In the context of single-BS NLoS single-epoch localization with the aid of a reflective RIS, this paper introduces a novel three-step algorithm that jointly estimates the position and velocity of a mobile UE, while compensating for the  Doppler effects observed in NF at the RIS elements over the short transmission duration of a sequence of DL pilot symbols. First, a low-complexity initialization procedure is proposed, relying in part on FF approximation and a static user assumption. Then, an alternating optimization procedure is designed to iteratively refine the velocity and position estimates, as well as the channel gain.
The refinement routines leverage small angle approximations and the linearization of the RIS response, accounting for both NF and mobility effects. We evaluate the performance of the proposed algorithm through extensive simulations under diverse operating conditions with regard to SNR, UE mobility, uncontrolled multipath and RIS-UE distance.

Our results reveal remarkable performance improvements over the SoTA mobility-agnostic benchmark algorithm, while indicating convergence of the proposed algorithm to respective theoretical bounds on position and velocity estimation.

§ INTRODUCTION

As the 5G wireless communication systems are revolutionizing the way we connect and communicate, the industry is now gearing up for the next leap in wireless technology, namely 6G. As we set our sights on this new frontier, a new set of challenges, KPI, and enabling technologies are emerging [1]. The vision for 6G encompasses even higher data rates, ultra-reliable and low-latency communication, ubiquitous connectivity, and intelligent network infrastructure to support diverse applications such as autonomous systems, virtual reality, and IoT [2, 3, 4]. Meeting these ambitious goals requires breakthroughs in various enabling technologies, including advanced antenna systems, efficient spectrum utilization, edge computing, and innovative signal processing techniques [5, 6, 7].

Among the promising enabling technologies for the realization of 6G,  RIS have emerged as a disruptive and transformative concept [8]. RIS, also known as intelligent reflecting surfaces or programmable metasurfaces, are planar structures comprising numerous passive elements that can control and manipulate electromagnetic waves [9].
By dynamically adjusting the phase, the polarization, and even possibly the amplitude,
of the incident signals, RIS can actively shape the wireless propagation environment [10]. This ability opens up a myriad of possibilities to enhance the performance and efficiency of wireless systems. RIS can indeed contribute to optimizing signal quality, improve coverage, mitigate interference, and extend the reach of wireless networks, offering a cost-effective, energy-efficient, easy-to-deploy, and scalable
solution compared to conventional approaches [11]. The integration of RIS into the wireless ecosystem holds great potential for changing
the way we design, deploy, and operate future communication networks, making it a subject of significant research interest and exploration [12, 13, 14, 15, 16, 17].

In addition to its impact on communication performance, the RIS technology also holds great potential for localization applications [18]. The capability to precisely position UE in wireless networks is indeed crucial for various emerging services, such as autonomous navigation, augmented reality, and other location-based services but also for network performance improvement [19, 20].
For instance, by leveraging the controllable element-wise phases distribution of a reflective RIS,
it is possible to enhance localization continuity, accuracy, and reliability, or even simply just to make localization feasible, in restrictive scenarios and challenging environments [21]. RIS can overcome harsh propagation conditions (e.g., NLoS)
or limited deployment settings for which conventional systems based on active BS would fail [21, 22]. More generally, they can contribute to optimizing signal strength and quality to better estimate the location-dependent radio parameters required for localization [23].
This natural synergy between RIS and localization techniques opens up new possibilities for high-precision positioning.
Related open research challenges are summarized in [18]. As concrete examples, reflective RIS have been considered for parametric multipath-aided static UE positioning in two cases: when the BS-UE direct path is present, namely LoS [23, 24, 25, 26, 27], and when it is blocked, i.e., NLoS [21, 28, 29]. In the former case, the BS and UE have at least two communication links, which facilitates the use of the traditional two-step localization approach and geometric FF model. In contrast, when only one link is established, the previous approach might fail and a direct one-step alternative is used where we exploit the wavefront curvature in case the UE lies in the NF region of the RIS.

A major drawback of the existing studies in the literature is that
only a few works so far have addressed the impact of mobility in the very context of RIS localization, and those can be classified into either snapshot estimation (i.e., single-epoch) methods or tracking. Among the contributions that follow the former case, the authors in [30] performed
position estimation of a mobile UE in the FF region of the RIS under spatial-wideband effects, considering the presence of both the LoS and the RIS-induced NLoS path.

While in [31], velocity is estimated in FF LoS conditions, via a reflective RIS, relying on both direct and RIS-reflected paths.
Then, shifting away from snapshot estimation, in [32], a UE transmitting a narrowband signal is tracked through filtering (i.e., including 3D velocity as estimated variable, besides 3D position), while exploiting phase and amplitude observations accounting for the curvature-of-arrival of the impinging wavefront at a RIS in receiving mode in the NF region.
Moreover, in [33] the authors present a tracking algorithm based on EKF to localize UE, in a LoS scenario, with the aid of a reflective RIS at the mmWave frequency domain,
while [34] addresses a joint RIS reflection coefficients and BS precoder optimization problem and estimates UE's trajectory in a single mobile UE multi-RIS MIMO scenario in the NF regime. It is true that [32, 33, 34] take the UE mobility into account, however, the estimation happens over multiple snapshots via a tracking filter and the mobility within individual snapshots is ignored.

In summary, the literature lacks studies where the position and/or the velocity of a mobile UE is estimated, with a single epoch, via a reflective surface while exploiting the small-scale Doppler effects resulting from the mobility.

Therefore, in this paper, we tackle the problem of snapshot position and velocity estimation of a mobile single-antenna UE under LoS blockage (i.e., in NLoS) based on multiple narrowband DL transmissions from a single-antenna BS, while benefiting from the collateral effects caused by the UE mobility at a reflective RIS in the geometric NF propagation regime.

More specifically, we show how the  Doppler effects induced by UE mobility (i.e., over the time duration of the sequence of pilot signals) at the different elements of a large reflective RIS can be taken into account and even compensated for in the estimation of position and/or velocity.

The main paper contributions can be summarized below as follows:

* Problem Formulation for RIS-Aided 6D Near-Field Snapshot Estimation: For the first time, we formulate the problem of single-snapshot near-field estimation of the 3D position and 3D velocity of a mobile UE using narrowband DL transmissions from a BS, assisted by a RIS in the challenging SISO scenario under LoS blockage.

* Static UE Position Estimation: We propose a practical initialization procedure that provides coarse estimates for both the 3D UE position and the complex channel gain of the RIS-reflected path, by relying on preliminary static UE assumption and FF approximation, before re-injecting these results into the generic NF RIS response formulation for further corrections.

* Mobile UE Position Estimation: Leveraging the previous initialization, we also solve the UE positioning problem with known velocity, which performs linearization and small angle approximation to compensate for position estimation residuals caused by mobility. Subsequently, we develop a global position and velocity estimation algorithm that assumes no prior information (i.e., neither on position nor on velocity) and iterates alternatively over both estimates to gradually correct their residuals.

* Extensive Performance Evaluation: We evaluate the performance of the proposed estimation framework through comprehensive simulations and benchmark against both the theoretical CRLB (namely PEB and VEB) and a state-of-the-art algorithm neglecting the effect of mobility on snapshot positioning, while illustrating the impact of key parameters such as the RIS-UE distance in light of geometric NF conditions, uncontrolled multipath, SNR, UE velocity and the prior knowledge of velocity (comparing both perfect and no velocity knowledge assumptions from the algorithm point of view).

§.§.§ Notations

Scalar variables, vectors, and matrices are respectively denoted by lower-case, lower-case bold, and upper-case bold letters (e.g., $a, \aa, \AA$) and their indices are denoted by subscripts. Moreover, $\II_N$ and $\boldzero_N$ represent the identity and the all-zero matrices of size $N\times N$, respectively. The symbols $(.)^\top$, $(.)^{*}$ and $(.)^{\mathsf{H}}$ respectively denote transpose, conjugate and hermitian conjugate, the $\mathrm{tr}(\AA)$ operator is the trace of matrix $\AA$, and $\text{diag}(\AA)$ denotes a diagonal matrix with diagonal elements defined by vector $\aa$. Furthermore, the operators $\Vert\cdot\Vert$, $\Re\{\cdot\}$ and $\Im\{\cdot\}$ represent the $l_{2}$-norm real and imaginary operators. Finally, $\mathbb{E}\{\cdot\}$ is the expectation of a random variable w.r.t. its distribution and $\projnull{\AA} = \Imatrix - \projrange{\AA}$ the the projection into the null-space of $\AA$ with $\projrange{\AA} = \AA (\AA^\h \AA)^{-1} \AA^\h$.

§ SYSTEM MODEL

Considered scenario and problem geometry. The LoS between the BS and UE is blocked and the UE is in the near-field of the RIS.

We consider a DL localization scenario with a single-antenna BS transmitting $L$ narrowband pilot symbols, a single-antenna UE and an $M$-element reflective RIS, visualized in Fig. <ref>.
The LoS path between the BS and the UE is considered to be blocked, thus localization is made feasible by relying on the RIS-reflected paths and the geometric NF model, as in [21]. In this section, we present the geometric and signal models, followed by the problem formulation.

§.§ Geometric Model

The 3D positions are all expressed in the same global reference coordinates system, as follows: $\pp_{b} \in \realset{3}{1}$ is a vector containing the known BS coordinates, $\pp_{m} \in \realset{3}{1}$ is a vector holding the known coordinates of the RIS $m$-th element where $m \in \{1,\dots, M\}$ and $r$ is the reference element which is the RIS center, in our case. Also, $\pp\in \realset{3}{1}$ holds the UE's coordinates and is expressed as $[p_x,p_y,p_z]^{\top}$ in the Cartesian coordinates system or as $[\rho, \theta, \phi]^{\top}$ in the spherical coordinates system. Fig. <ref> illustrates the scenario, its geometry, and the chosen coordinates axes.
Moreover, we denote by $\vv = [v_{x}, v_{y}, v_{z}]^\top \in \realset{3}{1}$ the UE velocity vector, and by
$v_{m}= \vv^\top \uu_{m}(\pp)$ the radial velocities along the $m$-th RIS element - UE direction.

We also define $\uu_{m}(\pp) = ({\pp_{m} - \pp})/ {\dm}$, and $\dm = \norm{\pp_{m} - \pp}$.

§.§ Signal Model

The position of the mobile UE at measurement instance $\ell$ (i.e., $\ell$-th pilot symbol) for $\ell \in \{1,\dots, L\}$ can be expressed $\pp_\ell= \pp + \vv \ell \Ts$ where $\vv \ell \Ts$ denotes the displacement induced by the velocity vector $\vv$.
Then, leveraging the model in [35], the received signal model at instance $\ell$ can be expressed in NF as[We assume the absence of uncontrolled multipath in algorithm derivation [30, 36]. In the numerical results of Section <ref>, we will investigate the impact of uncontrolled multipath on estimation performance.]

\begin{align}
y_\ell &=  \alpha \, \aa^{\top}(\pp_\ell) \Omegabl \aa(\ppb) s_\ell + n_\ell \label{eq:obs_model}
%\alpha \wwl^{\top} \aal(\pp, \vv)\xl + n_\ell
\end{align}

where $\pp_\ell$ is a function of $\pp$ and $\vv$,

and $s_\ell$ denotes the transmitted pilot (taken as $s_\ell = 1, \, \forall \ell$ hereafter for ease of exposition), $\alpha \in \mathbb{C}$ is the complex channel gain that involves the impact of the transmit power, attenuation in both BS-RIS and RIS-UE paths and global phase offset, $\Omegabl = {\rm{diag}}(\omegabl)$ with $\omegabl \in \mathbb{C}^{M\times1}$ denoting the vector of controlled RIS element-wise reflection coefficients, and  $n_\ell \sim \mtcn(0, N_0)$ is an independent and identically distributed additive zero-mean normal noise of standard deviation $N_0$. In (<ref>), the element-wise RIS response model can be expressed as follows:

\begin{align}
[\aa(\pp_\ell)]_{m} &=  \exp\left(-j\frac{2\pi}{\lambda} \flm (\pp, \vv)
%(\norm{\pp_{m} - \pp}
%\vv^\top \uu_{m}(\pp) \ell\Ts
%\norm{\pp_{r} - \pp}
%\vv^\top \uuris(\pp) \ell\Ts)
\right)~, \label{eq:nf_steer}\\
\flm (\pp, \vv) &= \norm{\pp_{m} - (\pp + \vv  \ell\Ts)}  - \norm{\ppr -  (\pp + \vv  \ell\Ts)} ~,\label{eq:flm_undeveloped}\\
& \approx \dm - \dr
(\uu_{m}(\pp) - \uuris(\pp))^\top\vv \ell \Ts~,\label{eq:flm}
\end{align}

where (<ref>) is obtained from (<ref>) after a Taylor series expansion and dropping higher-order terms (See Appendix <ref>). It is worth noting that $\aa(\pp_{b})$, which represents the RIS response vector corresponding to signals arriving from the BS, remains independent of $\ell$ since both the BS and the RIS are stationary entities.

To have a compact model for the observations, we stack (<ref>) over $L$ symbols and define the observation  vector $\yy \triangleq [y_1 \ \cdots \ y_L]^\top \in \complexset{L}{1}$, leading to

\begin{align}
\yy &= \alpha \, \hh(\pp, \vv) + \nn ~,\label{eq:yy}\\
\hh(\pp, \vv) &\triangleq [ h_1(\pp,\vv) \ \cdots \ h_L(\pp,\vv) ]^\top \in \complexset{L}{1}, \label{eq:hl_pv_vec}\\
h_{\ell}(\pp,\vv) &\triangleq \wwl^\top \aa(\pp_\ell) \label{eq:hl_pv} ~,
\end{align}

where $\nn \triangleq [n_1 \ \cdots \ n_L]^\top$ and $\wwl \triangleq \omegabl \odot \aa(\ppb)  \in \mathbb{C}^{M\times 1}$.

§.§ Problem Description and Maximum-Likelihood Estimator

Given the NLoS RIS-induced observations (<ref>) at the mobile UE, our goal is to perform joint estimation of its position $\pp$ and velocity $\vv$, along with the nuisance parameter $\alpha$.
For this problem, the ML estimator of the unknown parameter vector $\vect{\zeta} = [\pp^\top, \vv^\top, \alpha]^{\top}$ can be expressed as

\begin{align}
( \alphahat  ~, \pphat ~, \vvhat)
%&= \arg \max_{\alpha, \pp, \vv} -\sum_{\ell=1}^{L} \mid y_\ell - \alpha \vect{w}_\ell^{\top} \aal(\pp, \vv)  \mid ^2 \\
&= \arg \min_{\alpha, \pp, \vv}
\lVert \yy - \alpha \hh(\pp, \vv)  \rVert ^2,
\label{eq:ml_pos_vel_gain}
\end{align}

which involves a high-dimensional search.
Alternatively, the ML estimator of $\pp$ and $\vv$ can be obtained by the CF estimation of $\alpha$ in (<ref>):

\begin{align} \label{eq:alphahat_update}
\alphahat = \frac{  \hh^\her(\pp, \vv) \yy }{\norm{\hh(\pp, \vv)}^2} ~.
\end{align}

Plugging (<ref>) back into (<ref>), we obtain the concentrated cost function

\begin{align}\label{eq:ml_pos_vel}
(\pphat,\vvhat) = \arg \min_{\pp, \vv}  \norm{\projnull{\hh(\pp, \vv)} \yy }^2 ~,
\end{align}

leading to a 6D optimization problem. In the subsequent sections, our focus will be directed towards solving (<ref>), and when convenient, its concentrated form (<ref>), with an emphasis on achieving low computational complexity.

§ PROPOSED METHOD

In this section, we will elaborate on
the overall procedure of estimating the 6D position and velocity parameters of the mobile UE in the absence of prior knowledge.

§.§ Time Complexity Analysis

We now present the complexity scaling of the algorithms as a function of the RIS size $M$ and number of pilot transmissions $L$, as well as the number of iterations in Algorithm <ref>,  Algorithm <ref>, Algorithm <ref>, and Algorithm <ref>, denoted by $I_1$, $I_2$, $I_3$, and $I_4$, respectively. We denote the complexity of Algorithm $i$ by $\mathcal{C}_i$.

We find that $\mathcal{C}_3=\mathcal{O}(I_3 L M)$ and $\mathcal{C}_4=\mathcal{O}(I_4 L M)$, mainly due to the computation of line 5 (i.e., $\etab$, $\bxib$,   $\nub$, and $\Mb$) in both methods.  Moving this line before the while loop will turn out to not severely affect performance, but leads to a significant complexity reduction $\mathcal{C}_3=\mathcal{O}(I_3 L)$ and $\mathcal{C}_4=\mathcal{O}(I_4 L)$.
It is also readily verified that $C_2=\mathcal{O}(L M I_2 \max\{K_\theta \times K_\phi, K_\rho\})$, where  $K_\theta$ (resp. $K_\phi$ and $K_\rho$) denotes the grid size for the search over $\theta$ (resp. $\phi$, $\rho$).

Finally, introducing $I_{\text{GD}}$ as the number of gradient descent iterations,  we find that

\begin{align}
\mathcal{C}_1& =\mathcal{C}_2 + I_1 (\mathcal{C}_3+\mathcal{C}_4) + \mathcal{O}(I_{\text{GD}}LM)\\
& =\mathcal{O}(L M I_2 \max\{K_\theta \times K_\phi, K_\rho\}) + \mathcal{O}(I_1 I_3 L) \notag \\
&+ \mathcal{O}(I_1 I_4 L) + \mathcal{O}(I_{\text{GD}}LM),
\end{align}

provided line 5 is moved before the while loop in Algorithms  <ref>–<ref>.

Algorithm one:

* Grid search over angles then distance: depends on $M$? Maybe in the size of the matrices

* Loop over the two search routines

Algorithm two:

* Compute $\nu$: size of the matrix?

* Compute $\Xi$: size of the matrix?

* Update residual position and gain estimate: independnt of $M$?

Algorithm three: same as algorithm two.

Algorithm four:

* Algorithm one

* Repeat Algorithm three and two

§ NUMERICAL SIMULATIONS AND RESULTS

In order to evaluate the effectiveness of the proposed estimation framework, numerical simulations were conducted in a canonical indoor scenario.

§.§ Scenario Definition and Performance Metrics

In the following, we consider random RIS phase profile design and we list, in Table <ref>, the scenario parameters [28].

We evaluate the performance of the algorithmic variants introduced in section <ref> (under different prior knowledge assumptions), while comparing them with both theoretical bounds and that of a SoTA algorithm assuming zero velocity from [35].
In the following, we refer to the algorithms as follows:

* GS: This method refers to Algorithm <ref>, providing a position estimate assuming a zero velocity.

* CF Position Refinement: This method refers to the closed-form position refinement routine of Algorithm <ref>.

* CF Velocity Refinement: This method refers to the closed-form velocity refinement procedure of Algorithm <ref>.

* Global Refinement: This methods is the 6D gradient descent  in Algorithm <ref>, line 9, which is used to enhance the accuracy of both position and velocity estimation.

We also assess the sensitivity of the proposed approach to critical system parameters, such as the distance to the RIS, the norm of the velocity vector, or small-scale fading effects resulting from uncontrolled multipath components (i.e., besides the used RIS-reflected path).
In terms of geometry, the RIS is located at the xy-plane center and the UE's position is defined as
$\pp = \rho \vect{i}$ where $\rho \in [1,10]\unit{m}$ denotes the RIS-UE distance and $\vect{i} = \nicefrac{[-1,2,1]^\top}{\norm{[-1,2,1]^\top}}$ acts as a unit vector. Similarly, the UE's velocity is defined as $\vv = v \vect{i}$ where $v \in [0, 20]\unit{m/s}$ denotes the speed.[Note that the velocity vector utilized in the simulations is radial w.r.t. the RIS. We have verified that the proposed algorithms operate close to the corresponding bounds, for other velocity vectors as well.] According to the approximation in Appendix <ref>, the UE speed $v$ should be below the limit computed via (<ref>), given the parameters in Table <ref>. More precisely, a minimum distance condition $\norm{\bm{\delta}_m} > 1 \, \rm{m}, \,   \forall m$ is considered, as well as maximum values for $L = 50$ and $\Ts = 100\unit{\mu s}$. Under these settings, the condition $\norm{\vv} L\Ts \ll \norm{\bm{\delta}_m}, \forall m$ simplifies to $\norm{\vv} \ll 200 \, \rm{m/s}$, a condition that aligns with our simulations as we consider speeds up to $20 \, \rm{m/s}$.

Performance is  evaluated through the RMSE of the position and velocity, which is calculated over 1000 random measurement noise realizations for each tested UE position-velocity configuration. The RMSE are compared to the corresponding error bounds, PEB and VEB.

We found that in Algorithms <ref> and <ref>, when the linearization in line 5  is only executed once before the loop instead of every iteration, the performance is nearly identical to performing the linearization every iteration. Given the complexity analysis in Section <ref>, from now on we only use this low-complexity version the CF refinement methods.

parameter     value     parameter     value

Frequency     $f_{c} = 28\unit{GHz}$     Wavelength     $\lambda_c \approx 1.07\unit{cm}$

Bandwidth     $W = 1\unit{MHz}$     Power transmitted     $P_{\text{tx}} = 20\unit{dBm}$

Noise PSD     $N_0 = -174\unit{dBm/Hz}$     Noise figure     $n_f = 8\unit{dB}$

RIS loc.     $\ppr = [0, 0, 0]^\top\unit{m}$     BS loc.     $\ppb = [3, 3, 1]^\top\unit{m}$

RIS size     $M = 32\times 32$ elements      Transmissions     $L = 40$

General simulation parameters.

§.§ Performance of 3D Estimation

We first discuss the performance of the 3D estimation algorithms for position and velocity separately.

§.§.§ Position Estimation with Known Velocity

RMSE of position estimation with known UE velocity ($v = 1\unit{m/s}$) using Algorithms <ref> and <ref>, along with the corresponding PEB, as a function of the RIS-UE distance.

RMSE of velocity estimation with known position and an unkown velocity ($v = 1\unit{m/s}$) (i.e., Algorithm <ref>) and corresponding VEB as a function of RIS-UE distance.

First, in Fig. <ref>, we show the RMSE of the position estimation for the two algorithmic steps of Section <ref>, along with the corresponding PEB, as a function of the RIS-UE distance and with a fixed velocity of $v = 1\unit{m/s}$. One can notice that the PEB (in blue) remains at a sub-cm level at short RIS-UE distances (i.e., below $6\unit{m}$) but then increases up to a few cm as the UE moves away from the RIS. The first part of our estimation routine, which relies on GS (in red) routine in <ref>,
apparently suffers from a big performance gap compared to the bound, this is caused by the multiple approximations made during the development of this algorithm (See Section <ref>). However, feeding the output of this initial search routine into the CF position refinement routine (in amber), i.e., Algorithm <ref> enables to reach the performance bound, whatever the distance.
To better assess the performance of our algorithm, we have also benchmarked our algorithm with the closest contribution from the SoTA. More specifically, in [35], the authors devised a grid-search-based algorithm to estimate the position of the UE by exploiting the signal’s wavefront curvature. This approach accounts for the effects of geometric NF on positioning while assuming no velocity in the model.
The method from [35] involves
three distinct 1D search procedures over the three spherical dimensions. For this purpose, they performed a Jacobi Anger expansion to be able to separate both angles in the RIS model. Fig. <ref> shows that the method from [35] leads to a significant performance degradation.

§.§.§ Velocity Estimation with Known Position

In Fig. <ref>, we show the RMSE of velocity estimation for the refinement algorithm of Section <ref>, along with the corresponding VEB, still as a function of the RIS-UE distance. Without loss of generality, the actual velocity value was set to $1\unit{m/s}$ in this example, but further studies include higher values to evaluate the sensitivity to more extreme UE mobility (see Section <ref>). Note that the tested algorithm does not include an extensive grid search like before, but uniquely the CF velocity refinement routine, which is called with an initial velocity value of $0\unit{m/s}$. We thus observe
that the RMSE of this estimation almost coincides with the VEB curve, regardless of the RIS-UE distance.

RMSE of position estimation and related PEB versus RIS-UE distance.

RMSE of velocity estimation and related VEB versus RIS-UE distance.

6D estimation errors and related theoretical bounds versus RIS-UE distance, for a constant UE velocity ($v = 1\unit{m/s}$).

§.§ Performance of 6D estimation

After simulating the 3D position and 3D velocity estimation algorithms separately, we now evaluate the performance of 6D position and velocity estimation with no prior knowledge regarding the UE state.
Figs. <ref> and <ref> first show the RMSE of position and velocity estimation respectively, alongside their related theoretical bounds, versus the RIS-UE distance
for a static UE, i.e., with $v = 1\unit{m/s}$.

The curves in Fig. <ref> show that Algorithm <ref> (GS - in red) alone does not perform well compared to the PEB, especially at short distance,

whereas Algorithm <ref> (CF Refinement routine - in amber) significantly boosts the estimation performance at short distances, and even touches the PEB curve at short-to-mid ranges. However, as the RIS-UE distance increases, we notice a drop in performance, this is caused by implementing the approximation in (<ref>) whose error increases as the UE moves further from the RIS.
For this reason, a Global Refinement (in dashed black) routine has been implemented to enhance estimation mostly at far distances, where the previous algorithms fail.

RMSE of position estimation and related PEB versus RIS-UE distance.

RMSE of velocity estimation and related VEB versus RIS-UE distance.

6D estimation errors and related theoretical bounds versus RIS-UE distance, for $v = 0\unit{m/s}$.

Then we repeat the same experiment with a dynamic UE, setting the velocity amplitude to $v = 1\unit{m/s}$. The corresponding results are presented in Figs. <ref> and <ref>. The curves globally show a very similar behavior to that in the static case, with a very slight degradation of performance for Algorithm 1 at short distances, which motivated us to study further the effect of UE mobility (See section <ref> hereafter).

§.§ Benchmark with SoTA

To better asses the performance of our algorithm, we have also benchmarked our designed algorithm with the closest contribution from the SoTA. More specifically, in [35] the authors devised a grid search based algorithm to estimate the position of the UE by exploiting the signal’s wavefront curvature. This approach accounts for the effects of geometric NF on positioning, while assuming no velocity in the model. We applied the aforementioned algorithm to our scenario, i.e., with a mobile UE, with $v = 1\unit{m/s}$ and then compared it with our own grid search performance. In contrast to Algorithm <ref>, the authors in [35] relied on the NF modeling of the RIS response to apply three distinct 1D search procedures over the three spherical dimensions. For this purpose, they performed a Jacobi angers expansion to be able to separate both angles in the RIS model. Fig. <ref> shows the performance of both algorithms, where we can clearly notice a big gap. To enumerate, estimating UE’s position via a basic grid search algorithm relying on a simplistic model ignoring the effects of mobility yields significant performance degradation.

RMSE of position estimation through GS for the proposed Algorithm <ref> and the SoTA contribution from [35].

§.§ Sensitivity Analyses

Now that we have established the performance of the proposed method and its ability to attain the CRLB, we move on to study the sensitivity to the speed, to uncontrolled multipath, and to the SNR.

§.§.§ Sensitivity to the Speed

The same simulation setup is now used to study the effect of the UE's velocity on our algorithms.
Thus, we assume a dynamic UE, with a velocity varying from $0\unit{m/s}$ to around $50\unit{m/s}$,
while the RIS-UE distance is set to $2\unit{m}$. The corresponding results are displayed in Fig. <ref>. We can hence see clearly the negative effect of large velocity on the GS algorithm, while the rest of the other routines still continue performing well, in compliance with the PEB behavior, suggesting that our overall estimation framework is fairly robust against high UE mobility.

UE's mobility effect on the position estimation and PEB in NF. The RIS-UE distance is set to $\rho = 2\unit{m}$.

§.§.§ Sensitivity to Uncontrolled Multipath

We now discuss the performance in the presence of a multipath-rich channel. The BS is considered to be directive, and hence only one LoS path exists in the BS-RIS channel [37, 38]. On the other hand, the channel between the RIS and the UE is modeled as Rician [39, 40], which yields a change in the signal model of (<ref>) into

\begin{align}
y_\ell &=  \alpha
\left( \sqrt{\frac{\kfactor} {\kfactor+1}} \aa(\pp_\ell)  + \sqrt{\frac{1}{\kfactor+1}} \hhtilde \right)^\top
\Omegabl \aa(\pp_{b}) + n_\ell,
\label{eq:obs_model_multipath}
\end{align}

where $\hhtilde \in \mathbb{C}^{M\times1}$ is the NLoS[The NLoS is herein defined with respect to the direct path from the RIS to the UE and accounts for the effect of the uncontrolled multipath components besides this direct path.
component of the RIS-UE channel with $\hhtilde \sim \mtcn(\boldzero, \II)$ and $\kfactor$ is the Rician factor. From (<ref>), we notice that $\kfactor \rightarrow 0$ results in a Rayleigh channel which translates to losing the LoS path between the RIS and UE.
Fig. <ref> depicts the effect of the multipath profile introduction on the proposed position estimation algorithms, more precisely, Algorithms <ref> and <ref> as a function of $\kfactor \in [5, 10^3]$, while the RIS-UE distance is set to $2\unit{m}$. We also plot the PEB of the same scenario (in terms of velocity and distance) but without any multipath for the sake of benchmarking. We notice that,

as $\kfactor$ increases, the RMSE levels of both algorithms improve but coincide until around $\kfactor \approx 10^2$ where the refinement routine surpasses the GS output and starts operating normally again.
Finally, as $\kfactor \approx 10^3$, a significant drop in the RMSE of the refinement algorithm is noticed while that of the GS seems to decrease more slowly, but both algorithms follow the trend approaching the PEB values. Mitigating a bit the results of the previous sensitivity analysis, and in light of comments from [41] regarding the validity of small-scale fading models in the mmWave domain, we recall that the combined effects of spatial filtering (thanks to large antenna arrays) and higher power path losses would anyway lead to the reception of relatively sparse and weak secondary multipath components (besides the direct RIS-UE path) in our case.

RMSE of position estimation for the proposed GS and CF refinement algorithms versus the multipath Rician $\kfactor$-factor. The RIS-UE distance is set to $\rho = 2\unit{m}$ and the UE's velocity to $v = 1\unit{m/s}$, and $\kfactor \in [5, 10^3]$.

§.§.§ Sensitivity to SNR

We also evaluated the performance of our system w.r.t. SNR.

Thus, in Fig. <ref> we have plotted the PEB performance alongside our devised estimation routines as we change the SNR, all while fixing the RIS-UE distance and UE velocity to $5\unit{m}$ and $1\unit{m/s}$ respectively. We first notice that the PEB follows an expected trend of linearly decreasing with higher SNR values. On the other hand, Algorithm <ref> does not achieve acceptable levels of accuracy at very low SNR values, which means that no localization service could be guaranteed at such conditions. However, as the SNR hits $-20\unit{dB}$, the performance of the aforementioned algorithm is boosted to the point where it touches the PEB, and then continuous at that same level even if the SNR is significantly increased. This saturation is due to the finite grid resolution used in this algorithm.
Finally, it is noticeable that the CF position refinement algorithm fail completely at very low SNR, but then, also at $-20\unit{dB}$ its  performance comes back and it follows the trend of the PEB; and of course, since its output is fed as input to the global refinement algorithm, the latter follows the same trend as well.

RMSE of position estimation for the proposed Algorithms <ref> and <ref> versus the SNR. The RIS-UE distance is set to $\rho = 5\unit{m}$ and the UE's velocity to $v = 1\unit{m/s}$, as a function of the SNR.

§.§ Convergence

Finally, we empirically illustrate the average convergence rates (over all the Monte Carlo simulation trials) of both Algorithm <ref> (lines 4 through 7) and Algorithm <ref> (lines 6 through 8) in Fig. <ref> and <ref>, respectively, where the RIS-UE distance is set to $2\unit{m}$ and the UE's velocity is set to $1\unit{m/s}$.
Convergence is considered to be accomplished if the evolution of the respective objective functions (<ref>) and (<ref>) is smaller than an arbitrarily chosen threshold $e\approx10^{-15}$.

Fig. <ref> shows that (<ref>) converges in just $3$ iterations whereas Fig. <ref> shows that (<ref>) takes around $20$ iterations to finally converge, which is still acceptable from an algorithmic point-of-view. This practically shows how quick the two algorithmic steps can be in a typical application case.

Convergence of  Algorithm <ref> via objective function (<ref>)

Convergence of Algorithm <ref> via objective function (<ref>)

Convergence rate of Algorithms <ref> and <ref>, via objective functions (<ref>) and (<ref>) respectively, as a function of the number of iterations at $\rho = 2\unit{m}$ RIS-UE distance and $v= 1\unit{m/s}$ UE's velocity.

Position error and PEB versus UE velocity at distance.

Velocity error and PEB versus RIS-UE distance.

Velocity Estimation Error and theoretical bounds versus UE velocity at distance ($10\unit{m}$).

§ CONCLUSION

In this work, we have proposed a single-BS DL SISO positioning scenario, by considering the 6D state estimation of a mobile UE, while relying on one reflective RIS in the presence of LoS blockage.
We have accounted for small-scale velocity effects by modeling the RIS response in geometric NF (i.e., taking into account extra phase shifts at the RIS elements); and we have derived both theoretical position and velocity error bounds to be used as performance benchmarks. Then, we have designed a 6D estimation algorithm consisting of three subroutines. First, we assume the UE to be static and we coarsely estimate the position parameters via grid search which are then passed to an iterative subroutine that refines the results while assuming the knowledge of the UE's velocity. Then, we estimate the velocity parameters with the perfect knowledge assumption of the UE's position.
We have introduced algorithmic time complexity analysis and we have simulated our algorithms against different system parameters, including the RIS-UE distance and UE's velocity. The obtained results show that our final algorithmic results attain the theoretical performance bounds at close RIS-UE distances. For farther distances however, as our CF refinement routines are shown to experience much poorer performance, we have resolved this issue by introducing a global refinement process (as part of our algorithm), which restores a fine performance level. Furthermore, we have tested the resilience of our algorithm against both low SNR values and multipath interference.

This work is to be extended in multiple dimensions, including the integration of UE state tracking process with the help of Bayesian filters. Moreover, previous research works could be leveraged to include location-based RIS phase optimization routines as well as realistic RIS hardware prototypes.

§ DERIVATION OF $\FLM(\PP,\VV)$ IN (<REF>)

Let's define the new UE position $\pp_{\ell} = \pp + \bm{\Delta}_{\ell}$ after a displacement by a vector $\bm{\Delta}_{\ell} = \vv \ell \Ts$. We also introduce ${\pp_m - \pp}=\bm{\delta}_m$. Accordingly, one can write

\begin{align}
\norm{\pp_m - \pp_\ell}
& = \norm{\bm{\delta}_m}
\frac{\norm{\bm{\delta}_m - \bm{\Delta}_{\ell}}}{\norm{\bm{\delta}_m}} ~,\\
% & = \norm{\pp_d}
% \frac{\norm{\pp_m - \pp - \bm{\Delta}_{\ell}}}{\norm{\pp_d}}\\
& = \norm{\bm{\delta}_m}
\sqrt{\frac{{(\bm{\delta}_m - \bm{\Delta}_{\ell})^\top (\bm{\delta}_m - \bm{\Delta}_{\ell}})}{\norm{\bm{\delta}_m}^2}} ~,\\
%& = \norm{\pp_d}
%\frac{\sqrt{\norm{\pp_d}^2 + \norm{\bm{\Delta}_{\ell}}^2 - 2 \pp_d^\top\bm{\Delta}_{\ell}}}
& = \norm{\bm{\delta}_m}
\sqrt{1
- \frac{2 \bm{\delta}_m^\top\bm{\Delta}_{\ell}}{\norm{\bm{\delta}_m}^2}
+ \frac{\norm{\bm{\Delta}_{\ell}}^2 }{\norm{\bm{\delta}_m}^2}
}~. \label{eq:approx_3}
\end{align}

When the total displacement of the UE during the entire observation interval is negligible compared to the distance between the RIS and the UE, i.e.,

\begin{align}
\norm{\vv} &\ll \frac{\max_m \norm{{\pp_m - \pp}}}{L\Ts},  \label{eq:v_limit}
\end{align}

the last term under the square root in (<ref>) can be neglected. Under the same condition (<ref>), the second term under the square root in (<ref>) is much smaller than $1$, which allows us to apply the first-order Taylor series approximation $\sqrt{1-2x}\approx 1-x$ for  $x\ll 1$. Combined, this allows us to

We now assume that the total displacement of the UE during the entire observation interval is negligible compared to the distance between the RIS and the UE, i.e.,

\begin{align}
\frac{\norm{\bm{\Delta}_{\ell}}^2 }{\norm{\bm{\delta}_m}^2} \ll 1, \forall m, \ell, \label{eq:d_limit}
\end{align}

or, equivalently

\begin{align}
%    \norm{\vv} L\Ts & \ll \norm{\bm{\delta}_m},\\
\norm{\vv} &\ll \frac{\norm{\bm{\delta}_m}}{L\Ts}, \forall m. \label{eq:v_limit}
\end{align}

Based on (<ref>) and (<ref>), we consider the function

$f(x) = \sqrt{1 - 2x + e}$, where $x \triangleq \frac{\bm{\delta}_m^\top\bm{\Delta}_{\ell}}{\norm{\bm{\delta}_m}^2} $ and $e \triangleq \frac{\norm{\bm{\Delta}_{\ell}}^2 }{\norm{\bm{\delta}_m}^2}$ is  an arbitrarily small quantity, which yields the approximation $f(x) \approx \sqrt{1 - 2x}$. In addition, the condition in (<ref>) also implies that

\begin{align}
\frac{2 \bm{\delta}_m^\top\bm{\Delta}_{\ell}}{\norm{\bm{\delta}_m}^2} \ll 1 ~. \label{eq:d_limit2}
\end{align}

Following (<ref>), we apply the first order Taylor's expansion to the approximated $f(x)$ around $x=0$, which yields $f(x) \approx 1 - x$.

Finally, we

rewrite (<ref>) as

\begin{align} \nonumber
\norm{\pp_m - \pp_\ell}
& \approx \norm{\bm{\delta}_m} - \frac{\bm{\delta}_m^\top\bm{\Delta}_{\ell}}{\norm{\bm{\delta}_m}} ~,
\\ \nonumber
& = \norm{\pp_m - \pp} - \frac{(\pp_m - \pp)^\top}{\norm{\pp_m - \pp}}
\vv\ell\Ts ~,
\\
&= d_m - \uu_{m}^\top(\pp) \vv \ell \Ts ~.
\end{align}

§ FF APPROXIMATION

The NF RIS response form in (<ref>) can be in fact approximated to its equivalent FF form as following.

From the geometry of Fig. <ref> we have:

\begin{align}
d_{m} = \sqrt{\dr^{2} + \qm^{2} - 2\dr \qm\kappa
} ~,
\end{align}

where $\qm = \norm{\pp_m - \pp_r}$ and $\kappa \triangleq \sin{(\phi)}\cos{(\theta - \psi_{m})} $ is the constant holding the angular (azimuth and elevation) terms.

Using first-order Taylor's expansion, the function $f(x) = \sqrt{1 + x^{2} - 2x\kappa}$ can be approximated around $x = 0$ into $f(x) \approx 1 - x\kappa$; then given that the RIS is large enough and the UE is far from the surface, $d_{m}$ can be approximated as:

\begin{align}
d_{m}  & \approx \dr(1 - \frac{\qm}{\dr}\kappa) \\
&= \dr - \qm\kappa,
\end{align}

\begin{align}
\norm{\pp_{m} - \pp} -
\norm{\ppr - \pp} = \dm - \dr = -\qm \sin{(\phi)}\cos{(\theta - \psi_{m})}.
\end{align}

Following [35]
, we then approximate the steering vector in far-field as

\begin{align} \label{eq:ff}
[\aa(\theta, \phi)]_{m} \approx
\exp{
(- j (\pp_{m} - \ppr)^{\top}\kk(\phi, \theta))
} ~,
\end{align}

$\kk(\phi,\theta) = -\frac{2\pi}{\lambda}[\sin{\phi}\cos{\theta}, \hspace{1mm}
\sin{\phi}\sin{\theta}, \hspace{1mm}
\cos{\phi}]^\top$

§ GRADIENT OF $\FLM(\PP,\VV)$

In the linearized RIS response model from (<ref>), we have that

\begin{align}\label{eq:grad_flm}
\gradF(\pp, \vv) &= \frac{\pp_m-\pp}{\dm}-\frac{\ppris-\pp}{\dr} \nonumber \\
&+  \left( \partder{\uu_m(\pp)}{\pp} - \partder{\uuris(\pp)}{\pp} \right)^\top  \vv\ell\Ts \nonumber \\
&= \uu_m(\pp) - \uuris(\pp) + \gradGamT \vv\ell\Ts \in \realset{3}{1}~,\\
\gradGam
&\triangleq \left(
\frac
{\norm{\pp_m-\pp}^2 \boldone_{3\times3} - (\pp_m-\pp)(\pp_m-\pp)^\top }
{\norm{\pp_m-\pp}^3} \right.\nonumber\\
\frac
{ \norm{\ppris-\pp}^2 \boldone_{3\times3} - (\ppris-\pp)(\ppris-\pp)^\top }
\right) \in \realset{3}{3}.
\end{align}

§ DERIVATION OF $\PD$

The problem (<ref>) for fixed $\alpha = \alphahat$ can be reduced to

\begin{align}\label{eq:ml_pos_vel_gain_appr_pd}
\pdhat = \arg \min_{\pd \in \realset{3}{1}}  \norm{\yy - \alphahat \,  (\etab + j \bxib^\top \pd )  }^2 ~.
\end{align}

Opening up the terms in (<ref>)  , we have

\begin{align} \nonumber
\llk(\pd) &\triangleq \norm{\yy - \alphahat \,  (\etab + j \bxib^\top \pd )  }^2 ~,
\\ \nonumber
&= \norm{\yy}^2 - 2 \realp{\yy^\her \alphahat \,  (\etab + j \bxib^\top \pd ) } + \abs{\alphahat}^2 \norm{\etab + j \bxib^\top \pd}^2 ~,
\\ \nonumber
&= \norm{\yy}^2 - 2 \realp{ \alphahat \, \yy^\her  \etab } - 2 \realp{j \alphahat \, \yy^\her  \bxib^\top \pd  } + \abs{\alphahat}^2 \norm{\etab}^2 \\ \nonumber
& + 2 \abs{\alphahat}^2 \realp{j \etab^\her  \bxib^\top \pd} + \abs{\alphahat}^2 \norm{j \bxib^\top \pd}^2 ~,
\\ \label{eq:lpd}
&= \abs{\alphahat}^2 \pd^\top \bxib^\conj \bxib^\top \pd + \big( 2 \abs{\alphahat}^2 \realp{j \etab^\her  \bxib^\top} \nonumber \\
& - 2 \realp{j \alphahat \, \yy^\her  \bxib^\top  } \big) \pd
+\norm{\yy}^2 \nonumber \\
& - 2 \realp{ \alphahat \, \yy^\her  \etab } +\abs{\alphahat}^2 \norm{\etab}^2,
\end{align}

where we have used the fact that $\pd$ is a real vector. According to Lemma S1 in the supplementary material of [42], $\pd^\top \bxib^\conj \bxib^\top \pd = \pd^\top \realp{\bxib^\conj \bxib^\top} \pd$ since $\pd \in \realset{3}{1}$ and $\bxib^\conj \bxib^\top$ is Hermitian. Then, ignoring the constant terms in (<ref>), the problem (<ref>) can be re-written as

\begin{align}
\pdhat &= \arg \min_{\pd \in \realset{3}{1}}  \left\{ \abs{\alphahat}^2 \pd^\top \realp{\bxib^\conj \bxib^\top} \pd \right. \\
& \left. +  \big( 2 \abs{\alphahat}^2 \realp{j \etab^\her  \bxib^\top} - 2 \realp{j \alphahat \, \yy^\her  \bxib^\top  } \big) \pd \right\} ~. \label{eq:ml_pos_vel_gain_appr_pd2}
\end{align}

Since (<ref>) is an unconstrained quadratic optimization problem, the solution can readily be obtained in closed-form as

\begin{align} \label{eq:pdhat_2}
\pdhat = \frac{1}{\abs{\alphahat}^2} \big( \realp{\bxib^\conj \bxib^\top} \big)^{-1} \imp{ \bxib \big(    \abs{\alphahat}^2 \etab^\conj   -  \alphahat \, \yy^\conj  \big)    } ~.
\end{align}

that the $3 \times 3$ matrix

\begin{align}
\realp{\bxib^\conj \bxib^\top} = \sum_{\ell=1}^{L} \realp{ \bxi_{\ell}^\conj \bxi_{\ell}^\top }
\end{align}

is full-rank for $L \geq 3$ under mild conditions (e.g., if RIS phase profiles are random). Hence, if the number of transmissions is at least $3$, then the closed-form UE position can be computed using (<ref>).

Update $\alpha$ for fixed $\pd$: The solution of $\alpha$

for a fixed $\pd = \pdhat$ is given by

\begin{align} \label{eq:alphahat_p_2}
\alphahat = \frac{ (\etab + j \bxib^\top \pdhat )^\her \yy }{ \norm{\etab + j \bxib^\top \pdhat}^2 } ~.
\end{align}

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