposed tracking algorithm. With different parameter settings, the performance of our tracking algorithm against the BCRB is compared.

5.1. Simulation Scenario and Parameter Settings

Unless otherwise indicated, we focus on the tracking scenario as sketched in Figure 3.

For the channel settings, we consider a simple scenario where each single-antenna user transmits a single data symbol $p=1$ at a unique frequency. We set carrier frequency $f_c=28\mathrm{GHz}$, subcarrier spacing $\Delta f=120\mathrm{kHz}$, the frequency of the m-th user $f_m=f_c+m\Delta f-B/2$, and the signal bandwidth $B=M\Delta f$. The gain of the RIS unit element $G_p=\pi$, and the gain of the antenna of users and BS are $G_m=1$ and $G_b=1$, respectively. In our tracking scheme, the LoS paths between BS and users are presumed to be blocked. The interference power caused by large-scale fading and reflection loss is $20\mathrm{dB}$ weaker than the LoS path in general [51]. Thus, at the receiver, we fix the noise variance per subcarrier at ${\sigma }^2=-125\mathrm{dBm}$, and at the transmitter, we consider that transmitted power per subcarrier P varies from $0\mathrm{dBm}$ to $15\mathrm{dBm}$.

For system geometry, the only BS is located at ${\mathbf{p}}_{\mathrm{B}}=\left[\begin{array}{c}16,0,2\end{array}\right]\mathrm{m}$, and $K=3$ RISs are deployed at ${\left\{{\mathbf{p}}_k\right\}}_{k\in \mathcal{K}}$, including $\left[\begin{array}{c}0,0,3\end{array}\right]\mathrm{m}$, $\left[\begin{array}{c}8,-8,3\end{array}\right]\mathrm{m}$ and $\left[\begin{array}{c}8,8,3\end{array}\right]\mathrm{m}$. The BS consists of a uniform planar array of $8\times 3$ elements lying on the $YZ$ plane and the RISs are equipped with a uniform linear array of $60\times 1$ elements on the $YZ$, $XZ$, and $XZ$ planes, respectively. According to $f_c$, the antenna spaces on the BS array and the element spaces on the RISs’ array are $d_{space}=0.54\mathrm{cm}$. Notably, the elements in RISs are mainly deployed horizontally as long stripes, by which the radiating near-field scope is extended over about $18\mathrm{m}$.

For the kinematic model of users, $M=3$ single-antenna users move in a uniform linear pattern with ${\sigma }_x^2=0.5{\mathrm{m}}^2/{\mathrm{s}}^2$, ${\sigma }_y^2=0.5{\mathrm{m}}^2/{\mathrm{s}}^2$ and ${\sigma }_z^2=0$. The initial positions of users ${\left\{{\mathbf{p}}_m\right\}}_{m\in \mathcal{M}}^0$ are $\left[\begin{array}{c}10,0,1\end{array}\right]\mathrm{m}$, $\left[\begin{array}{c}4,-4,1\end{array}\right]\mathrm{m}$, and $\left[\begin{array}{c}4,4,1\end{array}\right]\mathrm{m}$, the first of which has an initial velocity of $-2\mathrm{m}/\mathrm{s}$ in the x direction, and the other two users both have initial velocities of $-1\mathrm{m}/\mathrm{s}$ in the x direction and $1\mathrm{m}/\mathrm{s}$ and $-1\mathrm{m}/\mathrm{s}$ in the y direction, respectively. The initial state is assumed to be relatively accurately measured with minor errors generated based on $\mathbf{Q}$ randomly.

For clearer declarations, the parameter settings of simulations are given in

Table 3. Simulation parameter settings.

Parameter	Value
$f_c$	$28\mathrm{GHz}$
$\Delta f$	$120\mathrm{kHz}$
P	$0\sim 15\mathrm{dBm}$
$G_p$	$\pi$
$G_b$	1
$G_m$	1
${\mathbf{p}}_{\mathrm{B}}$	$\left[\begin{array}{c}16,0,2\end{array}\right]\mathrm{m}$
$N_B$	$8\times 3$
$d_{space}$	$0.54\mathrm{cm}$
K	3
${\left\{{\mathbf{p}}_k\right\}}_{k\in \mathcal{K}}$	$\left[\begin{array}{c}0,0,3\end{array}\right]\mathrm{m}$, $\left[\begin{array}{c}8,-8,3\end{array}\right]\mathrm{m}$, $\left[\begin{array}{c}8,8,3\end{array}\right]\mathrm{m}$
$N_R$	$60\times 1$
M	3
${\left\{{\mathbf{p}}_m\right\}}_{m\in \mathcal{M}}^0$	$\left[\begin{array}{c}10,0,1\end{array}\right]\mathrm{m}$, $\left[\begin{array}{c}4,-4,1\end{array}\right]\mathrm{m}$, $\left[\begin{array}{c}4,4,1\end{array}\right]\mathrm{m}$
${\left\{{\dot{\mathbf{p}}}_m\right\}}_{m\in \mathcal{M}}^0$	$\left[\begin{array}{c}-2,0,0\end{array}\right]\mathrm{m}/\mathrm{s}$, $\left[\begin{array}{c}-1,1,0\end{array}\right]\mathrm{m}/\mathrm{s}$, $\left[\begin{array}{c}-1,-1,0\end{array}\right]\mathrm{m}/\mathrm{s}$
${\left\{{\sigma }_{x_m}^2\right\}}_{m\in \mathcal{M}}$	$0.5{\mathrm{m}}^2/{\mathrm{s}}^2$
${\left\{{\sigma }_{y_m}^2\right\}}_{m\in \mathcal{M}}$	$0.5{\mathrm{m}}^2/{\mathrm{s}}^2$
${\left\{{\sigma }_{z_m}^2\right\}}_{m\in \mathcal{M}}$	$0{\mathrm{m}}^2/{\mathrm{s}}^2$
${\sigma }^2$	$-125\mathrm{dBm}$

.

5.2. Simulation Results and Discussions

5.2.1. Convergence and Error Analysis of Tracking

We first study the convergence and error of our tracking scheme by executing a 3-second trajectory once with the experimental method and parameters in Section 5.1, which contains 100 discrete samplings of users’ positions. Other simulations conducted within the range of our parameter settings follow the same path.

The number of operations is hard to generalize as it depends on the choice of GD updating step size and other parameters. However, the convergence characteristics of the algorithm are worthy of study. As illustrated in Figure 4, this cluster of curves represents the convergence process of the optimization algorithm in 100 discrete sampling time slots, where the ordinate denotes the value of the objective function, and the abscissa denotes the number of iterations. The phase shift matrix of all RISs in each sampling time slot is randomly generated. We set the value of the objective function MSE at the end of the first tracking time slot as the metric for subsequent time slots, where the optimization procedure will repeat until it satisfies the stopping criteria. It can be found that the optimization of the phase shift design converges well, which validates the effectiveness of Algorithm 2. Further, our algorithm exploits temporal correlation and continuity during the tracking process, requiring fewer adjustments in subsequent time slots. Thus, except for the first optimization of RISs’ phase shift at the first tracking time slot, the optimization process takes much fewer operations in the subsequent time slots.

The accuracy of our proposed tracking scheme is evaluated by the root mean square error (RMSE) of tracking errors. As indicated in Figure 5, the abscissa denotes 100 consecutive time slots and the ordinate denotes the RMSE of the tracking errors. The RMSE of the tracking errors with random phase shift is divergent, as the LoS paths between users and the BS are presumed to be blocked, which is a relatively harsh tracking scenario. However, our tracking scheme with optimized phase shift can achieve significant accuracy and can approximate the theoretical BCRB, which validates the effectiveness of Algorithm 1. Further, in the optimized case, it can be found that at some certain sampling slots, the RMSE of tracking errors is larger than other slots. This indicates that even if our tracking scheme is temporarily underperforming, it will not cause major interference with the subsequent tracking process, since the EKF procedure is a data fusion process based on both model prediction and data measurement.

5.2.2. Effect of the Number of BS Antennas

We first investigate the effect of the number of BS antennas on the RMSE of tracking with varying transmit power. In this and all the subsequent simulations, the performance of our tracking scheme is evaluated by the RMSE of tracking errors averaging from 30 conducted 3-second trajectories, each containing 100 discrete samplings of users’ positions. The tracking error with random phase shift on RISs, the averaging of which has no practical significance, will not be further studied because it is divergent.

With other variable parameters fixed at $K=3$, $N_R=60$, ${\left\{{\sigma }_{x_m}^2\right\}}_{m\in \mathcal{M}}=0.5{\mathrm{m}}^2/{\mathrm{s}}^2$, ${\left\{{\sigma }_{y_m}^2\right\}}_{m\in \mathcal{M}}=0.5{\mathrm{m}}^2/{\mathrm{s}}^2$, ${\left\{{\sigma }_{z_m}^2\right\}}_{m\in \mathcal{M}}=0{\mathrm{m}}^2/{\mathrm{s}}^2$, the transmit power P varying from $0\mathrm{dBm}$ to $15\mathrm{dBm}$ and the number of BS antennas $N_B$ is set by $8\times 3$, $8\times 6$, and $8\times 9$ as a comparison, respectively.

As illustrated in Figure 6, from three groups of simulations with different numbers of BS antennas, it is easy to find that the accuracy of our tracking algorithm can approximate the theoretical BCRB. With the increment of BS antennas, our tracking algorithm performs significantly better and approaches closer to the BCRB. This is reasonable as the increment of BS antennas extends the dimension of measurements, which contributes to the accuracy of the estimation procedure. Moreover, the performance of our tracking scheme improves and approximates BCRB more closely with the increment of transmit power.

5.2.3. Effect of Users’ Mobility

To study the effect of the mobility of users, we have a fixed sampling frequency, which means that the localization and optimization procedure is conducted at the same time slot length. Thus, users’ mobility can be characterized by both covariance matrix $\mathbf{Q}$. With other variable parameters fixed at $K=3$, $N_R=60$, $N_B=24$, ${\left\{{\sigma }_{z_m}^2\right\}}_{m\in \mathcal{M}}=0{\mathrm{m}}^2/{\mathrm{s}}^2$, the transmit power P varies from $0\mathrm{dBm}$ to $15\mathrm{dBm}$. Controlled simulations are conducted where the velocities of users are fixed, ${\left\{{\sigma }_{x_m}^2\right\}}_{m\in \mathcal{M}}$ and ${\left\{{\sigma }_{y_m}^2\right\}}_{m\in \mathcal{M}}$ in $\mathbf{Q}$ vary within the range of $0.1{\mathrm{m}}^2/{\mathrm{s}}^2$, $0.5{\mathrm{m}}^2/{\mathrm{s}}^2$, and $1{\mathrm{m}}^2/{\mathrm{s}}^2$.

It is obvious in Figure 7 that the tracking of trajectories with minor covariance attains higher accuracy and can reach a relatively good accuracy even under a high-covariance circumstance. Moreover, the performance of our tracking scheme improves and approximates BCRB more closely with the increment of transmit power.

5.2.4. Effect of the Number of RISs and RIS Elements

We exploit the influence of both the number of RISs and RIS elements.

To assess the influence of the number of RISs, with other variable parameters fixed at $N_R=60$, $N_B=72$, ${\left\{{\sigma }_{x_m}^2\right\}}_{m\in \mathcal{M}}=0.5{\mathrm{m}}^2/{\mathrm{s}}^2$, ${\left\{{\sigma }_{y_m}^2\right\}}_{m\in \mathcal{M}}=0.5{\mathrm{m}}^2/{\mathrm{s}}^2$, ${\left\{{\sigma }_{z_m}^2\right\}}_{m\in \mathcal{M}}=0{\mathrm{m}}^2/{\mathrm{s}}^2$, the transmit power P varying from $0\mathrm{dBm}$ to $15\mathrm{dBm}$ and the RIS deployed at $\left[\begin{array}{c}0,0,3\end{array}\right]\mathrm{m}$ is removed in the $K=2$ case. As indicated in Figure 8, the performance of tracking in the $K=3$ case is significantly better than in the $K=2$ case. With the increment in the number of RISs, our tracking algorithm approaches much closer to the BCRB as well.

To exploit the effect of the number of RIS elements, we consider other variable parameters fixed at $K=3$, $N_B=24$, ${\left\{{\sigma }_{x_m}^2\right\}}_{m\in \mathcal{M}}=0.5{\mathrm{m}}^2/{\mathrm{s}}^2$, ${\left\{{\sigma }_{y_m}^2\right\}}_{m\in \mathcal{M}}=0.5{\mathrm{m}}^2/{\mathrm{s}}^2$, ${\left\{{\sigma }_{z_m}^2\right\}}_{m\in \mathcal{M}}=0{\mathrm{m}}^2/{\mathrm{s}}^2$, the transmit power P varying from $0\mathrm{dBm}$ to $15\mathrm{dBm}$, and the numbers of RISs’ elements $N_R$ varying in $60\times 1$, $60\times 2$, and $60\times 3$. As illustrated in Figure 9, with the increment in the number of RIS elements, the performance of our tracking algorithm improves. Further, we can find that with the increment in the number of elements, the tracking accuracy of the proposed scheme is closer to the BCRB. Moreover, as indicated by the gaps between three groups of lines with different $N_R$, the improvement of accuracy with the increment of $N_R$ becomes less significant when the same number of elements is added.

From the above two sets of experiments, it can be concluded that with abundant RISs deployed and optimization of their phase shift, high accuracy can be achieved in harsh localization environments. In all cases, the performance of our tracking scheme improves and approximates BCRB more closely with the increment of transmit power.

6. Conclusions

In summary, concerning the mobility of users and the potential obstruction of LoS paths, a multi-user tracking system based on EKF is designed by directly utilizing the non-linear relationship between the signals received at the BS antennas and the coordinates of the moving targets, where the temporal correlations between each user’s coordinate changes are fully exploited. An optimization scheme of passive phase shift design at the RISs is devised by minimizing the derived BCRB and solving the problem by leveraging the GD method.

Numerical results indicate that the accuracy of our tracking scheme is affected by a series of factors. With the increment of transmit power, the number of BS antennas, users’ mobility, and the number of RISs or RIS elements, the proposed method attains higher tracking accuracy and approaches closer to the BCRB in the conducted simulations. High-precision multi-user tracking via a single BS can be realized by abundant RISs deployed and optimized, even in harsh localization environments. Future works will entail the tracking of multiple users under more variable scenarios, i.e., monitoring the existence of the LoS path between the RISs and the users and switching the adopted tracking model under varied circumstances.