# Location-aware Channel Estimation for RIS-aided mmWave MIMO Systems via
Atomic Norm Minimization ††thanks: This research was supported by the MSIP
(Ministry of Science, ICT and Future Planning), Korea, under the ITRC
(Information Technology Research Center) support program
(IITP-2021-2017-0-01637) supervised by the IITP (Institute for Information &
communications Technology Promotion).
Hyeonjin Chung and Sunwoo Kim are with the Department of Electronics and
Computer Engineering, Hanyang University, Seoul, 04763, South Korea (email:
<EMAIL_ADDRESS>remero@hanyang.ac.kr).

Hyeonjin Chung and Sunwoo Kim

###### Abstract

In this paper, we propose a location-aware channel estimation based on the
atomic norm minimization (ANM) for the reconfigurable intelligent surface
(RIS)-aided millimeter-wave multiple-input-multiple-output (MIMO) systems. The
beam training overhead at the base station (BS) is reduced by the direct beam
steering towards the RIS with the location of the BS and the RIS. The RIS
beamwidth adaptation is proposed to reduce the beam training overhead at the
RIS, and also it enables accurate channel estimation by ensuring the user
equipment receives all the multipath components from the RIS. After the beam
training, the cascaded effective channel of the RIS-aided MIMO systems is
estimated by ANM. Depending on whether the beam training overhead at the BS or
at the RIS is reduced or not, the channel is represented as a linear
combination of either 1D atoms, 2D atoms, or 3D atoms, and the ANM is applied
to estimate the channel. Simulation results show that the proposed location-
aware channel estimation via 2D ANM and 3D ANM achieves superior estimation
accuracy to benchmarks.

###### Index Terms:

Reconfigurable intelligent surface, atomic norm minimization, multi-
dimensional atomic norm, beamwidth adaptation, channel estimation, low-
overhead, MIMO

##  I Introduction

A millimeter-wave (mmWave) has been considered as a solution to accommodate
increasing data traffic and frequency spectrum shortage [1]. Over few years,
5G NR and IEEE 802.11 have worked on a standard for mmWave communications, and
the standardization is still in progress up to date [2, 3]. Moreover, a
communication on terahertz (THz) frequency band has been pointed out as one of
the candidate technologies for 6G, where 6G is expected to support heavy
applications such as virtual reality, augmented reality, and holograms [4].
However, mmWave and THz signals suffer from high path loss and are susceptible
to blockages due to the weak diffraction. To address the high path loss, a
massive antenna array is deployed to form a highly directional beam, which can
extend the coverage [5]. On the other hand, the weak diffraction is still a
difficult issue to solve since the strength of the mmWave and the THz signal
significantly drops if the line-of-sight (LoS) path is blocked.

Recently, a reconfigurable intelligent surface (RIS) has been studied to
address the weak diffraction [6]. Normally, the RIS is made up of either
reflectarray or metasurface, and each passive element or unit cell can
programmably change the propagation characteristic of the incident wave, such
as phase, reflection angle, and refraction angle [7]. In the seminal works on
RIS-aided mmWave and THz communications, the RIS is located in the place where
a base station (BS) is visible and focuses the incident wave towards user
equipment (UE) via reflect beamforming [8, 9, 10]. By this way, the signal can
bypass the blockage between the BS and the UE so that the communication
becomes more reliable. A concept of RIS-aided mmWave or THz communications has
generated various new research area, including channel estimation for RIS-
aided systems [8], reflect beamforming design [9], RIS-aided UAV
communications [10], aerial RIS [11], and RIS-aided localization [12].

The channel for RIS-aided systems can be represented as a cascade of two
channels: BS-to-RIS channel and RIS-to-UE channel. Generally, estimating BS-
to-RIS channel and RIS-to-UE channel separately is difficult since the RIS
only consists of passive elements [13]. Thus, the most of works estimate the
cascaded effective channel instead, where the cascaded effective channel
denotes a combination between the BS-to-RIS channel and the RIS-to-UE channel.
To estimate the cascaded effective channel for RIS-aided systems, the BS, the
RIS, and the UE have to perform beam search over the entire angular domain,
and this results in excessive beam training overhead. For this reason, the
large beam training overhead has been considered as one of major issues of the
channel estimation for RIS-aided systems.

The algorithm in [14] utilizes the property of the BS-to-RIS channel and the
RIS-to-UE channel to reduce the beam training overhead, where the BS-to-RIS
channel is static compared to the RIS-to-UE channel. With the BS that supports
the full-duplex operation, the static BS-to-RIS channel is estimated less
frequently than the RIS-to-UE channel. In [8, 15], the channel estimation
algorithms based on the compressive sensing are proposed. Without excessive
beam training overhead, these algorithms can estimate channel parameters such
as angle-of-departures (AoDs), angle-of-arrivals (AoAs), and channel gains.
However, the channel estimation accuracy of [8, 15] is limited by a grid-
mismatch problem [16]. To enhance the channel estimation accuracy, the channel
estimation algorithm based on the 1D atomic norm minimization (ANM) is
proposed in [17], where the 1D ANM is free from the grid-mismatch problem
[18]. Here, BS-to-RIS AoDs and RIS-to-UE AoAs are estimated by 1D ANM. Then,
the remaining channel parameters are estimated. In [19], the channel
estimation based on the location information and the 1D ANM has been proposed.
Assuming the LoS path always exists, and the location of the BS, the RIS, the
UE, and the objects surrounding UE are given, the algorithm in [19] reduces
the beam training overhead by restricting the angular domain to search. The
AoDs/AoAs are estimated via 1D ANM after the beam training.

Still, there are some issues that have not been considered and addressed in
the related works. Unlike [19], the LoS path between the RIS and the UE may
not exist when the UE hides behind the structure, and obtaining the location
of objects surrounding UE may not be feasible in practice. Also, if the beam
training overhead at the RIS is reduced, the UE may not receive some multipath
components from the RIS. Although this issue makes the channel estimation
inaccurate, it is not properly discussed in [8, 15, 17], and its solution has
not been proposed. In [17, 19], the 1D ANM is employed to estimate the
AoDs/AoAs, however, the cascaded effective channel can be directly estimated
if the atoms and the atomic set are modeled with proper dimension. In this
paper, we propose a location-aware channel estimation based on the ANM, which
addresses aforementioned issues. Main distinctions and contributions of this
paper are summarized as follows:

  * •

The proposed algorithm exploits the location of the BS and the RIS to reduce
the beam training overhead. During the beam training, the BS directly steers
the beam towards the RIS.

  * •

When the beam training overhead at the RIS is reduced, a proposed RIS
beamwidth adaptation widens a reflect beam created by RIS to ensure the UE
receives all the multipath components from the RIS.

  * •

The proposed algorithm estimates the cascaded effective channel via multi-
dimensional ANM [20]. When the BS steers the beam towards the RIS using
location information, we reveal that the channel is represented as a linear
combination of 2D atoms or 3D atoms.

The rest of the paper is organized as follows. Section II defines the channel
model for the downlink RIS-aided systems. Section III presents the beam
training procedures for both non location-aware and location-aware scenarios.
Section IV introduces the RIS beamwidth adaptation which reduces beam training
overhead at the RIS. Section V introduces the atomic norm and proposes non
location-aware channel estimation and location-aware channel estimation which
are based on ANM. Section VI provides the simulation results and the analysis,
and Section VII concludes this paper.

Notations: We use lower-case and upper-case bold characters to respectively
represent vectors and matrices throughout this paper. $(\cdot)^{T}$,
$(\cdot)^{H}$, and $(\cdot)^{*}$ respectively denote transpose, conjugate
transpose, and complex conjugation. $(\cdot)^{-1}$ denotes the inverse of a
matrix. $\textrm{Tr}(\cdot)$ denotes the trace of a matrix, and
$\textrm{diag}(\cdot)$ denotes the diagonal matrix whose diagonal entries
equal to entries of given vector. $\textrm{vec}(\cdot)$ denotes vectorization
of given matrix. $\lVert\cdot\rVert_{2}$ and $\lVert\cdot\rVert_{\textrm{F}}$
respectively denote L2 norm and Frobenius norm. The curled inequality symbol
$\succeq$ denotes matrix inequality. If $\mathbf{A}\succeq\mathbf{B}$, a
matrix $\mathbf{A}-\mathbf{B}$ is positive semidefinite. $\otimes$ and
$\diamond$ respectively denote Kronecker product and Khatri-Rao product.
$\mathbf{0}_{N}$ and $\mathbf{I}_{N}$ respectively denote a $N\times 1$ zero
vector and a $N\times N$ identity matrix. $\mathcal{CN}(\bm{\mu},\bm{\Sigma})$
denotes a circularly-symmetric complex Gaussian distribution whose mean is
$\bm{\mu}$ and covariance is $\bm{\Sigma}$.

##  II Signal Model

We consider a downlink RIS-aided mmWave multiple-input-multiple-output (MIMO)
system, which means that the BS transmits the signal to the RIS and the RIS
bounces back the signal to the UE. The BS, the RIS, and the UE are equipped
with $M_{\textrm{B}}$, $M_{\textrm{R}}$, and $M_{\textrm{U}}$ antennas
respectively. Here, antenna arrays that BS, RIS, and UE use are uniform linear
arrays (ULAs) with half-wavelength spacing. In this paper, the BS and the UE
employ full-complexity hybrid beamforming structure [21], where the BS and the
UE are respectively equipped with $N_{\textrm{B}}$ and $N_{\textrm{U}}$ RF
chains. Owing to the property of the full-complexity hybrid beamforming, the
BS and the UE can simultaneously form up to $N_{\textrm{B}}$ beams and
$N_{\textrm{U}}$ beams.

The steering vector of the ULA with half-wavelength spacing,
$\mathbf{a}(\theta)$ is

$\mathbf{a}(\theta)=[1,e^{j\pi\cos\theta},\ldots,e^{j\pi(M-1)\cos\theta}]^{T}\in\mathbb{C}^{M\times
1},$ (1)

where $\theta$ denotes the steering direction, and $M$ denotes the number of
antennas. A scheme of RIS-aided mmWave MIMO system and propagation paths is
given in Fig. 1. Assuming all signal paths between BS and UE are blocked, a
channel for RIS-aided mmWave MIMO system can be represented as a cascade of
two separate channels: BS-to-RIS channel and RIS-to-UE channel. The BS-to-RIS
channel $\mathbf{H}_{\textrm{BR}}$ can be given by

$\begin{split}\mathbf{H}_{\textrm{BR}}&=\sum_{l=1}^{L_{\textrm{BR}}}\alpha_{\textrm{BR}}^{l}\mathbf{a}(\phi_{\textrm{BR}}^{l})\mathbf{a}(\theta_{\textrm{BR}}^{l})^{H}\\\
&=\mathbf{A}(\bm{\phi}_{\textrm{BR}})\textrm{diag}(\bm{\rho}_{\textrm{BR}})\mathbf{A}(\bm{\theta}_{\textrm{BR}})^{H}\in\mathbb{C}^{M_{\textrm{R}}\times
M_{\textrm{B}}},\end{split}$ (2)

where $L_{\textrm{BR}}$ denotes the number of signal paths between BS and RIS.
$\alpha_{\textrm{BR}}^{l}$, $\phi_{\textrm{BR}}^{l}$, and
$\theta_{\textrm{BR}}^{l}$ respectively denote the channel gain, the BS-to-RIS
AoA, and the BS-to-RIS AoD of the $l$-th signal path.
$\bm{\phi}_{\textrm{BR}}=\\{{\phi}^{1}_{\textrm{BR}},\ldots,{\phi}^{L_{\textrm{BR}}}_{\textrm{BR}}\\}$
and
$\bm{\theta}_{\textrm{BR}}=\\{{\theta}^{1}_{\textrm{BR}},\ldots,{\theta}^{L_{\textrm{BR}}}_{\textrm{BR}}\\}$.
$\mathbf{A}(\bm{\phi}_{\textrm{BR}})=[\mathbf{a}(\phi_{\textrm{BR}}^{1}),\ldots,\mathbf{a}(\phi_{\textrm{BR}}^{L_{\textrm{BR}}})]\in\mathbb{C}^{M_{\textrm{R}}\times
L_{\textrm{BR}}}$,
$\mathbf{A}(\bm{\theta}_{\textrm{BR}})=[\mathbf{a}(\theta_{\textrm{BR}}^{1}),\ldots,\mathbf{a}(\theta_{\textrm{BR}}^{L_{\textrm{BR}}})]\in\mathbb{C}^{M_{\textrm{B}}\times
L_{\textrm{BR}}}$, and
$\bm{\rho}_{\textrm{BR}}=[\alpha_{\textrm{BR}}^{1},\ldots,\alpha_{\textrm{BR}}^{L_{\textrm{BR}}}]^{T}$.
To bypass the blockage between BS and UE, we assume that the LoS path between
BS and RIS always exists. Also, BS and RIS are assumed to be placed where
there is no reflector or scatterer nearby. Thus, considering the surrounding
environment of BS and RIS and sparse propagation characteristic of mmWave
[22], there is only LoS path between BS and RIS so that $L_{\textrm{BR}}=1$.

The RIS-to-UE channel $\mathbf{H}_{\textrm{RU}}$ can be given by

$\begin{split}\mathbf{H}_{\textrm{RU}}&=\sum_{l=1}^{L_{\textrm{RU}}}\alpha_{\textrm{RU}}^{l}\mathbf{a}(\phi_{\textrm{RU}}^{l})\mathbf{a}(\theta_{\textrm{RU}}^{l})^{H}\\\
&=\mathbf{A}(\bm{\phi}_{\textrm{RU}})\textrm{diag}(\bm{\rho}_{\textrm{RU}})\mathbf{A}(\bm{\theta}_{\textrm{RU}})^{H}\in\mathbb{C}^{M_{\textrm{U}}\times
M_{\textrm{R}}},\end{split}$ (3)

where $L_{\textrm{RU}}$ denotes the number of signal paths between RIS and UE.
$\alpha_{\textrm{RU}}^{l}$, $\phi_{\textrm{RU}}^{l}$, and
$\theta_{\textrm{RU}}^{l}$ respectively denote the channel gain, the RIS-to-UE
AoA, and the RIS-to-UE AoD of the $l$-th signal path.
$\bm{\phi}_{\textrm{RU}}=\\{{\phi}^{1}_{\textrm{RU}},\ldots,{\phi}^{L_{\textrm{RU}}}_{\textrm{RU}}\\}$
and
$\bm{\theta}_{\textrm{RU}}=\\{{\theta}^{1}_{\textrm{RU}},\ldots,{\theta}^{L_{\textrm{RU}}}_{\textrm{RU}}\\}$.
$\mathbf{A}(\bm{\phi}_{\textrm{RU}})=[\mathbf{a}(\phi_{\textrm{RU}}^{1}),\ldots,\mathbf{a}(\phi_{\textrm{RU}}^{L_{\textrm{RU}}})]\in\mathbb{C}^{M_{\textrm{U}}\times
L_{\textrm{RU}}}$,
$\mathbf{A}(\bm{\theta}_{\textrm{RU}})=[\mathbf{a}(\theta_{\textrm{RU}}^{1}),\ldots,\mathbf{a}(\theta_{\textrm{RU}}^{L_{\textrm{RU}}})]\in\mathbb{C}^{M_{\textrm{R}}\times
L_{\textrm{RU}}}$, and
$\bm{\rho}_{\textrm{RU}}=[\alpha_{\textrm{RU}}^{1},\ldots,\alpha_{\textrm{RU}}^{L_{\textrm{RU}}}]^{T}$.
Unlike the assumptions related to the BS-to-RIS channel, we assume that there
can be multiple signal paths between RIS and UE since the UE is mobile and can
be located nearby reflectors or scatterers. Also, the LoS path between RIS and
UE may not exist when the UE is located behind the blockage. Due to these
reasons, $L_{\textrm{RU}}$ can be larger than $1$, where $L_{\textrm{RU}}$
depends on the surrounding environment.

Figure 1: A scheme of RIS-aided mmWave MIMO system and propagation paths. The
signal path between BS and UE is blocked.

A RIS control matrix $\bm{\Omega}$ can be given by

$\bm{\Omega}=\begin{bmatrix}\beta_{1}e^{j\vartheta_{1}}&0&\ldots&0\\\
0&\beta_{2}e^{j\vartheta_{2}}&\ldots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\ldots&\beta_{M_{\textrm{R}}}e^{j\vartheta_{M_{\textrm{R}}}}\end{bmatrix}\in\mathbb{C}^{M_{\textrm{R}}\times
M_{\textrm{R}}},$ (4)

where $\beta_{m}$ and $\vartheta_{m}$ respectively denote a reflection
coefficient and a phase shift of the $m$-th antenna in RIS.
$\vartheta_{m}\in[0,2\pi)$ and $\beta_{m}$ can be either $0$ or $1$, where $0$
and $1$ respectively denotes the deactivation and the activation of the $m$-th
antenna in the RIS. For a convenient representation of $\bm{\Omega}$, a RIS
control vector $\bm{\omega}$ is defined as

$\bm{\omega}=\left[\beta_{1}e^{j\vartheta_{1}},\beta_{2}e^{j\vartheta_{2}},\ldots,\beta_{M_{\textrm{R}}}e^{j\vartheta_{M_{\textrm{R}}}}\right]^{T}\in\mathbb{C}^{M_{\textrm{R}}\times
1}.$ (5)

Note that $\bm{\Omega}=\textrm{diag}(\bm{\omega})$. The cascaded channel for
RIS-aided MIMO system, $\mathbf{H}$ can be given by

$\mathbf{H}=\mathbf{H}_{\textrm{RU}}\bm{\Omega}\mathbf{H}_{\textrm{BR}}\in\mathbb{C}^{M_{\textrm{U}}\times
M_{\textrm{B}}}.$ (6)

##  III Beam Training Procedure for RIS-aided MIMO Systems

In this section, we introduce the location-aware beam training whose overhead
is reduced by the location of the BS and the RIS. Schemes of non location-
aware beam training and location-aware beam training are compared in Fig. 2.
Fig. 2(a) shows the procedure of the non location-aware beam training. Since
the BS does not know the BS-to-RIS AoD, the BS must perform exhaustive beam
search over entire angular domain. If the location information is given, the
BS-to-RIS AoD can be calculated so that the BS can form a beam towards RIS as
in Fig. 2(b).

After the beam training, the received pilot signals collected during the beam
training are organized for the channel estimation. Key properties for the
organization are given as follows.

  * •

Property 1:
$\textrm{vec}\left(\mathbf{A\textrm{diag}(\mathbf{b})C}\right)=(\mathbf{C}^{T}\diamond\mathbf{A})\mathbf{b}.$

  * •

Property 2:
$(\mathbf{AB}\diamond\mathbf{CD})=(\mathbf{A}\otimes\mathbf{C})(\mathbf{B}\diamond\mathbf{D}).$

  * •

Property 3:
$\textrm{vec}\left(\mathbf{ABC}\right)=(\mathbf{C}^{T}\otimes\mathbf{A})\textrm{vec}(\mathbf{B}).$

Definitions and properties of Kronecker product and Khatri-Rao product are
well-explained in [23].

###  III-A Non Location-aware Beam Training

A frame structure for non location-aware beam training in RIS-aided MIMO
systems is depicted in Fig. 3(a). To simplify notations,
$M_{\textrm{B}}/N_{\textrm{B}}$ and $M_{\textrm{U}}/N_{\textrm{U}}$ are
respectively defined as $P_{\textrm{B}}$ and $P_{\textrm{U}}$. Each frame
consists of multiple training symbols, and the RIS control matrix changes
frame by frame. There are $P_{\textrm{B}}$ precoding matrices and
$P_{\textrm{U}}$ combining matrices, and
$\mathbf{F}_{i}\in\mathbb{C}^{M_{\textrm{B}}\times N_{\textrm{B}}}$ and
$\mathbf{C}_{j}\in\mathbb{C}^{M_{\textrm{U}}\times N_{\textrm{U}}}$
respectively denote the $i$-th precoding matrix and the $j$-th combining
matrix. The precoding matrix and the combining matrix change by each training
symbol, so that one frame consists of $P_{\textrm{B}}P_{\textrm{U}}$ training
symbols. Here, the L2 norm of each precoding vector and combining vector is
$1$. Letting $B$ denotes the number of frames, the total number of training
symbols equals to $BP_{\textrm{B}}P_{\textrm{U}}$.

(a) Non location-aware beam training

(b) Location-aware beam training

Figure 2: Schemes of non location-aware beam training and location-aware beam
training.

$\mathbf{X}^{i,j}_{b}$, a received pilot signal at the $b$-th frame which uses
the $i$-th precoding matrix and the $j$-th combining matrix, can be given by

$\mathbf{X}^{i,j}_{b}=\mathbf{C}_{j}^{H}\mathbf{H}_{\textrm{RU}}\bm{\Omega}_{b}\mathbf{H}_{\textrm{BR}}\mathbf{F}_{i}\mathbf{S}+\mathbf{N}^{i,j}_{b}\in\mathbb{C}^{N_{\textrm{U}}\times
D},$ (7)

where $\bm{\Omega}_{b}$ denotes the RIS control matrix at the $b$-th frame,
and $D$ denotes the number of signal samples per one training symbol.
$\mathbf{S}=\left[\mathbf{s}_{1},\ldots,\mathbf{s}_{N_{\textrm{B}}}\right]^{T}\in\mathbb{C}^{N_{\textrm{B}}\times
D}$, where $\mathbf{s}_{n}$ is the $n$-th pilot signal that satisfies
$\mathbf{s}^{H}_{n}\mathbf{s}_{n}/D=P_{\textrm{Tx}}/N_{\textrm{B}}$. Here,
$P_{\textrm{Tx}}$ denotes the transmitting power of the BS, which is
equivalent to $\lVert\mathbf{S}\rVert^{2}_{\textrm{F}}/D$. The pilot signals
are orthogonal to each other so that
$\mathbf{S}\mathbf{S}^{H}/D=P_{\textrm{Tx}}\mathbf{I}_{N_{\textrm{B}}}/N_{\textrm{B}}$.
$\mathbf{N}^{i,j}_{b}\in\mathbb{C}^{N_{\textrm{U}}\times D}$ is a noise matrix
whose columns follow
$\mathcal{CN}(\mathbf{0}_{N_{\textrm{U}}},\sigma^{2}\mathbf{I}_{N_{\textrm{U}}})$,
where $\sigma^{2}$ denotes the power of the noise.

After receiving $\mathbf{X}^{i,j}_{b}$, $\mathbf{S}^{H}$ are multiplied to
$\mathbf{X}^{i,j}_{b}$ to filter the noise. We define this filtered signal
$\mathbf{Y}^{i,j}_{b}$ as

$\begin{split}\mathbf{Y}^{i,j}_{b}&=\frac{\mathbf{X}^{i,j}_{b}\mathbf{S}^{H}}{D}\\\
&=\frac{P_{\textrm{Tx}}}{N_{\textrm{B}}}\mathbf{C}_{j}^{H}\mathbf{H}_{\textrm{RU}}\bm{\Omega}_{b}\mathbf{H}_{\textrm{BR}}\mathbf{F}_{i}+\frac{\mathbf{N}^{i,j}_{b}\mathbf{S}^{H}}{D}\in\mathbb{C}^{N_{\textrm{U}}\times
N_{\textrm{B}}}.\end{split}$ (8)

A total $P_{\textrm{B}}P_{\textrm{U}}$ filtered signals received at the $b$-th
frame is organized as

$\mathbf{Y}_{b}=\begin{bmatrix}\mathbf{Y}^{1,1}_{b}&\mathbf{Y}^{2,1}_{b}&\cdots&\mathbf{Y}^{P_{\textrm{B}},1}_{b}\\\
\mathbf{Y}^{1,2}_{b}&\mathbf{Y}^{2,2}_{b}&\cdots&\mathbf{Y}^{P_{\textrm{B}},2}_{b}\\\
\vdots&\vdots&\cdots&\vdots\\\
\mathbf{Y}^{1,P_{\textrm{U}}}_{b}&\mathbf{Y}^{2,P_{\textrm{U}}}_{b}&\cdots&\mathbf{Y}^{P_{\textrm{B}},P_{\textrm{U}}}_{b}\end{bmatrix}\in\mathbb{C}^{M_{\textrm{U}}\times
M_{\textrm{B}}},$ (9)

where $\mathbf{Y}_{b}$ is a compilation of all filtered signals at the $b$-th
frame. $\mathbf{Y}_{b}$ can also be represented as

$\mathbf{Y}_{b}=\frac{P_{\textrm{Tx}}}{N_{\textrm{B}}}\mathbf{C}^{H}\mathbf{H}_{\textrm{RU}}\bm{\Omega}_{b}\mathbf{H}_{\textrm{BR}}\mathbf{F}+\mathbf{V}_{b},$
(10)

where $\mathbf{F}\in\mathbb{C}^{M_{\textrm{B}}\times M_{\textrm{B}}}$ and
$\mathbf{C}\in\mathbb{C}^{M_{\textrm{U}}\times M_{\textrm{U}}}$ respectively
denote a full-rank precoding matrix and a full-rank combining matrix.
$\mathbf{V}_{b}\in\mathbb{C}^{M_{\textrm{U}}\times M_{\textrm{B}}}$ is a
matrix that represents the remaining noise, and each column of
$\mathbf{V}_{b}$ follows
$\mathcal{CN}\left(\mathbf{0}_{M_{\textrm{U}}},(\sigma^{2}P_{\textrm{Tx}}/DN_{\textrm{B}})\mathbf{I}_{M_{\textrm{U}}}\right)$.

For the channel estimation, $\mathbf{Y}_{b}$ for $b=1,\ldots,B$ should be
organized to exploit the pilot signals from all directions. Letting a lengthy
column vector $\mathbf{y}_{b}$ equals $\textrm{vec}(\mathbf{Y}_{b})$,
$\mathbf{y}_{b}$ can be represented as follows by using Property 1.

$\begin{split}\mathbf{y}_{b}&=\textrm{vec}(\mathbf{Y}_{b})\\\
&=\frac{P_{\textrm{Tx}}}{N_{\textrm{B}}}\left(\mathbf{F}^{T}\mathbf{H}^{T}_{\textrm{BR}}\diamond\mathbf{C}^{H}\mathbf{H}_{\textrm{RU}}\right)\bm{\omega}_{b}+\mathbf{v}_{b}\in\mathbb{C}^{M_{\textrm{B}}M_{\textrm{U}}\times
1},\end{split}$ (11)

where $\bm{\omega}_{b}$ denotes the RIS control vector at the $b$-th frame,
and $\bm{\Omega}_{b}=\textrm{diag}(\bm{\omega}_{b})$.
$\mathbf{v}_{b}=\textrm{vec}(\mathbf{V}_{b})$, where $\mathbf{v}_{b}$ follows
$\mathcal{CN}\left(\mathbf{0}_{M_{\textrm{B}}M_{\textrm{U}}},(\sigma^{2}P_{\textrm{Tx}}/DN_{\textrm{B}})\mathbf{I}_{M_{\textrm{B}}M_{\textrm{U}}}\right)$.
Then, we form an organized matrix $\bm{\mathcal{Y}}$ which is constructed by
stacking $\mathbf{y}_{b}$ for $b=1,\ldots,B$ as follows.

$\begin{split}\bm{\mathcal{Y}}&=\left[\mathbf{y}_{1},\ldots,\mathbf{y}_{B}\right]\\\
&=\frac{P_{\textrm{Tx}}}{N_{\textrm{B}}}\left(\mathbf{F}^{T}\mathbf{H}^{T}_{\textrm{BR}}\diamond\mathbf{C}^{H}\mathbf{H}_{\textrm{RU}}\right)\mathbf{W}+\bm{\mathcal{V}}\in\mathbb{C}^{M_{\textrm{B}}M_{\textrm{U}}\times
B},\end{split}$ (12)

where
$\mathbf{W}=[\bm{\omega}_{1},\ldots,\bm{\omega}_{B}]\in\mathbb{C}^{M_{\textrm{R}}\times
B}$ and
$\bm{\mathcal{V}}=[\mathbf{v}_{1},\ldots,\mathbf{v}_{B}]\in\mathbb{C}^{M_{\textrm{B}}M_{\textrm{U}}\times
B}$. By using Property 2, $\bm{\mathcal{Y}}$ can be also represented as

$\bm{\mathcal{Y}}=\frac{P_{\textrm{Tx}}}{N_{\textrm{B}}}\left(\mathbf{F}^{T}\otimes\mathbf{C}^{H}\right)\left(\mathbf{H}^{T}_{\textrm{BR}}\diamond\mathbf{H}_{\textrm{RU}}\right)\mathbf{W}+\bm{\mathcal{V}}.$
(13)

Here,
$\mathbf{H}^{T}_{\textrm{BR}}\diamond\mathbf{H}_{\textrm{RU}}\in\mathbb{C}^{M_{\textrm{B}}M_{\textrm{U}}\times
M_{\textrm{R}}}$ contains a channel information that is independent of
$\mathbf{F}$, $\mathbf{C}$, and $\mathbf{W}$. Once
$\mathbf{H}^{T}_{\textrm{BR}}\diamond\mathbf{H}_{\textrm{RU}}$ is successfully
estimated, the optimal RIS control matrix that maximizes SNR can be derived by
conducting singular value decomposition (SVD) to
$\mathbf{H}^{T}_{\textrm{BR}}\diamond\mathbf{H}_{\textrm{RU}}$ [17].
Throughout this paper, we define
$\mathbf{H}^{T}_{\textrm{BR}}\diamond\mathbf{H}_{\textrm{RU}}$ as an effective
cascaded channel $\mathbf{H}_{\textrm{eff}}$, which is a goal of channel
estimation for RIS-aided MIMO systems.

With (2), (3), and Property 2, $\bm{\mathcal{Y}}$ can be fully unfolded as

$\begin{split}\bm{\mathcal{Y}}&=\frac{P_{\textrm{Tx}}}{N_{\textrm{B}}}\left(\mathbf{F}^{T}\otimes\mathbf{C}^{H}\right)\left(\mathbf{A}(\bm{\theta}_{\textrm{BR}})^{*}\otimes\mathbf{A}(\bm{\phi}_{\textrm{RU}})\right)\\\
&\left(\textrm{diag}(\bm{\rho}_{\textrm{BR}})\otimes\textrm{diag}(\bm{\rho}_{\textrm{RU}})\right)\left(\mathbf{A}(\bm{\phi}_{\textrm{BR}})^{T}\diamond\mathbf{A}(\bm{\theta}_{\textrm{RU}})^{H}\right)\mathbf{W}+\bm{\mathcal{V}}.\end{split}$
(14)

To simplify
$\mathbf{A}(\bm{\phi}_{\textrm{BR}})^{T}\diamond\mathbf{A}(\bm{\theta}_{\textrm{RU}})^{H}\in\mathbb{C}^{L_{\textrm{BR}}L_{\textrm{RU}}\times
M_{\textrm{R}}}$, $\bm{\varphi}$ is defined as

$\begin{split}\bm{\varphi}=\\{\varphi_{i,j}&:\cos^{-1}(\cos\theta^{j}_{\textrm{RU}}-\cos\phi^{i}_{\textrm{BR}}),\\\
&i=1,\ldots,L_{\textrm{BR}},j=1,\ldots,L_{\textrm{RU}}\\}.\end{split}$ (15)

Then,
$\mathbf{A}(\bm{\phi}_{\textrm{BR}})^{T}\diamond\mathbf{A}(\bm{\theta}_{\textrm{RU}})^{H}$
can be rewritten as

$\begin{split}&\mathbf{A}(\bm{\phi}_{\textrm{BR}})^{T}\diamond\mathbf{A}(\bm{\theta}_{\textrm{RU}})^{H}=\mathbf{A}(\bm{\varphi})^{H}\\\
&=\left[\mathbf{a}(\varphi_{1,1}),\ldots,\mathbf{a}(\varphi_{1,L_{\textrm{RU}}}),\ldots,\mathbf{a}(\varphi_{L_{\textrm{BR}},1}),\ldots,\mathbf{a}(\varphi_{L_{\textrm{BR}},L_{\textrm{RU}}})\right]^{H}.\end{split}$
(16)

To evaluate the efficiency of the beam training by quantifying the quality of
the pilot signals received during the beam training, we define an efficiency
of the non location-aware beam training, $\textrm{eBT}_{\textrm{NLA}}$ as
follows.

$\textrm{eBT}_{\textrm{NLA}}=\frac{\lVert\frac{P_{\textrm{Tx}}}{N_{\textrm{B}}}\left(\mathbf{F}^{T}\otimes\mathbf{C}^{H}\right)\left(\mathbf{H}^{T}_{\textrm{BR}}\diamond\mathbf{H}_{\textrm{RU}}\right)\mathbf{W}\rVert^{2}_{\textrm{F}}}{\lVert\bm{\mathcal{V}}\rVert^{2}_{\textrm{F}}}.$
(17)

The beam training efficiency defined in (17) is a ratio between the pilot
signal part and the noise part of $\bm{\mathcal{Y}}$. The beam training
efficiency is a factor that significantly affects the performance of the
channel estimation.

(a) Non location-aware beam training

(b) Location-aware beam training

Figure 3: A frame structure for non location-aware beam training and location-
aware beam training in RIS-aided MIMO communication.

###  III-B Location-aware Beam Training

A frame structure for location-aware beam training is depicted in Fig. 3(b).
Since $\theta^{1}_{\textrm{BR}}$ can be calculated by location of the BS and
the RIS, the beam training procedure can be simplified by setting a precoding
vector $\mathbf{f}$ as
$\mathbf{a}(\theta^{1}_{\textrm{BR}})/\sqrt{M_{\textrm{B}}}$ during the beam
training. When the location information of BS and RIS are given, there are
$P_{\textrm{U}}$ training symbols per frame, and the total number of training
symbols equals to $BP_{\textrm{U}}$. Note that one RF chain of the BS is used
during the beam training, and the UE still perform the exhaustive beam search
as in non location-aware beam training.

When the location information is given, a received pilot signal at the $b$-th
frame which uses the $i$-th combining matrix, $\mathbf{J}^{i}_{b}$ can be
given by

$\mathbf{J}^{i}_{b}=\mathbf{C}_{i}^{H}\mathbf{H}_{\textrm{RU}}\bm{\Omega}_{b}\mathbf{H}_{\textrm{BR}}\mathbf{f}\mathbf{s}+\mathbf{N}^{i}_{b}\in\mathbb{C}^{N_{\textrm{U}}\times
D},$ (18)

where $\mathbf{s}\in\mathbb{C}^{1\times D}$ is the pilot signal that satisfies
$\mathbf{s}\mathbf{s}^{H}/D=P_{\textrm{Tx}}$.
$\mathbf{N}^{i}_{b}\in\mathbb{C}^{N_{\textrm{U}}\times D}$ is a noise matrix
whose columns follow
$\mathcal{CN}(\mathbf{0}_{N_{\textrm{U}}},\sigma^{2}\mathbf{I}_{N_{\textrm{U}}})$.

After receiving $\mathbf{J}^{i}_{b}$, $\mathbf{s}^{H}$ are multiplied to
$\mathbf{J}^{i}_{b}$ to filter the noise, and the filtered signal
$\mathbf{j}^{i}_{b}$ can be given by

$\mathbf{j}^{i}_{b}=\frac{\mathbf{J}^{i}_{b}\mathbf{s}^{H}}{D}=P_{\textrm{Tx}}\mathbf{C}_{j}^{H}\mathbf{H}_{\textrm{RU}}\bm{\Omega}_{b}\mathbf{H}_{\textrm{BR}}\mathbf{f}+\frac{\mathbf{N}^{i}_{b}\mathbf{s}^{H}}{D}\in\mathbb{C}^{N_{\textrm{U}}\times
1}.$ (19)

Then, a total $P_{\textrm{U}}$ filtered signals received at the $b$-th frame
is organized as

$\mathbf{j}_{b}=\left[(\mathbf{j}^{1}_{b})^{T},(\mathbf{j}^{2}_{b})^{T},\ldots,(\mathbf{j}^{P_{\textrm{U}}}_{b})^{T}\right]^{T}\in\mathbb{C}^{M_{\textrm{U}}\times
1},$ (20)

where $\mathbf{j}_{b}$ is a compilation of all filtered signals at the $b$-th
frame. $\mathbf{j}_{b}$ can also be represented as

$\mathbf{j}_{b}=P_{\textrm{Tx}}\mathbf{C}^{H}\mathbf{H}_{\textrm{RU}}\bm{\Omega}_{b}\mathbf{H}_{\textrm{BR}}\mathbf{f}+\mathbf{u}_{b},$
(21)

where $\mathbf{u}_{b}\in\mathbb{C}^{M_{\textrm{U}}\times 1}$ is a vector that
represents the remaining noise and
$\mathbf{u}_{b}\sim\mathcal{CN}\left(\mathbf{0}_{M_{\textrm{U}}},(\sigma^{2}P_{\textrm{Tx}}/D)\mathbf{I}_{M_{\textrm{U}}}\right)$.

To exploit the received pilot signals from all directions, $\mathbf{j}_{b}$
for $b=1,\ldots,B$ are organized in the same manner that is introduced in
Section III-A. Since organizing $\mathbf{j}_{b}$ for $b=1,\ldots,B$ is not
much different from the organization in Section III-A, the detailed step of
organizing $\mathbf{j}_{b}$ for $b=1,\ldots,B$ is omitted. An organized matrix
$\bm{\mathcal{J}}$ is constructed by stacking $\mathbf{j}_{b}$ for
$b=1,\ldots,B$ as

$\begin{split}\bm{\mathcal{J}}&=\left[\mathbf{j}_{1},\ldots,\mathbf{j}_{B}\right]\\\
&=P_{\textrm{Tx}}\left(\mathbf{f}^{T}\mathbf{H}^{T}_{\textrm{BR}}\diamond\mathbf{C}^{H}\mathbf{H}_{\textrm{RU}}\right)\mathbf{W}+\mathbf{U}\\\
&=P_{\textrm{Tx}}\left(\mathbf{f}^{T}\otimes\mathbf{C}^{H}\right)\left(\mathbf{H}^{T}_{\textrm{BR}}\diamond\mathbf{H}_{\textrm{RU}}\right)\mathbf{W}+\mathbf{U}\in\mathbb{C}^{M_{\textrm{U}}\times
B},\end{split}$ (22)

where
$\mathbf{U}=[\mathbf{u}_{1},\ldots,\mathbf{u}_{B}]\in\mathbb{C}^{M_{\textrm{U}}\times
B}$. $\bm{\mathcal{J}}$ can be fully unfolded as

$\begin{split}\bm{\mathcal{J}}=&P_{\textrm{Tx}}\left(\mathbf{f}^{T}\otimes\mathbf{C}^{H}\right)\left(\mathbf{A}(\bm{\theta}_{\textrm{BR}})^{*}\otimes\mathbf{A}(\bm{\phi}_{\textrm{RU}})\right)\\\
&\left(\textrm{diag}(\bm{\rho}_{\textrm{BR}})\otimes\textrm{diag}(\bm{\rho}_{\textrm{RU}})\right)\mathbf{A}(\bm{\varphi})^{H}\mathbf{W}+\mathbf{U}.\end{split}$
(23)

As in (17), we define the efficiency of the location-aware beam training,
$\textrm{eBT}_{\textrm{LA}}$ as follows.

$\textrm{eBT}_{\textrm{LA}}=\frac{\lVert
P_{\textrm{Tx}}\left(\mathbf{f}^{T}\otimes\mathbf{C}^{H}\right)\left(\mathbf{H}^{T}_{\textrm{BR}}\diamond\mathbf{H}_{\textrm{RU}}\right)\mathbf{W}\rVert^{2}_{\textrm{F}}}{\lVert\mathbf{U}\rVert^{2}_{\textrm{F}}}.$
(24)

The beam training efficiency defined in (24) is a ratio between the pilot
signal part and the noise part of $\bm{\mathcal{J}}$.

(a) Without RIS beamwidth adaptation

(b) With RIS beamwidth adaptation

Figure 4: Reflecting radiation patterns created by RIS during the beam
training. $L_{\textrm{BR}}=1$, $L_{\textrm{RU}}=3$, $M_{\textrm{R}}=16$, and
$B=10$. For successful multipath signal reception, every RIS-to-UE AoD has to
be captured within one of reflect beams.

##  IV RIS Beamwidth Adaptation for Robust Low-Overhead Channel Estimation

During the beam training, the RIS must deliver the signal transmitted from the
BS to the UE by sweeping reflect beam over entire angular domain. If not,
there is a possibility that the UE does not receive the signal transmitted
from the BS, which leads to the channel estimation failure. A reflecting
radiation pattern of RIS, $\Gamma(\theta,\bar{\phi},\bm{\omega})$ can be given
by

$\Gamma(\theta,\bar{\phi},\bm{\omega})=\left|\mathbf{a}(\theta)^{H}\textrm{diag}(\bm{\omega})\mathbf{a}(\bar{\phi})\right|,\;\textrm{for}\;0^{\circ}<\bar{\phi}<180^{\circ},$
(25)

where $\theta$, $\bar{\phi}$, and $\bm{\omega}$ respectively denote the BS-to-
RIS AoA, the steering direction of the reflect beam, and the RIS control
vector. The received signal is boosted if the RIS-to-UE AoD is on the mainlobe
of the reflect beam. Otherwise, the signal is suppressed or nulled.

One of general ways to perform reflect beamforming is to use a discrete
Fourier transform (DFT) matrix [24]. $N\times N$ DFT matrix $\bm{\Psi}_{N}$
can be given by

$\bm{\Psi}_{N}=\begin{bmatrix}1&1&\cdots&1\\\
1&e^{j\frac{2\pi}{N}}&\cdots&e^{j\frac{2\pi(N-1)}{N}}\\\
1&e^{j\frac{4\pi}{N}}&\cdots&e^{j\frac{4\pi(N-1)}{N}}\\\
\vdots&\vdots&\cdots&\vdots\\\
1&e^{j2\pi}&\cdots&e^{j2\pi(N-1)}\end{bmatrix}\in\mathbb{C}^{N\times N}.$ (26)

For mainlobes of reflect beams to cover the entire angular domain during the
beam training, $\mathbf{W}$ should be equal to $\bm{\Psi}_{M_{\textrm{R}}}$ so
that $B=M_{\textrm{R}}$. However, considering the total number of training
symbol is proportional to $B$, $B$ should be reduced in order to prevent the
beam training overhead from getting excessively large. As in [15], a few
columns from $\bm{\Psi}_{M_{\textrm{R}}}$ can be selected to form
$\mathbf{W}$, but lack of reflect beams can cause an erroneous multipath
signal reception. Fig. 4(a) shows a case of the erroneous multipath signal
reception when none of RIS-to-UE signal paths falls onto the mainlobes of
reflect beams. In this case, the UE cannot receive multipath components during
the beam training, and the channel estimation fails since signals from all
paths are required for perfect channel estimation.

To address this issue, a RIS beamwidth adaptation is proposed to make
multipath signal reception robust when $B<M_{\textrm{R}}$. The width of
reflect beams can be widened by deactivating the part of RIS, and $\mathbf{W}$
that contains $B$ widened reflect beams can be given by

$\mathbf{W}=\begin{bmatrix}\bm{\Psi}_{B}\\\
\mathbf{O}_{M_{\textrm{R}}-B,B}\end{bmatrix}\in\mathbb{C}^{M_{\textrm{R}}\times
B},$ (27)

where $\mathbf{O}_{M,N}$ denotes a $M\times N$ zero matrix. Note that only $B$
antennas are activated, and other antennas are not. Fig. 4(b) shows beams that
are widened by RIS beamwidth adaptation when $M_{\textrm{R}}=16$ and $B=10$.
In Fig. 4(b), mainlobes of $B$ beams cover the entire angular domain so that
every RIS-to-UE signal path is captured within one of $B$ beams. However, the
deactivating the part of the RIS drops the power of the received signal and
may degrade an accuracy of the channel estimation.

For the following sections of this paper, notations $\mathbf{W}_{0}$ and
$\mathbf{W}_{1}$ are defined to distinguish whether the RIS beamwidth
adaptation is used or not. $\mathbf{W}_{0}$ stands for the set of the RIS
control vectors when the RIS beamwidth adaptation is not employed, so that
$\mathbf{W}_{0}=\bm{\Psi}_{M_{\textrm{R}}}$. $\mathbf{W}_{1}$ stands for the
set of the RIS control vectors when the RIS beamwidth adaptation is employed.
Also, $B$ is considered to be always less than $M_{\textrm{R}}$.

##  V Low-overhead Channel Estimation via Atomic Norm Minimization

In this section, we introduce the non location-aware channel estimation and
the location-aware channel estimation based on ANM. If the location
information of BS and RIS are not given, the channel can be estimated via 1D
ANM. When the location information of BS and RIS is given, there are two
options to estimate the channel depending on whether the RIS beamwidth
adaptation is used or not. Two options are:

  * •

Option 1: Perform exhaustive beam search at the RIS. In this case, 2D ANM is
employed for the channel estimation.

  * •

Option 2: Perform RIS beamwidth adaptation. If so, the beam training overhead
can be further reduced. In this case, 3D ANM is employed for the channel
estimation.

To ease the understanding of the atomic norm, an explanation of atom, atomic
set, and Toeplitz matrix is given in the following subsection.

###  V-A Atom, Atomic set, and Toeplitz Matrix

If the signal can be represented as the linear combination of basic
components, the atomic norm of the signal can be defined. Here, the basic
component is referred to as an atom, and a set of atoms is referred to as an
atomic set. The atomic norm has been studied in the seminal works, especially
for a case when the atom can be represented as the steering vector of the ULA
[18, 25]. In this case, the atomic set $\mathcal{A}_{\textrm{1D}}$ is
generally defined as

$\begin{split}\mathcal{A}_{\textrm{1D}}=&\big{\\{}\mathbf{a}(\theta)\mathbf{b}^{T}\in\mathbb{C}^{M\times
P}:0^{\circ}<\theta<180^{\circ},\\\
&\quad\mathbf{a}(\theta)\in\mathbb{C}^{M\times
1},\mathbf{b}\in\mathbb{C}^{P\times
1},\lVert\mathbf{b}\rVert_{2}=1\big{\\}},\end{split}$ (28)

where $\mathbf{a}(\theta)\mathbf{b}^{T}$ denotes the atom. A case when $P=1$
is referred to as single measurement vector (SMV) case [18], and a case when
$P>1$ is referred to as multiple measurement vector (MMV) case [25]. An atomic
norm that is derived based on $\mathcal{A}_{\textrm{1D}}$ is referred to as 1D
atomic norm. To represent the 1D atomic norm with SDP, a Hermitian Toeplitz
matrix is used, where a $M\times M$ Hermitian Toeplitz matrix $\mathbf{T}$ can
be given by

$\mathbf{T}=\begin{bmatrix}a_{1}&a^{*}_{2}&\ldots&a^{*}_{M-1}&a^{*}_{M}\\\
a_{2}&a_{1}&a^{*}_{2}&\ldots&a^{*}_{M-1}\\\
\vdots&a_{2}&\ddots&\ddots&\vdots\\\ a_{M-1}&\vdots&\ddots&\ddots&a_{2}\\\
a_{M}&a_{M-1}&\ldots&a_{2}&a_{1}\end{bmatrix}\in\mathbb{C}^{M\times M}.$ (29)

However, there are cases that the atom is modeled as a result of the Kronecker
product of multiple steering vectors [20]. In this paper, we refer the atomic
norm for such cases as a multi-dimensional atomic norm. Most well-known cases
that employ multi-dimensional atomic norm are joint azimuth/elevation
estimation [26] and MIMO channel estimation [27]. An atomic set comprised of
$N$-dimensional ($N$D) atoms can be defined as

$\begin{split}\mathcal{A}_{N\textrm{D}}=\big{\\{}\left(\mathbf{a}_{1}(\theta_{1})^{*}\otimes\ldots\otimes\mathbf{a}_{N}(\theta_{N})\right)\mathbf{b}^{T}\in\mathbb{C}^{(\prod_{n=1}^{N}M_{n})\times
P}:\\\ \mathbf{b}\in\mathbb{C}^{P\times
1},\lVert\mathbf{b}\rVert_{2}=1,0^{\circ}<\theta_{n}<180^{\circ},\;\textrm{for}\;n=1,\ldots,N\big{\\}},\end{split}$
(30)

where
$\left(\mathbf{a}_{1}(\theta_{1})^{*}\otimes\ldots\otimes\mathbf{a}_{N}(\theta_{N})\right)\mathbf{b}^{T}$
is the $N$D atom, and $\mathbf{a}_{n}(\theta_{n})\in\mathbb{C}^{M_{n}\times
1}$ denotes the $n$-th steering vector that composes the $N$D atom. The SDP
representation the $N$D atomic norm requires a $N$-level Toeplitz matrix
$\mathbf{T}_{N\textrm{D}}$, which can be given by follows [20]:

$\begin{split}\mathbf{T}_{N\textrm{D}}&=\sum_{r=1}^{R}d_{r}\mathbf{a}_{N\textrm{D},r}\mathbf{a}^{H}_{N\textrm{D},r}\\\
&=\mathbf{A}_{N\textrm{D}}\mathbf{D}\mathbf{A}^{H}_{N\textrm{D}}\in\mathbb{C}^{(\prod_{n=1}^{N}M_{n})\times(\prod_{n=1}^{N}M_{n})}.\end{split}$
(31)

$\mathbf{a}_{N\textrm{D},r}$ denotes the $r$-th $N$D atom, which can be given
by
$\left(\mathbf{a}_{1}(\theta_{1,r})^{*}\otimes\ldots\otimes\mathbf{a}_{N}(\theta_{N,r})\right)\mathbf{b}_{r}^{T}$.
Here, $\theta_{n,r}$ denotes the $r$-th angle of the $n$-th steering vector.
$d_{r}$ denotes the complex coefficient of the $r$-th $N$D atom.
$\mathbf{A}_{N\textrm{D}}=\left[\mathbf{a}_{N\textrm{D},1},\ldots,\mathbf{a}_{N\textrm{D},R}\right]$,
and $\mathbf{D}=\textrm{diag}(\mathbf{d})$, where
$\mathbf{d}=\left[d_{1},\ldots,d_{R}\right]^{T}\in\mathbb{C}^{R\times 1}$.

###  V-B Non Location-aware Channel Estimation via 1D ANM

When performing the RIS beamwidth adaptation without using the location
information, the effective cascaded channel can be estimated via 1D ANM. To
simplify equations and notations, we define $\mathbf{G}_{\textrm{1D}}$ and
$\mathbf{Z}_{\textrm{1D}}$ as follows.

$\begin{split}\mathbf{G}_{\textrm{1D}}=\left\\{\left(\mathbf{F}^{T}\otimes\mathbf{C}^{H}\right)^{-1}\bm{\mathcal{Y}}\right\\}^{H}=\mathbf{W}_{1}^{H}\mathbf{Z}_{\textrm{1D}}+\mathbf{E}_{\textrm{1D}},\end{split}$
(32)
$\begin{split}&\mathbf{Z}_{\textrm{1D}}=\frac{P_{\textrm{Tx}}}{N_{\textrm{B}}}\mathbf{H}_{\textrm{eff}}^{H}=\frac{P_{\textrm{Tx}}}{N_{\textrm{B}}}\left(\mathbf{H}^{T}_{\textrm{BR}}\diamond\mathbf{H}_{\textrm{RU}}\right)^{H}=\\\
&\frac{P_{\textrm{Tx}}}{N_{\textrm{B}}}\mathbf{A}(\bm{\varphi})\left(\textrm{diag}(\bm{\rho}_{\textrm{BR}})\otimes\textrm{diag}(\bm{\rho}_{\textrm{RU}})\right)^{H}\left(\mathbf{A}(\bm{\theta}_{\textrm{BR}})^{*}\otimes\mathbf{A}(\bm{\phi}_{\textrm{RU}})\right)^{H}.\end{split}$
(33)

Here,
$\mathbf{E}_{\textrm{1D}}=\\{\left(\mathbf{F}^{T}\otimes\mathbf{C}^{H}\right)^{-1}\bm{\mathcal{V}}\\}^{H}$,
and each column of $\mathbf{E}_{\textrm{1D}}$ follows
$\mathcal{CN}\left(\mathbf{0}_{M_{\textrm{B}}M_{\textrm{U}}},(\sigma^{2}P_{\textrm{Tx}}/DN_{\textrm{B}})\mathbf{I}_{M_{\textrm{B}}M_{\textrm{U}}}\right)$.
To derive the 1D atomic norm of $\mathbf{Z}_{\textrm{1D}}$, an atomic set
$\mathcal{A}_{\textrm{1D}}$ is defined as follows.

$\begin{split}\mathcal{A}_{\textrm{1D}}=&\big{\\{}\mathbf{a}(\theta)\mathbf{b}^{T}\in\mathbb{C}^{M_{\textrm{R}}\times
M_{\textrm{B}}M_{\textrm{U}}}:0^{\circ}<\theta<180^{\circ},\\\
&\mathbf{a}(\theta)\in\mathbb{C}^{M_{\textrm{R}}\times
1},\mathbf{b}\in\mathbb{C}^{M_{\textrm{B}}M_{\textrm{U}}\times
1},\lVert\mathbf{b}\rVert_{2}=1\big{\\}},\end{split}$ (34)

where $\mathbf{a}(\theta)\mathbf{b}^{T}$ is an atom of
$\mathbf{Z}_{\textrm{1D}}$ since $\mathbf{Z}_{\textrm{1D}}$ can be represented
as a linear combination of atoms. Since $M_{\textrm{B}}M_{\textrm{U}}>1$, the
1D atomic norm of $\mathbf{Z}_{\textrm{1D}}$ is considered as the MMV case.
The 1D atomic norm of $\mathbf{Z}_{\textrm{1D}}$ for the MMV case,
$\lVert\mathbf{Z}_{\textrm{1D}}\rVert_{\mathcal{A}_{\textrm{1D}}}$ can be
represented by following SDP [28]:

$\begin{split}\lVert\mathbf{Z}_{\textrm{1D}}\rVert_{\mathcal{A}_{\textrm{1D}}}=&\min_{\mathbf{T},\mathbf{P}_{\textrm{1D}}}\;\frac{1}{2M_{\textrm{R}}}\textrm{Tr}(\mathbf{T})+\frac{1}{2}\textrm{Tr}(\mathbf{P}_{\textrm{1D}})\\\
&\;\textrm{s.t.}\begin{bmatrix}\mathbf{T}&\mathbf{Z}_{\textrm{1D}}\\\
\mathbf{Z}_{\textrm{1D}}^{H}&\mathbf{P}_{\textrm{1D}}\end{bmatrix}\succeq
0.\end{split}$ (35)

With the ANM denoising theorem studied in [25], an equation that estimates
$\mathbf{Z}_{\textrm{1D}}$ from $\mathbf{G}_{\textrm{1D}}$ can be given by

$\hat{\mathbf{Z}}_{\textrm{1D}}=\operatorname*{argmin}_{\bar{\mathbf{Z}}_{\textrm{1D}}}\;\tau_{\textrm{1D}}\lVert\bar{\mathbf{Z}}_{\textrm{1D}}\rVert_{\mathcal{A}_{\textrm{1D}}}+\frac{\lVert\mathbf{G}_{\textrm{1D}}-\mathbf{W}_{1}^{H}\bar{\mathbf{Z}}_{\textrm{1D}}\rVert_{\textrm{F}}^{2}}{2},$
(36)

where $\hat{\mathbf{Z}}_{\textrm{1D}}$ and $\bar{\mathbf{Z}}_{\textrm{1D}}$
respectively denote the estimate of $\mathbf{Z}_{\textrm{1D}}$ and the
variable for estimation of $\mathbf{Z}_{\textrm{1D}}$. Considering the
covariance of each element of $\mathbf{E}_{\textrm{1D}}$, a regularization
parameter $\tau_{\textrm{1D}}$ is set as in [25]:

$\begin{split}\tau_{\textrm{1D}}=&\frac{\sigma\sqrt{P_{\textrm{Tx}}}}{\sqrt{DN_{\textrm{B}}}}\Big{(}1+\frac{1}{\log
M_{\textrm{R}}}\Big{)}^{\frac{1}{2}}\Big{(}M_{\textrm{B}}M_{\textrm{U}}+\log(\alpha_{\textrm{1D}}M_{\textrm{B}}M_{\textrm{U}})+\\\
&\sqrt{2M_{\textrm{B}}M_{\textrm{U}}\log(\alpha_{\textrm{1D}}M_{\textrm{B}}M_{\textrm{U}})}+\sqrt{\frac{\pi
M_{\textrm{B}}M_{\textrm{U}}}{2}}+1\Big{)}^{\frac{1}{2}},\end{split}$ (37)

where $\alpha_{\textrm{1D}}=8\pi M_{\textrm{R}}\log M_{\textrm{R}}$. (36) can
be fully unfolded as

$\begin{split}&\\{\hat{\mathbf{T}},\hat{\mathbf{P}}_{\textrm{1D}},\hat{\mathbf{Z}}_{\textrm{1D}}\\}=\\\
&\operatorname*{argmin}_{\mathbf{T},\mathbf{P}_{\textrm{1D}},\bar{\mathbf{Z}}_{\textrm{1D}}}\;\frac{\tau_{\textrm{1D}}\textrm{Tr}(\mathbf{T})}{2M_{\textrm{R}}}+\frac{\tau_{\textrm{1D}}\textrm{Tr}(\mathbf{P}_{\textrm{1D}})}{2}+\frac{\lVert\mathbf{G}_{\textrm{1D}}-\mathbf{W}_{1}^{H}\bar{\mathbf{Z}}_{\textrm{1D}}\rVert_{\textrm{F}}^{2}}{2}\\\
&\;\textrm{s.t.}\begin{bmatrix}\mathbf{T}&\bar{\mathbf{Z}}_{\textrm{1D}}\\\
\bar{\mathbf{Z}}_{\textrm{1D}}^{H}&\mathbf{P}_{\textrm{1D}}\end{bmatrix}\succeq
0.\end{split}$ (38)

Finally, $\mathbf{H}_{\textrm{eff}}$ can be approximated as
$(N_{\textrm{B}}/P_{\textrm{Tx}})\hat{\mathbf{Z}}_{\textrm{1D}}^{H}$.

A computational complexity of (38) can be derived by a theorem in [29] after
transcribing (38) into a general SDP form. If an influence of a stopping
criteria for the optimization is not considered,
$\mathcal{O}\left((M_{\textrm{B}}M_{\textrm{U}}+M_{\textrm{R}})^{0.5}\right)$
iterations are required, where each iteration costs
$\mathcal{O}\left((M_{\textrm{B}}M_{\textrm{U}}+M_{\textrm{R}})^{3}\right)$.
Thus, the computational complexity of the non location-aware channel
estimation via 1D ANM is
$\mathcal{O}\left((M_{\textrm{B}}M_{\textrm{U}}+M_{\textrm{R}})^{3.5}\right)$.

###  V-C Location-aware Channel Estimation via 2D ANM

If the location information is given and the RIS performs the exhaustive
search, the effective cascaded channel can be estimated via 2D ANM. To
simplify equations and notations, we define $\mathbf{G}_{\textrm{2D}}$ and
$\mathbf{Z}_{\textrm{2D}}$ as follows.

$\begin{split}\mathbf{G}_{\textrm{2D}}=\bm{\mathcal{J}}\mathbf{W}_{0}^{-1}=\left(\mathbf{f}^{T}\otimes\mathbf{C}^{H}\right)\mathbf{Z}_{\textrm{2D}}+\mathbf{E}_{\textrm{2D}}\in\mathbf{C}^{M_{\textrm{U}}\times
M_{\textrm{R}}},\end{split}$ (39)
$\begin{split}&\mathbf{Z}_{\textrm{2D}}=P_{\textrm{Tx}}\mathbf{H}_{\textrm{eff}}=\\\
&P_{\textrm{Tx}}\left(\mathbf{A}(\bm{\theta}_{\textrm{BR}})^{*}\otimes\mathbf{A}(\bm{\phi}_{\textrm{RU}})\right)\left(\textrm{diag}(\bm{\rho}_{\textrm{BR}})\otimes\textrm{diag}(\bm{\rho}_{\textrm{RU}})\right)\mathbf{A}(\bm{\varphi})^{H}.\end{split}$
(40)

Here, $\mathbf{E}_{\textrm{2D}}=\mathbf{U}\mathbf{W}_{0}^{-1}$, and each
column of $\mathbf{E}_{\textrm{2D}}$ follows
$\mathcal{CN}\left(\mathbf{0}_{M_{\textrm{U}}},(\sigma^{2}P_{\textrm{Tx}}/DM_{\textrm{R}})\mathbf{I}_{M_{\textrm{U}}}\right)$.
To estimate $\mathbf{Z}_{\textrm{2D}}$ using $\mathbf{G}_{\textrm{2D}}$, the
atomic norm and the atomic set need to be determined. An atomic norm of
$\mathbf{Z}_{\textrm{2D}}$, $\mathcal{A}_{\textrm{2D}}$ can be represented as
follows.

$\begin{split}\mathcal{A}_{\textrm{2D}}=&\left\\{\left(\mathbf{a}_{1}(\theta_{1})^{*}\otimes\mathbf{a}_{2}(\theta_{2})\right)\mathbf{b}^{T}\in\mathbb{C}^{M_{\textrm{B}}M_{\textrm{U}}\times
M_{\textrm{R}}}:\right.\\\
&\left.\mathbf{b}\in\mathbb{C}^{M_{\textrm{R}}\times
1},\;\lVert\mathbf{b}\rVert_{2}=1,\;0^{\circ}<\theta_{1},\theta_{2}<180^{\circ}\right\\},\end{split}$
(41)

where
$\left(\mathbf{a}_{1}(\theta_{1})^{*}\otimes\mathbf{a}_{2}(\theta_{2})\right)\mathbf{b}^{T}$
is an atom for $\mathbf{Z}_{\textrm{2D}}$, and $\mathbf{Z}_{\textrm{2D}}$ can
be represented as a linear combination of these atoms. If the atomic set of
$\mathbf{Z}_{\textrm{2D}}$ is $\mathcal{A}_{\textrm{2D}}$, the atomic norm for
$\mathbf{Z}_{\textrm{2D}}$,
$\lVert\mathbf{Z}_{\textrm{2D}}\rVert_{\mathcal{A}_{\textrm{2D}}}$ can be
given by following SDP [20]:

$\begin{split}\lVert\mathbf{Z}_{\textrm{2D}}\rVert_{\mathcal{A}_{\textrm{2D}}}=&\min_{\mathbf{T}_{\textrm{2D}},\mathbf{P}}\;\frac{1}{2M_{\textrm{B}}M_{\textrm{U}}}\textrm{Tr}(\mathbf{T}_{\textrm{2D}})+\frac{1}{2}\textrm{Tr}(\mathbf{P})\\\
&\;\textrm{s.t.}\begin{bmatrix}\mathbf{T}_{\textrm{2D}}&\mathbf{Z}_{\textrm{2D}}\\\
\mathbf{Z}_{\textrm{2D}}^{H}&\mathbf{P}\end{bmatrix}\succeq 0,\end{split}$
(42)

where $\mathbf{T}_{\textrm{2D}}$ denotes $2$-level Toeplitz matrix. The
structure of $\mathbf{T}_{\textrm{2D}}$ can be represented by Hermitian
Toeplitz matrices and Toeplitz matrices as follows.

$\mathbf{T}_{\textrm{2D}}=\begin{bmatrix}\mathbf{T}_{0}&\mathbf{T}_{1}^{H}&\cdots&\mathbf{T}_{M_{\textrm{B}}-1}^{H}&\mathbf{T}_{M_{\textrm{B}}}^{H}\\\
\mathbf{T}_{1}&\mathbf{T}_{0}&\mathbf{T}_{1}^{H}&\ddots&\mathbf{T}_{M_{\textrm{B}}-1}^{H}\\\
\vdots&\ddots&\ddots&\ddots&\vdots\\\
\mathbf{T}_{M_{\textrm{B}}-1}&\ddots&\mathbf{T}_{1}&\mathbf{T}_{0}&\mathbf{T}_{1}^{H}\\\
\mathbf{T}_{M_{\textrm{B}}}&\mathbf{T}_{M_{\textrm{B}}-1}&\cdots&\mathbf{T}_{1}&\mathbf{T}_{0}\end{bmatrix}.$
(43)

Here, $\mathbf{T}_{0}$ denotes a Hermitian Toeplitz matrix. For
$i=1,\ldots,M_{\textrm{B}}$, $\mathbf{T}_{i}$ denotes a Toeplitz matrix. The
structure of the Toeplitz matrix is different from that of the Hermitian
Toeplitz matrix, where the structure of $M_{\textrm{B}}$ Toeplitz matrices
that compose $\mathbf{T}_{\textrm{2D}}$ can be given by

$\begin{split}\mathbf{T}_{i}=\begin{bmatrix}a_{1}&a_{-2}&\ldots&a_{-M_{\textrm{U}}+1}&a_{-M_{\textrm{U}}}\\\
a_{2}&a_{1}&a_{-2}&\ldots&a_{-M_{\textrm{U}}+1}\\\
\vdots&a_{2}&\ddots&\ddots&\vdots\\\
a_{M_{\textrm{U}}-1}&\vdots&\ddots&\ddots&a_{-2}\\\
a_{M_{\textrm{U}}}&a_{M_{\textrm{U}}-1}&\ldots&a_{2}&a_{1}\end{bmatrix},\\\
\textrm{for}\;i=1,\ldots,M_{\textrm{B}}.\end{split}$ (44)

To simplify the representation of following equations, we define $\mathbf{Q}$
as $\mathbf{f}^{T}\otimes\mathbf{C}^{H}$. An equation that estimates
$\mathbf{Z}_{\textrm{2D}}$ from $\mathbf{G}_{\textrm{2D}}$ can be given by

$\hat{\mathbf{Z}}_{\textrm{2D}}=\operatorname*{argmin}_{\bar{\mathbf{Z}}_{\textrm{2D}}}\;\tau_{\textrm{2D}}\lVert\bar{\mathbf{Z}}_{\textrm{2D}}\rVert_{\mathcal{A}_{\textrm{2D}}}+\frac{1}{2}\lVert\mathbf{G}_{\textrm{2D}}-\mathbf{Q}\bar{\mathbf{Z}}_{\textrm{2D}}\rVert_{\textrm{F}}^{2},$
(45)

where $\hat{\mathbf{Z}}_{\textrm{2D}}$ and $\bar{\mathbf{Z}}_{\textrm{2D}}$
respectively denote the estimate of $\mathbf{Z}_{\textrm{2D}}$ and the
variable for estimation of $\mathbf{Z}_{\textrm{2D}}$. Considering the
covariance of each element of $\mathbf{E}_{\textrm{2D}}$, a regularization
parameter $\tau_{\textrm{2D}}$ is set as

$\begin{split}\tau_{\textrm{2D}}=&\frac{\sigma\sqrt{P_{\textrm{Tx}}}}{\sqrt{DM_{\textrm{R}}}}\Big{(}1+\frac{1}{\log
M_{\textrm{B}}M_{\textrm{U}}}\Big{)}^{\frac{1}{2}}\Big{(}M_{\textrm{R}}+\log(\alpha_{\textrm{2D}}M_{\textrm{R}})+\\\
&\sqrt{2M_{\textrm{R}}\log(\alpha_{\textrm{2D}}M_{\textrm{R}})}+\sqrt{\frac{\pi
M_{\textrm{R}}}{2}}+1\Big{)}^{\frac{1}{2}},\end{split}$ (46)

where $\alpha_{\textrm{2D}}=8\pi
M_{\textrm{B}}M_{\textrm{U}}\log\left(M_{\textrm{B}}M_{\textrm{U}}\right)$.
(45) can be fully unfolded as

$\begin{split}&\\{\hat{\mathbf{T}}_{\textrm{2D}},\hat{\mathbf{P}}_{\textrm{2D}},\hat{\mathbf{Z}}_{\textrm{2D}}\\}=\\\
&\operatorname*{argmin}_{\mathbf{T}_{\textrm{2D}},\mathbf{P}_{\textrm{2D}},\bar{\mathbf{Z}}_{\textrm{2D}}}\;\frac{\tau_{\textrm{2D}}\textrm{Tr}(\mathbf{T}_{\textrm{2D}})}{2M_{\textrm{B}}M_{\textrm{U}}}+\frac{\tau_{\textrm{2D}}\textrm{Tr}(\mathbf{P}_{\textrm{2D}})}{2}+\frac{\lVert\mathbf{G}_{\textrm{2D}}-\mathbf{Q}\bar{\mathbf{Z}}_{\textrm{2D}}\rVert_{\textrm{F}}^{2}}{2}\\\
&\;\textrm{s.t.}\begin{bmatrix}\mathbf{T}_{\textrm{2D}}&\bar{\mathbf{Z}}_{\textrm{2D}}\\\
\bar{\mathbf{Z}}_{\textrm{2D}}^{H}&\mathbf{P}_{\textrm{2D}}\end{bmatrix}\succeq
0.\end{split}$ (47)

After solving (47), $\mathbf{H}_{\textrm{eff}}$ can be approximated as
$\hat{\mathbf{Z}}_{\textrm{2D}}/P_{\textrm{Tx}}$.

A computational complexity of (47) can be derived by a theorem in [29] after
transcribing (47) into a general SDP form. If an influence of a stopping
criteria for the optimization is not considered,
$\mathcal{O}\left((M_{\textrm{B}}M_{\textrm{U}}+M_{\textrm{R}})^{0.5}\right)$
iterations are required, where each iteration costs
$\mathcal{O}\left((M_{\textrm{B}}M_{\textrm{U}}+M_{\textrm{R}})^{3}\right)$.
Thus, the computational complexity of the location-aware channel estimation
via 2D ANM is
$\mathcal{O}\left((M_{\textrm{B}}M_{\textrm{U}}+M_{\textrm{R}})^{3.5}\right)$.

###  V-D Location-aware Channel Estimation via 3D ANM

By performing the RIS beamwidth adaptation when the location information is
given, the beam training overhead can be further reduced, and the effective
cascaded channel can be estimated via 3D ANM. The 3D ANM-based low overhead
channel estimation requires the vectorization of $\bm{\mathcal{J}}$, where the
vectorized $\bm{\mathcal{J}}$, $\bm{\Upsilon}$ can be represented as

$\begin{split}\bm{\Upsilon}=&\textrm{vec}(\bm{\mathcal{J}})=P_{\textrm{Tx}}\left(\mathbf{W}_{1}^{T}\otimes\mathbf{f}^{T}\otimes\mathbf{C}^{H}\right)\textrm{vec}(\mathbf{H}_{\textrm{eff}})+\textrm{vec}(\mathbf{U})\\\
=&P_{\textrm{Tx}}\left(\mathbf{W}_{1}^{T}\otimes\mathbf{f}^{T}\otimes\mathbf{C}^{H}\right)\left(\mathbf{A}(\bm{\varphi})^{*}\otimes\mathbf{A}(\bm{\theta}_{\textrm{BR}})^{*}\otimes\mathbf{A}(\bm{\phi}_{\textrm{RU}})\right)\\\
&\textrm{vec}\left(\textrm{diag}(\bm{\rho}_{\textrm{BR}})\otimes\textrm{diag}(\bm{\rho}_{\textrm{RU}})\right)+\mathbf{u}\in\mathbb{C}^{BM_{\textrm{U}}\times
1}.\end{split}$ (48)

Here, $\mathbf{u}=\textrm{vec}(\mathbf{U})$ and
$\mathbf{u}\sim\mathcal{CN}\left(\mathbf{0}_{BM_{\textrm{U}}},(\sigma^{2}P_{\textrm{Tx}}/D)\mathbf{I}_{BM_{\textrm{U}}}\right)$.
Note that (48) can be derived by Property 3 in Section III. Letting
$\mathbf{z}_{\textrm{3D}}$ denotes
$P_{\textrm{Tx}}\textrm{vec}(\mathbf{H}_{\textrm{eff}})$, $\bm{\Upsilon}$ can
be rewritten as

$\begin{split}\bm{\Upsilon}=\left(\mathbf{W}_{1}^{T}\otimes\mathbf{f}^{T}\otimes\mathbf{C}^{H}\right)\mathbf{z}_{\textrm{3D}}+\mathbf{u}\in\mathbf{C}^{BM_{\textrm{U}}\times
1}.\end{split}$ (49)

To simplify the representation of following equations, we define $\mathbf{R}$
as $\mathbf{W}_{1}^{T}\otimes\mathbf{f}^{T}\otimes\mathbf{C}^{H}$ and
$\mathcal{M}$ as $M_{\textrm{B}}M_{\textrm{R}}M_{\textrm{U}}$.

To define the atomic norm of $\mathbf{z}_{\textrm{3D}}$, the atomic set
$\mathcal{A}_{\textrm{3D}}$ can be given by

$\begin{split}\mathcal{A}_{\textrm{3D}}=&\left\\{\mathbf{a}_{1}(\theta_{1})^{*}\otimes\mathbf{a}_{2}(\theta_{2})^{*}\otimes\mathbf{a}_{3}(\theta_{3})\in\mathbb{C}^{\mathcal{M}\times
1}:\right.\\\
&\quad\left.0^{\circ}<\theta_{1},\theta_{2},\theta_{3}<180^{\circ}\right\\},\end{split}$
(50)

where
$\mathbf{a}_{1}(\theta_{1})^{*}\otimes\mathbf{a}_{2}(\theta_{2})^{*}\otimes\mathbf{a}_{3}(\theta_{3})$
is an atom for $\mathbf{z}_{\textrm{3D}}$, and $\mathbf{z}_{\textrm{3D}}$ can
be represented as a linear combination of these atoms.
$\mathcal{A}_{\textrm{3D}}$ is a set of 3D atoms defined in (30) and belongs
to the SMV case. The atomic norm for $\mathbf{z}_{\textrm{3D}}$,
$\lVert\mathbf{z}_{\textrm{3D}}\rVert_{\mathcal{A}_{\textrm{3D}}}$ can be
given by follows.

$\begin{split}\lVert\mathbf{z}_{\textrm{3D}}\rVert_{\mathcal{A}_{\textrm{3D}}}=&\min_{\mathbf{T}_{\textrm{3D}},p}\;\frac{1}{2\mathcal{M}}\textrm{Tr}(\mathbf{T}_{\textrm{3D}})+\frac{1}{2}p\\\
&\;\textrm{s.t.}\begin{bmatrix}\mathbf{T}_{\textrm{3D}}&\mathbf{z}_{\textrm{3D}}\\\
\mathbf{z}_{\textrm{3D}}^{H}&p\end{bmatrix}\succeq 0.\end{split}$ (51)

Here, $\mathbf{T}_{\textrm{3D}}$ denotes 3D Toeplitz matrix and comprises of
multiple 2D Toeplitz matrices. A structure of $\mathbf{T}_{\textrm{3D}}$ is
given in (53). In (53), $\mathbf{T}^{0}_{\textrm{2D}}$ denotes
$M_{\textrm{B}}M_{\textrm{U}}\times M_{\textrm{B}}M_{\textrm{U}}$ 2-level
Toeplitz matrix whose structure follows that of (43). However, the structure
of $\mathbf{T}^{i}_{\textrm{2D}}$ for $i=1,\ldots,M_{\textrm{R}}$ is different
from that of (43), where the structure of $\mathbf{T}^{i}_{\textrm{2D}}$ for
$i=1,\ldots,M_{\textrm{R}}$ can be given by

$\begin{split}\mathbf{T}^{i}_{\textrm{2D}}=\begin{bmatrix}\mathbf{T}_{0}&\mathbf{T}_{-1}&\cdots&\mathbf{T}_{-M_{\textrm{B}}+1}&\mathbf{T}_{-M_{\textrm{B}}}\\\
\mathbf{T}_{1}&\mathbf{T}_{0}&\mathbf{T}_{-1}&\ddots&\mathbf{T}_{-M_{\textrm{B}}+1}\\\
\vdots&\ddots&\ddots&\ddots&\vdots\\\
\mathbf{T}_{M_{\textrm{B}}-1}&\ddots&\mathbf{T}_{1}&\mathbf{T}_{0}&\mathbf{T}_{-1}\\\
\mathbf{T}_{M_{\textrm{B}}}&\mathbf{T}_{M_{\textrm{B}}-1}&\cdots&\mathbf{T}_{1}&\mathbf{T}_{0}\end{bmatrix},\\\
\textrm{for}\;i=1,\ldots,M_{\textrm{R}}.\end{split}$ (52)

Note that Toeplitz matrices are not placed symmetrically as in (43).

$\displaystyle\mathbf{T}_{\textrm{3D}}=\begin{bmatrix}\mathbf{T}_{\textrm{2D}}^{0}&(\mathbf{T}_{\textrm{2D}}^{1})^{H}&\cdots&(\mathbf{T}_{\textrm{2D}}^{M_{\textrm{R}}-1})^{H}&(\mathbf{T}_{\textrm{2D}}^{M_{\textrm{R}}})^{H}\\\ \mathbf{T}_{\textrm{2D}}^{1}&\mathbf{T}_{\textrm{2D}}^{0}&(\mathbf{T}_{\textrm{2D}}^{1})^{H}&\ddots&(\mathbf{T}_{\textrm{2D}}^{M_{\textrm{R}}-1})^{H}\\\ \vdots&\ddots&\ddots&\ddots&\vdots\\\ \mathbf{T}_{\textrm{2D}}^{M_{\textrm{R}}-1}&\ddots&\mathbf{T}_{\textrm{2D}}^{1}&\mathbf{T}_{\textrm{2D}}^{0}&(\mathbf{T}_{\textrm{2D}}^{1})^{H}\\\ \mathbf{T}_{\textrm{2D}}^{M_{\textrm{R}}}&\mathbf{T}_{\textrm{2D}}^{M_{\textrm{R}}-1}&\cdots&\mathbf{T}_{\textrm{2D}}^{1}&\mathbf{T}_{\textrm{2D}}^{0}\end{bmatrix}\in\mathbb{C}^{\mathcal{M}\times\mathcal{M}}.$ (53) Table I: Summary of ANM-based channel estimation algorithms for RIS-aided MIMO systems | Non location-aware | Location-aware
---|---|---
| 1D ANM | 2D ANM | 3D ANM
Beam search at BS | Exhaustive search | Direct steering towards RIS | Direct steering towards RIS
RIS beamwidth adaptation | O | X | O
Number of training symbols (Beam training overhead) | $BP_{\textrm{B}}P_{\textrm{U}}$ | $M_{\textrm{R}}P_{\textrm{U}}$ | $BP_{\textrm{U}}$
Complexity | $\mathcal{O}\left((M_{\textrm{B}}M_{\textrm{U}}+M_{\textrm{R}})^{3.5}\right)$ | $\mathcal{O}\left((M_{\textrm{B}}M_{\textrm{U}}+M_{\textrm{R}})^{3.5}\right)$ | $\mathcal{O}\left((M_{\textrm{B}}M_{\textrm{R}}M_{\textrm{U}}+1)^{3.5}\right)$

An equation that estimates $\mathbf{z}_{\textrm{3D}}$ can be given by

$\hat{\mathbf{z}}_{\textrm{3D}}=\operatorname*{argmin}_{\bar{\mathbf{z}}_{\textrm{2D}}}\;\tau_{\textrm{3D}}\lVert\bar{\mathbf{z}}_{\textrm{3D}}\rVert_{\mathcal{A}_{\textrm{3D}}}+\frac{1}{2}\lVert\bm{\Upsilon}-\mathbf{R}\bar{\mathbf{z}}_{\textrm{3D}}\rVert_{2}^{2},$
(54)

where $\hat{\mathbf{z}}_{\textrm{3D}}$ and $\bar{\mathbf{z}}_{\textrm{3D}}$
respectively denote the estimate of $\mathbf{z}_{\textrm{3D}}$ and the
variable for estimation of $\mathbf{z}_{\textrm{3D}}$. For the SMV case, a
regularization parameter $\tau_{\textrm{3D}}$ can be set as in [30],
considering the covariance of each element of $\mathbf{u}$.

$\tau_{\textrm{3D}}=\frac{\sigma\sqrt{P_{\textrm{Tx}}}}{\sqrt{D}}\Big{(}1+\frac{1}{\log\mathcal{M}}\Big{)}\sqrt{\mathcal{M}\log\mathcal{M}+\mathcal{M}\log\left(4\pi\log\mathcal{M}\right)}.$
(55)

Finally, (54) can be fully unfolded as

$\begin{split}&\\{\hat{\mathbf{T}}_{\textrm{3D}},\hat{p},\hat{\mathbf{z}}_{\textrm{3D}}\\}=\\\
&\operatorname*{argmin}_{\mathbf{T}_{\textrm{3D}},p,\bar{\mathbf{z}}_{\textrm{2D}}}\;\frac{\tau_{\textrm{3D}}}{2\mathcal{M}}\textrm{Tr}(\mathbf{T}_{\textrm{3D}})+\frac{\tau_{\textrm{3D}}}{2}p+\frac{1}{2}\lVert\bm{\Upsilon}-\mathbf{R}\bar{\mathbf{z}}_{\textrm{3D}}\rVert_{2}^{2}\\\
&\;\textrm{s.t.}\begin{bmatrix}\mathbf{T}_{\textrm{3D}}&\bar{\mathbf{z}}_{\textrm{3D}}\\\
\bar{\mathbf{z}}_{\textrm{3D}}^{H}&p\end{bmatrix}\succeq 0.\end{split}$ (56)

$\mathbf{H}_{\textrm{eff}}$ can be estimated by reshaping a vector
$\hat{\mathbf{z}}_{\textrm{3D}}$ to $M_{\textrm{B}}M_{\textrm{U}}\times M_{R}$
matrix.

A computational complexity of (56) can be derived by a theorem in [29] after
transcribing (56) into a general SDP form. If an influence of a stopping
criteria for the optimization is not considered,
$\mathcal{O}\left((M_{\textrm{B}}M_{\textrm{R}}M_{\textrm{U}}+1)^{0.5}\right)$
iterations are required, where each iteration costs
$\mathcal{O}\left((M_{\textrm{B}}M_{\textrm{R}}M_{\textrm{U}}+1)^{3}\right)$.
Thus, the computational complexity of the location-aware channel estimation
via 3D ANM is
$\mathcal{O}\left((M_{\textrm{B}}M_{\textrm{R}}M_{\textrm{U}}+1)^{3.5}\right)$.
Characteristics of three ANM-based channel estimation algorithms are
summarized in Table I.

##  VI Simulation Results and Discussions

Figure 5: The location of the BS, the RIS, and the UE throughout simulation.

###  VI-A Simulation Environments

For the default simulation setting, $M_{\textrm{B}}$, $M_{\textrm{R}}$, and
$M_{\textrm{U}}$ are respectively set to 4, 16, and 4, and $N_{\textrm{B}}$
and $N_{\textrm{U}}$ are respectively set to 2 and 2. The number of snapshots
of pilot signal, $D$ is $100$. Precoding matrices and combining matrices are
set as parts of scaled DFT matrices. Thus, $\mathbf{F}$ and $\mathbf{C}$ are
respectively $\bm{\Psi}_{M_{\textrm{B}}}/\sqrt{M_{\textrm{B}}}$ and
$\bm{\Psi}_{M_{\textrm{U}}}/\sqrt{M_{\textrm{U}}}$. The transmission power is
$30$ dBm so that $P_{\textrm{Tx}}=1000$. The noise power is $-100$ dBm so that
$\sigma=10^{-5}$. For implementation of [15], a size of discretized grid is
set to $360$ so that the angular domain is divided by $0.5^{\circ}$.

The location of the BS, the RIS, and the UE are set as in Fig. 5. The BS is
located in $[0,0]$ m, and the RIS is located in $[50,40]$ m. If not noted, the
UE is randomly located in the area of which the center is $[80,0]$ m and the
radius is $10$ m. The BS, the RIS, and the UE are oriented in parallel. Since
there is only LoS path between the BS and the RIS, $\phi^{1}_{\textrm{BR}}$
and $\theta^{1}_{\textrm{BR}}$ are set as follows.

$\phi^{1}_{\textrm{BR}}=\theta^{1}_{\textrm{BR}}=\textrm{tan}^{-1}\left(\frac{40}{50}\right)=38.66^{\circ}.$
(57)

The magnitude of $\alpha^{1}_{\textrm{BR}}$ is determined based on the
distance between the BS and the RIS, the carrier frequency, and the path loss
exponent. Here, the carrier frequency is $28$ GHz, and the path loss exponent
is $2.1$. The phase of $\alpha^{1}_{\textrm{BR}}$ is randomly set between $0$
and $2\pi$.

To model the RIS-to-UE channel, the 28 GHz statistical channel model in [22]
is employed. The elevation of the RIS-to-UE AoD/AoA is not considered since
the RIS and the UE are using ULAs. Also, we assume that the time cluster
comprises of one signal path. $L_{\textrm{RU}}$ is randomly set between $1$
and $6$, and the LoS path between the RIS and the UE exists if the distance
between the RIS and the UE is less than $60$ m. The channel gains and the RIS-
to-UE AoDs/AoAs are determined randomly based on [22]. However, if the LoS
path exists between the RIS and the UE, $\phi^{1}_{\textrm{RU}}$ and
$\theta^{1}_{\textrm{RU}}$ are set as

$\phi^{1}_{\textrm{RU}}=\theta^{1}_{\textrm{RU}}=\textrm{tan}^{-1}\left(\frac{40-\textrm{y}_{\textrm{UE}}}{\textrm{x}_{\textrm{UE}}-50}\right),$
(58)

where $\textrm{x}_{\textrm{UE}}$ and $\textrm{y}_{\textrm{UE}}$ respectively
denote the x-coordinate and the y-coordinate of the UE.

The SNR is defined as

$\textrm{SNR}=10\log_{10}\frac{P_{\textrm{Tx}}\left|\left(\sum_{l=1}^{L_{\textrm{RU}}}\alpha_{\textrm{RU}}^{l}\right)\left(\sum_{l=1}^{L_{\textrm{BR}}}\alpha_{\textrm{BR}}^{l}\right)\right|^{2}}{\sigma^{2}}(\textrm{dB}).$
(59)

The SNR defined in (59) is a ratio of signal power and noise power when
$M_{\textrm{B}}=M_{\textrm{R}}=M_{\textrm{U}}=1$. Note that the SNR defined in
(59) does not depend on precoding matrix, combining matrix, and RIS control
matrix. The normalized mean square error (NMSE) is defined as

$\textrm{NMSE}=\frac{1}{Q}\sum_{q=1}^{Q}\frac{\lVert\hat{\mathbf{H}}^{q}_{\textrm{eff}}-\mathbf{H}^{q}_{\textrm{eff}}\rVert^{2}_{\textrm{F}}}{\lVert\mathbf{H}^{q}_{\textrm{eff}}\rVert^{2}_{\textrm{F}}},$
(60)

where $Q$ is the number of Monte Carlo trials for NMSE calculation and is set
to 300. $\hat{\mathbf{H}}^{q}_{\textrm{eff}}$ and
$\mathbf{H}^{q}_{\textrm{eff}}$ respectively denote the estimated effective
cascaded channel and the actual effective cascaded channel on the $q$-th Monte
Carlo trial. For computation, Intel CPU i5-7500 (3.40 GHz) and 16 GB RAM are
used, and CVX [31] is used to solve (38), (47), (56), and the algorithm in
[17].

(a) Beam training efficiency versus Tx power

(b) Beam training efficiency versus number of training symbols

(c) Beam training efficiency versus number of RIS antennas

Figure 6: Beam training efficiency according to the use of location
information. In (a), the number of training symbols is set to $32$ for both
beam training methods. In (c), the number of training symbols is set to twice
the number of the RIS antennas.

###  VI-B Benefit and Limitation of Location-aware Beam Training

In this subsection, the benefit and the limitation of using location are
presented by analyzing the beam training efficiency defined in (17) and (24).
Three subfigures in Fig. 6 respectively present the beam training efficiency
versus the transmission power, the number of training symbols, and the number
of RIS antennas. The number of training symbols is set to $32$ for both beam
training methods in Fig. 6(a). In Fig. 6(b), the maximum number of training
symbols for the location-aware beam training is $32$, and the number of
training symbols is set to twice the number of the RIS antennas in Fig. 6(c).
In Fig. 6, the beam training efficiency increases by $12$ dB when using
location. Main reasons for the significant improvement can be summarized as
follows:

  * •

Since the BS steers the beam towards the RIS when using location, the power of
the pilot signal received during the beam training increases. Also, the BS
focuses the transmission power to the single beam.

  * •

If the number of training symbols is identical for both beam training methods,
the RIS can use sharper reflect beams when performing the location-aware beam
training.

According to the results in Fig. 6, the location-aware channel estimation is
expected to be superior to the non location-aware channel estimation when
there is no location error.

Figure 7: Beam training efficiency according to the BS-to-RIS AoD error.
$M_{\textrm{B}}=8$, and the number of training symbols is set to $32$ for both
beam training methods.

Although the location of the BS and the RIS are expected to be accurate when
the BS and the RIS are immobile, there are scenarios that the location can be
erroneous. For example, the location of the RIS can be inaccurate when using
mobile RIS, which has been studied in seminal works [11, 32]. In this
scenario, the BS-to-RIS AoD does not match with the beam steering direction of
the BS when performing location-aware beam training. Fig. 7 shows the beam
training efficiency according to the BS-to-AoD error, which is induced by the
location error of the RIS. In this simulation, the BS-to-RIS distance and the
RIS-to-AoD distance are respectively fixed to $\sqrt{4100}$ m and $50$ m. The
number of training symbols is set to $32$ for both beam training methods, and
$M_{\textrm{B}}=8$. In Fig. 7, the beam training efficiency of the location-
aware beam training becomes minimum when the BS-to-RIS AoD error reaches the
half of the null-to-null beamwidth, where the half of the null-to-null
beamwidth is the distance from the steering direction to the first null [33].
On the other hand, the beam training efficiency of the non location-aware beam
training is not affected by the BS-to-RIS AoD error.

###  VI-C Analysis on Channel Estimation Performance

In this subsection, we analyze the NMSE of the effective cascaded channel with
respect to various factors that can affect the performance-transmission power,
number of training symbols, number of RIS antennas, RIS-to-UE distance, and
BS-to-RIS AoD error. The former studies on the channel estimation for RIS-
aided MIMO systems, [15] and [17], are benchmarks for performance evaluation.
Note that [15] and [17] also employ the RIS beamwidth adaptation. The
algorithm in [8] is excluded from the comparison since the computation time
becomes excessively high when reforming the algorithm to work under RIS-aided
MIMO systems.

Figure 8: NMSE versus transmission power. The average SNR is $-5.63$ dB when
the transmission power is $30$ dBm.

Fig. 8 shows the NMSE according to the transmission power. The average SNR is
$-5.63$ dB when the transmission power is $30$ dBm, and the average SNR is
proportional to the transmission power. In this simulation, the number of
training symbols is set to $32$ except for the location-aware channel
estimation via 3D ANM, where $16$ training symbols are used for the location-
aware channel estimation via 3D ANM. This is because there is no need to
employ the 3D ANM-based algorithm since the 2D ANM-based algorithm is
available if using $32$ training symbols. In Fig. 8, the NMSE of the location-
aware channel estimation via 2D ANM and 3D ANM are lower than those of the non
location-aware channel estimation algorithms. The NMSE of the 3D ANM-based
algorithm is higher than that of the 2D ANM since the 3D ANM-based algorithm
uses only $16$ training symbols. Still, the 3D ANM-based algorithm shows
superior estimation accuracy over other non location-aware algorithms even
when the beam training overhead is further reduced.

Figure 9: NMSE versus number of training symbols.

Fig. 9 shows the NMSE according to the number of training symbols. The number
of active RIS antennas is adjusted based on the number of training symbols.
The only available number of training symbols for the location-aware channel
estimation via 2D ANM is $32$, and the maximum number of training symbols for
the location-aware channel estimation via 3D ANM is $32$. Fig. 9 presents that
the NMSE of all algorithms decrease as the number of training symbols
increases, and the location-aware channel estimation is superior to the non
location-aware channel estimation. As shown in Fig. 6(b), the beam training
efficiency increases when using location if the number of training symbols is
identical. For this reason, the location-aware channel estimation algorithms
achieve lower NMSE than others when consuming the same beam training overhead.
Among the non location-aware channel estimation algorithms, the 1D ANM-based
channel estimation has the lowest NMSE.

(a) NMSE versus number of RIS antennas

(b) Average computation time versus number of RIS antennas

Figure 10: NMSE and average computation time according to the number of RIS
antennas. Since the computing system cannot afford the 3D ANM-based algorithm
if $M_{\textrm{R}}>16$, the analysis on the 3D ANM-based algorithm is omitted
when $M_{\textrm{R}}>16$.

Fig. 10 shows the NMSE and the average computation time according to the
number of the RIS antennas. Since the computing system cannot afford the 3D
ANM-based algorithm if $M_{\textrm{R}}>16$, the analysis of the 3D ANM-based
algorithm is omitted when $M_{\textrm{R}}>16$. For all algorithms except the
3D ANM-based algorithm, the number of the training symbols is set to twice the
number of the RIS antennas. For the 3D ANM-based algorithm, the number of the
training symbols equals to the number of the RIS antennas. In Fig. 10, the
NMSE of all algorithms are inversely proportional to the number of the RIS
antennas except [15]. In case of [15], the AoD/AoA estimation accuracy does
not improve with the number of the RIS antennas due to the grid-mismatch. If
the number of the RIS antennas increases, the channel estimation accuracy
becomes more dependent on the AoD/AoA estimation accuracy since even the small
angle variation significantly affects the last elements of the steering
vector. In Fig. 10(b), the computation time of the 3D-ANM based algorithm is
exceptionally high, where this result can be inferred by the complexity
analysis in Table I. Other algorithms consume a similar level of the average
computation time except for [15], which takes about $0.2$ seconds.

Fig. 11 shows the NMSE according to the RIS-to-UE distance. In this
simulation, the UE is located so that it is separated by a specified distance
from the RIS. A number of training symbols is set to $32$ except for the
location-aware channel estimation via 3D ANM, where the location-aware channel
estimation via 3D ANM uses $16$ training symbols. When using the statistical
mmWave channel model in [22], the dominant LoS path between RIS and UE
disappears if the RIS-to-UE distance goes beyond $60$ m, and the path loss
coefficient for LoS environment and NLoS environment are respectively $2.1$
and $3.4$. For this reason, the SNR changes abruptly around $60$ m. In Fig.
11, the non location-aware channel estimation algorithms fail under NLoS
environment since the NMSE are around or beyond $1$. When the NMSE is around
$1$, it means that elements of the estimated effective cascaded channel are
near $0$. If the NMSE is beyond $1$, it means that the estimation is
completely wrong. On the other hand, the location-aware channel estimation
algorithms frequently succeed in estimating the effective cascaded channel
even under the NLoS environment, and this results in lower NMSE.

Figure 11: NMSE versus RIS-to-UE distance. The average SNR are $-3.54$ dB and
$-31.2$ dB when the RIS-to-UE distance are $40$ m and $70$ m. Figure 12: NMSE
versus BS-to-RIS AoD error. Colored dotted lines denote the half of the null-
to-null beamwidth according to the number of antennas on the BS.

Fig. 12 shows the NMSE according to the BS-to-RIS AoD error, which is induced
by the location error of the RIS. The BS-to-RIS distance and the RIS-to-UE
distance are respectively fixed to $\sqrt{4100}$ m and $50$ m. Colored dotted
lines denote the half of the null-to-null beamwidth according to the number of
antennas on the BS. Analysis of the location-aware channel estimation via 3D
ANM is omitted since the computation time becomes excessive when
$M_{\textrm{B}}>4$. Fig. 12 shows that the performance of the location-aware
channel estimation degrades as the location error increases. To be more
specific, the NMSE of the location-aware channel estimation becomes $1$ when
the BS-to-RIS AoD reaches the null. This is because the strength of the pilot
signal decreases when the BS-to-RIS AoD reaches closer to the null. Thus, the
location-aware algorithms may not perform well when the location of the RIS is
erroneous and the BS forms highly directional beam.

##  VII Conclusions

In this paper, we propose a location-aware channel estimation based on the ANM
for the RIS-aided MIMO systems. With the location of the BS and the RIS, the
beam training overhead at the BS is reduced by the beam steering towards the
RIS. Also, the RIS beamwidth adaptation reduces the beam training overhead at
the RIS while ensuring the UE receives all the multipath components from the
RIS. When the BS steers the beam towards the RIS during beam training, the
effective cascaded channel is estimated via either 2D ANM or 3D ANM. Here, if
the beam training overhead is further reduced by RIS beamwidth adaptation, the
effective cascaded channel is represented as a linear combination of 3D atoms,
hence is estimated via 3D ANM. The simulation results present that the
location-aware channel estimation via 2D ANM and 3D ANM achieves superior
estimation accuracy to benchmarks. Still, the location-aware channel
estimation via 3D ANM is expected to be more useful if its computational
complexity is reduced. Also, the location-aware channel estimation that are
robust against the location error remains for further study.

## References

  * [1] T. S. Rappaport et al., “Millimeter wave mobile communications for 5G cellular: It will work!,” IEEE Access, vol. 1, pp. 335–349, 2013\.
  * [2] M. Giordani et al., “A tutorial on beam management for 3GPP NR at mmwave frequencies,” IEEE Commun. Surveys Tut., vol. 21, pp. 173–196, 1st qtr. 2019.
  * [3] T. Nitsche et al., “IEEE 802.11ad: directional 60 GHz communication for multi-gigabit-per-second Wi-Fi [invited paper],” IEEE Commun. Mag., vol. 52, pp. 132–141, Dec. 2014.
  * [4] Samsung Research, “6G: The next hyper-connected experience for all,” White paper, 2020.
  * [5] E. G. Larsson, O. Edfors, F. Tufvesson, and T. L. Marzetta, “Massive MIMO for next generation wireless systems,” IEEE Commun. Mag., vol. 52, pp. 186–195, Feb. 2014.
  * [6] M. Di Renzo et al., “Smart radio environments empowered by reconfigurable intelligent surfaces: How it works, state of research, and the road ahead,” IEEE J. Sel. Areas Commun., vol. 38, pp. 2450–2525, Jul. 2020\.
  * [7] M. A. ElMossallamy et al., “Reconfigurable intelligent surfaces for wireless communications: Principles, challenges, and opportunities,” IEEE Trans. Cogn. Commun. Netw., vol. 6, pp. 990–1002, Sep. 2020.
  * [8] P. Wang, J. Fang, H. Duan, and H. Li, “Compressed channel estimation for intelligent reflecting surface-assisted millimeter wave systems,” IEEE Signal Process. Lett., vol. 27, pp. 905–909, May 2020.
  * [9] G. Zhou et al., “Stochastic learning-based robust beamforming design for RIS-aided millimeter-wave systems in the presence of random blockages,” IEEE Trans. Veh. Technol., vol. 70, pp. 1057–1061, Jan. 2021\.
  * [10] S. Li, B. Duo, X. Yuan, Y. Liang, and M. Di Renzo, “Reconfigurable intelligent surface assisted UAV communication: Joint trajectory design and passive beamforming,” IEEE Wireless Commun. Lett., vol. 9, pp. 716–720, May 2020.
  * [11] S. Alfattani et al., “Aerial platforms with reconfigurable smart surfaces for 5G and beyond,” IEEE Commun. Mag., vol. 59, pp. 96–102, Jan. 2021.
  * [12] H. Wymeersch et al., “Radio localization and mapping with reconfigurable intelligent surfaces: Challenges, opportunities, and research directions,” IEEE Veh. Technol. Mag., vol. 15, pp. 52–61, Oct. 2020.
  * [13] X. Wei, D. Shen, and L. Dai, “Channel estimation for RIS assisted wireless communications—part i: Fundamentals, solutions, and future opportunities,” IEEE Commun. Lett., vol. 25, pp. 1398–1402, May 2021.
  * [14] C. Hu, L. Dai, S. Han, and X. Wang, “Two-timescale channel estimation for reconfigurable intelligent surface aided wireless communications,” IEEE Trans. Commun., pp. 1–1, Apr. 2021.
  * [15] K. Ardah, S. Gherekhloo, A. Andre L. F. de, and M. Haardt, “TRICE: A channel estimation framework for RIS-aided millimeter-wave MIMO systems,” IEEE Signal Process. Lett., pp. 1–1, Feb. 2021.
  * [16] Y. Chi, L. L. Scharf, A. Pezeshki, and A. R. Calderbank, “Sensitivity to basis mismatch in compressed sensing,” IEEE Trans. Signal Process., vol. 59, pp. 2182–2195, May 2011.
  * [17] J. He, H. Wymeersch, and M. Juntti, “Channel estimation for RIS-aided mmWave MIMO systems via atomic norm minimization,” Arxiv, Jan. 2021.
  * [18] G. Tang, B. N. Bhaskar, P. Shah, and B. Recht, “Compressed sensing off the grid,” IEEE Trans. Inf. Theory, vol. 59, pp. 7465–7490, Aug. 2013\.
  * [19] J. He, H. Wymeersch, and M. Juntti, “Leveraging location information for RIS-aided mmWave MIMO communications,” IEEE Wireless Commun. Lett., pp. 1–1, Mar. 2021.
  * [20] Z. Yang, L. Xie, and P. Stoica, “Vandermonde decomposition of multilevel Toeplitz matrices with application to multidimensional super-resolution,” IEEE Trans. Inf. Theory, vol. 62, pp. 3685–3701, Apr. 2016.
  * [21] A. F. Molisch et al., “Hybrid beamforming for massive MIMO: A survey,” IEEE Commun. Mag., vol. 55, pp. 134–141, Sep. 2017.
  * [22] M. K. Samimi and T. S. Rappaport, “3-D millimeter-wave statistical channel model for 5G wireless system design,” IEEE Trans. Microw. Theory Tech., vol. 64, pp. 2207–2225, Jun. 2016.
  * [23] C. R. Rao and M. B. Rao, Matrix algebra and its applications to statistics and econometrics.  Singapore: World Scientific, 1998.
  * [24] A. Sayeed and N. Behdad, “Continuous aperture phased MIMO: Basic theory and applications,” in 2010 48th Annu. Allerton Conf. Commun., Control, Comput. (Allerton), pp. 1196–1203, Oct. 2010.
  * [25] Y. Li and Y. Chi, “Off-the-grid line spectrum denoising and estimation with multiple measurement vectors,” IEEE Trans. Signal Process., vol. 64, pp. 1257–1269, Mar. 2016.
  * [26] Z. Tian, Z. Zhang, and Y. Wang, “Low-complexity optimization for two-dimensional direction-of-arrival estimation via decoupled atomic norm minimization,” in 2017 IEEE Int. Conf. Acoust., Speech, Signal Process. (ICASSP), pp. 3071–3075, Mar. 2017.
  * [27] Y. Tsai, L. Zheng, and X. Wang, “Millimeter-wave beamformed full-dimensional MIMO channel estimation based on atomic norm minimization,” IEEE Trans. Commun., vol. 66, pp. 6150–6163, Aug. 2018\.
  * [28] Y. Chi and M. Ferreira Da Costa, “Harnessing sparsity over the continuum: Atomic norm minimization for superresolution,” IEEE Signal Process. Mag., vol. 37, pp. 39–57, Mar. 2020.
  * [29] I. Pólik and T. Terlaky, “Interior point methods for nonlinear optimization,” in Nonlinear Optimization (G. Di Pillo and F. Schoen, eds.), pp. 215–276, Berlin, Heidelberg: Springer, 2010.
  * [30] B. N. Bhaskar, G. Tang, and B. Recht, “Atomic norm denoising with applications to line spectral estimation,” IEEE Trans. Signal Process., vol. 61, pp. 5987–5999, Jul. 2013.
  * [31] M. Grant and S. Boyd, “CVX: Matlab software for disciplined convex programming, version 2.1.” http://cvxr.com/cvx.
  * [32] M. Samir et al., “Optimizing age of information through aerial reconfigurable intelligent surfaces: A deep reinforcement learning approach,” IEEE Trans. Veh. Technol., vol. 70, pp. 3978–3983, Apr. 2021\.
  * [33] H. L. V. Trees, Optimum Array Processing: Detection, Estimation, and Modulation Theory.  New York: Wiley, 2002.