channels), the relationship can be treated as a reference for deriving the
scaling laws and designing localization systems.

###  IV-C Geometry-based Algorithms

####  IV-C1 Direct Localization

Geometry-based localization algorithms can be categorized into direct
localization and multi-stage algorithms. In direct localization, the state
vector ${\bf s}$ is estimated directly from the received signals, without
estimating any intermediate parameters [168, 37]. Given the transmitted
signal, and assuming no prior information is available on the UE’s positions,
the direct localization problem can be formulated as a maximization of the
likelihood function

$\begin{split}\hat{\bf s}_{\mathrm{direct}}&=\operatorname*{arg\,max}_{{\bf
s}}p(\hat{\bf Y}|{\bf s})=\operatorname*{arg\,max}_{{\bf s}}\ln(p(\hat{\bf
Y}|{\bf s}))\\\ &=\operatorname*{arg\,min}_{{\bf s}}[(\hat{\bf
Y}-\hbox{\boldmath$\mu$}({\bf s}))^{H}\hbox{\boldmath$\Sigma$}^{-1}_{\hat{\bf
Y}}(\hat{\bf Y}-\hbox{\boldmath$\mu$}({\bf s}))].\end{split}$ (80)

Here, $\hbox{\boldmath$\Sigma$}_{\hat{\bf Y}}$ is the covariance matrix of the
received symbols, which can be ignored if independent identical Gaussian
distribution of noise vector in (29) is assumed for different antennas and
measurements.

Direct localization is applicable to both quasi-synchronous (only BSs are
synchronized) and asynchronous (none of the devices are synchronized) systems
[9]. However, the computational complexity of solving the optimization problem
formulated in (80) is high due to non-convexity and the large search space.
Prior information is important in this case to limit the search area.

####  IV-C2 Multi-stage Localization

The multi-stage localization procedure divides direct localization into a
geometry information estimation stage and a position/orientation estimation
stage, which reduces the complexity of calculating all the unknowns from the
received data. More specifically, the measurement vector $\gamma$ is first
estimated and then the state vector is extracted from it [60, 69]. Similar to
(80), a multi-stage localization problem can be formulated as

$\begin{split}\hat{\bf s}_{\mathrm{multi-
stage}}&=\operatorname*{arg\,max}_{{\bf
s}}p(\hat{\hbox{\boldmath$\gamma$}}|{\bf s})\\\
&=\operatorname*{arg\,min}_{{\bf
s}}[(\hat{\hbox{\boldmath$\gamma$}}-\hbox{\boldmath$\gamma$}({\bf
s}))^{H}\hbox{\boldmath$\Sigma$}^{-1}_{\hat{\gamma}}(\hat{\hbox{\boldmath$\gamma$}}-\hbox{\boldmath$\gamma$}({\bf
s}))],\end{split}$ (81)

where $\hbox{\boldmath$\Sigma$}_{\mathrm{\hat{\gamma}}}$ is the covariance
matrix of the measurement vector and $\hat{\hbox{\boldmath$\gamma$}}$ is the
estimated measurement vector from channel estimation. Multi-stage approaches
are inherently sub-optimal [168] and usually inferior to direct localization.
However, by considering all multipath components, multi-stage localization can
reduce the performance gap with direct localization, and it is hence pursued
in many works [60, 69].

In multi-stage localization, the parameter vector $\gamma$ needs to be
estimated first. Each element in $\gamma$ (e.g., AOA/AOD, channel gains, and
signal delay for each path) can be obtained independently or jointly. The
channel gain can be estimated by solving a least-squares (LS) problem [60].
AOA/AOD can be estimated using subspace-based methods (e.g,. MUSIC) [186],
compressed sensing (CS) [187], deep learning (DL) [188], or Bayesian inference
[87]. TOA can be estimated using correlation-based [189] or energy-based
methods [190]. The channel parameters can also be estimated jointly using
multidimensional channel parameter estimation via rotational invariance
techniques (MD-ESPRIT) [191, 192]. In general, the performance of AOA/AOD
estimation depends on the array size of the device, while TOA estimation
benefits from synchronization and wideband signals. In addition, the error in
different stages propagates and may affect the localization performance, which
should be considered in system design for a better tradeoff of processing time
and performance.

####  IV-C3 Practical Algorithms for Geometry-based Localization

With the geometry information $\hat{\hbox{\boldmath$\gamma$}}$ obtained from
channel estimation, multi-stage localization problems can be formulated using
(81) (direct localization only requires $\hat{\bf Y}$ as in (73)). An
analytical closed-form solution might be obtained by setting the derivative of
an objective function equal to zero and solving for the position parameters.
However, this approach is impractical considering the non-convexity of the
cost function. We discuss two practical categories of optimization algorithms:
convergent iterative methods and heuristic methods [193, 48].

  * •

Convergent iterative methods: If the gradient information from the signal
model is known, gradient- or Hessian-based algorithms can be implemented.
Other iterative algorithms such as alternative projection [194] and
expectation-maximization [60] are also practical solutions to reduce the
computational burden. Within a few iterations, such deterministic algorithms
converge to an optimum of the objective function. The convergence depends on
the formulation of objective functions and iteration parameters (e.g., step
size), where local solutions can be reached.

  * •

Heuristic/metaheuristics methods: Heuristic methods are capable of dealing
with non-differential nonlinear objective functions and reaching near-optimal
solutions faster. Popular algorithms include swarm intelligence, tabu search,
simulated annealing, genetic algorithms, and so on [55, 195, 196].

Given the sparsity of high-frequency channels and a large number of
measurements (due to large bandwidth and array sizes/RFCs), we expect that
multi-stage localization will be favored in THz localization. However, for
applications that require high localization accuracy, a practical approach is
to determine an initial position via multi-stage algorithms and then refine it
using direct localization. Next, we discuss learning-based localization.

###  IV-D Learning-based Algorithms

In the previous subsection, we discussed geometry-based localization. In
challenging environments where geometric models cannot be formulated (e.g.,
many non-resolvable NLOS paths), or when geometry-based localization cannot
handle the processing speed requirements of the system, learning-based methods
can be used. In this subsection, we briefly describe the implementations of
ML-based localization algorithms in two categories, namely, direct
localization and multi-stage localization. Then, practical ML-based algorithms
will be discussed.

####  IV-D1 Direct Localization

ML-based localization involves two phases, offline training of the model
$f(\cdot)$ and online processing of the observation to obtain a position
estimation $\hat{\bf p}=f({\bf y})$. During the training phase, a training
data set $\mathcal{D}=\langle\mathcal{D}_{\bf y},\mathcal{D}_{\bf p}\rangle$
(including $|\mathcal{D}|$ signal-position pairs $\langle{\bf
y}^{\text{train}}_{i},{\bf p}^{\text{train}}_{i}\rangle$, ($1\leq
i\leq|\mathcal{D}|$, ${\bf y}^{\text{train}}_{i}\in\mathcal{D}_{\bf y}$, ${\bf
p}^{\text{train}}_{i}\in\mathcal{D}_{\bf p}$) is needed to train the model
$f(\cdot)$ (optimize the parameters of this function) in order to reduce the
loss function $\mathcal{L}(f({\bf y}^{\text{train}}),{\bf p}^{\text{train}})$.
Take the mean squared error (MSE) cost function for example, we can have

$\mathcal{L}(f(\mathcal{D}_{\bf y}),\mathcal{D}_{\bf
p})=\sum_{i}^{|\mathcal{D}|}\|f({\bf p}^{\text{train}}_{i})-{\bf
p}^{\text{train}}_{i}\|^{2}.$ (82)

After the training, the model $f(\cdot)$ can be used to output end-to-end
location information by taking the raw observation data as the input.

Fingerprinting (or pattern matching) is an approach that utilizes a database
of fingerprints to find the best position match for a particular signal
measurement [6]. The csi and RSS could be used as the entries to construct the
database. While RSS suffers from limited accuracy and CSI requires high
computational power, spatial beam SNRs are adopted as a mid-grained
intermediate channel measurement [176]. For the retrieval process, deep
learning methods such as deep neural networks (DNN) and convolutional neural
networks (CNN) are valuable tools to obtain effective models for location
estimations [176, 167].

In this category, all the information is maintained and will provide accurate
results if the data in the implementation scenario matches the training data
set. However, the drawbacks are the data collection in the training phase, and
the scalability issue as one model only works for a specific scenario.

####  IV-D2 Multi-stage Localization

Similar to the geometry-based localization, the direct localization task can
be decomposed into several sub-tasks (e.g., signal pre-processing,
intermediate geometry parameters estimation, and localization). Each sub-task
can be solved using learning-based methods with a much smaller training
dataset. In the first stage, learning-based methods can be used to reduce the
effect of the hwi such as antenna spacing error [197], iqi [198], mc [199],
and pan [200]. The distorted signal due to the impairments can be recovered or
compensated during the data pre-processing stage. In channel parameter
estimation, learning-based methods have been implemented to estimate the angle
[201, 202] and delay [203, 204]. In terms of the localization stage, machine
learning has shown the potential to improve localization performance via NLOS
identification [205], and global fusion profile [206].

Considering the high dimension of the system parameters and the complexity of
the environment, the training of an end-to-end localization model may not be
practical. The design of learning-based algorithms for sub-tasks reduces the
training cost. These trained models are also flexible to adapt to different
scenarios (e.g., a trained model in MIMO systems may not fit a MISO system,
but range or angle estimation are more general). Nevertheless, the propagation
of the error caused at each stage needs to be considered while adopting
learning-based methods.

####  IV-D3 Practical ML algorithms

Machine learning algorithms are usually classified into supervised learning
(used for solving classification and regression problems) and unsupervised
learning (used for data clustering) [38]. Other approaches, including semi-
supervised learning, reinforcement learning, transfer learning, and federated
learning, are designed to solve the issues faced by the supervised and
unsupervised learning algorithms, which will be discussed as follows.

  * •

Supervised Learning: Traditional machine learning algorithms, such as random
forest, support vector machine, and recent popular deep learning, belong to
supervised learning. Due to the wide application scenarios in many fields, a
lot of toolboxes such as Tensorflow [207] and PyTorch [208] make the
implementation simple for the researchers. However, two challenges exist. One
is the data collection of the offline phase, where sufficient real data are
not easy to obtain, and synthesized data may not be accurate. Another is the
selection of model parameters; for example, the number of layers and neurons,
as well as model structures, make deep learning often an art rather than a
science.

  * •

Unsupervised learning: Without the need for well-labeled datasets,
unsupervised learning is widely used for clustering, and dimension reduction
(or feature extraction). A novel framework called channel charting is proposed
in [170], which learns CSI in a fully unsupervised manner and can map a high-
dimensional point set (the channel features) into a low-dimensional point set
(the channel chart). However, this category can only perform data pre-
processing, and location information cannot be obtained.

  * •

Other approaches: By combining the two above-mentioned categories, semi-
supervised learning can train the model with partially labeled data (e.g.,
$|\mathcal{D}_{\bf p}|<<|\mathcal{D}_{\bf y}|$ in the training dataset
$\mathcal{D}$). For the scenario without a clear objective function (only a
reward is known after taking action) reinforcement learning is preferred,
which is suitable for training without a clear cost function using online
data. Transfer learning is able to take advantage of the existing model to
reduce the training time, and federated learning works in a distributed manner
and hence protects user privacy. More details of ML-based localization can be
found in [209, 38, 210, 39, 42, 40].

In summary, despite the channel at THz frequencies being more deterministic
than at lower frequencies, which suits geometry-based methods well, we argue
that learning-based methods still have advantages in two aspects. Firstly,
processing large volumes of data (due to a wide bandwidth) necessitates faster
algorithms for localization, and ML algorithms are efficient at feature
extraction and hence speed up the processing. Secondly, hardware impairments
are severe in high-frequency systems, and the mismatch between the theoretical
and actual system models affects the performance, which needs to be mitigated
by learning-based methods with onsite data.

###  IV-E Tracking and SLAM Algorithms

While this paper is focused on the snapshot localization problem, this is
generally part of a wider tracking [211] or SLAM [212] routine, which the UE
performs sequentially, based on its own mobility model and periodic
measurements. For completeness, we briefly describe their operation in the
following sections.

####  IV-E1 Tracking

In mobile applications, initial access is only needed for the first several
frames or when the communication link is lost. Once initial access is
completed, the UE goes into tracking mode. Mathematically, the model for a
tracking problem can be expressed as

$\displaystyle{\bf s}_{\mathrm{U},t}$ $\displaystyle\sim p({\bf
s}_{\mathrm{U},t}|{\bf s}_{\mathrm{U},t-1})$ (83)
$\displaystyle\hat{\hbox{\boldmath$\gamma$}}_{t}$ $\displaystyle\sim
p(\hbox{\boldmath$\gamma$}_{t}|{\bf s}_{\mathrm{U},t}),$ (84)

where ${\bf s}_{\mathrm{U},t}$ is the UE state vector at time $t$, which
depends on the previous state ${\bf s}_{\mathrm{U},t-1}$ via a stochastic
mobility model, and $\hat{\hbox{\boldmath$\gamma$}}_{t}$ is the measurement
vector at time $t$ as defined in Sec. IV-A, which depends on the UE state at
time $t$. The observation contains the estimated angles and delays related to
LOS and RIS paths. In addition, a prior $p({\bf s}_{\mathrm{U},0})$ is assumed
to be given.

Solving the tracking problem refers to determining the posterior of the state
$p({\bf s}_{\mathrm{U},t}|\hat{\hbox{\boldmath$\gamma$}}_{1:t})$ given all the
collected measurements up till the current time. Several filters exist to
solve the tracking problem, though they are all approximate (unless the
mobility and measurement models are linear and Gaussian). These filters
include:

  * •

_Filters based on the kf:_ the KF provides a recursive solution for linear
filtering problems. For nonlinear problems, an ekf can be used, which
approximates the state distribution using a Gaussian random variable and
propagates analytically through the first-order linearization [213]. Other
extensions of the KF families include the ukf and the ckf, where the former
addresses the approximation issues of the EKF [213], and the ckf suits high-
dimensional state estimation [214]. The kf-based filters generally have low
complexity but are unable to cope with highly nonlinear models or multi-modal
distributions.

  * •

_Filters based on the pf:_ the PF is another widely-used filter that exploits
the representation of an arbitrary pdf by a set of particles [215]. PFs have
the advantage of dealing with highly nonlinear and non-Gaussian models, but at
the cost of high computational complexity, as the number of particles grows
exponentially in the state dimensionality.

An added advantage of tracking is that the transmitted signals and the
precoders, combiners, and RIS coefficients can be optimized to account for the
a priori information on the UE state. This topic will be covered in more
detail in Section V.

####  IV-E2 SLAM

While not considered in this work, the measurements
$\hat{\hbox{\boldmath$\gamma$}}_{t}$ at each time step $t$ also provide
information about the location of the scatter points (_landmarks_ in SLAM
parlance), shown in Fig. 3. In turn, this knowledge can improve estimating the
UE state, which is the main idea behind SLAM. SLAM has been widely applied in
robotics [216, 217] and autonomous driving [2], where an agent locates itself
and constructs the unknown map at the same time [173]. With the wide bandwidth
and MIMO structure implemented in 5G/6G systems, this topic draws the
attention of the communication community with several mmWave systems proposed.
The SLAM systems can broadly be classified into two categories,
infrastructure-based [173] and non-infrastructure-based systems [218]. In the
infrastructure-based systems, the positions of UE and scatters are estimated
from the signals transmitted from the BS, as mentioned in (64). In a situation
where no BSs are deployed, the UE sends a sequence of signals and then
processes the received signal reflected from the surrounding environments
[218]. The SLAM problem is inherently challenging since the data association
between the landmarks and measurements is unknown (i.e., which landmark
generated which delay or angle measurements).

In our THz context, to infer the locations of scatter points and execute SLAM,
the following modifications are needed. First of all, the channel model ${\bf
H}_{N}^{{\scriptscriptstyle{({l})}}}$ from (25) should be expressed as a
function of the scatter location, say ${\bf
p}_{\mathrm{N}}^{{\scriptscriptstyle{({l})}}}$ [78, 219, 220]. Secondly, the
local and global data associations between the angles and delays in
$\hat{\hbox{\boldmath$\gamma$}}_{t}$ and the landmark locations ${\bf
p}_{\mathrm{N}}^{{\scriptscriptstyle{({l})}}}$ should be enumerated and their
likelihoods should be calculated. This calculation should account for the
hidden UE state as well as the possibility of false alarms (spurious
measurements) and missed detections (landmarks without measurements at the
current time). Finally, the joint posterior of the UE state and the landmark
state should be computed in an iterative manner, with well-defined prediction
and correction steps, accounting for all or a subset of most likely data
associations. Common methods in this field are FastSLAM [221], GraphSLAM
[222], belief propagation SLAM [223], and random finite set theory-based SLAM
[220, 224]. These mainly differ in how the data associations are computed, how
the prediction and correction steps are performed, how the UE state is
represented (e.g., particles or a parametric density), and how the map is
represented (e.g., parametric, grid maps, feature maps, topological maps,
semantic maps, appearance maps, and hybrid maps [217]).

In summary, tracking in THz systems is challenging due to the narrow
beamwidths resulting from beamforming with large array sizes. Adaptive
beamwidth design could thus be adopted for different tracking scenarios (e.g.,
high speed, confident prior information). Nevertheless, the accuracy of both
tracking and SLAM improves with narrow beamwidths resulting in a high angular
resolution. Furthermore, with dense network deployments and wide bandwidths,
an unparalleled SLAM performance can be achieved in THz systems.

###  IV-F Summary

In this section, we formulate the localization problems, describe the CRB, and
detail some localization techniques. In particular:

  * •

We describe the localization parameters as a state vector and a measurement
vector. Different vectors can be defined based on application scenarios and
algorithm selections.

  * •

We introduce the CRB for position and orientation estimation based on the
state and measurement vectors.

  * •

We formulate geometry-based methods, namely, direct localization and multi-
stage localization, and discuss several channel estimation and localization
algorithms.

  * •

We discuss several THz localization and sensing extensions, namely, learning-
based localization, tracking, and SLAM. These techniques can deal with
different localization scenarios and improve localization performance.

In the next section, we formulate system design and optimization problems and
discuss the relationships between the variables to be optimized and the
affected objectives.

##  V Localization System Design and Optimization

System design and optimization are essential for determining the fundamental
limits of attainable localization performance. We start by presenting the
optimization problem formulation based on the desired system objectives. Then,
we discuss the high-level design considerations of the system. Afterward, we
detail two groups of system design problems: offline optimization and online
optimization. Finally, we conclude this section with simulations and system
evaluation.

###  V-A Optimization Problem Statement

####  V-A1 Motivation

Optimization is essential in communication systems to meet different
objectives of signal-to-interference-plus-noise ratio (SINR), energy
efficiency, maximum throughput, Etc. For localization purposes, the PEB and
OEB defined in IV-B2 are used when accuracy is chosen as an objective in
system design. Although this criterion is valid only when the estimator is
efficient, it is still a tractable and effective tool for analyzing
performance in the asymptotic region. Other objectives described in Sec. II-C
are also important in certain scenarios. However, the definition of an
objective is not always straightforward and is not unique; objectives need to
be defined based on the application scenario. Due to different formulations of
objective functions, we have to make compromises, especially when optimizing
joint communication and localization systems.

In low-frequency localization systems, prs are broadcast by the BSs, and the
corresponding system design is mainly offline (such as BS layout and antenna
array design). In mmWave MIMO systems, localization performance benefits from
the beamforming gain. However, beamforming requires the location knowledge of
receivers. Hence, online design of precoding and combining matrices, as well
as resource allocation, are of great importance. Such knowledge of transceiver
locations is crucial in UM-MIMO THz systems with narrow beams. For AOSA-based
THz systems, the optimization of precoder/combiner is at the SA level instead
of the antenna level. Thus, in addition to the data symbols from the RFCs, the
SA beamforming angles should also be well-designed. Furthermore, the
optimization of RIS coefficients and resource allocation inside a dense
network (possibly in near-field scenarios) requires effective algorithms. In
summary, offline and online optimizations are equally important in future
communication systems. We next formulate the optimization problem and discuss
the effect of different variables on system objectives.

####  V-A2 Problem Formulation

Different localization scenarios have different performance requirements (or
objectives, such as accuracy, coverage, and so on, as defined in Sec. II-C).
In most cases, these objectives are related, and tradeoffs have to be made.
For example, increased coverage may increase Latency, and increased update
rate may affect accuracy. A system may seek one or several objectives to be
optimized while meeting other practical constraints.

A general optimization problem formulation of THz localization systems,
consisting of an objective function ${\bf f}(\mathcal{V})$ and a constraint
function ${\bf g}(\mathcal{V})$, can be expressed as

$\begin{split}\mathcal{V}=&\operatorname*{arg\,min}_{\mathcal{V}}{{\bf
f}(\mathcal{V})},\\\ \mathrm{s.t.}\ \ &{\bf g}(\mathcal{V})\leq 0.\end{split}$
(85)

Here, $\mathcal{V}$ is a set of variables that could be chosen from the number
of devices $L_{\mathrm{Q}}$, positions ${\bf p}_{\mathrm{Q}}$, antennas per
array $N_{\mathrm{Q}}$ and SA $\mathring{N}_{\mathrm{Q}}$, SA spacing
$\Delta$, AE spacing $\mathring{\Delta}$, beamforming angles
$\mathring{\boldsymbol{\varphi}}$, RIS coefficient $\Omega$, number of
transmissions $\mathcal{G}$, etc. Rather than choosing a single objective or
constraint, multiple objective optimization (MOO) problems can also be
considered, implying that ${\bf f}(\mathcal{V})$ and ${\bf g}(\mathcal{V})$
could comprise a set of objectives and constraints.

The objective functions depend on the system requirements for localization
performance discussed in Sec. II-C, while the constraints reflect the types of
variables (e.g., discrete variables or continuous variables) and the search
space (e.g., positions within a specific area) of the variables to be
optimized. In different scenarios, a parameter could either be an objective or
a constraint. For example, localization accuracy can be used as an objective
to be optimized, but it could also be accounted for as a constraint to be met
(e.g., the minimum required accuracy) alongside other objectives to be
optimized (e.g., energy efficiency).

To achieve the system objectives while sustaining the constraints, we classify
the system design and optimization into offline and online. The corresponding
variables and the effect on the system objectives are summarized in Table
VIII. Before discussing these two categories, we detail design considerations.

TABLE VIII: Objectives of Different Design/Optimization Considerations (and the Corresponding Variables of Optimization) |  |
---|---|---
|  Considerations |  Variables |   Accuracy |   Coverage |   IA Delay |   Update Rate |   Stability |   Scalability |   Mobility |   Complexity
|  |
 Offline |  #./Pos. of BS/RIS  |  $L_{\mathrm{Q}}$, ${{\bf p}_{\mathrm{Q}}}$ |  $\\!\\!\checkmark$ |  $\\!\\!\checkmark$ |  |  |  $\\!\\!\checkmark$ |  $\\!\\!\checkmark$ |  $\\!\\!\checkmark$ |  $\\!\\!\checkmark$
Array Size  |  $N_{\mathrm{Q}}$, $\mathring{N}_{\mathrm{Q}}$ |  $\\!\\!\checkmark$ |  $\\!\\!\checkmark$ |  |  |  |  $\\!\\!\checkmark$ |  |  $\\!\\!\checkmark$
Directionality  | ${G_{0}}$ |  $\\!\\!\checkmark$ |  $\\!\\!\checkmark$ |  $\\!\\!\checkmark$ |  |  |  |  |
Quantization  | ${\mathcal{Q}}$ |  $\\!\\!\checkmark$ |  |  |  |  |  |  |  $\\!\\!\checkmark$
Codebook  | ${\mathcal{C}}$ |  |  $\\!\\!\checkmark$ |  $\\!\\!\checkmark$ |  |  |  $\\!\\!\checkmark$ |  |
 Online |  Time/ #. of Meas.  | ${T,G}$ |  $\\!\\!\checkmark$ |  |  |  $\\!\\!\checkmark$ |  $\\!\\!\checkmark$ |  $\\!\\!\checkmark$ |  $\\!\\!\checkmark$ |
Bandwidth  | ${B,K}$ |  $\\!\\!\checkmark$ |  $\\!\\!\checkmark$ |  |  $\\!\\!\checkmark$ |  |  $\\!\\!\checkmark$ |  $\\!\\!\checkmark$ |  $\\!\\!\checkmark$
Power  | ${P,{\bf s}}$ |  $\\!\\!\checkmark$ |  $\\!\\!\checkmark$ |  |  |  $\\!\\!\checkmark$ |  $\\!\\!\checkmark$ |  |
Beamforming Angles  | ${\mathring{\boldsymbol{\boldsymbol{\varphi}}}}$ |  $\\!\\!\checkmark$ |  $\\!\\!\checkmark$ |  |  |  $\\!\\!\checkmark$ |  $\\!\\!\checkmark$ |  $\\!\\!\checkmark$ |
RIS Coefficients  | ${\Omega}$ |  $\\!\\!\checkmark$ |  $\\!\\!\checkmark$ |  $\\!\\!\checkmark$ |  |  $\\!\\!\checkmark$ |  $\\!\\!\checkmark$ |  $\\!\\!\checkmark$ |
|

###  V-B Design Considerations

When designing a localization system, we consider aspects such as the
selection of network structures, the cooperative strategy, and algorithms
determined by the application scenarios.

####  V-B1 Network Topology

In previous sections, we described a communication system consisting of an LOS
channel, an RIS channel, and multiple NLOS channels. However, multiple
BSs/RISs/UEs (e.g., $L_{\mathrm{B}}/L_{\mathrm{R}}/L_{\mathrm{U}}$) should be
involved as densification is one of the main features in future communication
systems. THz communication system topologies can be classified into three
types: centralized, distributed, and clustered. The clustered architecture is
mainly seen in nanonetwork environments where short communication distances
and high energy efficiency are favored [26]. For macro scenarios where the
communication distance is large, centralized and distributed structures are
usually used. The centralized structures can yield better overall performance
with proper scheduling, while the distributed ones protect user privacy.

####  V-B2 Network Structure

For macro scenarios, three structures can be considered to improve system
performance:

  * •

Heterogeneous network: Future networks are likely to be heterogeneous where
different wireless (and wired) protocols coexist [225]. Such a multi-band
network can sufficiently alleviate the deafness issue and reduce the initial
access delay.

  * •

RIS-assisted network: Passive RISs can reshape the channel and increase
coverage. The footprints of RISs operating at THz frequencies are expected to
be small due to short wavelengths, which can provide extra flexibility in
deployment.

  * •

Cell-free network: UM-MIMO systems provide beamforming gains and energy
efficiency [226]. However, performance is limited by the THz channel due to
its lower-rank and poorer, sparser structure of multipath propagation [31]. By
adopting a distributed MIMO system with multiple BSs (probably with a smaller
array size) without cell boundaries, UE could have a high coverage probability
[227], and the geometrical diversity of the BSs can also improve the
localization performance.

Such infrastructure enablers can improve the localization performance,
assuming proper protocols, useful network management overheads, and efficient
real-time processing.

####  V-B3 Cooperative Strategies

Although frequent communications between the UEs cause overheads and energy
consumption, cooperative localization improves the localization accuracy and
the localization coverage [8]. The corresponding performance metrics should
thus be defined for a reasonable tradeoff. In addition to cooperation between
BSs, RISs, and UEs, other types of cooperation, including UAV-assisted
localization [228] and data fusion from other types of sensors such as IMUs
[229] and cameras [230], are also important.

####  V-B4 Hardware Selection

In order to achieve a good tradeoff between hardware cost and system
performance, hardware selection is involved in offline system design. Hardware
selection considers the directionality of antennas, the quantization of phase-
shifters (or RIS coefficients), and the effect of hardware imperfection. In
[231], the effect of the antenna model, blockage, absorption, density on the
interference, and SNR are analyzed for THz systems. This analysis provides
insight into device density and antenna directionality selection for THz
network design. In general, omnidirectional antennas are used at the service
discovery phase, and directional antennas are used for message transmissions
and localization [232]. In addition, the amplitude and phase control of RISs
are not continuous in practice, where a quantized model should be considered
in system design [233].

####  V-B5 Signal Design

Implementing single-carrier versus multi-carrier modulation in THz systems is
still not conclusive. Wideband single-carrier modulation has low complexity
and could be used in scenarios with frequency-flat channels (e.g., limited
multipath components). However, due to the frequency-dependent molecular
absorption loss and multipath (mainly in indoor environments), multi-carrier
systems are still preferred at the cost of high complexity and low power
efficiency. OFDM can serve as a direct off-the-shelf solution, and dftsofdm
[75, 76] can be used to reduce the PAPR effect. Other multi-carrier
modulations such as otfs modulation (suitable for highly dynamic channels)
[234], hierarchical bandwidth modulations [235] (mitigate the effect of
molecular absorption), spatial [236] and index modulations [237] (improve
spectral efficiency) are also considered for certain scenarios. THz non-
orthogonal multiple access is also being studied [238, 239]. In this work, we
want to compare THz systems and mmWave systems directly; hence, OFDM
modulation is assumed.

From a communication point of view, the selection of signal parameters, such
as carrier frequency, bandwidth, and packet length, affects the data rate or
spectrum efficiency. These parameters are also crucial for localization to
obtain specific objectives. A large bandwidth is helpful to separate paths in
the delay domain, but the increased sampling rate and data size should not
exceed the hardware limit. The packets should be long enough to capture enough
energy but short enough to meet delay constraints, especially in mobile
scenarios. The design considerations directly affect the performance of a
localization system; we evaluate some signal parameters via simulations in
Sec. VI.

###  V-C Offline Optimization

In an offline design, no knowledge of the position/orientation information of
UEs is available. However, the surrounding environment information could be
available. The offline design includes layout optimization, array design, and
codebook optimization.

####  V-C1 Layout Optimization

If the number of BSs/RISs/UEs is determined, their positions can be optimized
based on the CRB derived using a predefined codebook. Environmental
information (e.g., the geometry of the detection area and position of the
blockage) can also be used to optimize the layouts and achieve the best
localization performance. For the BSs with antenna arrays or directional
antennas, the orientation should also be optimized.

####  V-C2 Array Design

Increasing the number of antennas in an array yields higher angular resolution
and beamforming gains. However, more antennas indicate higher system
complexity and power cost. When adopting an AOSA structure, the design of SA
size is also important. A large number of AE per SA increases beamforming gain
and improves accuracy, but narrow beamwidths reduce coverage and cause
deafness issues.

####  V-C3 Codebook Optimization

IA is the procedure in which a new UE establishes a physical link with a BS to
switch from an idle mode to a connected mode [240]. We can treat the IA
procedure as localization without UE prior information. The narrow beams in
THz systems make IA challenging due to deafness (transmit-receive beams do not
point to each other) and blockage (channel drop caused by obstacles, device
movement, or rotation) [241]. Hence, effective initial access procedures and
dedicated codebook design are needed [242].

The design of codebooks depends on the search strategies, which can be broadly
classified into several categories:

  * •

Exhaustive search: The BS/UE transmits/receives data symbols by beamforming in
different directions [243].

  * •

Iterative search: Hierarchical codebooks can be designed to transmit pilots
over wider sectors at the beginning and then narrow down the beams to find the
best angular space [244, 245, 246].

  * •

Scene-aware search: If the position prior or environmental information is
available, the beams can be learned for each partitioned area to reduce IA
delay [240, 247].

For these strategies, an exhaustive search provides the best coverage and
hardware feasibility, but the discovery delay grows linearly with beamforming
gain [248, 249]. Iterative search reduces the discovery delay at the expense
of limited coverage. Considering the potential of THz SLAM, we expect a scene-
aware search to be used.

####  V-C4 Offline Design Example

Consider an RIS placement problem in which we want to minimize the
localization coverage area with UE’s PEB greater than an error threshold
$\epsilon$ (e.g., 0.1 u), given BS locations and orientations. For each
possible RIS placement (position ${\bf p}_{\mathrm{R}}$ and orientation ${\bf
o}_{\mathrm{R}}$) and each possible UE location ${\bf p}_{\mathrm{U}}$
(assuming an omnidirectional antenna), there exist a FIM ${\bf J}({\bf
p}_{\mathrm{U}},\hbox{\boldmath$\eta$}|{\bf p}_{\mathrm{R}},{\bf
o}_{\mathrm{R}})$ and corresponding $\mathrm{PEB}({\bf
p}_{\mathrm{U}},\hbox{\boldmath$\eta$}|{\bf p}_{\mathrm{R}},{\bf
o}_{\mathrm{R}})$ that can be obtained from (70). Here, $\eta$ contains
nuisance parameters (e.g., channel gains, clock biases), which are replaced
with nominal values (e.g., $\hbox{\boldmath$\eta$}({\bf p}_{\mathrm{U}},{\bf
p}_{\mathrm{R}},{\bf o}_{\mathrm{R}})$ obtained from channel models). Similar
assumptions need to be made for other variables such as precoders, combiners,
and RIS coefficients. We can then formulate the RIS placement problem as

$\begin{split}\mathrm{{maximize}\ \ }&|\mathcal{R}({\bf p}_{\mathrm{R}},{\bf
o}_{\mathrm{R}})|\\\ \mathrm{s.t.\ \ }&{\bf
p}_{\mathrm{R}}\in\mathbb{R}^{3},{\bf
o}_{\mathrm{R}}\in\mathrm{SO(3)},\end{split}$ (86)

where $\mathcal{R}({\bf p}_{\mathrm{R}},{\bf o}_{\mathrm{R}})=\\{{\bf
p}_{\mathrm{U}}\\!\in\\!\mathbb{R}^{3}|\mathrm{PEB}({\bf
p}_{\mathrm{U}},\hbox{\boldmath$\eta$}|{\bf p}_{\mathrm{R}},{\bf
o}_{\mathrm{R}})\leq\epsilon\\}$ is the localization coverage area, and
$|\mathcal{R}({\bf p}_{\mathrm{R}},{\bf o}_{\mathrm{R}})|$ denotes the volume
of the coverage area (e.g., a set of discrete UE positions). Such a problem is
generally non-convex and grid-search techniques can be applied.

Offline optimization in THz systems differs from low-frequency systems in
several aspects. Firstly, the precoders/combiners and RIS coefficients need to
be optimized first before layout optimization. Furthermore, optimization with
multiple BSs/RISs is highly non-convex, and it is thus hard to obtain globally
optimal solutions. Heuristic algorithms could be alternative time-saving
options to get satisfactory sub-optimal results. Note that the optimization
problem formulated in (86) is a simplified case in which the antenna at the UE
is assumed to be omnidirectional. In general, however, the orientation of the
UE needs to be considered when optimizing the layout.

###  V-D Online Optimization

Unlike the offline design, where no prior information is available, online
optimization is performed with known UE position/orientation information (or
with prior information in the tracking scenario). Online optimization can be
formulated as minimizing the worst-case localization performance (e.g., PEB)
[250]. We consider online optimization in three aspects: resource allocation,
active beamforming optimization, and RIS coefficient optimization.

####  V-D1 Resource Allocation

Resource allocation is an essential phase in the operation of a communication
network serving multiple UEs or conducting multiple tasks. This subsection
focuses on three types of resources: time, bandwidth, and power (the space
resource is discussed in Sec. V-D2 and Sec. V-D3.

  * •

Time Resource: For single-user communication, a tradeoff between transmission
time and overhead needs to be made. Intuitively, more
transmissions/measurements yield better localization accuracy at the cost of
increasing the latency and overhead. The allocation strategy should also
consider UE speed and channel coherence. The allocation of time slots (or
transmissions) for multiple users aims at meeting the positioning quality-of-
service (QoS) within the served area.

  * •

Frequency Resource: Due to the variation of vapor absorption coefficients at
different frequencies, the THz spectrum is divided into a set of distance-
varying spectral windows [130]. The size of the effective bandwidth window
gets smaller with increasing link distance. In THz communications, the
effective bandwidth is expected to support hierarchical bandwidth modulation
[235], optimizing device density to maximize capacity [251]. From a
localization point of view, suitable subbands and subcarriers need to be
selected and assigned to the UEs at different distances.

  * •

Power Allocation: For most applications, localization accuracy is a constraint
rather than an objective to be optimized. For example, an accuracy of, say
$1\,\mathrm{cm}$ is sufficient for a mobile user to know its location inside
an office building. Hence, there is no reason to increase the transmission
power to achieve an accuracy of $1\,\mathrm{mm}$. With proper power
allocation, different performance requirements of different UEs can be met
with minimal resource utilization.

Resource allocation is usually expressed as constraints for active and passive
beamforming optimizations, as we describe shortly.

####  V-D2 BS/UE Beamforming Optimization (Active Beamforming)

With prior location information, setting the beamforming angles to point to
the receiver increases the power of the received signal, which is beneficial
for communication. However, this SNR increment does not guarantee an
improvement in localization performance. More practical solutions utilize the
CRB as an indicator. Given the uncertainty range of the target directions, the
optimal precoders for tracking the DOA and DOD are derived in [252] by solving
a formulated convex optimization problem. With multiple measurements,
iterative location estimation and beamforming optimization can also be
performed [253].

By implementing AOSA structures and directional antennas, space resource
allocation reduces the assignment of beams (SAs). The optimization of the
analog beamforming angles directly affects the accuracy and coverage of the
system. For scenarios with multiple UEs, the SAs need to be assigned wisely to
different UEs, completing both the localization and communication tasks. The
SA selection is especially important in localization, where accuracy depends
on the array layout. For a communication network with multiple BSs, joint
beamforming optimization is also needed to achieve a better overall system
performance.

####  V-D3 RIS Coefficients Optimization (Passive Beamforming)

The optimization of RIS coefficients is as crucial as active beamforming at
the BS/UE arrays to enhance signal gain in RIS-assisted systems. When the UE
position is unknown, multiple transmissions with random symbols or beamforming
angles can be used for localization purposes. With prior information of the UE
position/orientation, beamforming angles at the BS/RIS/UE can be jointly
optimized.

The coefficients of RIS elements can be optimized to serve communication or
localization purposes [250]. The elements in an RIS can be optimized to
maximize the SNR at the receiver for a higher data rate. However, a high SNR
does not indicate a lower CRB. By analyzing the FIM, [250] optimizes the RIS
elements in a 2D SISO localization system to reduce the PEB. However, the
optimization algorithms for 3D MIMO systems are not yet available.

####  V-D4 Online Optimization Example

We again consider the case with RIS, where we aim to optimize the RIS phase
profiles
$\hbox{\boldmath$\Omega$}_{1},\ldots,\hbox{\boldmath$\Omega$}_{\mathcal{G}}$
for different transmissions, given a certain precoder at the UE and a combiner
at the BS. We assume that the UE location is known to be in some region with
${\bf p}_{\mathrm{U}}\in\mathcal{R}^{*}\subset\mathbb{R}^{3}$. By introducing
$\hbox{\boldmath$\omega$}_{g}\\!=\\!\mathrm{diag}(\hbox{\boldmath$\Omega$}_{g})\in\mathbb{C}^{N_{\mathrm{R}}\times
N_{\mathrm{R}}}$ and using $\eta$ to indicate the estimated values of other
nuisance parameters, we can compute the FIM ${\bf J}(\hat{\bf
p}_{\mathrm{U}},\hbox{\boldmath$\eta$}|\hbox{\boldmath$\omega$}_{1},\ldots,\hbox{\boldmath$\omega$}_{\mathcal{G}})$
and its corresponding PEB. An online optimization can then be formulated as

$\begin{split}\mathrm{minimize\ \ }&\varepsilon\\\ \mathrm{s.t.\ \
}&\mathrm{PEB}({\bf
p}_{\mathrm{U}},\hbox{\boldmath$\eta$}|\hbox{\boldmath$\omega$}_{1},\ldots,\hbox{\boldmath$\omega$}_{\mathcal{G}})\leq\varepsilon\\\
&{\bf p}_{\mathrm{U}}\in\mathcal{R}^{*}\\\ &|\omega_{r,g}|=1,\forall
r,g.\end{split}$ (87)

This problem can be solved by first removing the unit norm constraint so that
a convex problem can be obtained. Then the solution needs to be projected onto
the appropriate manifold.

In THz systems with wide bandwidths, more time-frequency blocks will be
available. These resources need to be allocated wisely to serve a large number
of users with different communication and localization performance
requirements. In addition, joint beamforming optimization for BS/RIS/UE is
crucial since THz systems are expected to rely heavily on RISs due to severe
blockage. Furthermore, the optimization at the SA level in an AOSA structure
is different from the optimization at the antenna level in traditional MIMO
systems. Consequently, novel online optimization algorithms are called for in
THz systems to assist in accurate tracking.

###  V-E Summary

In this section, we formulate the optimization problems and discuss several
aspects of localization system design and optimization, which can be
summarized as follows:

  * •

We start by motivating system optimization and formulating the optimization
problem with a set of variables related to different objective functions.

  * •

We discuss high-level system considerations such as network topology, network
structure, cooperative strategy, hardware selection, and signal design, which
is dependent on the application scenario.

  * •

We divide system design into offline design and online optimization, and
discuss the challenges for THz systems compared with low-frequency systems.

The next section provides simulations to show the effect of parameters on the
system CRB.

##  VI Simulation and Evaluation

In this section, we provide several simulations to evaluate the effect of
system parameters on localization performance. A $0.3\,\mathrm{THz}$ sub-THz
system and a $60\,\mathrm{GHz}$ mmWave system in an uplink scenario are
considered with the default parameters listed in Table IX. These parameters
are the default setting for the rest of the simulations unless otherwise
specified. Simulations A/B/C/D (Sec. VI-A to VI-D) consider only LOS channels,
while the effects of RIS and NLOS channels are discussed later in simulations
E/F/G (Sec. VI-E to VI-G). Matlab code is available in [254].

TABLE IX: Default Simulation Parameters Parameters | Simulation Values
---|---
mmWave / THz Frequency ${f_{c}}$ |  $60\,\mathrm{GHz}$ / $0.3\,\mathrm{THz}$
Transmission Power ${P}$ | $10\,\mathrm{dBm}$
Noise PSD  | $-173.86\,\mathrm{dBm/Hz}$
Noise Figure  | $13\,\mathrm{dBm}$
Array Footprint (BS/RIS/UE)  |  $2\\!\\!\times\\!\\!2\,\mathrm{cm^{2}}$ / $10\\!\\!\times\\!\\!10\,\mathrm{cm^{2}}$ / $1\\!\\!\times\\!\\!1\,\mathrm{cm^{2}}$
AE Spacing ${\mathring{\Delta}}$ | $\lambda_{c}/2$
Bandwidth ${W}$ | $100\,\mathrm{MHz}$
Number of Transmissions $\mathcal{G}$ | 10
Synchronization Offset $B$ | 10 us (for Simulation D/E/F/G)
Number of Subcarriers $K$ | 10
Signal Wave Model  | SWM (near-field)
Localization Scenario  | 2D Position, 1D Orientation
mmWave Array Dim ${N_{\mathrm{Q}}}$ (BS/RIS/UE)  |  $4\\!\\!\times\\!\\!4$/ $20\\!\\!\times\\!\\!20$ / $2\\!\\!\times\\!\\!2$
THz Array Dim ${N_{\mathrm{Q}}}\mathring{N}_{\mathrm{Q}}$ (BS/RIS/UE)  |  $20\\!\\!\times\\!\\!20$/ $100\\!\\!\times\\!\\!100$ / $10\\!\\!\times\\!\\!10$
THz SA Dim ${\mathring{N}_{\mathrm{B}}}$ /$\mathring{N}_{\mathrm{R}}$/ ${\mathring{N}_{\mathrm{U}}}$ |  $5\\!\\!\times\\!\\!5$ / $1\\!\\!\times\\!\\!1$ / $5\\!\\!\times\\!\\!5$
Position ${{\bf p}_{\mathrm{B}}}$ / ${{\bf p}_{\mathrm{R}}}$ / ${{\bf p}_{\mathrm{U}}}$ |  $[0,0,0]^{T}$ / $[5,5,0]^{T}$ / $[10,0,0]^{T}$
Orientation ${{\bf o}_{\mathrm{B}}}$ / ${{\bf o}_{\mathrm{R}}}$ / ${{\bf o}_{\mathrm{U}}}$ |  $[0,0,0]^{T}$ / $[-\frac{\pi}{2},0,0]^{T}$ / $[\frac{5\pi}{6},0,0]^{T}$

###  VI-A A Comparison between mmWave and THz Systems

We first compare the PEB and OEB between two systems with different array
configurations. To make a fair comparison, we fix variables such as
transmission power, time, and maximum footprint. A fully connected antenna
array is adopted in the mmWave system, while an AOSA structure is used for the
THz system. All the RFCs send different random data symbols with normalized
energy.

In this comparison, the system is assumed to be synchronized 111111If only an
LOS channel is considered in a far-field model, synchronization between the BS
and the UE is needed for delay estimation. However, if more paths are
available, e.g., extra LOS paths provided by other BSs, RIS, or NLOS paths,
synchronization is not a requirement. One typical example is TDOA-based
localization., while for simulations D/E/F/G a synchronization offset is
assumed. We also assume that no prior information of the UE is available, and
hence the beamforming angles at the SA (AOSA systems) are set randomly as
$\tilde{\phi},\tilde{\theta}\in(-90^{\circ},90^{\circ})$ for different
transmissions. We use CRB (PEB/OEB) to evaluate the fundamental limit of the
localization systems as shown in Fig. 5. This figure illustrates the potential
of THz localization (lower PEB and OEB), where $\sim\\!\\!5$ ($\sim\\!\\!20$)
times better positioning performance without (with) prior information is
expected to be achieved with the same power and footprint compared with mmWave
systems. In other words, to achieve the same performance as that of mmWave
localization systems, less resources (e.g., power, footprint) are needed.
However, to fully exploit the potential of THz localization systems, multiple
transmissions and prior knowledge are required to solve the deafness issue. We
evaluate the effect of transmission numbers on the CRB in the next subsection.

(a) PEB

(b) OEB

(c) Illustration of different arrays ($N_{\mathrm{B}}=4\times 4$)

Figure 5: PEB/OEB vs. the number of BS antennas/SAs. (a) PEB; (b) OEB; (c)
Illustration of different array configurations. By moving the carrier
frequency from 60 Ghz to 300 GHz, the error bound increases due to high path
losses. Furthermore, increasing the antenna spacing from $0.5\lambda$ to
$2.5\lambda$ (maintaining the same footprint as the mmWave system) improves
the performance. Adopting an AOSA structure with an SA size of $5\times 5$
introduces a beamforming gain, and the error bounds outperform the benchmark
mmWave system. The bounds can be even lower with a larger bandwidth (1 GHz
rather than 100 MHz) and prior information (e.g., setting the beamforming
angle of the BS to the direction of a UE).

###  VI-B The Effect of Transmission Numbers on CRB

We simulate the effect of transmission numbers ($\mathcal{G}$) on the PEB with
different SA dimensions using fixed total transmission energy and the number
of RFCs. Although the AOSA structure can also be adopted in mmWave systems, we
use a fully digital array with $4\times 4$ antennas for benchmarking purposes.
For THz systems, we simulate different SA dimensions with
$\mathring{N}_{\mathrm{B}}\\!=\\!\mathring{N}_{\mathrm{U}}\\!=\\!2\times 2$,
$\mathring{N}_{\mathrm{B}}\\!=\\!\mathring{N}_{\mathrm{U}}\\!=\\!5\times 5$,
and $\mathring{N}_{\mathrm{B}}\\!=\\!\mathring{N}_{\mathrm{U}}\\!=\\!10\times
10$, respectively. The results are shown in Fig. 6.

We notice that the number of transmissions has a minor effect on the benchmark
mmWave systems because of the large beamwidth generated from a small array
size. By incorporating the AOSA structures, the computational and hardware
cost can be reduced. However, multiple transmissions are needed to obtain a
lower bound, where the $\mathcal{G}$ needed for the bound to converge
increases with the SA dimension. With more transmissions, the deafness issues
can be solved, and a UE can thus be located more accurately. In the next
subsection, we discuss the effects of PWM/SWM and Syn/Asyn on the CRB of the
system.

Figure 6: PEB vs. number of transmissions with a fixed total transmission
energy. The number of transmissions has a minor effect on the benchmark mmWave
systems. However, multiple transmissions are needed for the AOSA structures
adopting analog beamformings, especially when the SA dimension is large. ($N$
is the number of SA, and $\mathring{N}$ is the number of AE per SA.)

###  VI-C The Evaluation of PWM/SWM for Different Channel Models

We have discussed several system assumptions, namely synchronized or
asynchronized systems (Remark 1), unknown or partially-known channel gains
(Remark 2), and SWM/PWM (Sec. III-E1). These assumptions affect the channel
realization, system model, and CRB. The PEB for Asyn/Unknown/PWM are not
included because the position cannot be estimated in this scenario. For
simulation purposes, we assume a single-antenna UE and evaluate the PEBs for
different channel models by changing the position of the UE on the x-axis
$x_{\mathrm{U}}$ with fixed $y_{\mathrm{U}}=z_{\mathrm{U}}=0$. The simulation
results are shown in Fig. 7.

We can see that the boundary between the far-field and near-field is at around
1 m. With the increasing distance, the PEBs of SWM and PWM models are
converging. However, the SWM is a more accurate signal model (at the expense
of high computational complexity), which is advantageous when the UE is close
to the BS. In addition, the SWM can help in synchronization (by exploiting the
curvature of arrival) and the PEBs of asynchronized and synchronized systems
converge in the near-field.

Figure 7: PEB vs. distance for different localization models assumptions. The
PEBs of the systems using different signal models (e.g., SWM and PWM) have a
mismatch when the distance from the BS is smaller than 1 m. In addition, a
synchronized system performs better, especially when the distance is large.

###  VI-D Evaluation of the Beam Split Effect

To evaluate the effect of beam split on the CRB, we assume that prior position
information is known, and hence the beamforming angle at the BS is set as
pointing to the UE. The orientation of the BS is set as $15^{\circ}$ and
$45^{\circ}$, and an asynchronized UE is located at $[2,0,0]^{T}$. The PEBs
for channel models with and without (w/o) the beam split effect are shown in
Fig. 8. Beam split affects the signal gain and is not preferred in
communications. However, this ‘split’ phenomenon may provide extra geometry
information (e.g., beams at different subcarriers are pointing to different
directions) that can lower the error bound, especially in wide bandwidth
systems.

So far, our discussions are limited to LOS channel. In the following, we
discuss the effect of RIS and NLOS channels on localization performance.

Figure 8: CRB vs. bandwidth for the systems with/without the beam split
effect. The mismatch between different models (solid vs. dashed curves) is
larger with increased beamwidth, array size (blue dot vs. green triangle), and
angles (blue dot vs. red square).

###  VI-E The Effect of RIS on CRB

We simulate the effect of RIS dimensions on the PEB of a THz system. For a
better visualization of the convergence, we use scaled positions of BS/RIS/UE
as $[0,0,0]^{T}/[0.5,0.5,0.1]^{T}/[0.5,0.4,0.05]^{T}$, and keep other
parameters as in Table IX. Assuming prior information is known, the RIS
coefficients can be optimized using the method in [70] to maximize the SNR of
the received signal. The beamforming angles of each SA at the BS (UE) are set
to the directions of either the RIS or UE (BS). For a BS with $N_{\mathrm{B}}$
SAs, the total beams assigned to RIS ($b_{\mathrm{R}}$) and UE
($b_{\mathrm{U}}$) satisfies $N_{\mathrm{B}}=b_{\mathrm{R}}+b_{\mathrm{U}}$.
Note that such directional beams and SNR-based RIS coefficient optimization is
not optimal for localization purposes; however, we use it as a benchmark for
further optimization algorithms. Other scenarios also include quantized
coefficients (1-bit and 2-bit quantization on the RIS coefficients), and the
corresponding simulation results are shown in Fig. 9.

Figure 9: CRB vs. the number of RIS elements ($N_{\mathrm{R}}$) for different
scenarios (‘QTZ’ is short for ‘quantization’). ‘$b_{\mathrm{R}}$=0’ means the
SAs at BS/UE are beamforming to UE/BS with prior information;
‘$b_{\mathrm{R}}$=16’ indicates all the $N_{B}/N_{U}$ SAs at BS/UE are
beamforming to the RIS; ‘Adaptive’ utilizes an optimal beam assignment with
grid search).

The figure shows that a large RIS with optimized coefficients improves the
performance, and a 2-bit quantization on the RIS coefficient is sufficient to
assist localization (PEB is close to the setup with continuous phases).
Without beamforming to the RIS, an increased number of RIS elements has almost
no effect on the PEB of the system. However, with the AOSA structure, the PEB
is also affected by the SA beamforming angles (e.g., assigned beams to RIS
$b_{\mathrm{R}}$) and more efforts are needed to jointly optimize active and
passive beamforming.

###  VI-F The Effect of NLOS Paths

We also evaluate the effect of NLOS signals on the CRB with different number
of reflectors ($L_{N}=0,1,2,3$) as shown in Fig. 10. Four layouts of the
landmarks (incident points) are considered, namely, $\\{l_{1}\\}$,
$\\{l_{1},l_{2}\\}$, $\\{l_{1},l_{2},l_{3}\\}$ and $\\{l_{1},l_{2},l_{4}\\}$,
where $l_{1}$-$l_{4}$ are located at $[5,-5,0]^{T}$, $[1,4,0]^{T}$,
$[9,-4,0]^{T}$ and $[5.1,-5,0]^{T}$, respectively. By changing the reflection
coefficients of all the reflectors (assumed to be equal) from 0 to 1, the
PEB/OEB of the UE and the reflector position error bound (RPEB) are shown in
Fig. 10 (b)-(d).

This figure shows that the NLOS paths are helpful if they are resolvable and
the reflection coefficient is large (which depends on the shape and the
material of an object). THz channels are expected to have fewer NLOS paths due
to the high path loss and narrow beamwidth. With fewer NLOS paths, the number
of localization parameters (e.g., the position of reflectors) and non-
resolvable paths are reduced. Fewer NLOS paths reduce the computational
complexity, but lose geometrical diversity. Hence, more transmission times are
needed to improve the localization performance and to map the whole
environment.

(a) Layout

(b) PEB

(c) OEB

(d) RPEB

Figure 10: CRB vs. NLOS coefficients. The layout of the BS, UE and possible
reflectors is shown in (a), where $l_{1}$ and $l_{4}$ are not resolvable. The
NLOS paths are harmful to the CRB when the reflection coefficients are small.
When increasing the coefficients, PEB, OEB, and RPEB get lower. We also notice
that even if the positions of the reflectors are not resolvable
($\mathcal{L}=\\{l_{1},l_{2},l_{4}\\}$), the OEB is still lower than in a
setup with a weaker NLOS path ($\mathcal{L}=\\{l_{1},l_{2},l_{3}\\}$).
However, the RPEB of $l_{1}$ in the unresolvable scenario is much higher than
in other layouts.

###  VI-G The Visualziation of PEB for Different UE Positions

We visualize the 2D PEB ($z_{\mathrm{U}}=0$) for different setups by changing
the position of UE within a $5\times 5\,\mathrm{m^{2}}$ area. The positions of
the BS/RIS/reflector are $[0,0,0]^{T}$, $[2.5,2.5,0]^{T}$ and
$[2.5,-2.5,0]^{T}$, respectively. The number of the BS/UE elements is
$N_{\mathrm{B}}=4\times 4$/$N_{\mathrm{U}}=2\times 2$, and the reflection
coefficient is set as $0.9$. The PEB is obtained with a single snapshot
($\mathcal{G}=1$), which only works for systems with multiple RFCs.

In Fig. 11 (a) and Fig. 11 (b), the transmitted symbols, beamforming angles,
and RIS coefficients are chosen randomly. The THz system generally shows a
lower PEB, and even lower PEBs appear when the UE is close to the BS or RIS.
Due to the implementation of analog beamforming in the AOSA structure, there
exist a ‘blind area’ in (b) (e.g., $y_{\mathrm{U}}=0$). In Fig. 11 (c) and
Fig. 11 (d), fixed beamforming angles are utilized with prior information.
This beam allocation strategy is obviously not optimal, and efficient
beamforming optimization algorithms are needed.

(a) mmWave (no prior)

(b) THz-1 (no prior)

(c) THz-1 (with prior)

(d) THz-2 (with prior)

Figure 11: 2D PEB visualization for different setups. (a) A conventional
$4\times 4$ MIMO system and a $20\times 20$ RIS ($5\times 5\,\mathrm{cm}$
footprint); (b) An AOSA-based THz system with $20\times 20$ antennas
($\mathring{N}_{\mathrm{B}}=\mathring{N}_{\mathrm{U}}=5\times 5$) and an
$100\times 100$ RIS (same footprint as scenario (a)); (c) The same parameters
as in (b) are used. By assuming the prior information of the UE position is
known, the coefficients of the RIS elements are chosen to maximize the SNR,
and 1/4 of the SA beams at the BS/UE are set to point to the RIS; (d) The same
setup as (c) THz-1 by changing the UE orientation from ${\bf
o}_{\mathrm{U}}=[0,0,0]^{T}$ to ${\bf o}_{\mathrm{U}}=[5\pi/6,0,0]^{T}$.

###  VI-H Summary

In this section, we perform extensive simulations to illustrate the potential
of THz systems in localization and sensing. With the incorporation of the RIS,
a better localization performance is expected. However, joint optimization of
AOSA active beamforming and RIS components is a challenging problem that
requires more investigation in the future. Next, we discuss several potential
research directions for THz localization, which can assist in algorithm and
system design and further improve localization performance.

##  VII Lessons Learned and Future Directions

Until now, we have discussed important topics of THz localization and
performed extensive simulations with some interesting results observed. In
this section, we would like to highlight the lessons learned from the
simulations in Section VI, and discuss the future directions from the aspects
of channel modeling, localization, and optimization in Sections III-V.

###  VII-A Lessons Learned

  1. 1.

Deal with SWM and PWM wisely in channel modeling and performance analysis. SWM
requires high computational complexity since the phase change cannot be
described using a simple steering vector, and there could be amplitude
variations across the array. As a result, PWM is usually preferred, with
possible performance loss in the near-field. From the CRB analysis point of
view, the SWM and PWM are also different. In the far-field, the local AOA and
AOD can be estimated directly, whereas the UE state is integrated into the
channel model as shown in (62) and (65).

  2. 2.

A similar concept of AOSA can be used in the RIS channel realization. A large
number of RIS elements is needed to combat the high path loss of the RIS
channel. This is even more challenging in high-frequency signals (e.g.,
$200\times 200$ RIS elements can be fitted into a $10\times
10\,\mathrm{cm^{2}}$ area in a $0.3\,\mathrm{THz}$ system). With segmented
sub-RIS and equivalent array response, the complexity can be reduced.

  3. 3.

Be aware of the model mismatch. We have seen the CRB mismatch between the SWM
and the PWM models. Since PWM is an approximation, the model is inaccurate in
the near field and should be avoided if possible. In addition to the channel
model mismatch, the mismatch caused by the mobility of the UE and hwi should
also be considered. The mcrb could be used as a tool to analyze such
mismatches.

  4. 4.

Tradeoff between coverage and beamforming gain needs to be taken care of in
system design. When designing a system, the directionality of the antenna and
the size of SA should be considered to achieve a high beamforming gain.
However, the gain in SNR will affect the coverage of the system, and severe
misalignment will occur. These two aspects need to be well-designed depending
on the application scenarios.

  5. 5.

Performance (CRB) analysis is an important step for algorithm evaluation and
system optimization. However, the lower bound may not be reached in some
scenarios, for example, low SNR scenarios and the existence of multiple non-
resolvable paths. In addition to the CRB, other types of bound such as CCRB,
and MCRB are also important in different scenarios.

###  VII-B Future Directions

By moving from mmWave to THz, a better localization performance is expected.
However, new issues and challenges need to be rethought to benefit from the
system in this band. First, new KPIs may need to be defined for specific
applications (e.g., quality of service rather than localization error). Also,
position integrity121212Positioning integrity: is a measure of the trust in
the accuracy of the position-related data provided by the positioning system
and the ability to provide timely and valid warnings to the location service
client when the positioning system does not fulfill the condition for the
intended operation [255]. and availability131313Availability: is defined as
the fraction of the time that the estimated localization error is below the
alert limit [83]. may become more important in localization, especially for
the scenarios that need highly reliable position information. In addition, we
need to have methods that are scalable, given a large number of antennas/RIS
elements and a large volume of data. Regarding the RIS, this new enabler
brings topics such as the synchronization to other network elements,
information-sharing between operators, and the roles at different frequencies
(should RIS operate in the same way at mmWave-band and THz-band or not). More
problems will need to be tackled when we are moving to a higher frequency.

The research on THz localization is still at the early stage, with many
directions to be explored. We list a few directions from the aspects of the
channel model (1-2), localization performance analysis and algorithm design
(3-6), and system optimization (7-9):

  1. 1.

Stochastic model analysis: We have formulated a deterministic channel model in
this work. In realistic scenarios, however, the AOAs and amplitudes of
scattered signals are stochastic. The effect of randomized NLOS signals needs
to be modeled, and the effect of scatters on the localization performance can
be evaluated. The stochastic model is also helpful for object/reflector
detection and classification in sensing tasks.

  2. 2.

Accurate channel modeling: Channel model is the foundation of geometry-based
localization. Currently, we use an extrapolation of mmWave models by
introducing features in high-frequency systems such as atmosphere attenuation,
SWM, wideband effect and AOSA structure. However, the effects of HWIs and
other THz-specific aspects may not be captured in the channel model (including
the LOS, RIS, and NLOS channel models). These model mismatches degrade the
localization performance, and thus, a more accurate channel model is
important.

  3. 3.

BS/RIS calibration. In most of the localization tasks, we are interested in
the position and orientation of UEs by assuming the known anchor information
(e.g., BS/RIS position and orientation). For the scenarios with more than one
anchor, there could be calibration errors in the rest of the anchors compared
to the reference anchor. In this case, jointly UE localization and BS/RIS
calibration would be of great interest.

  4. 4.

Doppler estimation: In addition to position and orientation, the Doppler
effect of the UE is not discussed in this work, which is crucial for mobile
scenarios. This additional type of channel parameter can contribute to the
tracking and SLAM tasks. In addition, by introducing the Doppler effect,
localization can be done within a longer integration time.

  5. 5.

Cooperative localization. Coverage is one of the challenging issues in high-
frequency communication and localization. As a result, D2D communication and
cooperative localization can help even if there is an outage between the UE
and the BS. As for SLAM, the collaboration between UEs can complete the
mapping tasks within a shorter period of time.

  6. 6.

Advanced performance analysis tools: We have discussed various types of CRBs
(e.g., CRB, CCRB) for position and orientation estimation. Nevertheless, other
bounds should be studied and enter more widespread use to account for phase
ambiguities (e.g., in carrier phase-based localization), low-SNR operation,
model mismatch, and with prior information. These advanced performance
analysis tools will enable the algorithm development and system design for
THz-band localization.

  7. 7.

Scene-aware localization: From the SLAM algorithm, surrounding map information
could be available. In addition to the map, the probability density functions
of past access locations can also be used to perform scene-aware localization.
Beamforming vectors at the BS/UE and RIS element coefficients can be optimized
to avoid obstacles and take advantage of the strong reflectors for
localization purposes. In this topic, how to maintain a map with minimum
resources and update the map with time is a problem to be solved.

  8. 8.

Dynamic deployment optimization. It is straightforward to optimize the
deployment of static BSs for coverage or accuracy purposes. In temporal high
traffic situations as in stadiums or conference halls, the BS could also be
dynamically deployed, e.g., attached to UAVs. The location and route of the
UAVs need to be optimized to meet the communication and localization
requirements of the UEs. However, the connectivity in dynamic THz UAV networks
is also challenging, which should be addressed carefully.

  9. 9.

AI-based methods: Model-based methods are easy to analyze, but when unknown
model mismatches exist, AI-based methods are more preferred to learn or to
mitigate the effect of such mismatches. In the latter case, access to common
databases is needed to compare and evaluate different approaches. Furthermore,
the collection, sharing, and storage of a large amount of data, transfer a
learned model into another domain to reduce the training duration, and the
protection of user privacy are urgent issues that need to be solved.

In summary, we need to put more effort in three aspects in order to improve
THz localization accuracy and efficiency: (a) develop a more accurate system
model (directions 1, 2, 3), (b) utilize other types of information (directions
4, 5, 7, 8), and (c) develop more advanced tools for analysis and optimization
(directions 6, 9).

##  VIII Conclusion

This work explores the potential of the 6G THz system from a localization
point of view, emphasizing comparisons with 5G mmWave localization systems.
Comparisons include system and signal properties, channel modeling and
assumptions, localization problem formulation, and system design and
optimization. Preliminary simulations are carried out to show the potential of
THz localization compared with mmWave systems, in terms of the PEB and OEB.
This tutorial outlines recommendations on efficient and practical localization
algorithm design for RIS-assisted AOSA-based MIMO systems, providing insights
into other research directions. With joint localization and communication
systems operating at the terahertz band, data-hungry and high localization
accuracy demanding applications such as intelligent networks, autonomous
transportation, and tactile internet are anticipated in future communication
systems.

## Abbreviations

[ display = used, heading = none, template = supertabular ]

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Hui Chen received his B.S. degree in electrical engineering from Beijing
Forestry University, Beijing, China, in 2013, the M.S. degree in computer
application technology from the University of Chinese Academy of Sciences
(UCAS), Beijing, China, in 2016, and the Ph.D. degree in electrical & computer
engineering from King Abdullah University of Science and Technology