Reconfigurable Intelligent Surface Aided
Wireless Localization

The fifth generation (5G) wireless technology offers great opportunities for accurate wireless localization due to the very high carrier frequency and large antenna arrays. Moreover, the sixth generation (6G) systems continue to develop towards even higher frequency ranges, e.g., at the millimeter-wave (mm-wave) as well as Tera Hertz (THz) ranges. The 6G white paper on localization and sensing released by University of Oulu [1] points out that 6G systems will not only provide ubiquitous communication but also empower high accuracy localization due to the very fine angular resolutions. Decimeter-level or even centimeter-level positioning will be achieved by taking advantages of the new 6G key technology enablers.

The advantages of mm-wave and large antenna arrays for accurate wireless localization have been extensively studied. Jeong, et al., propose an active beamforming method to enhance the localization accuracy of distributed antenna systems [2]. Shahmansoori, et al., study the position and orientation estimation through mm-wave massive multiple-input multiple-output (MIMO) systems [3]. Wang, Wu, and Shen prove the asymptotic spatial orthogonality of large-scale MIMO localization with general non-orthogonal waveforms [4]. They also derive the Fisher information matrices for the position and orientation to characterize the performance bounds of MIMO localization. Zhou, et al., focus on active beamforming to reduce the localization error, and propose a successive localization and beamforming scheme [5].

However, how to further improve the accuracy of wireless localization, even in the case of obstructed line-of-sight, is largely undiscovered. Thus, in this paper, we introduce the reconfigurable intelligent surface (RIS), which is an efficient method to control the wavefront of the impinging signals, into the system to make the positioning more accurate and energy efficient. Our prior work shows that an RIS-assisted wireless communication system can achieve both high spectral and energy efficiency [6]. Wu and Zhang verify that the RIS is able to drastically enhance the link quality over the conventional setup without the RIS [7].

In this paper, we introduce the RIS into the wireless localization system, and focus on the (passive) reflect beamforming to reduce the localization error. We first build the three-dimensional RIS-assisted wireless localization system model. After that, we derive the Fisher information matrix and the Cramer-Rao lower bound for the estimate of absolute mobile station (MS) position. For the sake of convenience in reflect beamforming design, we separate the response of RIS in the Fisher information matrix. Finally, we propose an alternative optimization method and a gradient decent method to optimize the reflect beamforming, which aims to minimize the Cramer-Rao lower bound. Our results show that the proposed methods significantly improve the accuracy of positioning, and decimeter-level or even centimeter-level positioning can be achieved by utilizing the RIS with a large number of reflecting elements.

Encouraged by the potential of mm-wave signals and large antenna arrays to improve the accuracy of positioning, many standardized channel models have started to emerge [3, 8, 5]. We base the evaluation of our work on these models while introducing the RIS into the system to make the MS positioning more accurate. Introducing RIS can provide extra reflection paths which help in the estimation of MS position. On the one hand, it makes the positioning become possible when the line-of-sight is obstructed; On the other hand, it adds more position information for estimation.

Based on the channel model, the unknown parameters to be estimated can be denoted as a $3N$ dimensional vector $\boldsymbol{\eta}$,

We denote the unbiased estimator of $\boldsymbol{\eta}$ as $\hat{\boldsymbol{\eta}}$, and based on the Cramer-Rao theorem [9, 10], the mean-square error is bounded as

where $\mathbf{J}_{\boldsymbol{\eta}}$ is the $3N\times 3N$ Fisher information matrix, and $[\mathbf{J}_{\boldsymbol{\eta}}^{-1}]_{m,m}$ is the Cramer-Rao lower bound for the $m$th parameter estimate. The $(m,n)$th entry of $\mathbf{J}_{\boldsymbol{\eta}}$ is defined as

where $p(\boldsymbol{y};\boldsymbol{\eta})$ is the likelihood function of the random vector $\boldsymbol{y}$ conditioned on $\boldsymbol{\eta}$, and $\eta_{m}$ is the $m$th entry of $\boldsymbol{\eta}$. The proof of the identity in Eq. (9) is given in [10]. Based on the received signal model, $p(\boldsymbol{y};\boldsymbol{\eta})$ can be written as

where the vector $\boldsymbol{\mu}$ is the mean vector of the random vector $\boldsymbol{y}$ conditioned on $\boldsymbol{\eta}$.

To further obtain the Fisher information matrix $\mathbf{J}_{\boldsymbol{\eta}}$, we first present the following Lemma from Eq. (6) in [11]:

By using the above Lemma, the $(m,n)$th entry of $\mathbf{J}_{\boldsymbol{\eta}}$ in Eq. (9) can be rewritten as

where the mean vector $\boldsymbol{\mu}(l)=\tilde{\mathbf{H}}\boldsymbol{x}(l)$. Eq. (12) demonstrates that the Fisher information matrix $\mathbf{J}_{\boldsymbol{\eta}}$ is the sum of the Fisher information matrix $\mathbf{J}_{\boldsymbol{\eta}}(l)$ at each time slot. The rest in this section gives the solving process of $\mathbf{J}_{\boldsymbol{\eta}}(l)$.

We rewrite the Fisher information matrix $\mathbf{J}_{\boldsymbol{\eta}}(l)$ as a partitioned matrix in Eq. (13) which is for later use.

where the entries in submatrix $\bm{\mathcal{A}}$ contain the partial derivatives with respect to the azimuth/elevation AoA, while the entries in submatrices $\bm{\mathcal{B}}$, $\bm{\mathcal{C}}$ and $\bm{\mathcal{D}}$ contain the partial derivatives with respect to the propagation gains.

Next, we give the partial derivatives of $\boldsymbol{\mu}(l)$ with respect to the azimuth/elevation AoA. Before giving it, we first give the partial derivatives of the matrix $\mathbf{\Lambda}_{\mathrm{r}}(\mathbf{H}\mathbf{\Phi})\mathbf{\Lambda}_{\mathrm{t}}^{\mathrm{H}}$. The $(m,n)$th entry of the matrix $\mathbf{\Lambda}_{\mathrm{r}}(\mathbf{H}\mathbf{\Phi})\mathbf{\Lambda}_{\mathrm{t}}^{\mathrm{H}}$ is given by

where $\xi_{i}=k\sin\vartheta_{i}\sin\varphi_{i}$ and $\zeta_{i}=k\sin\theta_{i}\sin\phi_{i}$. To separate the response of RIS, the partial derivatives of the matrix $\mathbf{\Lambda}_{\mathrm{r}}(\mathbf{H}\mathbf{\Phi})\mathbf{\Lambda}_{\mathrm{t}}^{\mathrm{H}}$ with respect to the azimuth/elevation AoA are given by

where

Then, we obtain the partial derivatives of $\boldsymbol{\mu}(l)$ with respect to the azimuth/elevation AoA as follows:

where $\boldsymbol{\dot{\varpi}}_{i}(l)=\boldsymbol{\dot{\Xi}}_{i}\boldsymbol{x}(l)$ and $\boldsymbol{\ddot{\varpi}}_{i}(l)=\boldsymbol{\ddot{\Xi}}_{i}\boldsymbol{x}(l)$.

Compared with the azimuth/elevation AoA, what we are more interested in is the absolute MS position $\boldsymbol{q}_{1:2}=[q_{x},q_{y}]^{\mathrm{T}}$. The Fisher information matrix of it can be obtained by means of the $2\times 3N$ transformation matrix $\mathbf{T}$, which is expressed as

where the transformation matrix $\mathbf{T}$ is defined as

The entries of the transformation matrix $\mathbf{T}$ can be obtained by the geometrical relationship between the azimuth/elevation AoA and the MS position which is shown as follows:

Then, by computing the partial derivatives of the azimuth and elevation AoA with respect to the MS position, we obtain the transformation matrix $\mathbf{T}$ as follows:

where the $i$th column vector of the submatrices $\bm{\mathcal{E}}$, $\bm{\mathcal{F}}$ and $\bm{\mathcal{G}}$ respectively are

The Eqs. (26) - (28) are derived in subsection A of Appendix. Because the $2\times N$ submatrix $\bm{\mathcal{G}}$ is a zero matrix, then based on the multiplication principle of partitioned matrix, Eq. (21) can be simplified as

where the matrix $\tilde{\mathbf{T}}$ consists of the submatrices $\bm{\mathcal{E}}$ and $\bm{\mathcal{F}}$, and the matrix $\bm{\mathcal{A}}$ is the $2N\times 2N$ submatrix in Eq. (13).

We rewrite the matrix $\bm{\mathcal{A}}$ as a partitioned matrix as follows:

Based on the definition of the $(m,n)$th entry of $\mathbf{J}_{\boldsymbol{\eta}}(l)$ in Eq. (12) and the partial derivatives of $\boldsymbol{\mu}(l)$ with respect to the azimuth/elevation AoA in Eqs. (19) and (20), we find that the response of RIS in the $(m,n)$th entry of the submatrix $\bm{\mathcal{H}}$ is same with those in the submatrices $\bm{\mathcal{I}}$, $\bm{\mathcal{J}}$ and $\bm{\mathcal{K}}$, which is $e^{j({\color[rgb]{0,0,0}\varrho_{n}}-{\color[rgb]{0,0,0}\varrho_{m}})}$. Then, the entries in the $2\times 2$ Fisher information matrix $\mathbf{J}_{\boldsymbol{q}}(l)$ can be given as

where the scalars $\kappa_{m,n}^{1,1}(l)$, $\kappa_{m,n}^{1,2}(l)$, $\kappa_{m,n}^{2,1}(l)$ and $\kappa_{m,n}^{2,2}(l)$ are

and the parameters $\alpha_{m}^{x}$, $\alpha_{m}^{y}$, $\beta_{m}^{x}$ and $\beta_{m}^{y}$ respectively are the first and second entries of $[\bm{\mathcal{E}}]_{m}$ and $[\bm{\mathcal{F}}]_{m}$. Then, by using Eq. (12), we obtain the entries of the Fisher information matrix $\mathbf{J}_{\boldsymbol{q}}$ as follows:

Finally, we obtain the Cramer-Rao lower bound for the MS position estimate which is the trace of the inverse matrix of the Fisher information matrix $\mathbf{J}_{\boldsymbol{q}}$. The above result can be summarized as a theorem:

In this paper, we aim to minimize the Cramer-Rao lower bound for the MS position estimate by optimizing the reflect beamforming at the RIS. Accordingly, the optimization problem can be formulated as

Because the objective function in Eq. (41) is a non-convex fractional function, it is difficult to obtain the global optimal solution. To further analyze this problem, we utilize the gradient decent method (GDM) to make the reflect beamforming design. In the gradient flow direction which is defined as the direction of the negative gradient $-\nabla_{\boldsymbol{\varrho}}f(\boldsymbol{\varrho})$, the function $f(\boldsymbol{\varrho})$ decreases at the maximum descent rate. And each component of the gradient vector gives the rate of change of the scalar function in the component direction [12].

The function $f(\boldsymbol{\varrho})$ is a scalar function with respect to an $N\times 1$ vector-variable. Thus, the gradient of it is expressed as

where the partial derivative of $f(\boldsymbol{\varrho})$ with respect to $\varrho_{i}$ is given in Eq. (43) at the top of the next page.

The parameters $\mathfrak{Nu}$ and $\mathfrak{De}$ in it represent the numerator and denominator of $f(\boldsymbol{\varrho})$, respectively, and the partial derivatives of the entries in the Fisher information matrix $\mathbf{J}_{\boldsymbol{q}}$ with respect to $\varrho_{i}$ is given as follows:

The Eqs. (44) - (47) are derived in subsection B of Appendix.

It should be noted that the objective function $f({\boldsymbol{\varrho}})$, i.e., the Cramer-Rao lower bound for the MS position estimate is depend on the unknown parameter $\boldsymbol{\eta}$, and we cannot directly utilize the GDM to optimize the objective function. To address this challenge, we adopt alternative optimization, as illustrated in Algorithm 1. More specifically, we start at an initial phase shift vector, and alternately update the estimator of $\boldsymbol{\eta}$ and optimize the phase shift vector $\boldsymbol{\varrho}$, until they converge. The estimation method used in Algorithm 1 is provided in the journal version, and will not be discussed here due to space limitation, and the detailed step of GDM-based reflect beamforming design is illustrated in Algorithm 2.

This paper focuses on the impact of reflect beamforming on the Cramer-Rao lower bound for the MS position estimate, and aims to minimize it ¹¹1 Some prior works, like [3, 13], only provide the values of the Cramer-Rao lower bound for the estimate of AoA/AoD. But a small deviation in angle can result in a large deviation in absolute position, and the positioning accuracy can not be reflected directly. Thus, in this section, we present the values of the Cramer-Rao lower bound for the estimate of absolute MS position. . The estimation method is provided in the journal version, and will not be discussed here due to space limitation. Thus, we set up the system parameters to be known or have been estimated: the wavelength of mm-wave signal is set up as $0.006$, $\boldsymbol{p}=[0,0,0]^{\mathrm{T}}$, $\boldsymbol{q}=[50,100,0]^{\mathrm{T}}$, and the RIS is a uniform planar array in the vertical plane where $\boldsymbol{s}^{1}=[-20,50,20]^{\mathrm{T}}$ and the separation between each reflecting element is $0.1$, all in meters. The propagation gains $h_{i}\;(i=1,\cdots,N)$ of $N$ reflection paths are set up as random complex numbers which obey the distribution of $\mathcal{C}\mathcal{N}(0,1)$.

In this paper, we utilize the advantages of mm-wave signals and large antenna arrays technologies, and introduce the RIS into the system to make the MS positioning more accurate. Our first contribution is to build the three-dimensional RIS-assisted wireless localization system model. After that, we derive the Fisher information matrix and the Cramer-Rao lower bound for the estimate of absolute MS position. Finally, we propose an alternative optimization method and a GDM-based algorithm to make the reflect beamforming design which aims to minimize the Cramer-Rao lower bound. The simulation results show that introducing RIS can make the positioning become possible when the line-of-sight is obstructed, and the proposed algorithm can make the estimation of MS position more accurate. The decimeter-level or even centimeter-level positioning can be achieved by utilizing the RIS with a large number of reflecting elements.