found in the literature. The single-user case was studied in [198] and [199]
for Rician fading channels and correlated Rician fading channels,
respectively. Given the beamforming vectors, the optimal phase shifts were
derived in closed form in [198], and the SDR method was used in [199] to solve
the phase shift optimization problem. The authors of [200] derived an upper
bound for the data rate in both downlink and uplink multiuser scenarios, based
on which the ADMM method was used for solving the phase shift optimization
problem. The more complex multicell networks scenario in the presence of
interference channels was studied in [201] and [202] assuming Rician fading
channels, and the phase shifts were optimized using the GA and CCM methods,
respectively. Jensen’s inequality was used in [203] to derive an upper bound
of the ergodic data rate in RIS-aided UAV communication systems, based on
which the trajectory of the UAV and the phase shifts of the RIS were jointly
optimized.

(2) _Large system analysis_ : Another widely used method to tackle the
computation of the expectation in (116) relies on large system analysis [204,
205, 206, 207]. The main idea is to apply the replica method in large
dimension random matrix theory to derive a deterministic approximation of the
ergodic data rate, based on which various optimization techniques can be used
to optimize the phase shifts. Specifically, the authors of [204] and [205]
considered the RIS-aided single-user case under correlated Rician fading
channels and the double-scattering channel model, respectively. The gradient
decent method and element-wise BCD method were used for determining the RIS
phase shifts in [204] and [205], respectively. Recently, the authors of [206]
and [207] studied RIS-aided MIMO multiple-access channels. Specifically, in
[206], the BS-RIS channel is assumed to be deterministic and the RIS-user
channels are distributed according to a spatially correlated Rayleigh fading
distribution. The MM method is used to solve the phase shift optimization
problem. In [207], all the channels are assumed to be subject to correlated
Rician fading and the gradient decent method is used to solve the phase shift
optimization problem.

(3) _Deep reinforcement learning_ : For example, a novel deep deterministic
policy gradient (DDPG) based algorithm was proposed in [208] for solving the
optimization problem in (116). The idea is that the BS first estimates the
long-term CSI, based on which the BS randomly generates a set of instantaneous
CSI samples based on the channel distribution in an offline manner. Then, the
generated data set is employed for DDPG training. The final trained solutions
can be used in subsequent coherence blocks. Compared with the above two
methods, the appealing feature of the DDPG approach is the low training
overhead, while providing almost the same performance as the existing methods.

In [118] and [209], the authors considered the case study in the presence of
imperfect long-term CSI. Specifically, in [118], the effective angles from the
BS to the user are first derived, and are then used for optimizing the BS
beamforming and the RIS phase shifts by accounting for the errors in angle
estimation. In [209], the authors exploited user location information to
obtain the angle information and derived the corresponding angle error
distribution based on the distribution of estimated location information. The
achievable data rate was then derived based on the statistical information of
the angle error distribution.

Figure 6: Comparison of achievable rate under three different levels of CSI
availability in an RIS-aided massive MIMO system, where $K=4$, $U=50$,
$\delta=1$, $\varepsilon_{k}=20,\forall k$ and $N=100$. The number of time
slots in each coherence block is $T_{c}=196$[210, 211]. For brevity, we set
$\varpi_{k}=0,\forall k$ and consider the rank-$1$ LoS BS-RIS channel
$\overline{\mathbf{H}}=\mathbf{a}_{N}\mathbf{a}_{M}^{H}$.

Simulation results: Fig. 6 illustrates the performance of transmission schemes
based on instantaneous CSI, two-timescale CSI, and fully long-term CSI. We
consider the uplink transmission in massive MIMO systems where low-complexity
MRC is employed at the BS. The channel estimation overhead for long-term CSI
estimation is ignored. The pilot lengths required for acquiring the necessary
instantaneous CSI and two-timescale CSI are equal to
$T_{\rm{ins}}=K+M+\max\left({K-1,(K-1)\left\lceil{M/N}\right\rceil}\right)$
[52] and $T_{\rm{two}}=K$[130], respectively. The fully long-term CSI scheme
does not require estimation of the instantaneous (effective) CSI, which leads
to a pilot length equal to $T_{\rm{ful}}=0$.

As far as the instantaneous CSI case is concerned, in the $u$-th coherence
block, the cascaded and direct channels of user $k$ and the phase shifts of
the RIS are denoted by $\mathbf{G}_{k}^{u}$, $\mathbf{h}_{d,k}^{u}$ and
$\bm{\theta}^{u}$, respectively. Then, in the $u$-th coherence block, the
instantaneous CSI-based MRC decoding vector is set as
$(\mathbf{w}_{k}^{u}{(\bm{\theta}^{u})})^{H}=(\mathbf{G}_{k}^{u}\bm{\theta}^{u}+\mathbf{h}_{d,k}^{u})^{H}$
and the corresponding SINR of user $k$ is

$\displaystyle\mathrm{SINR}_{{\rm{ins}},k}^{u}\left(\bm{\theta}^{u}\right)=$
$\displaystyle\frac{P_{k}\left|\left(\mathbf{w}_{k}^{u}{\left(\bm{\theta}^{u}\right)}\right)^{H}\left({\mathbf{G}}_{k}^{u}\bm{\theta}^{u}+{\mathbf{h}}_{d,k}^{u}\right)\right|^{2}}{\sum\limits_{i=1,i\neq
k}^{K}P_{i}\left|\left(\mathbf{w}_{k}^{u}{\left(\bm{\theta}^{u}\right)}\right)^{H}\left({\mathbf{G}}_{i}^{u}\bm{\theta}^{u}+{\mathbf{h}}_{d,i}^{u}\right)\right|^{2}+\sigma^{2}\left\|\mathbf{w}_{k}^{u}{\left(\bm{\theta}^{u}\right)}\right\|^{2}},$
(117)

and the corresponding average sum user rate is given by

$\displaystyle
R_{\rm{ins}}\\!\\!=\\!\\!\left(1\\!-\\!\frac{T_{\rm{ins}}}{T_{c}}\right)\\!\\!\frac{1}{U}\\!\\!\sum_{u=1}^{U}\sum_{k=1}^{K}\log_{2}\left(1\\!+\\!\mathrm{SINR}_{{\rm{ins}},k}^{u}(\bm{\theta}^{u})\right),$
(118)

where the factor $1-\frac{T_{\rm{ins}}}{T_{c}}$ accounts for the rate loss due
to the pilot overhead.

As far as the two-timescale CSI scheme is concerned, the effective channel of
user $k$ in the $u$-th coherence block is denoted by
$\mathbf{q}_{k}^{u}={\bf{H}}^{u}{\bm{\Theta}}{{\bf{h}}^{u}_{r,k}}+{{\bf{h}}^{u}_{d,k}}$,
where the phase shift matrix ${\bm{\Theta}}$ remains constant for all
coherence blocks $1\leq u\leq U$. In the $u$-th coherence block, the
instantaneous CSI-based MRC decoding vector is set equal to
$(\mathbf{w}_{k}^{u}{(\bm{\Theta})})^{H}=({\bf{H}}^{u}{\bm{\Theta}}{{\bf{h}}^{u}_{r,k}}+{{\bf{h}}^{u}_{d,k}})^{H}$
and then the SINR of user $k$ is given by

$\displaystyle\mathrm{SINR}_{{\rm{two}},k}^{u}(\bm{\Theta})=$
$\displaystyle\frac{P_{k}\left|\left(\mathbf{w}_{k}^{u}{\left(\bm{\Theta}\right)}\right)^{H}\left({\bf{H}}^{u}{\bm{\Theta}}{{\bf{h}}^{u}_{r,k}}+{{\bf{h}}^{u}_{d,k}}\right)\right|^{2}}{\sum\limits_{i=1,i\neq
k}^{K}P_{i}\left|\left(\mathbf{w}_{k}^{u}{\left(\bm{\Theta}\right)}\right)^{H}\left({\bf{H}}^{u}{\bm{\Theta}}{{\bf{h}}^{u}_{r,i}}+{{\bf{h}}^{u}_{d,i}}\right)\right|^{2}+\sigma^{2}\left\|\mathbf{w}_{k}^{u}{\left(\bm{\Theta}\right)}\right\|^{2}}.$
(119)

It is worth noting that the phase shift ${\bm{\Theta}}$ needs to be optimized
only once in the considered series of coherence blocks. However, in the
instantaneous CSI case, the phase shift needs to be optimized $U$ times. The
average sum user rate of the two-timescale scheme is given by

$\displaystyle
R_{\rm{two}}=\left(1-\frac{K}{T_{c}}\right)\frac{1}{U}\sum_{u=1}^{U}\sum_{k=1}^{K}\log_{2}\left(1+\mathrm{SINR}_{{\rm{two}},k}^{u}(\bm{\Theta})\right).$
(120)

As for the fully long-term CSI case is concerned, the long-term CSI-based MRC
beamformer is set equal to
$(\mathbf{w}_{k}{(\bm{\Theta})})^{H}=\left(\mathbb{E}\left\\{{\bf{H}}^{u}{\bm{\Theta}}{{\bf{h}}^{u}_{r,k}}+{{\bf{h}}^{u}_{d,k}}\right\\}\right)^{H}$
which only contains large-scale LoS channel components and remains constant
for all coherence blocks. Therefore, in the $u$-th coherence block, the SINR
of user $k$ is given by

$\displaystyle\mathrm{SINR}_{{\rm{ful}},k}^{u}(\bm{\Theta})=$
$\displaystyle\frac{P_{k}\left|\left(\mathbf{w}_{k}{\left(\bm{\Theta}\right)}\right)^{H}\left({\bf{H}}^{u}{\bm{\Theta}}{{\bf{h}}^{u}_{r,k}}+{{\bf{h}}^{u}_{d,k}}\right)\right|^{2}}{\sum\limits_{i=1,i\neq
k}^{K}P_{i}\left|\left(\mathbf{w}_{k}{\left(\bm{\Theta}\right)}\right)^{H}\left({\bf{H}}^{u}{\bm{\Theta}}{{\bf{h}}^{u}_{r,i}}+{{\bf{h}}^{u}_{d,i}}\right)\right|^{2}+\sigma^{2}\left\|\mathbf{w}_{k}{\left(\bm{\Theta}\right)}\right\|^{2}},$
(121)

and the corresponding average sum rate is

$\displaystyle
R_{\rm{ful}}=\frac{1}{U}\sum_{u=1}^{U}\sum_{k=1}^{K}\log_{2}\left(1+\mathrm{SINR}_{{\rm{ful}},k}^{u}(\bm{\Theta})\right).$
(122)

The phase shifts of the above three schemes are obtained by using the GA
method with the objective of optimizing the sum data rate. However, in the
instantaneous CSI scheme, the phase shift of the RIS, ${\bm{\theta}}^{u}$,
$1\leq u\leq U$, needs to be designed in each coherence block. In the two-
timescale and the fully long-term CSI schemes, we only need to design the
variable $\mathbf{\Theta}$ once.

Fig. 6 clearly unveils the performance tradeoffs of the considered schemes as
a function of the level of CSI available. The instantaneous CSI scheme with an
ideal pilot length equal to $T_{\rm{ins}}^{(\rm{ideal})}=K=T_{\rm{two}}$
offers, as expected, the best performance. If, the actual pilot training
overhead is taken into account, however, we note that the average rate first
increases and then decreases as a function of the number $M$ of RIS elements.
On the other hand, the fully long-term CSI scheme has the lowest pilot
overhead, but it offers a relatively low achievable rate since the decoding
vector at the BS cannot be adjusted according to the instantaneous effective
CSI. By contrast, the two-timescale scheme offers good performance while
maintaining a low pilot overhead.

##  IV RIS-aided Radio Localization

In wireless communication networks, radio localization [212] offers a viable
alternative for obtaining user location information in GPS-denied environments
[213]. Radio localization techniques are based on the general idea that the
radio signals provide information on the position of network nodes. More
precisely, the location of agent nodes (e.g., mobile devices or vehicles) are
estimated with the aid of known-position anchor nodes (e.g., BSs or APs) and
the exchange of radio signal between the anchor and agent nodes. In general,
the position of the agent node is estimated by using a two-step approach[214].
First, the distance/angle-related measurements are extracted from the received
signal, then the location is estimated from the measurements, such as the time
of arrival (ToA), the time difference of arrival (TDoA), the received signal
strength (RSS), the AoA, and the AoD.

Third generation (3G) communication systems can provide a positioning accuracy
of the order of tens of meters by using TDoA measurements[215]. This accuracy
is improved to some extent in fourth generation (4G) systems. Existing studies
have shown that the position errors in 5G mmWave communication systems are on
the order of centimeters[216, 217, 213]. Due to thriving new applications such
as smart factories, automated/assisted driving, and augmented reality, the
requirements in terms of positioning accuracy for 5G/6G communication networks
are becoming increasingly stringent. In addition, the reliability of the
localization provided by 5G/6G communications is of particular importance.
Since 5G/6G systems can be deployed in high-frequency mmWave and THz bands,
the links are vulnerable to obstacles. Since LoS propagation is usually
required for accurately estimating the location, existing localization methods
result in prohibitively large estimation errors if the LoS link is blocked.

The use of RISs can yield reliable and high-precision position estimates at a
low cost and high energy efficiency. The RISs can be integrated into existing
radio localization systems to co-work with other anchor nodes, and thus
provide better positioning estimates of the agent nodes. The benefits of RISs
for positioning are as follows. First, an RIS can establish a virtual LoS link
when the LoS link is blocked. Thus, the RIS can restore the positioning
capability of the network when the GPS or the BS signals are weak. Second,
RISs can be regarded, unlike active anchor nodes, as quasi-passive anchor
nodes that need no power amplifiers and radio frequency chains. Therefore,
they can provide high-precision estimate of the locations with low hardware
cost and low energy consumption. Third, RISs can be constructed with a large
physical aperture. Hence, they can offer a higher angular resolution, which is
appealing for radio localization. Fourth, unlike a non-reconfigurable
scatterer in the environment, RISs can provide a high beamforming gain by
tuning the phase shifts of the reflecting elements. As a result, the use of
RISs provides several promising opportunities for assisting localization
systems in next generation wireless networks.

Current state-of-the-art research on RIS-aided radio localization has mainly
focused on three aspects: performance analysis, development of algorithms, and
the interplay between communication and localization. As far as the
performance analysis is concerned, the authors of [218] studied the
theoretical performance of a 2D RIS-aided mmWave positioning system by
computing the CRB in the far field region. The authors of [219] derived the
Fisher information matrix (FIM) and the CRB for evaluating the performance of
a three-dimensional (3D) RIS-assisted positioning system, where a near-field
channel model was utilized. The authors of [220] derived the CRB for
evaluating the localization and orientation performance of synchronous and
asynchronous signalling schemes in an RIS-assisted localization system in the
near-field region. As far as the localization algorithms, the authors of [221]
considered an RIS-assisted 3D localization system, where the AoAs and AoDs
were estimated by using the MLE algorithm, and user localization was obtained
via a Taylor series algorithm. The localization accuracy of an indoor
localization system was improved in [222] and [223] by choosing proper RIS
reflection coefficients that maximize the differences of RSS values among
adjacent locations. As far as the interplay between communication and
localization is concerned, the localization accuracy and the data rate were
balanced by optimizing the time allocation of an RIS-aided wireless
communication system in [224]. In particular, the CRB and the effective
achievable data rate were used as the performance metrics. The authors of
[225] proposed an adaptive RIS phase shifter design based on hierarchical
codebooks and feedback from the mobile user for enabling accurate localization
and high data rate transmission.

Under the spherical wave-front channel model, there exist only a few
contributions that studied the RIS-aided radio localization problem. A near-
field codebook was developed in [226] for extremely large-scale RIS (XL-RIS)
beam training by dividing the two-dimensional (2D) plane into several sampled
points in the $x$-$y$ coordinate system. Also, the authors of [227] showed
that the characteristics of the transmitted signal (transmit antenna type,
size, orientation etc.) may profoundly affect the received signals in the near
field, which needs to be taken into consideration for high-precision
localization. In [228], the FIM was analysed for an uplink localization system
using an RIS-based lens, and the PEB and OEB were also evaluated by exploiting
the wavefront curvature. The theoretical localization performance of
multipath-aided localization in both LoS and NLoS conditions was characterized
in [229], where it was shown that using the RIS’s reflected signal wavefront
curvature in the near field is enough to deduce the user’s position even in
the absence of a direct path.

In the following, we discuss RIS-aided localization with focus on two channel
conditions: 1) Far-field channel model; 2) Near-field channel model.

###  IV-A Far-Field Localization Techniques

Figure 7: Illustration of an RIS-aided localization system.

####  IV-A1 System Model

In Fig. 7, as an illustrative example we consider a mmWave MIMO system for
user localization in a 3D space. The considered system model consists of a BS
equipped with a ULA with $N_{t}$ antennas, a mobile user (MU) with a ULA with
$N_{r}$ antennas, and $K-1$ RISs. Each RIS is equipped with a square UPA with
$L^{2}$ reflecting elements. The carrier frequency is $f_{c}$ with a
corresponding wavelength equal to $\lambda$ and bandwidth equal to $B$. A 3D
Cartesian coordinate system is used to describe the locations of the anchor
nodes. The ULA of the BS is placed parallel to the $x$ axis without rotation,
and its center is located at $\mathbf{q}=[q_{x},q_{y},q_{z}]^{\rm
T}\in\mathbb{R}^{3\times 1}$. The UPA of the $k$-th RIS is placed parallel to
the $yz$-plane, and its center is located at
$\mathbf{s}_{k}=[s_{kx},s_{ky},s_{kz}]^{\rm T}\in\mathbb{R}^{3\times 1}$. The
ULA of the MU is placed on the $xy$-plane, and is rotated by
$\alpha\in[0,\pi)$ radians relative to the $x$-axis. The center of the ULA of
the MU is located at $\mathbf{p}=[p_{x},p_{y},0]^{\rm T}\in\mathbb{R}^{3\times
1}$. The symbols $q_{z}$ and $s_{kz}$ denote the heights of the BS and the
$k$-th RIS with respect to the MU on the ground. The locations of the BS and
the RIS are assumed to be known. Then, the objective of the positioning system
is to estimate the location $(p_{x},p_{y})$ of the MU and its rotation angle
$\alpha$. To simplify the notation and to facilitate the analysis, we collect
the parameters to be estimated in a vector
$\tilde{\bm{{\eta}}}=[p_{x},p_{y},\alpha]^{\rm T}$.

A narrow-band channel model is considered. The overall channel from the MU to
the BS is expressed as the combination of the direct BS-MU channel and the
channels reflected by all the RISs. In mathematical terms, we have

$\mathbf{H}=\mathbf{H}_{\rm{BM}}+\sum\nolimits_{k=1}^{K-1}\mathbf{H}_{{\rm{RIS}},k},$
(123)

where $\mathbf{H}_{\rm{BM}}\in\mathbb{C}^{N_{r}\times N_{t}}$ is the BS-MU
channel and $\mathbf{H}_{{\rm{RIS}},k}\in\mathbb{C}^{N_{r}\times N_{t}}$ is
the $k$-th RIS’s reflected channel. For the sake of analysis, we only consider
the LoS paths in the channels of $\mathbf{H}_{{\rm{BM}}}$ and
$\mathbf{H}_{{\rm{RIS}},k}$, which can be further expressed as

$\displaystyle\mathbf{H}_{{\rm{BM}}}$
$\displaystyle=h_{0}\mathbf{a}_{{\rm{RX}}}(\theta_{{\rm{RX},0}})\mathbf{a}_{{\rm{TX}}}^{{\rm{H}}}(\theta_{{\rm{TX},0}}),$
(124a) $\displaystyle\mathbf{H}_{{\rm{RIS}},k}$
$\displaystyle=h_{k}\mathbf{a}_{{\rm{RX}}}(\theta_{{\rm{RX}},k})\mathbf{a}_{{\rm{RIS}},{\rm{OUT}}}^{\rm
H}(\varphi_{{\rm{out}},k}^{a},\varphi_{{\rm{out}},k}^{e})\bm{{\Theta}}_{k}$
$\displaystyle\quad\cdot\mathbf{a}_{{\rm{RIS}},{\rm{IN}}}(\varphi_{{\rm{in}},k}^{a},\varphi_{{\rm{in}},k}^{e})\mathbf{a}_{{\rm{TX}}}^{\rm
H}(\theta_{{\rm{TX}},k}),$ (124b)

where $h_{0}$ and $\\{h_{1},\cdots,h_{K-1}\\}$ are the complex channel gains
of the BS-MU direct link and the BS-RIS-MU reflected links, respectively.
$\bm{{\Theta}}_{k}$ is the reflection coefficient matrix of the $k$-th RIS.
$\mathbf{a}_{{\rm{TX}}}(\theta_{{\rm{TX},0}})\in\mathbb{C}^{N_{t}\times 1}$
and $\mathbf{a}_{{\rm{RX}}}(\theta_{{\rm{RX},0}})\in\mathbb{C}^{N_{r}\times
1}$ are the antenna response vectors of the transmitter and the receiver, in
which $\theta_{{\rm{TX},0}}$ denotes the AoD and $\theta_{{\rm{RX},0}}$
denotes the AoA of the BS-MU direct link. Since the transmitter and the
receiver use ULAs, $\mathbf{a}_{{\rm{TX}}}(\theta)$ and
$\mathbf{a}_{{\rm{RX}}}(\theta)$ can be expressed as

$\mathbf{a}_{{\rm{TX}}}(\theta)=[1,e^{j\frac{2\pi}{\lambda}d\sin(\theta)},\cdots,e^{j(N_{t}-1)\frac{2\pi}{\lambda}d\sin(\theta)}]^{{\rm{T}}},$
(125a)
$\mathbf{a}_{{\rm{RX}}}(\theta)=[1,e^{j\frac{2\pi}{\lambda}d\sin(\theta)},\cdots,e^{j(N_{r}-1)\frac{2\pi}{\lambda}d\sin(\theta)}]^{{\rm{T}}},$
(125b)

where $d$ denotes the distance between adjacent antennas and $\theta$ is the
AoA or AoD. The
$\mathbf{a}_{X}(\varphi_{{\rm{x}},k}^{a},\varphi_{{\rm{x}},k}^{e})$ in (124b)
denotes the array response vector of the RIS array, which is equal to
$\mathbf{a}_{{\rm{RIS}},{\rm{IN}}}(\varphi_{{\rm{in}},k}^{a},\varphi_{{\rm{in}},k}^{e})$
and
$\mathbf{a}_{{\rm{RIS}},{\rm{OUT}}}(\varphi_{{\rm{out}},k}^{a},\varphi_{{\rm{out}},k}^{e})$
when $x$ is equal to IN and OUT, respectively. Since the RISs use UPAs,
$\mathbf{a}_{X}(\varphi_{{\rm{x}},k}^{a},\varphi_{{\rm{x}},k}^{e})$ can be
expressed as

$\displaystyle\mathbf{a}_{X}(\varphi_{{\rm{x}},k}^{a},\varphi_{{\rm{x}},k}^{e})=[1,e^{j\frac{2\pi}{\lambda}d\cos(\varphi_{{\rm{x}},k}^{e})},\cdots,e^{j\frac{2\pi}{\lambda}\left(L-1\right)d\cos(\varphi_{{\rm{x}},k}^{e})}]^{{\rm{T}}}$
$\displaystyle\otimes[1,e^{j\frac{2\pi}{\lambda}d\sin(\varphi_{{\rm{x}},k}^{e})\sin(\varphi_{{\rm{x}},k}^{a})}\cdots,e^{j\frac{2\pi}{\lambda}\left(L-1\right)d\sin(\varphi_{{\rm{x}},k}^{e})\sin(\varphi_{{\rm{x}},k}^{a})}]^{{\rm{T}}},$

where $\varphi_{{\rm{in}},k}^{a}$ and $\varphi_{{\rm{in}},k}^{e}$ are the
azimuth AoA and elevation AoA at the $k$-th RIS from the BS, while
$\varphi_{{\rm{out}},k}^{a}$ and $\varphi_{{\rm{out}},k}^{e}$ are the azimuth
AoD and elevation AoD at the $k$-th RIS to the MU.

The complex channel gains $h_{k},k=0,1,\cdots,K-1$ in (124) follow the
distribution of $\mathcal{CN}(0,10^{\frac{-{\rm{PL}}_{k}}{10}})$, where
${\rm{PL}}_{0},{\rm{PL}}_{1},\cdots,{\rm{PL}}_{K-1}$ are the corresponding
path losses. Based on [21] and [230], the path loss of the BS-MU channel in
(124a) can be expressed in dB as

$\displaystyle{\rm{PL}}_{0}=$ $\displaystyle
10\log_{10}(64\pi^{3})+10\alpha_{0}\log_{10}d_{BM}$
$\displaystyle+20\log_{10}f_{c}+\xi_{0},$ (126)

where $d_{BM}$ is the distance (in meters) between the BS and the MU,
$\alpha_{0}$ is the path loss exponent.
$\xi_{0}\sim\mathcal{N}(0,\sigma_{SF_{0}}^{2})$ is the log-normal term
accounting for the shadow fading, where $\sigma_{SF_{0}}^{2}$ denotes the log-
normal shadowing variance.

The path loss of the RIS-reflected channel via the $k$-th RIS can be expressed
in dB as

$\displaystyle{\rm{PL}}_{k}=$ $\displaystyle
10\log_{10}(64\pi^{3})+10\alpha_{k}\log_{10}(d_{\rm{BR}}\cdot d_{\rm{RM}})$
$\displaystyle+40\log_{10}f_{c}+\xi_{k},$ (127)

where $d_{\rm{BR}}$ and $d_{\rm{RM}}$ are the distances (in meters) of the BS-
RIS $k$ link and RIS $k$-MU link, respectively. $\alpha_{k}$ is the path loss
exponent of the reflected channel, and $\xi_{k}$ is the log-normal term
accounting for the shadow fading following distribution
$\mathcal{N}(0,\sigma_{SF_{k}}^{2})$ with variance $\sigma_{SF_{k}}^{2}$.

For estimating the locations of users, we assume a pilot-based transmission
scheme. The transmitted pilot signal is a continuous time-domain waveform
$x(t)$ with bandwidth $B$ and duration of $T_{o}$ seconds. Assuming $M_{t}$
pilot signals, i.e.,
$\mathbf{x}(t)=[x_{1}(t),x_{2}(t),\cdots,x_{M_{t}}(t)]^{\rm
T}\in\mathbb{C}^{M_{t}\times 1}$, is transmitted at time $t$ with
corresponding beamforming matrix denoted by
$\mathbf{F}=[\mathbf{f}_{1},\mathbf{f}_{2},\cdots,\mathbf{f}_{M_{t}}]\in\mathbb{C}^{N_{t}\times
M_{t}}$ where $M_{t}\ll N_{t}$, then the signal
$\mathbf{y}(t)\in\mathbb{C}^{N_{r}\times 1}$ received at the MU is given by

$\mathbf{y}(t)=\mathbf{H}_{\rm{BM}}\mathbf{F}\mathbf{x}(t-\tau_{0})+\sum_{k=1}^{K-1}\mathbf{H}_{\rm{RIS},k}\mathbf{F}\mathbf{x}(t-\tau_{k})+\mathbf{n}(t),$

where $\tau_{0}$ is the propagation delay of the direct BS-MU path, and
$\tau_{k},k=1,2,\cdots,K-1$ is the propagation delay from the BS to the MU via
the $k$-th RIS. The delays $\tau_{k},k=0,1,2,\cdots,K-1$ is also called TOAs.
The received noise $\mathbf{n}(t)$ is Gaussian with zero mean and two-sided
power spectral density equal to $N_{0}/2$.

####  IV-A2 Two-Step Localization Scheme

The position and orientation of the MU can be estimated based on the observed
signal $\mathbf{y}(t)$ by using a two-step localization scheme. In the first
step, the angles, channel gains, and time delays are estimated using some
existing methods. For example, the channel gains
$\\{\hat{h}_{k},\hat{h}_{0}\\}$ can be estimated with the LS approach, while
the angles
$\\{\hat{\theta}_{TX,0},\hat{\varphi}_{out,k}^{a},\hat{\varphi}_{out,k}^{e},\hat{\theta}_{RX,0},\hat{\theta}_{RX,k}\\}$
can be estimated by using array signal processing algorithms in the spatial
domain, such as MUSIC and ESPRIT. The estimated time delays
$\\{\hat{\tau}_{k},\hat{\tau}_{0}\\}$ can be extracted from the pilot signal.
In the second step, the position of the MU can be obtained by using multi-
angulation (or triangulation) from the angle-related measurements, or by using
multi-lateration (or trilateration) methods from distance-related
measurements, or by a combination of both.

As shown in Fig. 7, the localization measurements are closely related with the
coordinates and rotation angle of MU, which means that there exists a mapping
from the MU’s location to the measurements obtained in the first step, which
can be described as follows.

#### ToA

The propagation delay of the BS-MU path, which is related with the MU
coordinate $\mathbf{p}$, is given by

$\displaystyle\hat{\tau}_{0}$
$\displaystyle=\left\|\mathbf{q}-\mathbf{p}\right\|/c+n_{t_{0}},$ (128)

where $c$ is the speed of light and $n_{t_{0}}$ denotes the ToA error. The ToA
$\hat{\tau}_{k0}$ from the MU to the $k$-th RIS can be obtained by subtracting
the propagation delay of the $k$-th RIS-BS path from the propagation delay of
the path from BS to MU via the $k$-th RIS, which are expressed as

$\displaystyle\hat{\tau}_{k0}=\left\|\mathbf{p}-\mathbf{s}_{k}\right\|/c+n_{t_{k}}=\hat{\tau}_{k}-\left\|\mathbf{q}-\mathbf{s}_{k}\right\|/c+n_{t_{k}},$
(129)

where $n_{t_{k}}$ denotes the ToA error.

The ToA measurements are related with the path lengths, which are further
related with the MU position. Specifically, the $\hat{\tau}_{0}$ and
$\hat{\tau}_{k0}$ can describe the estimated distance of the direct path as
well as the estimated distance from the MU to the $k$-th RIS, which are
expressed as follows

$\displaystyle\hat{d}_{0}$
$\displaystyle=c\hat{\tau}_{0}=\left\|\mathbf{q}-\mathbf{p}\right\|+n_{d_{0}},$
(130a) $\displaystyle\hat{d}_{k}$
$\displaystyle=c\hat{\tau}_{k0}=\left\|\mathbf{p}-\mathbf{s}_{k}\right\|+n_{d_{k}},$
(130b)

where $n_{d_{0}}$ and $n_{d_{k}}$ are the estimation errors.

#### TDoA

The location of the MU can be derived from the TDoA, which describes the
distance differences between different pairs of the direct MU-BS path and the
$k$ MU-RIS paths.

Specifically, by taking the direct path from the BS to MU as the reference
path, the TDoA can be expressed as

$\displaystyle\hat{\tau}_{td,k}=\hat{\tau}_{k0}-\hat{\tau}_{0}.$ (131)

Then, the distance difference corresponding to the TDoA is represented as

$\displaystyle c\hat{\tau}_{td,k}$
$\displaystyle=c\hat{\tau}_{k0}-c\hat{\tau}_{0}=\hat{d}_{d,k}=(\hat{d}_{k}-\hat{d}_{0})$
$\displaystyle=\left\|\mathbf{p}-\mathbf{s}_{k}\right\|-\left\|\mathbf{q}-\mathbf{p}\right\|+n_{td_{k}},$
(132)

where $n_{td_{k}}$ is the estimation error of $d_{d,k}$.

#### AoA and AoD

The angle-related measurements (AoA and AoD) are estimated at the arrays of
BS, MU, and RIS using angle estimation algorithms. They are also closely
related with the MU position, which can be described as

$\displaystyle\hat{\theta}_{TX,0}$
$\displaystyle=\arcsin\left(\frac{p_{x}-q_{x}}{\left\|\mathbf{p}-\mathbf{q}\right\|_{2}}\right)+n_{\theta_{TX,0}},$
(133) $\displaystyle\hat{\varphi}_{out,k}^{a}$
$\displaystyle=\arcsin\left(\frac{p_{y}-s_{ky}}{\sqrt{\left(p_{x}-s_{kx}\right)^{2}+\left(p_{y}-s_{ky}\right)^{2}}}\right)+n_{a_{k}},$
(134) $\displaystyle\hat{\varphi}_{out,k}^{e}$
$\displaystyle=\arccos\left(\frac{-s_{kz}}{\left\|\mathbf{p}-\mathbf{s}_{k}\right\|_{2}}\right)+n_{e_{k}},$
(135) $\displaystyle\hat{\theta}_{RX,0}$
$\displaystyle=\arcsin\left(\frac{\left(p_{x}-q_{x}\right)\cos\alpha-\left(p_{y}-q_{y}\right)\sin\alpha}{\left\|\mathbf{p}-\mathbf{q}\right\|_{2}}\right)$
$\displaystyle\quad+n_{\theta_{RX,0}},$ (136)
$\displaystyle\hat{\theta}_{RX,k}$
$\displaystyle=\arcsin\left(\frac{\left(p_{x}-s_{kx}\right)\cos\alpha-\left(p_{y}-s_{ky}\right)\sin\alpha}{\left\|\mathbf{p}-\mathbf{s}_{k}\right\|_{2}}\right)$
$\displaystyle\quad+n_{\theta_{RX,k}}.$ (137)

where $n_{\theta_{TX,0}}$,$n_{a_{k}}$,$n_{e_{k}}$, $n_{\theta_{RX,0}}$, and
$n_{\theta_{RX,k}}$ denote error associated to the AoA and AoD estimations.
Besides, $\theta_{TX,k}$ are known values, which can be calculated in advance
from the known coordinates of the BS and RISs.

#### Channel Gains

Some recent works, e.g., [218, 220], extract the distance information from the
channel gains $h_{k}$ according to the relations
$h_{0}=\frac{\lambda}{4\pi}\frac{1}{d_{BM}}$ and
$h_{k}=\frac{\lambda}{4\pi}\frac{1}{d_{BR}d_{RM}}$. However, the channel gains
depend on the distance-dependent large-scale path loss as well as the
distance-independent shadowing fading. Thus, it may not be accurate to extract
the distance information from the channel gains. Therefore, this measurement
is not considered in the following numerical results.

In the second step, by collecting the equations mapping the MU position into
the measurements illustrated above, a set of interlinked equations can be
formulated. The position and rotation angle of the MU, i.e., the vector
$\tilde{\bm{{\eta}}}=[p_{x},p_{y},\alpha]^{\rm T}$ can be retrieved by jointly
solving the set of equations. The larger the number of measurements
(equations) is available, the better the estimate of the position is. The
obtained equations are, however, nonlinear, and the measurements are affected
by errors. Consequently, they are difficult to solve. To circumvent this
difficulty, wireless localization algorithms, such as the Chan algorithm [231]
and the Taylor series algorithm [232] can be utilized.

####  IV-A3 Position Error Bound (PEB) and Rotation Error Bound (REB) Analysis

To evaluate the performance of the position and orientation estimation, we
analyze the FIM and CRB. To provide a benchmark for the accuracy of the
position estimation, the CRB and the PEB/REB are investigated in [218, 225],
by assuming a 2D localization problem formulation. In the following, we derive
the CRB and PEB/REB in a general 3D space. Further details can be found in
[233].

First, we construct the FIM based on the channel parameters, including the
ToA, AoA, AoD, and the complex channel gains. The unknown parameters are
collected in the following vector

$\bm{\eta}=[\bm{\tau},\theta_{{\rm{TX}},0},\bm{\theta}_{{\rm{RX}}},\bm{\varphi}_{{\rm{out}}}^{a},\bm{\varphi}_{{\rm{out}}}^{e},\mathbf{h}_{R},\mathbf{h}_{I}]^{\rm
T},$ (138)

where

$\displaystyle\bm{\tau}$
$\displaystyle=[\tau_{0},\tau_{1},\cdots,\tau_{K-1}]^{\rm T},$ (139a)
$\displaystyle\bm{\theta}_{{\rm{RX}}}$
$\displaystyle=[\theta_{{\rm{RX}},0},\theta_{{\rm{RX}},1},\cdots,\theta_{{\rm{RX}},K-1}]^{\rm
T},$ (139b) $\displaystyle\bm{\varphi}_{{\rm{out}}}^{a}$
$\displaystyle=[\varphi_{{\rm{out}},1}^{a},\varphi_{{\rm{out}},2}^{a},\cdots,\varphi_{{\rm{out}},K-1}^{a}]^{\rm
T},$ (139c) $\displaystyle\bm{\varphi}_{{\rm{out}}}^{e}$
$\displaystyle=[\varphi_{{\rm{out}},1}^{e},\varphi_{{\rm{out}},2}^{e},\cdots,\varphi_{{\rm{out}},K-1}^{e}]^{\rm
T},$ (139d) $\displaystyle\mathbf{h}_{R}$
$\displaystyle=[h_{R,0},h_{R,1},\cdots,h_{R,K-1}]^{\rm T},$ (139e)
$\displaystyle\mathbf{h}_{I}$
$\displaystyle=[h_{I,0},h_{I,1},\cdots,h_{I,K-1}]^{\rm T},$ (139f)

and $\theta_{{\rm{TX}},0}$ denotes the AOD from the BS to the MU. The real
parts of the channel gains $h_{R,k}={\rm Re}\\{h_{k}\\}$ are collected in the
vector $\mathbf{h}_{R}$ and the imaginary parts $h_{I,k}={\rm Im}(h_{k})$ are
collected in the vector $\mathbf{h}_{I}$. We denote the unbiased estimator of
the unknown channel parameters as $\hat{\bm{\eta}}$. Then, the mean-square
error bound is

$\mathbb{E}[(\hat{\bm{\eta}}-\bm{\eta})(\hat{\bm{\eta}}-\bm{\eta})^{\rm
H}]\geq\mathbf{J}_{{\rm\mathbf{\bm{\eta}}}}^{-1},$ (140)

where $\mathbf{J}_{\bm{{\eta}}}$ is the $(6K-1)\times(6K-1)$ dimensional FIM.
Assuming that the noise is a wide-sense stationary (WSS) Gaussian random
vector and is written as $\bm{n}(t)$, the element in the $m$-th row and $n$-th
column of matrix $\mathbf{J}_{{\rm\mathbf{\bm{\eta}}}}$ can be written as
[234]

$\displaystyle[\mathbf{J}_{{\rm\mathbf{\bm{\eta}}}}]_{mn}$
$\displaystyle=\mathbb{E}_{\mathbf{y}(t)\left|\mathbf{\bm{\eta}}\right.}\left[-\frac{\partial^{2}\ln
p(\mathbf{y}(t)\left|\bm{{\eta}}\right.)}{\partial\eta_{m}\partial\eta_{n}}\right]$
$\displaystyle\approx\frac{2}{N_{0}}\int_{0}^{T_{0}}\textrm{Re}\left\\{\frac{\partial\bm{\mu}^{\rm
H}(t)}{\partial\eta_{m}}\frac{\partial\bm{\mu}(t)}{\partial\eta_{n}}\right\\}.$
(141)

where $\eta_{m}$ denotes the $m$-th element of $\bm{\eta}$, $T_{0}$ denotes
the observation time, $\bm{\mu}(t)=\mathbf{y}(t)-\bm{n}(t)$, and
$p(\mathbf{y}(t)\left|\bm{{\eta}}\right.)$ denotes the probability density
function (PDF) (i.e, the likelihood function) of the random vector
$\mathbf{y}(t)$ conditioned on the parameter vector $\bm{{\eta}}$.

The FIM of the position and orientation of the MU, i.e., $\bf\tilde{\eta}$,
can be obtained by applying a transformation of variables from the channel
parameters $\bm{\eta}$ to the location parameters $\tilde{\bm{{\eta}}}$.
Specifically, the FIM of $\tilde{\bm{{\eta}}}$ is obtained by applying the
transformation matrix $\mathbf{T}$ defined as

$[\mathbf{T}]_{1:3,m}=[\frac{\partial\eta_{m}}{\partial
p_{x}},\frac{\partial\eta_{m}}{\partial
p_{y}},\frac{\partial\eta_{m}}{\partial\alpha}]^{\rm T}.$ (142)

The equation for the FIM of the position and orientation parameters in
$\tilde{\bm{{\eta}}}$ is given by [234]

$\mathbf{J}=\mathbf{T}\mathbf{J}_{\bm{\eta}}\mathbf{T}^{\rm H}.$ (143)

Once the FIM is obtained, the CRB is obtained by inverting the matrix. Thus,
the PEB is equal to the square root of the trace of the first $2\times 2$ sub-
matrix and can be formulated as

${\rm PEB}=\sqrt{\mathrm{tr}\left(\mathbf{J}_{1:2,1:2}^{-1}\right)},$ (144)

and the REB is equal to the square root of the third entry of diagonal

${\rm REB}=\sqrt{\mathrm{tr}\left(\mathbf{J}_{3,3}^{-1}\right)}.$ (145)

Simulation results: To illustrate the position accuracy that can be obtained
in the presence of RISs, we report some numerical results based on the
measurements of the TDoA in (132). Specifically, we compare the root mean
square error (RMSE) of the position estimation obtained by using Taylor and
Chan’s algorithms against the the CRB. The locations in metre of the BS and MU
are ${\bf q}=[0,0,40]^{T}$ and ${\bf p}=[90,30,0]^{T}$, respectively, and the
locations of three RISs are ${\bf s1}=[60,45,15]^{T}$, ${\bf
s2}=[50,50,5]^{T}$ and ${\bf s3}=[40,20,10]^{T}$. The setup with RISs is
considered as a case study. The phase shift matrix of the RIS is set to a unit
identity matrix for the sake of analysis, if the matrix is optimized, the
estimation accuracy will be improved. The other simulation parameters are
given in the caption of the figure, where $\sigma_{td}$ denotes the variance
of the Gaussian estimation error $n_{td_{k}}$. It can be seen that Taylor’s
and Chan’s algorithms achieve almost the same performance and have negligible
performance loss compared with the CRB.

Figure 8: RMSE versus the standard variance $\sigma_{td}$, when $N_{t}=32$,
$N_{r}=8$, $L^{2}=16\times 16$, $f_{c}=9$ GHz, and $K=4$.

###  IV-B Near-Field Localization Techniques

Thanks to the absence of power amplifiers, digital processing units, and RF
chains for each scattering element, a large number of quasi passive reflecting
elements can be integrated on the RIS panel at an affordable cost and power
consumption. Consequently, an RIS may evolve into a panel with a very large
size, which is sometimes called XL-RIS.

In this case, as illustrated in Fig. 9, the RIS may be large enough and the
observation point may be sufficiently close to it that the far-field planar
wave assumption may not hold anymore. Specifically, the spherical curvature of
the wavefront may not be ignored. In these cases, specifically, a signal
formulation that accounts for the near-field is needed. According to [235],
the far-field of the RIS can be defined as the set of observation distances
$R$ that are greater than the Fraunhofer distance $R_{f}$, i.e.,

$R\geq R_{f}=\frac{2L^{2}}{\lambda},$ (146)

where $L$ is the maximum aperture of the RIS UPA and $\lambda$ is the carrier
wavelength. This conventional definition of far-field region corresponds to a
maximum phase error equal to $\pi/8$ over the entire length $L$.

Figure 9: Illustration of the Fraunhofer distance and the far-field planar
wave approximation.

According to (146), the Fraunhofer distance can be quite large if an XL-RIS is
utilized and its size is sufficiently large with respect to the wavelength. If
the RIS is deployed at high-frequency bands, such as mmwave/sub-THz, the
Fraunhofer distance also increases due to the smaller wavelength [236, 237,
238]. For instance, let us consider a $100\times 100$-UPA RIS panel that
operates at the carrier frequency of $200$ GHz. The RIS elements are assumed
to be spaced by one half-wavelength, i.e. $\frac{\lambda}{2}=0.75$ mm, and,
thus, the maximum aperture (the diagonal) of the RIS panel is $\sqrt{2}\times
100\times\frac{\lambda}{2}\approx 10.6$ cm. Then, the Fraunhofer distance for
this scenario is $R_{f}=7.5$ meters. Therefore, it is likely that some users
are in the near-field when the RIS is deployed for indoor scenarios. In this
case, the spherical wavefront of the electromagnetic waves cannot be ignored.
Recent system-level simulations have shown that the near-field region may not
be ignored in outdoor scenarios either [239].

####  IV-B1 Near-Field Channel Model

In the near-field region, spherical wave-front provide an underlying generic
parametric model for estimating the positions of MUs and scatterers [240].
Under the assumption of a spherical wave-front, the steering responses of the
RIS and the BS arrays need to be parameterized by taking into account the 3D
location of the signal source rather than by simply utilizing the AoDs and the
AoAs.

As an example, let us consider the uplink transmission where MU is equipped
with one antenna, the RIS is an $M_{x}\times M_{z}$ UPA that lies on the XOZ
plane, and the BS is equipped with an $N$-element ULA. In the near-field, the
channel from the MU to the RIS is

$\displaystyle\bm{h}_{r}=\sum\nolimits_{l=0}^{L_{\mathrm{RU}}}\tilde{\beta}_{l}\bm{a}_{R}(\bm{r}_{l})=\bm{A}_{\mathrm{R}}(\bm{r}_{l})\bm{\tilde{\beta}},$
(147)

where $L_{\mathrm{RU}}$ is the number of scatterers or spatial paths, the
vector
$\bm{\tilde{\beta}}=[\tilde{\beta}_{0},\ldots,\tilde{\beta}_{L_{\mathrm{RU}}}]^{\mathrm{T}}$
contains the complex path gains, and the matrix
$\bm{A}_{\mathrm{R}}(\bm{r}_{l})=[\bm{a}_{\mathrm{R}}\left(\bm{r}_{0}\right),\ldots,\bm{a}_{R}(\bm{r}_{L_{\mathrm{RU}}})]$
collects the steering vectors associated with the scatterers that produce the
multiple propagation paths. The LoS path from the MU to the RIS is denoted as
path $0$, and the location of the MU is denoted as
$\bm{r}_{0}=[x_{0},y_{0},z_{0}]^{T}$. The steering vector associated with the
$l$-th scatterer located at ${\bm{p}}_{l}=[x_{l},y_{l},z_{l}]^{T}$ is denoted
as $\bm{a}({\bm{p}}_{l})$, and is given by

$\displaystyle\\!\\!\\!\\!\\!\\!\bm{a}({\bm{p}}_{l})$
$\displaystyle=[e^{-j\frac{2\pi}{\lambda}d_{1,1}({\bm{p}}_{l})},\cdots,e^{-j\frac{2\pi}{\lambda}d_{1,M_{z}}({\bm{p}}_{l})},\cdots,$
$\displaystyle\qquad\quad
e^{-j\frac{2\pi}{\lambda}d_{M_{x},1}({\bm{p}}_{l})},\cdots,e^{-j\frac{2\pi}{\lambda}d_{M_{x},M_{z}}({\bm{p}}_{l})}]^{\rm
T},$ (148)

where $d_{m,n}({\bm{p}}_{l})$ represents the distance from the scatterer
located at ${\bm{p}}_{l}$ to the $(m,n)$-th RIS element.

We concentrate on the LoS channel between the RIS and the AP and assume that
the communication channel is slowly varying and frequency-flat. By taking the
spherical nature of the wave propagation, the path length from each antenna
elements of the BS to each elements of the RIS determines the phase shifts of
the received signals. Similar to [241] and [242], we assume that the path loss
of all the received paths is the same and is denoted by $\alpha$. Under this
assumption, the elements of the BS-RIS LoS channel are collected in the matrix
$\bm{\bm{H}}$ whose elements are

$\bm{\bm{H}}(m,n)=\alpha\exp\left(j\frac{2\pi}{\lambda}r_{m,n}\right),$ (149)

where $r_{m,n}$ denotes the path length between the $m$-th antenna element of
the BS and the $n$-th RIS element.

The signal transmitted by the MU is denoted by $s$ and the received data at
the BS is

$\bm{y}=\bm{H}{\bf{\Theta}}\bm{h}_{r}s+\bm{n},$ (150)

where $\bm{n}\sim{\cal C}{\cal N}\left({{\bf{0}},{\sigma^{2}}\bf{I}}\right)$
and ${\bf{\Theta}}$ denote the AWGN and the reflection coefficient matrix
defined in Section II, respectively.

The set of unknown parameters are

$\displaystyle\bm{\eta}$
$\displaystyle=[{\bm{p}}_{0},{\bm{p}}_{1}\cdots,{\bm{p}}_{L_{\mathrm{RU}}},\bm{\beta}^{T}]^{T},$
(151)

where
$\bm{\beta}=[\alpha\tilde{\beta}_{0},\ldots,\alpha\tilde{\beta}_{L_{\mathrm{RU}}}]^{\mathrm{T}}$
denotes the vector of cascaded channel fading coefficients.

A commonly adopted localization method is the maximum-likelihood estimation.
The log-likelihood function of the received signal ${\bf y}$ is given by
$f\left(\bm{y}|\bf{\Theta},\bm{\eta}\right)\propto-||\bm{y}-{\bf{H}}{\bf{\Theta}}{\bf{h}}_{r}^{\rm{H}}(\bm{\eta})s||^{2}$,
where irrelevant constant terms are ignored. Then, the maximum likehood
estimation problem for the locations of the MU and the scatterers is given by

$\displaystyle\underset{\bm{\eta},\bf{\Theta}}{\max}\;\;f\left(\bm{y}|\bf{\Theta},\bm{\eta}\right)=\underset{\bm{\eta},\bf{\Theta}}{\min}\;\;||\bm{y}-{\bm{H}}{\bf{\Theta}}{\bm{h}}_{r}(\bm{\eta})s||^{2}$
(152) $\displaystyle\text{s.t.}\quad\text{C1}:\bm{\eta}\in{{\cal S}_{\eta}},$
$\displaystyle\quad\quad\;\text{C2}:{\theta_{m}}\in{{\cal S}_{1}}\ {\rm{or}}\
{{\cal S}_{2}},\forall m=1,2,\dots,M,$

where ${\cal S}_{\eta}$ denotes the set of feasible locations.

####  IV-B2 Near-Field Localization Scheme

The location of the MU can be estimated via a two-stage near-field
localization scheme. As the phase shift of the RIS can be obtained as
$\hat{\bf{\Theta}}$ by leveraging the methods discussed in Section III, here
we only focus on the localization problem. In the first stage, the coarse
estimation of the positions of MU and scatterers is derived [226, 243]. In the
second stage, the finer estimation of the positions of MU and scatterers can
be estimated with the iterative methods, e.g., two-dimensional search [244].
The details of the two-stage near-field localization scheme are outlined
below.

_Stage 1_ : Suppose that the transmit signal is $s=1$. As the received power
from the scatterer-reflected paths is much weaker than the LoS path, we first
try to find the location of MU based on the LoS path with the largest received
power. In the first stage, the location of the MU is estimated as follows:

$\hat{{\bm{p}}}_{0}=\arg\underset{{\bm{p}}_{0}}{\min}\;\;||\bm{y}-{\bm{H}}\hat{\bf{\Theta}}\bm{a}_{R}({\bm{p}}_{0})||^{2}.$
(153)

Without any prior information for the source location, the objective function
in (153) needs to be evaluated over a set of grid locations. For the sake of
reducing complexity, a non-uniform grid may be used. Specifically, the angle
and distance are sampled in the polar domain using a uniform grid in [226]. In
general, the grids should be dense near the array and sparse away from the
array[243].

With the obtained location, the corresponding channel gain can be obtained by
using the projection method as

$\hat{\beta}_{0}=\frac{({\bm{H}}\hat{\bf{\Theta}}\bm{a}_{R}(\hat{{\bm{p}}}_{0}))^{H}\bm{y}}{||\bm{H}\hat{\bf{\Theta}}\bm{a}_{R}(\hat{{\bm{p}}}_{0})||^{2}}.$
(154)

Then, the locations of the scatterers are estimated as

$\hat{{\bm{p}}}_{l}=\arg\underset{{\bm{p}}_{l}}{\min}\;\;\left\|\
\bm{y}_{-l}-{\bm{H}}\hat{\bf{\Theta}}\bm{a}_{R}({\bm{p}}_{l})\right\|^{2},$
(155)

where
$\bm{y}_{-l}=\bm{y}-\sum_{s=1}^{l-1}{\bm{H}}\hat{\bf{\Theta}}\hat{\beta}_{s}\bm{a}_{R}(\hat{{\bm{p}}}_{s})$,
and the corresponding channel gain is obtained as

$\hat{\beta}_{l}=\frac{({\bm{H}}\hat{\bf{\Theta}}\bm{a}_{R}(\hat{{\bm{p}}}_{l}))^{H}\bm{y}_{-l}}{||\bm{H}\hat{\bf{\Theta}}\bm{a}_{R}(\hat{{\bm{p}}}_{l})||^{2}}.$
(156)

_Stage 2_ : With the coarse stimation, a refined estimation can be further
conducted in the second stage. The position of the MU and the scatterers can
be refined using a finer grid while keeping the estimation of other locations
fixed. Specifically,

$\displaystyle\underset{{{\bm{p}}}_{l}}{\min}\;\;||\bm{y}-{\bm{H}}{\bf{\Theta}}{\bm{h}}_{r}({{\bm{p}}}_{l},\bm{\eta}_{-l})||^{2},$
(157)

where
$\bm{\eta}_{-l}=[\bm{\hat{r}}_{0},\cdots,\bm{\hat{r}}_{l-1},\bm{\hat{r}}_{l+1},\cdots,\bm{\hat{r}}_{L_{\mathrm{RU}}},\hat{\bm{\beta}}^{T}]^{T}$.
Several iterations are needed for high-resolution location estimates.

Simulation results: For the near field localization case, we evaluate the
localization performance in terms of the MSE, which is obtained by the
proposed two-stage near-field localization scheme. The locations of BS and RIS
are ${\bf q}=[3,0,3]^{T}$ and ${\bf p}=[0,0.5,2]^{T}$, while the locations of
the MU and scatterers are randomly generated in the $5$ m $\times$ $5$ m cell
range. The number of RIS elements is $21\times 21$, and the phase shift matrix
is obtained by maximizing the received signal power. It can be seen that the
MSE performance degrades with the number of scatterers due to the more
scattered energy and the feedback estimation error propagation, while
increasing the SNR can help improve the localization accuracy.

Figure 10: MSE versus the SNR, when $N_{t}=32$ and $f_{c}=240$ GHz.

##  V Future Directions

In this section, we summarize some open research issues that are focused on
the modeling, the analysis, and the optimization of RIS-aided wireless
systems.

###  V-A Mobility

Most existing works focus on quasi-static scenarios where the users and the
RISs are assumed to be nearly stationary. However, it is also imperative to
investigate scenarios with mobility [245, 246, 247, 248, 70], such as wireless
systems in the presence of vehicles, trains, and UAVs. In these scenarios, in
particular, the RISs may even be mobile as they may be carried by UAVs or they
may be deployed in trains. In these scenarios, several challenging research
problems emerge. For example, the estimation of rapidly time-varying channels
becomes more challenging. Therefore, new channel estimation methods are
required to quickly and accurately track the channels, in order to enable the
design of accurate phase shifts at the RIS. Also, since the channel is rapidly
changing, the channel estimation and the RIS phase shift configuration need to
be executed more frequently. As a result, it is crucial to design effective
signal processing schemes to avoid a prohibitively high overhead in high-
mobility scenarios. Furthermore, the RIS has the potential to reduce the
Doppler and delay spread [247], which deserves further research in more
practical scenarios.

###  V-B Near-Field Channel

To ensure a sufficiently large coverage, the aperture of the RIS planar array
is typically large in order to compensate for the absence of power amplifiers
and digital signal processing units for regenerating the signals. Depending on
the size of the RIS, the operating frequency, and its distance from the
transmitter and the receiver, it may be necessary to utilize a near-field
communication model [22, 249, 250, 251]. Near-field communication models are,
however, less understood than their far-field counterparts. Therefore, it is
essential to consider accurate channel models to unveil the fundamental
performance limits and the scaling laws of RISs in this context. It is well
known, for example, that the scaling laws of the electromagnetic field
radiated by the RIS is different in the near-field and far-field regions[252].
Also, new low-overhead channel estimation strategies are needed since the
channel sparsity in the angle domain, which is based on the far-field planar
wavefront assumption, may no longer hold when using the near-field spherical
wavefront assumption [226].

###  V-C Active RIS

Quasi Passive RIS architectures have attracted significant attention due to
their low hardware cost and energy consumption. Nevertheless, quasi passive
architectures have their limitations. Since the signals reflected by the RIS
is determined by the product of the distances from the transmitter to the RIS
and from the RIS to the receiver in the far-field region, the received signal
strength is relatively small compared with that from direct links, especially
when the direct links are strong. The recently proposed active RIS
architecture is a promising solution to overcome this issue [253, 254, 255,
256]. After integrating active reflection-type amplifiers[253], in addition to
adjusting the phase shifts, active RISs can simultaneously amplify the
magnitude of the reflected signals. However, the amplification at the active
RIS requires an additional power supply. Therefore, more dedicated beamforming
at the BS and phase shifts at the RIS should be designed to balance the
performance and energy consumption. Meanwhile, channel estimation with active
RISs needs to incorporate the statistical properties of amplification-induced
noise, which is more challenging and worth further investigation. Besides, how
to decide the optimal deployment of the active RISs is still an open question.

###  V-D Double/multi-RIS

Most of the existing works have considered quasi passive beamforming designs
and channel estimation schemes in systems with a single RIS. In some
scenarios, however, it may be convenient to enable the transmission of signals
through reflections from multiple RISs in order to route the signals and
bypass the blocking objects in a smart manner, directly at the electromagnetic
level (electromagnetic routing)[5]. Thus, double/multiple RISs may be utilized
to realize a blockage-free communication network via multiple signal
reflections. Moreover, the proper design of the cooperative quasi passive
beamforming can eliminate the inter-RIS interference, and achieve
multiplicative beamforming gain from the inter-RIS reflection channel [257,
258, 259, 260]. However, these methods require highly accurate CSI, which is
challenging to obtain due to the coupling of the different reflected signal
links and more channel coefficients to be estimated [261, 262, 263]. In this
context, channel estimation methods in double/multi-RIS aided communication
scenarios in the presence of single/double/multi-reflection links need to be
investigated. In addition, existing research works on double/multi-RIS aided
systems often ignore the impact of the secondary reflections among the RISs,
which may be a reasonable approximation if the RISs are in the far-field of
each other but it may not hold anymore if the RISs are closely located.
Finally, the analysis and design of multi-RIS communications at high frequency
bands is an open research issue as well.

###  V-E Multifunction RISs

In the existing literature, most of the research works concentrate on RISs
that operate as anomalous reflectors or as reflecting lenses. An RIS, however,
can realize multiple signal transformations depending on how the scattering
matrix (or equivalently the surface impedance) is designed[5]. Recently,
notably, a few research attempts have been made to design RISs that operate as
anomalous refracting mirrors or as anomalous refracting lenses[264] as well
RISs that can simultaneously realize reflections and refractions in order to
guarantee omni-coverage performance[265, 266, 267, 268]. Multifunction RISs
are an emerging research topic, and the corresponding modeling, performance
evaluation, and optimization are still at its infancy.

##  VI Conclusions

In this paper, we provided a comprehensive overview of state-of-the-art
research for new and revolutionary RIS/IRS-aided wireless systems, with an
emphasis on signal processing techniques for solving various channel
estimation, transmission design and radio localization problems. Specifically,
we first reviewed existing results on channel estimation under unstructured
and structured channel models. Next, we provided a detailed overview of the
research results from the perspective of different optimization techniques and
availability of CSI. In particular, several optimization techniques were
described for optimizing RIS with discrete and continuous time shifts. As far
the availability of CSI is concerned, three main cases were considered,
namely, fully instantaneous CSI, two-timescale CSI, and fully long-term CSI.
Simulation results demonstrated that the two-timescale CSI scheme constitutes
a promising approach when the pilot overhead is taken into account.
Furthermore, radio localization is an important application of RISs, and it
has been thoroughly reviewed by considering deployment scenarios and channel
models that account for far-field and near-field propagation. Finally, several
open research problems have been discussed.

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