# Near-Field Mobile Tracking: A Framework of Using XL-RIS Information

Tuo Wu, Cunhua Pan, _Senior Member, IEEE_ , Kangda Zhi, Hong Ren, Maged
Elkashlan, _Senior Member, IEEE_ , Chau Yuen, _Fellow, IEEE_ _(Corresponding
author: Cunhua Pan.)_ T. Wu, K. Zhi and M. Elkashlan are with the School of
Electronic Engineering and Computer Science at Queen Mary University of
London, London E1 4NS, U.K. (Email:{tuo.wu,k.zhi,
maged.elkashlan}@qmul.ac.uk). C. Pan, H. Ren are with the National Mobile
Communications Research Laboratory, Southeast University, Nanjing 210096,
China. (e-mail: {cpan, hren }@seu.edu.cn). Chau Yuen is with the School of
Electrical and Electronic Engineering, Nanyang Technological University,
639798, Singapore (e-mail: chau.yuen@ntu.edu.sg).

###### Abstract

This paper introduces a novel mobile tracking framework leveraging the high-
dimensional signal received from extremely large-scale (XL) reconfigurable
intelligent surfaces (RIS). This received signal, named XL-RIS information,
has a much larger data dimension and therefore offers a richer feature set
compared to the traditional base station (BS) received signal, i.e., BS
information, enabling more accurate tracking of mobile users (MUs). As the
first step, we present an XL-RIS information reconstruction (XL-RIS-IR)
algorithm to reconstruct the high-dimensional XL-RIS information from the low-
dimensional BS information. Building on this, this paper proposes a
comprehensive framework for mobile tracking, consisting of a _Feature
Extraction Module_ and a _Mobile Tracking Module_. The _Feature Extraction
Module_ incorporates a convolutional neural network (CNN) extractor for
spatial features, a time and frequency (T$\&$F) extractor for domain features,
and a near-field angles of arrival (AoAs) extractor for capturing AoA features
within the XL-RIS. These features are combined into a comprehensive feature
vector, forming a time-varying sequence fed into the _Mobile Tracking Module_
, which employs an Auto-encoder (AE) with a stacked bidirectional long short-
term memory (Bi-LSTM) encoder and a standard LSTM decoder to predict MUs’
positions in the upcoming time slot. Simulation results confirm that the
tracking accuracy of our proposed framework is significantly enhanced by using
reconstructed XL-RIS information and exhibits substantial robustness to
signal-to-noise ratio (SNR) variations.

###### Index Terms:

Reconfigurable intelligent surface (RIS), mobile tracking, RIS information.

##  I Introduction

The upcoming introduction of sixth generation (6G) Internet of Things (IoT)
wireless networks marks a pivotal moment in our technological journey [1, 2,
3]. Enhanced localization accuracy becomes central in this new era [4]. The
heightened demand for accuracy challenges us to not only innovate but to
fundamentally rethink our approach to localization [5, 6, 7, 8]. Recently,
reconfigurable intelligent surfaces (RIS) [9, 10, 11, 12, 13] have been
introduced as a revolutionary technology to enhance the localization accuracy
[14, 15, 16] with several advantages. First, RIS offers a cost-effective and
energy-efficient alternative to traditional base stations (BS) for maintaining
consistent connectivity, especially in challenging environments. Additionally,
their slim and adaptable design easily deploys in the urban structures, making
them ideal for 6G networks.

RIS-aided localization algorithms can be categorized into two main types:
_two-step method_ [17] and _fingerprint-based_ _method_ [18]. The _two-step
method_ estimates parameters like angle of arrival (AoA) and time of arrival
(ToA) to deduce the location by exploiting geometric relationships [19, 20,
21, 22]. The _fingerprint-based method_ [23, 24], on the other hand, relies on
a database of pre-recorded signal characteristics, such as received signal
strength (RSS), and compares real-time signals to the database for pinpointing
a location. Building upon these two methods, algorithm design for RIS-aided
localization has been widely investigated, focusing on various applications
and different perspective, such as multi-user [25], multi-RIS [26], and
sidelink scenarios [17].

It is worth noting that the above studies were built based on the far-field
channel model, with the help of tractable planar wavefront assumptions. To
overcome the high path-loss attenuation and fully unleash the gain of RIS, the
RIS is expected to be constructed on an extremely large scale and equipped
with large numbers of passive and low-cost reflecting elements. However, with
a large physical dimension, the communication between an RIS and users will be
located in the near field, rather than the conventional far field [27, 28, 6].
Recognizing this, recent research [29, 30, 31, 32, 33, 34] has significantly
advanced the design of RIS-aided localization algorithms in the presence of
the near-field effect. These studies have introduced various methodologies to
enhance localization accuracy, from creating virtual line-of-sight (VLoS)
paths using RIS [29] to employing advanced optimization techniques for
precision improvement [30]. Efforts have also been made to explore the
theoretical limits of holographic RIS-aided localization system [32] and
address specific challenges such as phase-dependent amplitude variations [33]
and bi-static sensing for fixed transmitter and receiver systems [34].

Besides the research limitation concerning the near field, another important
challenge is about moible localization aiming to accommodate the movement of
mobile user (MU). While the static approaches developed in earlier works [29,
30, 31, 32, 33, 34] laid a strong foundation, their applicability is somewhat
constrained in scenarios featuring mobile users. Addressing this need, more
recent research [35, 36, 37] have switched the focus towards designing the
mobile tracking algorithms. Specifically, the authors in [35] delved into
jointly optimizing the RIS reflection coefficients and BS precoding to adeptly
track MUs in scenarios with multi-RIS setups. Additionally, recognizing
challenge that integrated the mobility and LoS obstructions, Mei _et al._ in
[36] developed a multi-user tracking system using the extended Kalman filter
(EKF). Meanwhile, Rahal _et al._ in [37] presented a three-step algorithm for
simultaneously estimating MU position and velocity, accounting for Doppler
effects in the near-field.

However, the above-mentioned mobile tracking algorithms mainly rely on the
signals received at the BS, i.e., BS information. Nevertheless, the high cost
and power consumption associated with radio-frequency chains often result in a
limited number of antennas at the BS. This limitation reduces the
dimensionality of the BS information, thereby diminishes the localization
accuracy. However, an important yet unexplored potential is the use of rich
location-related information embedded in the signals received by the RIS,
i.e., RIS information. Due to its composition of a vast number of reflecting
elements, the RIS panel can catch a much higher dimension of useful
information. Actually, the idea of using RIS information for localization is
perfectly matched with the newly proposed concept, the extremely large-scale
RIS (XL-RIS). With an extremely large size, near-field effect exhibits and
therefore each individual reflecting element on XL-RIS could collect unique
AoA and ToA information. Exploiting the large number of diverse AoA and ToA
patterns across the entire XL-RIS, we are able to establish a detailed and
nuanced dataset and realize localization with high granularity and accuracy.

Despite the attractive benefits, the localization design based on XL-RIS
information is indeed a highly challenging task [10]. This challenge arises
from the complexity of reconstructing high-dimensional XL-RIS information. On
the one hand, the passive nature of XL-RIS poses a challenge in actively
acquiring the signals. On the other hand, when employing the low-dimensional
BS information to obtain the high-dimensional XL-RIS information, the result
is not unique. This is because the same BS information corresponds to multiple
distinct XL-RIS information. The uncertainty in the signal reconstruction
process requires innovative algorithms to accurately capture the XL-RIS
information.

Furthermore, for mobile tracking tasks, the works in [35, 36, 37] utilized the
velocity to model the trajectory of the MUs, employing traditional tracking
algorithms like the EKF for predicting trajectory of the MU. However,
traditional tracking algorithms face challenges in complex environments,
reducing localization accuracy. Specifically, the EKF linearizes MUs’
nonlinear trajectories and thereby introduces the linearization error,
especially exacerbated in indoor settings with multi-path and near-field
effects, significantly lowering precision. Additionally, traditional tracking
algorithms inadequately leverage time-series data, particularly in recognizing
long-term dependencies, restricting prediction accuracy and model
adaptability.

Against the above-mentioned background, this paper investigates the mobile
tracking problem with complex environments and near-field effects, and
proposes an effective algorithm to enable the use of XL-RIS information. We
first employ the data-driven methods. i.e., convolution neural network (CNN),
to design an XL-RIS information reconstruction (XL-RIS-IR) algorithm. Building
on this, we propose a comprehensive data-driven framework for MU tracking,
which includes a _Feature Extraction Module_ and a _Mobile Tracking Module_.
Our contributions are outlined as follows:

  * 1)

_XL-RIS information reconstruction_ : We propose an XL-RIS-IR algorithm for
reconstructing XL-RIS information from BS information. The XL-RIS-IR algorithm
includes a data processing module for preprocessing low-dimensional BS
information, a DenseNet module for feature extraction, and an output module to
produce the reconstructed high-dimensional RIS information.

  * 2)

_Comprehensive feature extraction_ : As the first component of the proposed
tacking framework, we design a _Feature Extraction Module_ for extracting the
essential features from the reconstructed XL-RIS information. This module
encompasses three distinct extractors: a CNN extractor for extracting the
spatial features, a time and frequency (T$\&$F) extractor aimed at extracting
time and frequency domains features, and a near-field AoAs extractor for
capturing the AoAs features within XL-RIS. Finally, these features are then
combined into a final feature vector for mobile tracking.

  * 3)

_Accurate mobile tracking algorithm_ : The _Mobile Tracking Module_ , serving
as the second component of the proposed tacking framework, employs an Auto-
encoder (AE) configured with a stacked bidirectional long short-term memory
(Bi-LSTM) encoder and a conventional LSTM decoder. This sophisticated AE
processes the time-varying sequence derived from the final feature vector to
accurately predict the MU’s three-dimensional (3D) coordinates for the
upcoming time slot.

  * 4)

_Confirm the effectiveness of the framework_ : Simulation results reveal that
the tracking accuracy of the proposed tacking framework is enhanced by
utilizing the high-dimensional XL-RIS information. Moreover, the proposed
framework exhibits considerable robustness against variations in SNR.
Additionally, numerical analyses confirm that our framework outperforms
established benchmark models in mean squared error (MSE) metrics, highlighting
its superior tracking precision.

##  II System Model

Consider an XL-RIS aided uplink (UL) localization system where an MU transmits
the pilot signals to the BS via the XL-RIS. The BS is equipped with a uniform
planar array (UPA) comprising $M_{1}\times M_{2}=M$ antennas, while the MU has
a single antenna. Additionally, the XL-RIS is equipped with a UPA comprising
$N_{1}\times N_{2}=N$ reflecting elements.

###  II-A Geometry and Channel Model

We now introduce the geometry configuration of the system, for the BS, the XL-
RIS, and the MU. The BS is positioned at a known location
$\bm{p}_{BS}\in\mathbb{R}^{3}$, while the center of the XL-RIS is denoted by
$\bm{p}_{RIS}\in\mathbb{R}^{3}$, and the location of the $n$-th RIS element is
represented by $\bm{p}_{n}\in\mathbb{R}^{3}$ for $1\leq n\leq N$. The MU’s
location, denoted by $\bm{p}\in\mathbb{R}^{3}$, is unknown and needs to be
estimated.

In this system, the MU is assumed to lie in the Fresnel (radiative) near-field
region of the XL-RIS. This region is defined by the distance $d$ between the
MU and the XL-RIS, which satisfies the following condition:

$0.62\sqrt{\frac{D^{3}}{\lambda}}\leq d\leq\frac{2D^{2}}{\lambda},$ (1)

where $\lambda$ is the carrier wavelength, and $D$ represents the XL-RIS
aperture size, defined as the largest distance between any two elements of the
XL-RIS. This condition ensures that the MU is located within a specific range
that allows for effective interaction with the radiative fields emitted by the
XL-RIS, facilitating precise estimation of the MU’s location.

Given the near-field condition, we next model the communication channel
between the MU and XL-RIS. This channel is characterized by a multipath
scenario, where the signal propagation can be divided into two primary
components: Line-of-Sight (LoS) and Non-Line-of-Sight (NLoS). The LoS
component represents the direct propagation path from the RIS to the MU
without any obstruction. Conversely, the NLoS component encompasses multiple
indirect paths where the signal is reflected by various scatterers in the
environment, such as walls, furniture, and human bodies.

For the LoS path from the MU to the XL-RIS, we introduce the steering vector
$\bm{a}(\bm{p})\in\mathbb{C}^{N\times 1}$, which models the signal phase
variations across the RIS elements, given the MU’s position $\bm{p}$. This
steering vector is derived with respect to the XL-RIS’s center, denoted by
$\bm{p}_{RIS}$, which is given as

$[\bm{a}(\bm{p})]_{n}=\textrm{exp}\left(-j\frac{2\pi}{\lambda}(\|\bm{p}-\bm{p}_{n}\|-\|\bm{p}-\bm{p}_{RIS}\|)\right),$
(2)

for $n\in\\{1,\cdots,N\\}$, where $\bm{p}_{n}$ represents the position of the
$n$-th element on the XL-RIS.

In addition, the NLoS paths, which result from reflections of $M_{s}$
scatterers, are modeled through steering vectors that account for the
additional path lengths. Specifically, for the $m_{s}$-th scatterer, the
steering vector is given by

$\displaystyle[\bm{a}_{m_{s}}(\bm{p})]_{n}$
$\displaystyle=\textrm{exp}\bigg{(}-j\frac{2\pi}{\lambda}(\|\bm{p}-\bm{p}_{n}\|+\|\bm{q}_{m_{s}}-\bm{p}_{n}\|$
$\displaystyle\qquad\qquad\qquad+\|\bm{p}-\bm{q}_{m_{s}}\|-\|\bm{p}-\bm{p}_{RIS}\|)\bigg{)},$
(3)

for ${m_{s}}\in\\{1,\cdots,M_{s}\\}$ and $n\in\\{1,\cdots,N\\}$, where
$\bm{q}_{m_{s}}$ denotes the position of the $m_{s}$-th scatterer.

Therefore, the overall complex channel from the MU to the XL-RIS,
incorporating both LoS and NLoS components, is thus represented as

$\bm{h}(\bm{p})=\alpha\bm{a}(\bm{p})+\sum_{m_{s}=1}^{M_{s}}\alpha_{m_{s}}\bm{a}_{m_{s}}(\bm{p}),$
(4)

where $\alpha$ and $\alpha_{m_{s}}$ denote the complex channel attenuation
coefficients for the LoS path and each NLoS path through the ${m_{s}}$-th
scatterer, respectively.

For the communication channel between the BS and the XL-RIS, it is assumed
that no NLoS paths exist due to the BS being situated close to the XL-RIS
[38]. The normalized channel gain for the LoS path between BS and the XL-RIS,
is represented by ${\bm{G}}$, whose $(m,n)$-th element is given by [39]

$\displaystyle{\bm{G}}(m,n)=\textrm{exp}\bigg{(}j\frac{2\pi}{\lambda}r_{m,n}\bigg{)},$
(5)

where $r_{m,n}$ denotes the path length between the $m$-th antenna element of
the BS and the $n$-th reflecting element. Letting $\beta$ denote the common
large-scale path loss attenuation, the complex channel gain of the channel
between the BS and the XL-RIS is then given by

$\displaystyle{\bm{H}}=\beta{\bm{G}}.$ (6)

###  II-B Signal Model

To motivate the deployment of the XL-RIS, we assume that the LoS path between
the BS and the MU is blocked, e.g., due to the buildings, cars or trees.
Letting ${\bm{y}}_{r}$ denote the XL-RIS received signal, we have

$\displaystyle{\bm{y}}_{r}={\bm{h}}({\bm{p}}){s},$ (7)

where ${s}$ denotes the pilot signal transmitted by the MU. Then, defining the
phase shift vector of the XL-RIS as
${\bm{\omega}}=[\omega_{1},\cdots,\omega_{N}]^{T}\in\mathbb{C}^{N\times 1}$,
the UL signal received by the BS can be expressed as

$\displaystyle{\bm{y}}={\bm{H}}\textrm{diag}({\bm{\omega}}){\bm{h}}({\bm{p}}){s}+{\bm{n}},$
(8)

where $\textrm{diag}({\bm{\omega}})$ and ${\bm{n}}$ denote the phase shift
matrix of the XL-RIS and the zero-mean additive white Gaussian noise,
respectively.

##  III XL-RIS Information Reconstruction (XL-RIS-IR) Algorithm

Figure 1: The structure of the XL-RIS-IR algorithm.

Traditionally, localization algorithm design are based on the BS information,
e.g., ${\bm{y}}$. Let us further define the estimated position of
$\bm{{p}}_{u}$ as $\bm{\hat{p}}_{u}$, and then the localization problem is
formulated as

$\displaystyle\underset{\bm{\hat{p}}_{u}}{\min}\ $
$\displaystyle||\mathcal{H}({\bm{y}})-\bm{{p}}_{u}||^{2},$ (9)

where $\mathcal{H}(\cdot)$ denotes the non-linear function between ${\bm{y}}$
and $\bm{\hat{p}}_{u}$. The above problem can be solved with several
localization algorithms which have been extensively investigated [29, 30, 18].
However, constraints such as cost and deployment conditions often limit the
number of antennas, resulting in low-dimensional BS information that provides
insufficient localization features, thereby potentially reducing localization
accuracy. In contrast, XL-RIS can achieve larger sizes at a lower cost,
enabling them to contain more sufficient location information, which can be
used to enhance localization precision. However, due to the passive nature of
RISs, it is not feasible to directly obtain XL-RIS information. This
necessitates the reconstruction of XL-RIS information to fully leverage its
potential for improving localization accuracy.

By defining the reconstructed high-dimensional XL-RIS information as
$\hat{\bm{y}}_{r}$, the RIS information reconstruction problem can be
formulated as

$\min_{\hat{\bm{y}}_{r}}\ \lVert{\bm{y}}_{r}-\hat{\bm{y}}_{r}\rVert^{2}_{2},$
(10)

which cannot be solved with traditional convex optimization methods, since
${\bm{y}}_{r}$ is unknown. Hence, we design a deep learning-based XL-RIS
information reconstruction (XL-RIS-IR) algorithm to solve Problem (10). As
shown in Fig. 1, the XL-RIS-IR algorithm includes three modules: data
processing module, DenseNet module, and output module, whose details are given
in the following.

###  III-A Data Processing Module

The data processing module is designed to process the input BS information
$\bm{y}$. Since $\bm{y}$ is a complex vector, we divide it into its real and
imaginary parts, represented as $\mathfrak{R}({\bm{y}})$ and
$\mathfrak{I}({\bm{y}})$, respectively. Then, since $\bm{y}$ consists of the
features of the horizontal angle domain and the vertical angle domain of the
BS, we reshape the two vectors $\mathfrak{R}({\bm{y}})\in\mathbb{C}^{M\times
1}$ and $\mathfrak{I}({\bm{y}})\in\mathbb{C}^{M\times 1}$ into matrices,
denoted as ${\bm{M}}_{Ry}\in\mathbb{R}^{M_{1}\times M_{2}}$ and
${\bm{M}}_{Iy}\in\mathbb{R}^{M_{1}\times M_{2}}$, respectively. Then, we
concatenate the reshaped matrices of the real and imaginary parts, resulting
in a 3D tensor represented as

$\bm{T}_{y}=\begin{bmatrix}{\bm{M}}_{Ry}\\\
{\bm{M}}_{Iy}\end{bmatrix}\in\mathbb{R}^{2\times M_{1}\times M_{2}}.$ (11)

However, the values of $M_{1}$ and $M_{2}$ may not be directly compatible with
CNNs’ architectures. Hence, an upsampling technique is applied:

$\bm{T}_{\text{up}}=\text{Upsample}(\bm{T}_{y})\in\mathbb{R}^{2\times
256\times 256}.$ (12)

The upsampling technique is employed to adjust the dimensions of the tensor
$\bm{T}_{y}$ to the desired dimensions of $2\times 256\times 256$. It
typically involves pixel value interpolation to expand the image or tensor
size, redistributing pixel values accordingly. To make $\bm{T}_{\text{up}}$
suitable for processing by CNNs, which expects an input tensor of the form
$(3,256,256)$, a convolutional (Con) layer is introduced to transform the bi-
channel tensor to a tri-channel one:

$\bm{T}_{\text{Con}}=\text{Con}(\bm{T}_{\text{up}}).$ (13)

Figure 2: Network structure of DenseNet. Figure 3: Network structure of a
Dense Block.

###  III-B DenseNet Module

DenseNet [40], inspired by the foundational principles of ResNet [41], reuses
the features from earlier layers. However, different from the skip connections
which might bypass a couple of layers, DenseNet establishes connections
between every layer and every other layer, ensuring a more robust feature
propagation. This approach, while reducing the parameter count, allows
DenseNet to potentially surpass the performance of ResNet.

DenseNet is distinguished by their excellent capability in feature extraction
and is used for extracting features from the low-solution images. Besides, the
intrinsic hierarchical structure of DenseNet allows for the map from the low-
level features to high-level features. Therefore, we adopt DenseNet as a main
module for reconstructing RIS information.

The design of DenseNet is illustrated in Fig. 2, and its architecture
comprises two main components: Dense Modules and Transition Modules, which
will be introduced as follows.

####  III-B1 Dense Module

As depicted in Fig. 2, the Dense Module comprises multiple Dense Blocks. The
details of a Dense Block are shown in Fig. 3. Each Dense Block starts with a
batch normalization (BN) layer, followed by a layer rectified linear unit
(ReLU) activation function layer, and concludes with a $3\times 3$ Con layer.
A notable characteristic of the Dense Module is its connectivity: Every block
incorporates the feature maps of all preceding blocks within the module. This
design ensures efficient feature reuse and a rich flow of information
throughout the network [40].

####  III-B2 Transition Module

From Fig. 2, we observe that the _Transition Module_ , which acts as a
connector between _Dense Modules_ , starts with a $1\times 1$ Con layer,
followed by a $2\times 2$ average pooling layer, effectively reducing the
spatial dimensions of the feature maps and streamlining them for either the
next _Dense Module_ or the concluding output layer.

###  III-C Output Module

After the Dense module, we design an output module to output the reconstructed
XL-RIS information. Within the output module, a ReLU layer is used to
introduce the non-linearity, enabling the network to capture and express the
complex patterns of the high-dimensional RIS information. Besides, the ReLU
layer can address the vanishing gradient problem, facilitating better learning
in deep neural networks.

Following the ReLU layer, a linear layer is employed for directly outputting
the desired high-dimensional XL-RIS information. This layer serves to
consolidate the complex features extracted by prior layers and transforms them
into a specific high-dimensional format, such as the required XL-RIS
information, effectively aligning the network’s output with the target data
structure.

###  III-D Training of XL-RIS-IR Algorithm

Building upon the detailed structure of the XL-RIS-IR algorithm, we initiate
the direct training process. This involves inputting the received signal
${\bm{y}}$ into the proposed algorithm. The training aims to adjust the
algorithm’s parameters to achieve an optimized output, namely best
reconstructing the XL-RIS information. As training progresses and converges,
the algorithm efficiently transforms the low-dimensional BS information into
the high-dimensional XL-RIS information.

##  IV Employing XL-RIS Information For Tracking

Once obtaining the XL-RIS information, we can use it estimate the 3D
coordinates of the MU. Hence, in this section, we employ the time-varying
sequence consisted by the reconstructed RIS Information, $\hat{\bm{y}}_{r}$,
to determine the MU’s location when the MU is moving within a 3D indoor space.

###  IV-A Tracking Problem Formulation

The movement of the MU causes variations in the XL-RIS information,
$\hat{\bm{y}}_{r}$, acting as a unique fingerprint that aids in determining
the MU’s 3D coordinates. Therefore, in this paper, we aim to exploit the
temporal dynamics of the RIS information to accurately track the MU’s
movements.

To accurately document the dynamic nature of the MU’s movement, we assemble a
time-varying sequence that reflects the changes in $\hat{\bm{y}}_{r}$ across a
specified interval. This sequence, indexed by time slot $t$, represents the
moments when the MU transitions to a subsequent position. Formally, the
sequence is expressed as
$\left[\hat{\bm{y}}_{r}(t),\hat{\bm{y}}_{r}(t+1),\ldots,\hat{\bm{y}}_{r}(t+T-1)\right]$,
covering $T$ discrete time slots. It serves to chronicle the evolution of RIS
information as the MU traverses various locations.

For predicting the MU’s future location at time step $T$, we consider the XL-
RIS information, $\hat{\bm{y}}_{r}$, as a dynamic fingerprint that evolves
over time with the MU’s movement. Therefore, we have

$\displaystyle\bm{\hat{p}}_{u}(T)$
$\displaystyle=\mathcal{F}([\hat{\bm{y}}_{r}(t),\hat{\bm{y}}_{r}(t+1),\ldots,\hat{\bm{y}}_{r}(t+T-1)]),$
(14)

where $\mathcal{F}(\cdot)$ is the nonlinear mapping function that transforms
the time-varying sequence of $\hat{\bm{y}}_{r}$ into the estimated position,
$\bm{\hat{p}}(T)$, of the MU at the subsequent time slot. Accordingly, the
mobile localization problem can be formulated as

$\displaystyle\underset{\bm{\hat{p}}_{u}(T)}{\min}\ $
$\displaystyle||\mathcal{Q}(\bm{\hat{p}}_{u}(T)-\bm{{p}}_{u}(T)||^{2}.$ (15)

aiming to minimize the discrepancy between the estimated position of the MU
and its actual position at time $T$, thereby enhancing the accuracy of MU
tracking.

###  IV-B Tracking Framework Assisted by XL-RIS Information

In addressing the challenges of tracking the MU, this paper introduces a
comprehensive framework designed to accurately track the MU location in the
near-field scenario. The proposed framework comprises two main modules: the
_Feature Extraction Module_ and the _Localization Module_.

####  IV-B1  Feature Extraction Module

The _Feature Extraction Module_ is pivotal in transforming raw XL-RIS
information into a meaningful representation that encapsulates spatial and
temporal cues essential for localization. This module integrates three key
components: a CNN extractor, a T$\&$F feature extractor and a near-field AoAs
extractor.

##### CNN extractor

The CNN extractor is adept at capturing spatial features from the XL-RIS
information by leveraging its hierarchical structure. Through Con layers, the
CNN learns to identify patterns and textures in $\hat{\bm{y}}_{r}$ that are
indicative of the MU’s 3D coordinates, making it a powerful tool for spatial
feature extraction.

##### Time $\&$ frequency (T$\&$F) feature extractor

The T$\&$F feature extractor is designed to extract time domain and frequency
domain characteristics to capture a comprehensive representation of the XL-RIS
information.

##### Near-field AoAs extractor

To better leverage the rich AoA information available in near-field scenarios,
we have designed a specialized AoAs extractor. This extractor is aimed at
capturing AoAs features within the XL-RIS panel.

####  IV-B2 Localization Module

Within the _Localization Module_ , an innovative architecture featuring an
Auto-encoder (AE) with a stacked bidirectional LSTM (Bi-LSTM) in the encoder
and a standard LSTM in the decoder is utilized to forecast the MU’s future
locations.

The AE architecture employs a stacked Bi-LSTM layer within the encoder to
delve into the temporal patterns and dependencies in the XL-RIS information,
capturing both past and future contexts for a comprehensive understanding. The
decoder, equipped with an LSTM layer, then reconstructs the future position
from this encoded representation, facilitating precise prediction of the MU’s
location at any given time $T$.

By integrating these two modules, our framework is specifically designed to
enhance the utilization of XL-RIS information for MU tracking. Further details
and insights into the XL-RIS-assisted tracking framework will be elaborated in
the following two sections.

##  V XL-RIS Information Based Feature Extraction

As the first module of the proposed tracking framework, the _Feature
Extraction Module_ includes three kinds of extractors, which are CNN
extractor, T$\&$F extractor, AoA extractor, respectively, whose details are
given as follows.

###  V-A CNN Extractor

To efficiently handle the high-dimensional reconstructed XL-RIS information
$\hat{\bm{y}}_{r}$, and extract potential location-related features, we design
a CNN-based extractor in this subsection, which includes several blocks. The
details of the designed module are given as follows.

####  V-A1 Data Processing Block

To accurately handle the preprocessing step for the reconstruction of high-
dimensional XL-RIS information $\hat{\bm{y}}_{r}$, before feature extraction
via a CNN block, it is crucial to detail the preparation process of the input
data. This preprocessing involves separating the XL-RIS information into its
real and imaginary components, denoted as $\mathfrak{R}(\hat{\bm{y}}_{r})$ and
$\mathfrak{I}(\hat{\bm{y}}_{r})$, respectively. Given that $\bm{y}_{r}$
encapsulates features from both the horizontal and vertical angle domains of
the XL-RIS, we first reshape the vectors
$\mathfrak{R}({\bm{y}}_{r})\in\mathbb{C}^{N\times 1}$ and
$\mathfrak{I}({\bm{y}}_{r})\in\mathbb{C}^{N\times 1}$ into matrices, denoted
as ${\bm{M}}_{Ry_{r}}\in\mathbb{R}^{N_{1}\times N_{2}}$ and
${\bm{M}}_{Iy_{r}}\in\mathbb{R}^{N_{1}\times N_{2}}$, respectively. These
matrices are then concatenated to form a 3D tensor represented as

$\bm{T}_{y_{r}}=\begin{bmatrix}{\bm{M}}_{Ry_{r}}\\\
{\bm{M}}_{Iy_{r}}\end{bmatrix}\in\mathbb{R}^{2\times N_{1}\times N_{2}}.$ (16)

This 3D tensor, $\bm{T}_{y_{r}}$, serves as the input to the CNN, which is
designed to extract location-related features from the complex XL-RIS
information.

####  V-A2 CNN Block

The CNN architecture processes the input tensor $\bm{T}_{y_{r}}$ through
several layers, each designed to perform specific operations that contribute
to feature extraction. The operation in a Con layer can be described as

$\displaystyle\bm{F}_{l}=f_{\mathrm{ReLU}}({\bm{W}}_{l}*\bm{T}_{y_{r}}+{\bm{b}}_{l}),$
(17)

where $\bm{F}_{l}$ is the feature map produced by the $l$-th layer,
$\bm{W}_{l}$ represents the weights of the filters, $\bm{b}_{l}$ is the bias
term, $*$ denotes the convolution operation, and $f_{\mathrm{ReLU}}=\max(0,x)$
denotes the ReLU function, which is widely used as a nonlinear activation
function in CNNs.

Consequently, the pooling layer is employed to reduce the dimensionality of
each feature map while retaining the most important information, typically
using max pooling. This procedure can be formulated as

$\displaystyle\bm{P}_{l}=f_{\mathrm{MaxPool}}(\bm{F}_{l}),$ (18)

where $\bm{P}_{l}$ represents the pooled feature map.

Then, the flattening layer is employed to convert the 2D feature maps into a
1D feature vector, preparing the data for processing by fully connected
layers, which can be expressed as

$\displaystyle\bm{g}_{\textrm{flattened}}=f_{\mathrm{Flatten}}(\bm{P}_{L}),$
(19)

where $\bm{g}_{\textrm{flattened}}$ is the flattened feature vector, and $L$
is the last Con layer.

Finally, the fully connected layer is applied to allocate weights to the
flattened feature vector to produce the final feature vector $\bm{T}$,
encapsulating the critical information extracted from the input. This can be
represented as

$\displaystyle\bm{f}_{\textrm{CNN}}=\bm{W}_{\textrm{FC}}\cdot\bm{g}_{\textrm{flattened}}+\bm{b}_{\textrm{FC}},$
(20)

where $\bm{W}_{\textrm{FC}}$ and $\bm{b}_{\textrm{FC}}$ are the weights and
bias of the fully connected layer, respectively.

To provide a clear understanding of the CNN block used for feature extraction
from the preprocessed XL-RIS information, the following Table I outlines each
layer’s definition and corresponding dimensions.

TABLE I: CNN Architecture for Feature Extraction Layer | Definition | Dimension
---|---|---
Input | Preprocessed Tensor | $2\times N_{1}\times N_{2}$
Conv Layer 1 | Convolution + ReLU Activation | $N_{f}^{(1)}\times H_{1}\times W_{1}$
Pooling Layer 1 | Max Pooling | $N_{f}^{(1)}\times H_{1}^{\prime}\times W_{1}^{\prime}$
Conv Layer 2 | Convolution + ReLU Activation | $N_{f}^{(2)}\times H_{2}\times W_{2}$
Pooling Layer 2 | Max Pooling | $N_{f}^{(2)}\times H_{2}^{\prime}\times W_{2}^{\prime}$
Flattening Layer | Flatten Feature Maps | $N_{f}^{(2)}\cdot H_{2}^{\prime}\cdot W_{2}^{\prime}$
Fully Connected Layer | Dense Layer | $N_{f}$

In this table, $N_{f}^{(1)}$, $N_{f}^{(2)}$ denote the number of filters in
the first and second convolutional layers, respectively. $H_{1}$, $W_{1}$,
$H_{2}$, $W_{2}$ represent the dimensions of the feature maps after each
convolutional layer before pooling. $H_{1}^{\prime}$, $W_{1}^{\prime}$,
$H_{2}^{\prime}$, $W_{2}^{\prime}$ are the dimensions of the feature maps
after pooling. $N_{f}$ is the dimension of the final feature vector
$\bm{f}_{\textrm{CNN}}$.

###  V-B T$\&$ F Feature Extractor

In addition to the spatial features extracted by the CNN, we enhance our
feature set with frequency domain characteristics and time statistical
measures to capture a comprehensive representation of the XL-RIS information.
Herein, we detail the feature extraction steps, including normalization,
frequency domain transformation, and time statistical analysis.

####  V-B1 Normalization

Before feature extraction, normalization is a critical preprocessing step to
ensure consistent scale across different signals, facilitating more effective
analysis and comparison. Normalization adjusts the amplitude of
$\hat{\bm{y}}_{r}$ to a common scale, enhancing the robustness of subsequent
feature extraction steps. The process can be mathematically represented as

$\hat{\bm{y}}_{r}^{\prime}=\frac{\hat{\bm{y}}_{r}-\min(\hat{\bm{y}}_{r})\cdot\mathbf{1}}{\max(\hat{\bm{y}}_{r})-\min(\hat{\bm{y}}_{r})},$
(21)

where $\hat{\bm{y}}_{r}^{\prime}$ is the normalized signal, $\mathbf{1}$ is a
all-ones vector with the same dimensions as $\hat{\bm{y}}_{r}$, and
$\min(\hat{\bm{y}}_{r})$ and $\max(\hat{\bm{y}}_{r})$ are the minimum and
maximum values of $\hat{\bm{y}}_{r}$, respectively.

####  V-B2 Frequency Domain Transformation

Performing the fast fourier transform (FFT) and extracting information in the
frequency domain are essential for wireless localization. Firstly, the
frequency domain analysis helps in identifying the presence of distinct path
components in a multi-path environment, where signals from a single source
reach the receiver via multiple paths due to reflections, diffractions, and
scattering. Each path can have different lengths, and consequently, different
phase shifts when observed in the frequency domain, providing valuable
information about the relative distances and angles of arrival, which are
directly related to the position of the source. Besides, the spectral content
of a signal, as revealed by the FFT, can be used to extract features that are
sensitive to the positioning of the MU. For example, the Doppler shift, a
change in frequency due to the relative motion between the transmitter and
receiver, can be precisely identified in the frequency domain, offering
insights into the velocity and direction of movement of the source relative to
the MU.

Hence, FFT is applied to $\hat{\bm{y}}_{r}^{\prime}$ to analyze its frequency
components, crucial for understanding the signal’s spectral properties. This
procedure is formulated as

$\hat{\bm{y}}^{\text{(FFT)}}_{r}=f_{\text{FFT}}(\hat{\bm{y}}_{r}^{\prime})\in\mathbb{C}^{N_{\text{yf}}\times
1},$ (22)

where $\hat{\bm{y}}^{\text{(FFT)}}_{r}$ denotes the frequency domain
representation of the normalized signal $\hat{\bm{y}}_{r}^{\prime}$,
$N_{\text{yf}}$ denotes frequency points of the
$\hat{\bm{y}}^{\text{(FFT)}}_{r}$. To quantify the signal’s energy
distribution across the frequency spectrum, we compute the spectral energy
using the squared magnitude of the FFT output, which is expressed as

$\bm{e}=|\hat{\bm{y}}^{\text{(FFT)}}_{r}|^{\circ
2}\in\mathbb{C}^{N_{\text{yf}}\times 1},$ (23)

where $(\cdot)^{\circ 2}$ denotes the element-wise square operation,
transforming the complex signal into a vector of real-valued energy
measurements across the frequency spectrum.

This energy vector $\bm{e}$ is then normalized to a probability distribution,
which is given by

$p(\hat{\bm{y}}^{\text{(FFT)}}_{r})=\frac{\bm{e}}{\sum^{N_{\text{yf}}}_{n_{\text{yf}}=1}\bm{e}},$
(24)

allowing for the calculation of spectral entropy, which can be used as an
essential feature of the reconstructed RIS information.

####  V-B3 Feature Extraction

Several features are extracted from both the time-domain and frequency-domain
representations of the signal to capture its essential characteristics.

##### Frequency Domain Features

We compute the spectral energy and entropy as key features to capture the
signal’s frequency distribution. First, the spectral energy is given by

$E_{\text{FFT}}=\sum^{N_{\text{yf}}}_{n_{\text{yf}}=1}\bm{e}.$ (25)

Then, the spectral entropy is formulated as

$H_{\text{FFT}}=-\sum^{N_{\text{yf}}}_{n_{\text{yf}}=1}p(\hat{\bm{y}}^{\text{(FFT)}}_{r})\log
p(\hat{\bm{y}}^{\text{(FFT)}}_{r}).$ (26)

#####  Time Domain Features

Except the frequency domain features, time domain features are also essential
to the localization, especially for the MU is moving with time. To extract the
time domain features, we employ the statistical time domain features,
including mean, variance. They are calculated directly from
$\hat{\bm{y}}_{r}^{\prime}$, which is given as

$\displaystyle\mu=f_{\text{mean}}(\hat{\bm{y}}_{r}^{\prime}),$
$\displaystyle\sigma^{2}=f_{\text{var}}(\hat{\bm{y}}_{r}^{\prime}),$ (27)

where $\mu$ is the mean, $\sigma^{2}$ is the variance.

####  V-B4 Feature Vector Construction

Finally, the extracted features are concatenated to form a comprehensive
feature vector $\bm{f}_{\textrm{T\&F}}$, which is written as

$\bm{f}_{\textrm{T\&F}}=[\mu,\sigma^{2},E_{\text{FFT}},H_{\text{FFT}}]^{T},$
(28)

providing a multi-dimensional representation of $\hat{\bm{y}}_{r}$ for further
localization.

###  V-C AoAs Feature Extraction

In near-field localization applications, extracting AoA information within the
XL-RIS panel is crucial as the near-field effects offer richer and more
detailed AoA data compared to far-field scenarios. Hence, this subsection
provide an approach to estimate the AoAs from the reconstructed XL-RIS
information $\hat{\bm{y}}_{r}$.

####  V-C1 Angle Estimation

Given the large number of different AoAs at different reflecting elements
within the XL-RIS to be estimated in near-field conditions, which increases
the estimation complexity and causes signal energy to be dispersed, it is
challenging to accurately estimate the AoA across the entire XL-RIS panel.
Consequently, we partition the XL-RIS into sub-arrays for angle estimation to
manage the computational complexity and concentrate the signal energy within
each sub-array, thus enhancing the accuracy and efficiency of AoA estimations.
By doing so, the wavefront impinging on each sub-array can be approximated as
a planar wavefront due to the limited physical dimension of each sub-array,
minimizing the impact of wavefront curvature. This approximation allows for
the effective use of high-resolution AoA estimation techniques like MUSIC,
which are predicated on the assumption of planar wavefronts, thus enabling
accurate AoA estimation within each sub-array’s constrained spatial domain.

Let us assume that there are $K$ sub-arrays within the XL-RIS. To estimate $K$
AoAs using the MUSIC algorithm across $K$ sub-arrays. For the $k$-th sub-
array, where $k\in\\{1,\cdots,K\\}$, we have the steering vector of the $k$-th
sub-array expressed as

$\displaystyle\bm{a}_{k}(\theta_{k},\phi_{k})={\bm{a}}_{k}^{(e)}(\theta_{k})\otimes{\bm{a}}_{k}^{(a)}(\theta_{k},\phi_{k}),$
(29)

where

$\displaystyle{\bm{a}}_{k}^{(e)}(\theta_{k})$
$\displaystyle=\left[1,\ldots,e^{\frac{-j2\pi(N_{k,e}-1)d_{x}\cos\theta_{k}}{\lambda_{c}}}\right]^{T},$
(30) $\displaystyle{\bm{a}}_{k}^{(a)}(\theta_{k},\phi_{k})$
$\displaystyle=\left[1,\ldots,e^{\frac{-j2\pi(N_{k,a}-1)d_{x}\sin\theta_{k}\cos\phi_{k}}{\lambda_{c}}}\right]^{T},$
(31)

where $N_{k,e}$ and $N_{k,a}$ denote the numbers of elements within the $k$-th
sub-array in the elevation and azimuth directions, respectively, $d_{x}$
denotes the distance between adjacent array elements, and $\lambda_{c}$ is the
carrier wavelength.

The received signal at the $k$-th sub-array of the XL-RIS can be expressed as

${\bm{y}}_{r,k}=\alpha\bm{a}_{k}(\theta_{k},\phi_{k})s,$ (32)

where $\bm{a}_{k}(\theta_{k},\phi_{k})$ denotes the array response vector. The
signal $s$ corresponds to the transmitted signal. Due to the passive nature of
XL-RIS, ${\bm{y}}_{r,k}$ cannot be directly observed. To address this
challenge, we employ the reconstructed XL-RIS information of the $k$-th sub-
array, which is denoted as $\hat{\bm{y}}_{r,k}$. Utilizing this information,
we are able to estimate the AoAs (e.g., $\theta_{k}$, $\phi_{k}$) across the
$K$ sub-arrays. The details are given as follows.

##### Signal Covariance Matrix

First, by employing the FFT technique to $\hat{\bm{y}}_{r,k}$, we have

$\displaystyle\hat{\bm{y}}^{\text{(FFT)}}_{r,k}$
$\displaystyle=f_{\text{FFT}}(\hat{\bm{y}}_{r,k}^{\prime}),$
$\displaystyle\hat{\bm{y}}_{r,k}^{\prime}$
$\displaystyle=\frac{\hat{\bm{y}}_{r,k}-\min(\hat{\bm{y}}_{r,k})\cdot\mathbf{1}}{\max(\hat{\bm{y}}_{r,k})-\min(\hat{\bm{y}}_{r,k})},$
(33)

Then, we calculate the frequency covariance matrix, which is given as follows:

$\displaystyle\bm{R}_{f,k}$
$\displaystyle=\frac{1}{N_{\text{yf}}}\sum_{n_{\text{yf}}=1}^{N_{\text{yf}}}(\hat{\bm{y}}^{\text{(FFT)}}_{r,k}[n]-\bar{\bm{y}}^{\text{(FFT)}}_{r,k})(\hat{\bm{y}}^{\text{(FFT)}}_{r,k}[n]-\bar{\bm{y}}^{\text{(FFT)}}_{r,k})^{H},$
(34)

where $\bar{\bm{y}}^{\text{(FFT)}}_{r,k}$ is the mean vector of
$\hat{\bm{y}}^{\text{(FFT)}}_{r,k}$.

##### Eigen Decomposition

For each sub-array $k$, we perform an eigen decomposition on $\bm{R}_{f,k}$,
which is formulated as

$\displaystyle\bm{R}_{f,k}=\bm{E}\bm{\Lambda}\bm{E}^{H},$ (35)

where $\bm{E}$ is a matrix whose columns are the eigenvectors of
$\bm{R}_{f,k}$, and $\bm{\Lambda}$ is a diagonal matrix containing the
eigenvalues in descending order. $H$ denotes the conjugate transpose.

##### Identifying Signal and Noise Subspaces

Following the results of decomposition, we proceed to determine the signal and
noise subspaces, represented by $\bm{E}_{\text{signal}}$ and
$\bm{E}_{\text{noise}}$, respectively.

##### MUSIC Spectrum

The MUSIC spectrum for estimating $\theta_{k}$ and $\phi_{k}$ is given by the
inverse of the projection of the steering vector onto $\bm{E}_{\text{noise}}$.
For each potential pair of $(\Theta_{k},\Phi_{k})$, MUSIC spectrum can be
derived as

$P_{\text{MUSIC},k}(\Theta_{k},\Phi_{k})=\frac{1}{\bm{a}_{k}(\Theta_{k},\Phi_{k})^{H}\bm{E}_{\text{noise},k}\bm{E}_{\text{noise},k}^{H}\bm{a}_{k}(\Theta_{k},\Phi_{k})}.$
(36)

##### Estimation of $\theta_{k}$ and $\phi_{k}$

The AoAs, $\theta_{k}$ and $\phi_{k}$, are estimated by identifying the peaks
in the MUSIC spectrum, which is given as

$(\hat{\theta}_{k},\hat{\phi}_{k})=\arg\max_{(\Theta_{k},\Phi_{k})}P_{\text{MUSIC},k}(\Theta_{k},\Phi_{k}),$
(37)

where $(\hat{\theta}_{k},\hat{\phi}_{k})$ are the estimated elevation and
azimuth angles for the $k$-th sub-array.

####  V-C2 Feature Vector Construction

Upon obtaining the AoAs estimates, $(\hat{\theta}_{k},\hat{\phi}_{k})$, from
the MUSIC algorithm for each of the $K$ sub-arrays, we aggregate these
estimations into a structured feature set. The collective feature set
$\mathcal{F}$, representing the AoA estimates across all sub-arrays, can be
constructed as:

$\mathcal{F}_{\textrm{AoA}}=\left\\{(\hat{\theta}_{1},\hat{\phi}_{1}),(\hat{\theta}_{2},\hat{\phi}_{2}),\ldots,(\hat{\theta}_{K},\hat{\phi}_{K})\right\\}.$
(38)

Alternatively, for numerical and analytical convenience, these AoA estimates
can be organized into a vector form $\bm{f}_{\textrm{AoA}}$, which is given as

$\bm{f}_{\textrm{AoA}}=\left[\hat{\theta}_{1},\hat{\phi}_{1},\hat{\theta}_{2},\hat{\phi}_{2},\cdots,\hat{\theta}_{K},\hat{\phi}_{K}\right].$
(39)

###  V-D Final Feature Vector Construction

After combing the feature vector ${\bm{f}}_{\text{CNN}}$, the T$\&$F feature
vector ${\bm{f}}_{\text{T$\&$F}}$ and the AoAs feature vector
${\bm{f}}_{\text{AoA}}$, we have the final feature vector, denoted as
${\bm{f}}_{\textrm{Final}}$, which is given by

$\bm{f}_{\text{Final}}=\left[{\bm{f}}_{\text{CNN}},{\bm{f}}_{\text{T$\&$F}},{\bm{f}}_{\text{AoA}}\right],$
(40)

which will be fed into the following _Mobile Tracking Module_.

##  VI XL-RIS Information Based Tracking Algorithm

In this section, we employ the final feature vector
${\bm{f}}_{\textrm{Final}}$ for tracking the MU using the data-driven DL based
framework. Data-driven DL based framework is now widely designed and employed
for MU localization and channel estimation. However, all these DL-based
localization work ignore the time sequence of the recieved signal when the MU
is moving on a specific trajectory. In another word, the adjacent observations
of the time-varying recieved signal can be further utilized for more precise
estimation.

There are two widely employed techniques called recurrent neural network (RNN)
and long short term memory (LSTM) on solving the time-varying tasks, such as
natural language processing. Both techniques consider the information from the
previous entered data and the currently entered data to estimate.
Specifically, RNN has feedback loops to maintain information over time.
However, it is difficult for RNN on learning long-term temporal dependencies
due to the vanishing gradient problem. Differently, LSTM introduces input and
forget gates for better preservation of long-term dependencies on dealing
gradient flow.

Therefore, in this section, we aim to develop an Autoencoder (AE) equipped
with a stacked bidirectional long short-term memory (Bi-LSTM) encoder and a
standard LSTM decoder, referred to as the Stacked Bi-LSTM AE algorithm. This
design is intended to accurately predict the 3D coordinates of the mobile user
(MU) in the subsequent time slot.

###  VI-A LSTM

Before presenting the details of the Stacked Bi-LSTM AE algorithm, the details
of LSTM network are given as follows.

The LSTM network, an enhancement over traditional RNN, introduces a gradient-
based learning algorithm capable of connecting past information to the current
task. LSTMs process time-sequence data in a chain-like structure, moving in a
forward direction. The unique aspect of LSTM architecture compared to standard
RNNs is its hidden layers within each LSTM cell. The details of LSTM network
are given as follows.

####  VI-A1 Forget Gate

The first layer of the LSTM network is called forget gate which is also named
forget layer, which is formulated as

${\bm{f}}_{t}=\sigma\left({\bm{W}}_{f}{\bm{x}}(t)+\tilde{\bm{W}}_{f}\tilde{\bm{x}}(t-1)+{\bm{b}}_{f}\right),$
(41)

where $\sigma$ denotes the sigmoid activation function, $\tilde{\bm{x}}(t-1)$
denotes the information passed from previous layer, ${\bm{x}}(t)$ represents
the input feature vector at time step $t$, while $\tilde{\bm{W}}_{f}$,
${\bm{W}}_{f}$ and ${\bm{b}}_{f}$ denote the weight and bias of forget gate,
respectively.

####  VI-A2 Input Gate

Consequently, the second layer is called input layer which is also named input
gate, which can be expressed as

${\bm{i}}_{t}=\sigma\left({\bm{W}}_{i}{\bm{x}}(t)+\tilde{\bm{W}}_{i}\tilde{\bm{x}}(t-1)+{\bm{b}}_{i}\right),$
(42)

where $\tilde{\bm{W}}_{i}$, ${\bm{W}}_{i}$ and ${\bm{b}}_{i}$ denote the
weight and bias of input gate, respectively.

####  VI-A3 Cell Input State

Then, to update the cell input state, e.g., ${\bm{c}}_{t}$, $\tanh$ function
is employed and calculated as

${\bm{c}}_{t}=\tanh\left({\bm{W}}_{c}{\bm{x}}(t)+\tilde{\bm{W}}_{c}\tilde{\bm{x}}(t-1)+{\bm{b}}_{c}\right).$
(43)

where $\tilde{\bm{W}}_{c}$, ${\bm{W}}_{c}$ and ${\bm{b}}_{c}$ denote the
weight and bias of cell input state, respectively.

####  VI-A4 Output Gate

To decide the next hidden state, the output gate ${\bm{o}}_{t}$ is utilized,
which is written as

${\bm{o}}_{t}=\sigma\left({\bm{W}}_{o}{\bm{x}}(t)+\tilde{\bm{W}}_{o}\tilde{\bm{x}}(t-1)+{\bm{b}}_{o}\right),$
(44)

where $\tilde{\bm{W}}_{o}$, ${\bm{W}}_{o}$ and ${\bm{b}}_{o}$ denote the
weight and bias of output layer, respectively.

####  VI-A5 Output State

Apart from the cell input state ${\bm{c}}_{g,t}$ in the hidden layers, there
are two other cell states in the structure: the previous cell output state
${\bm{c}}(t-1)$ fed in the current LSTM cell, and the current cell output
state ${\bm{c}}(t)$ passed to the next LSTM cell. The output state at the
current time $t$ can be formulated as

${\bm{c}}(t)={\bm{f}}_{t}\odot{\bm{c}}(t-1)+{\bm{i}}_{t}\odot{\bm{c}}_{t}.$
(45)

where $\odot$ denotes the Kronecker product.

####  VI-A6 Output Layer

The conventional input layer is ${\bm{x}}(t)$ at the time slot $t$. Finally,
the output layer can be formulated as

$\tilde{\bm{x}}(t)={\bm{o}}_{t}\odot\tanh({\bm{c}}(t)).$ (46)

###  VI-B Stacked Bi-LSTM

To accurately capture the dynamic nature of the input time-varying feature
sequence, the Bi-LSTM is employed when constructing encoder. This design
choice is pivotal because it allows the model to maintain the temporal
continuity and dependencies between each time step in the sequence. By
integrating both forward and backward LSTM cells, the architecture ensures
that information from both past and future contexts is utilized, facilitating
a more comprehensive understanding of the temporal patterns. This
bidirectional approach is especially beneficial for tasks like mobile
localization tracking, which recognizing the sequence’s temporal evolution.

Furthermore, it has been proved that by stacking multiple hierarchical models,
the performance can be improved progressively. Hence, we adopt a stacked
structure where the output from the lower layer is then fed as the input to
the upper layer with $L_{s}\geq 2$ Bi-LSTM layers. The workflow of the stacked
Bi-LSTM considers both forward and backward directions and deeper structure
with $T$ time slots. During $T$ time slots, we have the time sequence given by
${\bm{F}}_{\textrm{fea}}=[{\bm{f}}_{\textrm{Final}}(t),{\bm{f}}_{\textrm{Final}}(t+1),\cdots,{\bm{f}}_{\textrm{Final}}(t+T)]$.
By employing this time varying sequence from the past $T$ time slots, the
location of the MU at $T+1$ time slot can be estimated via the design AEBi-
LSTM. The details of the AE Bi-LSTM are given as follows subsections.

###  VI-C Encoder Construction for Stacked Bi-LSTM AE

In this subsection, we present the details of the encoder constructed by the
stacked Bi-LSTM, which is designed to extract the time-varying sequence
${\bm{F}}_{\textrm{fea}}$.

####  VI-C1 The first Bi-LSTM Layer

In a Bi-LSTM layer, the input time-varying feature sequence is processed to
capture both past (forward) and future (backward) dependencies within the
sequence. The first Bi-LSTM is employed for estimating the location of the MU
at the $T+1$ time slot by using the time-varying sequence
${\bm{F}}_{\textrm{fea}}$, whose details are given as follows.

##### Forward Pass in Bi-LSTM

The forward pass processes the input sequence from time $t$ to time $t+T$,
capturing information as it moves forward in time.

##### Backward Pass in Bi-LSTM

The backward pass processes the sequence in reverse, from time $t+T$ to time
$t$, capturing future context for each time step. The operations are analogous
to those of the forward pass but applied in reverse order.

####  VI-C2 Stacked $L_{s}$ Bi-LSTM Layers

After introducing the first Bi-LSTM layer, the construction of the Encoder in
an AE Bi-LSTM setup involves stacking additional Bi-LSTM layers to further
refine and process the features. This stacked structure enables the model to
capture more complex temporal dependencies in the sequence data, enhancing its
predictive capabilities for tasks like estimating the MU location based on
historical data.

Following the first Bi-LSTM layer, we incorporate $\mathrm{L}_{s}$ Bi-LSTM
layers. Each subsequent layer receives the output from all time slots of its
predecessor as input. Consequently, each Bi-LSTM layer builds upon the
patterns and dependencies identified in the time-varying sequence by the layer
before it, extracting increasingly abstract features from the sequence.

Within each Bi-LSTM layer, temporal steps are processed akin to the first
layer, employing forward and backward LSTM units to assimilate information
from past and future contexts, respectively. Thus, the model’s output at each
time step synthesizes current input features with contextual insights drawn
from both preceding and succeeding steps.

To mitigate overfitting, Dropout layers can be interspersed between Bi-LSTM
layers. These Dropout layers randomly nullify a fraction of the neurons’
activations during training, thereby enhancing the model’s generalization
ability.

The output generated by the last Bi-LSTM layer in the Encoder is typically
used to represent a compressed and high-level feature representation of the
entire input sequence. This representation captures the core information and
patterns of the input sequence, providing necessary context for the Decoder.

(a) 3D random walk example.

(b) 3D wave example.

(c) 3D spiral example.

Figure 4: Examples of different types of trajectory.

(a) Test and train losses of random walk trajectory.

(b) Test and train losses loss of spiral trajectory.

(c) Test and train losses of wave trajectory.

Figure 5: Losses of different types of trajectory.

###  VI-D Decoder Construction for Stacked Bi-LSTM AE

After constructing the encoder of the Stacked Bi-LSTM AE, the next step is to
build the decoder to predict the location of the MU at the $T+1$ time slot.
The decoder utilizes the high-level features generated by the encoder to
perform this prediction.

####  VI-D1 Initialization

The encoder processes the entire input sequence and abstracts the input
through its hierarchy into a high-level representation that captures the
essence of the input data, including timing dependencies and patterns. Hence,
the final hidden state generated by the encoder is employed for the initial
hidden state of the decoder. By initializing the decoder with this state, we
provide a strong starting point for the generation process, ensuring that the
decoder has all the context needed to generate accurate and relevant output.

####  VI-D2 Layers Construction for Decoder

The decoder consists of one LSTM layers, one dropout layer, one dense layer,
where the details of each layers are given as follows.

##### LSTM Layer

In the context of predicting the location at time $T+1$, a single LSTM layer
is chosen for the decoder instead of a Bi-LSTM or a Stacked Bi-LSTM
configuration due to the nature of the prediction task. Bi-LSTM layers, which
process information in both forward and backward directions, are more suited
to situations where the entire sequence is known and the context from future
steps is as important as the past, which is not the case when predicting
future locations based solely on past and current information. Stacked LSTMs,
while powerful for capturing complex patterns in more intricate sequences,
increase model complexity and computational cost, which might not be necessary
for tasks where the encoder has already provided a condensed and meaningful
representation of the input sequence. Thus, a single LSTM layer is efficient
and sufficient for leveraging the encoded context to predict the next
location, minimizing the risk of overfitting while maintaining model
simplicity and interpretability.

##### Dropout Layer

Dropout is included as a regularization technique. By randomly omitting
portions of the layer’s inputs, dropout forces the network to learn more
robust features that are useful in conjunction with many different random
subsets of the other neurons. This reduces the likelihood of overfitting,
especially when training data is limited or when the architecture is complex
relative to the amount of training data.

##### Dense Layer

Dense layer is used to produce a specific output size from the LSTM layer’s
output, matching the dimensionality of the prediction task (e.g., 3D
coordinates for location). The linear activation in the output Dense layer
allows for the direct prediction of continuous values. The linear activation
in the output Dense layer allows for the direct prediction of continuous
values.

##  VII Simulation Results

In this simulation, we consider a setup in a 3D space, involving an MU, an
RIS, and a BS. The RIS is composed of $N_{1}\times N_{2}=10\times 10=100$
elements. The center of the RIS is located at $\bm{p}_{r}=(6,0,2)$ m. The BS
comprises $M_{1}\times M_{2}=4\times 4=16$ antennas. The center of the BS is
positioned at $\bm{p}_{b}=(0,5,1.5)$ m. The MU-RIS link includes $P=10$
propagation paths. The transmit power of the MU is $30$ dBm.

To simulate the movement trajectories of objects, we have defined three types
of paths: random walk, wave, and spiral, to model the possible trajectories
that objects may follow in reality. Examples of these three distinct
trajectories are illustrated in Fig. 4. These trajectories are generated with
a random starting point within a space limited to $10\times 10\times 3$
$\text{m}^{3}$. Over $10000$ such paths are generated, each comprising $11$
steps. For each step along a trajectory, we generate the signal received at
both the BS and RIS, denoted by ${\bm{y}}_{t}$ and ${\bm{y}}_{r}$,
respectively. Besides, the principles behind the generation of the three types
of trajectories are as follows:

  1. 1.

Random Walk Trajectory: We start by selecting a random starting point within
the indoor space. For each step in the trajectory, we generate a unit vector
pointing in a random direction within 3D space, and move along this unit
vector’s direction by a step size randomly chosen from the range $(0,1]$. An
example of random walk is given in Fig. 4(a).

  2. 2.

Wave Trajectory: To generate a wave trajectory, we start by picking a random
point. We then draw this path along the x-axis, setting its length and
dividing it into steps. Then, based on the wave’s amplitude of $2$ and
wavelength of $5$, a sine function is employed to modify the y-coordinates.
This creates the familiar peaks and troughs of a wave. Throughout, we keep the
z-coordinate constant, ensuring the wave moves in a two-dimensional plane.
Fig. 4(b) presents an example of the wave trajectory.

  3. 3.

Spiral Trajectory: We create a spiral path by using the formula $r=a+b\theta$.
Here, $r$ is how far each point is from the random start point, $a=0.1$ is the
starting distance from the center, and $b=3$ decides how much this distance
increases. $\theta$ increases step by step, making the spiral trajectory. By
changing from polar coordinates (using $r$ and $\theta$) to regular 3D
coordinates, we can draw this spiral in a 3D space. Fig. 4(c) presents an
example of the generated spiral trajectory.

Figure 6: The MSE comparison of different inputs for proposed algorithm versus
SNR (dB). Figure 7: The MSE comparison of different types of trajectory versus
SNR (dB).

Fig. 5 showcases the training and test loss curves of three different kinds of
trajectories. Specifically, Fig. 5(a) displays the iterative convergence
behaviors of training and test losses for the random walk trajectory. It is
evident that by approximately $100$ epochs, both training and test losses for
the random walk trajectory have converged. Although there are minor
fluctuations in the test loss afterwards, their magnitude is negligible.
Furthermore, Fig. 5(b) and Fig. 5(c) depict the convergence trends of training
and testing losses for the spiral and wave trajectories, respectively. For
both trajectory types, the training and testing losses converge by the $15$-th
epoch, exhibiting minimal fluctuations thereafter. As we can see from Fig. 5,
due to the regular patterns of the spiral and wave trajectories, the proposed
algorithm is capable of quickly learning their movement patterns. In contrast,
the random walk, characterized by its unpredictable nature, requires more
epochs for the neural network to learn. However, the eventual convergence of
the random walk as well demonstrates the effectiveness of the proposed
algorithm.

Fig. 6 shows the mean squared error (MSE) comparison among three scenarios:
without near-field feature extraction, without CNN feature extraction, and
with feature extraction. Fig. 6 reveals that utilizing near-field feature
extraction alone yields better results than relying solely on CNN feature
extraction. This underscores the necessity of extracting near-field features,
suggesting that CNN may not fully capture all relevant localization
information on its own. Besides, as shown in Fig. 6, compared to the curve
without near feature extraction, incorporating our near-field feature
extraction significantly enhances performance, highlighting the value of
tailored near-field feature extraction in improving localization accuracy.

Fig. 6 also displays the MSE performance curve for localization using actual
RIS information, labeled as ‘Real RIS inf.’. The closeness of this curve to
the ‘Proposed, RIS inf.’ curve, which represents localization using
reconstructed RIS information, validates the effectiveness of our proposed
algorithm for XL-RIS information reconstruction. It shows that the
localization accuracy achieved with our reconstructed information closely
approximates that achieved with the actual RIS information, supporting the
reliability of our reconstruction approach in mimicking real RIS information
for localization purposes.

Fig. 7 presents the comparison of MSE for different trajectories versus the
SNR in dB. As depicted in Fig. 7, the MSE values associated with both the wave
and spiral trajectories are significantly lower than those of the random walk
trajectory. This distinction arises because the wave and spiral trajectories
exhibit structured movement patterns, so that the deployed Stacked Bi-LSTM AE
network can analyze more effectively. Besides, it is also noteworthy that
utilizing the received signal at the BS, e.g., BS information, results in
substantially poorer performance compared to the algorithms when XL-RIS
information is used. This further validates the efficacy of employing high-
dimensional RIS information for near-field tracking, as proposed in our study.
Moreover, as shown in Fig. 7, when employing the high-dimensional RIS
information, the accuracy maintains a certain level of accuracy even under low
SNR conditions with a random walk trajectory, which demonstrates robustness of
proposed algorithms against low SNR.

Figure 8: The MSE comparison of different number of reflecting elements within
the RIS versus SNR (dB).

As shown Fig. 8, it is evident that increasing the number of reflective
elements in the RIS panel enhances positioning accuracy, for both cases
employing BS information and RIS information. However, it can be observed that
increasing the number of reflecting elements from $100$ to $400$, the
improvement of localization accuracy when using BS information is not
significantly pronounced. Besides, it necessitates a higher SNR to achieve
better localization results. In contrast, when employing RIS information, as
shown in Fig. 8, the accuracy with $400$ reflective elements is significantly
higher than that with $100$. This emphasizes the importance of using XL-RIS
information for improved positioning accuracy in near-field scenarios.

Figure 9: MSE performance across SNR (dB) for localization algorithms with and
without feature extraction.

Fig. 9 showcases a comparison of MSE across SNR for different algorithms. As
shown in Fig. 9, ‘AE-Stacked-LSTM’ refers to an architecture that incorporates
multiple layers (stacked) but does not employ bidirectional processing, while
‘AE-LSTM’ utilizes a simpler, single-layer structure without the stacking or
bidirectional enhancements. The observed results clearly demonstrate that the
proposed algorithm, referred to as ‘AE-Stacked-Bi-LSTM’, significantly surpass
the other algorithms in localization performance. This superiority suggests
that the bidirectional processing, when combined with a stacked approach,
effectively captures the temporal dependencies and spatial characteristics of
the RIS information, thus enhancing the predictive accuracy and proving the
proposed methodology’s efficacy.

##  VIII Conclusion

This paper has introduced a mobile tracking framework leveraging the high-
dimensional signal received from XL-RIS information. We have presented an XL-
RIS-IR algorithm to reconstruct the high-dimensional XL-RIS information from
the low-dimensional BS information. Building on this, we have proposed a
comprehensive framework for mobile tracking, consisting of a _Feature
Extraction Module_ and a _Mobile Tracking Module_. Simulation results have
demonstrated that the tracking accuracy of our proposed framework is enhanced
by the reconstructed XL-RIS information. Besides, it has been validated that
the proposed framework exhibited the robustness to SNR variation.

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