# Blockage-Aware Robust Beamforming in RIS-Aided Mobile Millimeter Wave MIMO
Systems

Yan Yang, Shuping Dang, Miaowen Wen, Bo Ai, and Rose Qingyang Hu, Part of this
manuscript has been accepted for publication in IEEE ICC 2024 [1]. This work
was funded by the National Nature Science Foundation of China under Grants
62271041 and 62221001. (Corresponding author: Yan Yang)Yan Yang and Bo Ai are
with the School of Electronic and Information Engineering, Beijing Jiaotong
University, Beijing 100044, China (email<EMAIL_ADDRESS>boai@bjtu.edu.cn). Shuping Dang is with the School of Electrical, Electronic
and Mechanical Engineering, University of Bristol, Bristol BS8 1UB, U.K.
(e-mail: shuping.dang@bristol.ac.uk). Miaowen Wen is with the School of
Electronic and Information Engineering, South China University of Technology,
Guangzhou 510640, China (e-mail: eemwwen@scut.edu.cn).Rose Qingyang Hu is with
the Department of Electrical and Computer Engineering, Utah State University,
Logan, UT 84322, USA (e-mail: rose.hu@usu.edu).

###### Abstract

Millimeter wave (mmWave) communications are sensitive to blockage over radio
propagation paths. The emerging paradigm of reconfigurable intelligent surface
(RIS) has the potential to overcome this issue by its ability to arbitrarily
reflect the incident signals toward desired directions. This paper proposes a
Neyman-Pearson (NP) criterion-based blockage-aware algorithm to improve
communication resilience against blockage in mobile mmWave multiple input
multiple output (MIMO) systems. By virtue of this pragmatic blockage-aware
technique, we further propose an outage-constrained beamforming design for
downlink mmWave MIMO transmission to achieve outage probability minimization
and achievable rate maximization. To minimize the outage probability, a robust
RIS beamformer with variant beamwidth is designed to combat uncertain channel
state information (CSI). For the rate maximization problem, an accelerated
projected gradient descent (PGD) algorithm is developed to solve the
computational challenge of high-dimensional RIS phase-shift matrix (PSM)
optimization. Particularly, we leverage a subspace constraint to reduce the
scope of the projection operation and formulate a new Nesterov momentum
acceleration scheme to speed up the convergence process of PGD. Extensive
experiments confirm the effectiveness of the proposed blockage-aware approach,
and the proposed accelerated PGD algorithm outperforms a number of
representative baseline algorithms in terms of the achievable rate.

###### Index Terms:

Blockage detection, millimeter wave (mmWave) communications, reconfigurable
intelligent surfaces (RIS), robust beamforming, projected gradient descent,
momentum acceleration.

##  I Introduction

As the front runner of future wireless techniques, millimeter wave (mmWave)
communication has attracted considerable research attention since a large
chunk of potentially usable bandwidth can be unleashed to support the ever-
increasing data-intensive applications[2]. Meanwhile, signal transmissions
over mmWave bands are susceptible to high path and penetration loss. By virtue
of large-scale antenna arrays, path loss at mmWave bands can be compensated by
highly directional beamforming at both transmitter and receiver, by which a
very narrow beam pattern is used to provide a high directional antenna gain.
At present, hybrid analog-digital (HAD) beamforming architecture is a
favorable solution, which is capable of achieving almost the same performance
as the full-digital beamforming scheme[3, 4]. However, the highly directional
mmWave transmission is sensitive to the blockage over the radio propagation
path. The presence of random blockage, as well as self-blockage, e.g., human-
body blockage, normally incurs a sharp degradation of signal strength,
resulting in a high outage probability[5]. Hence, mmWave communications are
subject to a stringent limitation imposed by the blockages and are likely to
only operate in line-of-sight (LoS) scenarios. In mobile mmWave communication
networks, it has been proven that intermittent connectivity is rather
detrimental[6].
To overcome such a challenge, the recent emergence of reconfigurable
intelligent surfaces (RISs), a.k.a. intelligent reflecting surfaces (IRSs),
provides a potential solution by utilizing their unnatural reflection
property[7, 8, 9]. Specifically, the surface is composed of a large number of
low-cost and nearly passive elements, each capable of electronically
controlling the phase of incident electromagnetic waves (EMs). By actively
manipulating phase shifts (PSs), the reflection behavior of EMs can be
purposefully regulated. Therefore, RIS can reflect incident signals toward
desired directions, thereby endowing radio channels with programmability[10,
11, 12, 13]. Unlike traditional amplify-and-forward (AF) relays and
coordinated multi-point (CoMP) transmission, RIS is almost passive. As such,
even though massive RISs are densely deployed, sophisticated interference
management could be exempted. Such a deployment has the potential to give rise
to a dramatic improvement in the overall network’s signal-to-interference-
plus-noise ratio (SINR) performance.
For highly directional mmWave communications, the validity of an RIS
substantially depends on whether the blockages over the radio propagation path
can be effectively detected. More precisely, the receiver should be
sufficiently aware of the blockage status of a link, i.e., whether the
blockage is present and under what condition. In this context, when a complete
link obstruction arises, a base station (BS) can electronically steer the beam
toward a desired RIS, thereby providing an indirect link from the RIS to the
corresponding mobile station (MS). Prior works in the literature have focused
on blockage prediction, robust beamforming, and outage minimization to
overcome such challenges. With the aid of CoMP transmission, a stochastic
learning approach was proposed to capture crucial blockage patterns, and a
robust beamforming design was established to combat uncertain path blockages,
e.g.,[14, 15]. Similarly, a learning-based robust beamforming design was
proposed for RIS-aided mmWave systems in the presence of random blockages
[16].
In this new paradigm, because the RIS comprises only nearly passive elements
and has limited signal processing capability, the acquisition of channel state
information (CSI) is much more challenging compared with the case without an
RIS[17, 18, 19, 20]. On the one hand, RIS-aided communications involve the
cascaded BS-to-RIS and RIS-to-MS channels, where the number of channel
coefficients is proportional to the number of elements. Therefore, channel
estimation becomes a complicated problem due to the involvement of thousands
of interrelated variables[17, 19]. In most existing works, the perfect CSI is
assumed to be obtained for optimally designing the precoding vectors at the BS
and the RIS phase-shift matrix (PSM). Nevertheless, such an idealistic
assumption is practically infeasible. Many recent research efforts have
attempted to develop new approaches for channel estimation, reflection
optimization, and robust beamforming. Notably, the authors in [21] has
considered these critical issues, as well as pursue rate maximization with
lower complexity. On the other hand, in [15], the authors verified that by
exploiting the sparse mmWave channel structure, the channel coefficients of
reflective channels can be derived based on the estimation of angle-of-
departure (AoD), angle-of-arrival (AoA), and corresponding channel gains. At
present, despite these advances, relatively scarce studies have considered the
uncertainties of mmWave radio channel and random blockages, as well as the
impact of imperfect CSI on the performance of RIS-aided mmWave communication
systems. It is worth noting that the presence of random blockages may lead to
more unpredictable channel uncertainties, exacerbating the impact of the
imperfect CSI.
Motivated by this, we focus on tackling the impact of the blockages on RIS-
aided mmWave mobile communication systems. Specifically, we develop a
pragmatic blockage-aware algorithm to substantially reap the benefits of the
RIS, such that when the LoS link is blocked, the BS can immediately steer the
dominant beam towards the RIS and create an alternative reflection link. In
contrast, if the blockage status cannot be proactively identified, the role of
RIS would be futile, since effective blockage detection is a fundamental
prerequisite for RIS reflection to take effect. Furthermore, we propose a
robust beamforming design for downlink mmWave MIMO transmission to achieve
outage probability minimization and achievable rate maximization. The main
contributions of this work are summarized as follows:

  * •

From the perspective of the statistical decision theory, we transform the
blockage detection problem into a Neyman-Pearson (NP) criterion-based binary
hypothesis testing problem. In this way, we develop an NP criterion-based
blockage detector and a trusted NP decision rule. The goal is to maximize the
probability of correct detection and exclude the worst-case system outage.
Experiments validate our theoretical findings and show that without a priori
knowledge of randomly distributed blockers, the proposed blockage-aware
approach can accurately distinguish whether a channel is LoS or non-line-of-
sight (NLoS), giving the opportunity to exclude potential outages
preemptively. To the best of our knowledge, the proposed NP decision rule is
the first trustworthy solution to the random blockage detection problem in
mmWave communication systems.

  * •

Additionally, by taking advantage of angle reciprocity, we propose a mechanism
of link availability indication and formulate an easy-to-handle geometric
transformation approach, which can be integrated into the transceiver and
helps to mitigate the impact of the uncertain CSI introduced by cascaded BS-
RIS-MS channels.

  * •

By virtue of this blockage-aware technique, we further propose an outage-
constrained robust beamforming scheme. Specifically, a beamwidth-variant RIS
beamformer is designed to combat the uncertainty of cascaded BS-RIS-MS
channels. Compared to the constant-beamwidth strategy, our key observation is
that even though the estimates of the CSI and/or PSs are relatively coarse,
the RIS can provide more resilient connectivity for blocked users. The
simulation results show that even though a broadened beam may temporarily
reduce the achievable rate, the sum rate is higher than those of the baseline
algorithms and is close to the optimum achievable rate due to significant
suppression of the outage probability and beam-training overhead.

  * •

To solve the computational challenge of high-dimensional PSM optimization for
large-scale RIS reflection arrays, we propose an accelerated projected
gradient descent (PGD) method with a best-fit subspace constraint.
Essentially, this accelerated version of PGD employs a simplex Euclidean
projection onto a feasible convex set for the constrained PSM optimization
problem. We further leverage a subspace constraint to reduce the scope of the
projection operation, making the aforementioned intractable problem solvable.
To speed up the convergence process, we propose a new Nesterov momentum
acceleration scheme. We demonstrate the superiority of this scheme that can
avoid local minima by preventing the PGD algorithm from getting stuck at the
vicinity saddle points. Unlike the traditional alternating optimization (AO)
algorithm, which only updates a single variable in each iteration, the
accelerated PGD algorithm proposed in this paper can simultaneously update all
optimization variables, resulting in faster convergence and a higher
achievable rate.

The remainder of the paper is organized as follows. In Section II, we
introduce the system and channel models and formulate the problems of the
outage probability minimization and achievable rate optimization. In Section
III, we provide a general statistical decision-making framework to enable the
awareness of blockage and devise a mechanism for link availability indication.
In Section IV, a robust beamformer is developed to mitigate the impact of the
channel uncertainty, and a PGD-based solution is derived to maximize the
achievable rate. Simulation results are shown and discussed in Section V, and
the paper draws conclusions in Section VI.
Notations: Throughout the paper, bold upper and lower case letters denote
matrices and vectors, respectively; normal lower case letters are used for
scalars; Superscripts $(\cdot)^{*}$, $(\cdot)^{T}$ and $(\cdot)^{H}$ stand for
the complex conjugation, transpose, and Hermitian transpose, respectively. We
use $\mathbb{C}$ to denote the field of complex numbers and
$\mathbb{C}^{m\times n}$ to denote an $m$ by $n$ dimensional complex space.
Further, we use ${\text{Tr}}(\cdot)$, ${\text{log}}(\cdot)$, and $|\cdot|$ to
denote the trace, the natural logarithm, and the determinant of a matrix or
the absolute value depending on the context, respectively.
${\mathcal{CN}}(0;\sigma^{2})$ represents the zero-mean complex Gaussian
distribution with variance $\sigma^{2}$. Finally,
$\nabla_{\bm{\theta}}\mathcal{R}(\bm{\theta})$ denotes the partial gradient of
$\mathcal{R}(\cdot)$ with respect to parameter $\bm{\theta}$.

##  II System Model and Problem Formualtion

###  II-A Signal Transmission and Reception

Figure 1: RIS-aided mmWave MIMO communication system with HAD beamforming: A
downlink transmission subject to blockages.

We consider an RIS-aided mmWave MIMO communication system in the downlink, as
illustrated in Fig. 1, where the RIS, with $\mathcal{N}$ reflecting elements
arranged in an ${\mathcal{I}}\times{\mathcal{J}}$ uniform rectangular array
(URA) layout, is utilized to provide a substitutable link in the presence of
random link blockages. A BS with a two-stage HAD beamforming architecture uses
a uniform linear array (ULA) of $N_{t}$ transmit antennas and
$N_{t}^{\text{RF}}$ RF chains to serve $K$ MSs. Unless otherwise specified, we
assume that each MS is equipped with $N_{r}$-antenna ULA and
$N_{r}^{\text{RF}}$ RF chains, and these $N_{t,k}^{\text{RF}}$ RF chains can
be configured for a specific MS to transmit varying $N_{s}$ data streams given
$N_{s}\leq N_{t}^{\text{RF}}$. During downlink transmission, the data stream
${\textbf{s}}=\\{{\textbf{s}}_{k}\\}_{k=1}^{K},where{\textbf{s}}_{k}\in{\mathbb{C}}^{N_{s}\times
1}$, is processed by a set of digital baseband precoders
$\\{{\textbf{F}}_{\text{BB}_{k}}\\}_{k=1}^{K},where{\textbf{F}}_{\text{BB}_{k}}\in\mathbb{C}^{N_{t,k}^{\text{RF}}\times
N_{s}}$, and followed by a set of analog RF precoders
$\\{{\textbf{F}}_{\text{RF}_{k}}\\}_{k=1}^{K},where{\textbf{F}}_{\text{RF}_{k}}\in\mathbb{C}^{N_{t,k}\times
N_{s}}$. Likewise, a hybrid combiner is deployed at the receiver by the
concatenation of an RF combiner
${{\textbf{W}}_{{\text{RF}}_{k}}}\in\mathbb{C}^{N_{r,k}\times
N_{r}^{\text{RF}}}$ and a low-dimensional digital baseband combiner
${\textbf{W}}_{{\text{BB}}_{k}}\in\mathbb{C}^{N_{r}^{\text{RF}}\times N_{s}}$
to sequentially process the digitally precoded streams.
Under such a setting, the RIS associated with an RIS controller is installed
at a known location111The passive beamforming at the RIS is generally
manipulated by the BS. As such, the BS computes the PSs at the RIS, and the
corresponding results can be transferred to the RIS controller over dedicated
control channels [19].. We define $\mathcal{K}=\\{1,2,...,K\\}$ to be the set
of $K$ active MSs. For an arbitrary MS $k\in\mathcal{K}$, when the direct link
is blocked, we suppose that $\mathcal{N}_{k}$ reflecting element of the RIS
can be partitioned into multiple rectangular subarrays of size
${{\mathcal{I}_{k}}\times{\mathcal{J}_{k}}}$, given
$\sum_{k=1}^{K}\mathcal{N}_{k}\leq\mathcal{N}$,
${{\mathcal{I}}_{k}}\leq{\mathcal{I}}$ and
${{\mathcal{J}}_{k}}\leq{\mathcal{J}}$.
${\bm{\Psi}}_{k}({\bm{\theta}})={\textrm{diag}}({\bm{\theta}})\in\mathbb{C}^{{\mathcal{N}}_{k}\times{\mathcal{N}}k}$
is the PSM associated to MS $k$, where
${\bm{\theta}}=\left[e^{j\theta_{1,k}},\dots,e^{j\theta_{{\imath\jmath},k}},\dots\right]^{T}\in\mathbb{C}^{{{\mathcal{N}_{k}}}\times
1}$, $\\{\theta_{\imath\jmath,k}\\}_{\imath\jmath=1}^{\mathcal{N}_{k}}$
represents a set of PSs induced by these $\mathcal{N}_{k}$ RIS elements, and
$\theta_{\imath\jmath,k}$ is the PS of the $(\imath,\jmath)$th entry of the
URA, given $1\leq\imath\leq\mathcal{I}_{k}$ and
$1\leq\jmath\leq\mathcal{J}_{k}$. Due to the sparse nature of mmWave channels,
we can make a dichotomous assumption that the mmWave channel is either
unblocked or blocked. This strong duality suggests that $\mathcal{K}$ could be
partitioned into two disjoint subsets, denoted by $\mathcal{K}^{\text{NLoS}}$
and $\mathcal{K}^{\text{LoS}}$, corresponding to those MSs under the NLoS and
the LoS conditions, respectively.
In the downlink, the transmitted signal to the $k$th MS in a generic HAD
mmWave system is given by

${\textbf{x}}_{k}={\textbf{F}}_{k}{{\textbf{s}}_{k}},\quad k\in\mathcal{K},$
(1)

where
${\textbf{F}}_{k}={\textbf{F}}_{\text{RF}_{k}}{\textbf{F}}_{\text{BB}_{k}}\in\mathbb{C}^{N_{r}\times
N_{t,k}}$ is the hybrid precoder and is in nature different for each
individual MS; ${{\textbf{s}}_{k}}\in\mathbb{C}^{N_{s}\times 1}$ is a
normalized transmit signal vector, i.e.,
$\mathbb{E}\\{||{{\textbf{s}}_{k}}||^{2}\\}=1$.
Adopting the Rician fading channel model, both the direct path and the
reflection path are considered, such that the channel matrix between the BS
and the $k$th MS can be described as

$\overline{\textbf{H}}_{k}({\bm{\theta}})=\varepsilon_{k}\sqrt{\frac{\kappa\overline{\varrho}_{{\text{d}},k}}{\kappa+1}}{{{\textbf{H}}}_{{\text{d}},k}}+\sqrt{\frac{\overline{\varrho}_{\text{c},k}}{\kappa+1}}{{\textbf{H}}}_{\text{c},k}({\bm{\theta}}),\;k\in{\mathcal{K}}^{\text{LoS}}$
(2a)
$\widetilde{\textbf{H}}_{k}({\bm{\theta}})=\varepsilon_{k}\sqrt{\frac{\widetilde{\varrho}_{{\text{d}},k}}{\kappa+1}}{{\textbf{H}}_{{\text{d}},k}}+\sqrt{\frac{\kappa\widetilde{\varrho}_{\text{c},k}}{\kappa+1}}{{\textbf{H}}}_{{\text{c}},k}({\bm{\theta}}),\;k\in{\mathcal{K}}^{\text{NLoS}}$
(2b)

where ${{\textbf{H}}_{{\text{d}},k}}\in\mathbb{C}^{N_{r}\times N_{t,k}}$
represents the direct BS-MS channel, and
${\textbf{H}}_{\text{c},k}({\bm{\theta}})={{{\textbf{H}}_{\text{r},k}^{H}}{{\bm{\Psi}}_{k}}}(\bm{\theta}){\textbf{H}}_{\text{i},k}\in\mathbb{C}^{N_{r}\times
N_{t,k}}$ represents the cascaded BS-RIS-MS channel, both associated with MS
$k$. Here ${{\textbf{H}}_{\text{r},k}}\in\mathbb{C}^{\mathcal{N}_{k}\times
N_{t}}$ and
${\textbf{H}}_{\text{i},k}\in\mathbb{C}^{\mathcal{N}_{r}\times{\mathcal{N}}_{k}}$
are the matrices corresponding to the indirect channels between the BS and the
RIS (BS-RIS) and the RIS and MS (RIS-MS), respectively. In addition,
$\kappa\in[0,\infty)$ is the non-centrality parameter of the Rician
distribution; $\varepsilon_{k}\in[0,1]$ is a normalized random variable that
represents the link loss rate caused by a blocker, and $\varepsilon_{k}=0$
corresponds to the worst-case channel condition, that is, the case where the
$k$th MS is fully blocked. $\varrho_{\text{d},k}$ and $\varrho_{\text{r},k}$
are the fading coefficients of the direct and indirect links, respectively. To
simplify the notations and facilitate the following analysis, we denote
$\upsilon_{0}=\sqrt{\frac{\kappa\varrho_{\textup{d},k}}{\kappa+1}}$ and
$\upsilon_{1}=\sqrt{\frac{\varrho_{\text{c},k}}{\kappa+1}}$ by the two random
variables accounting for the effects of both the path loss and the
Ricianfactor factor in sparse scattering environments.
Accordingly, the $N_{r}\times 1$ received signal vector at the antenna array
of the $k$th MS can be expressed as

$\displaystyle{\overline{\textbf{y}}}_{k}={\underbrace{\varepsilon_{k}\overline{\upsilon}_{0}{{\textbf{W}}^{H}_{k}}{{\textbf{H}}_{{\text{d}},k}}{{\textbf{x}}_{k}}}_{\text{Dominant
signal}:\,{\overline{\textbf{y}}}_{{\text{d}},k}}}+{\underbrace{\overline{\upsilon}_{1}{{\textbf{W}}^{H}_{k}}{{{\textbf{H}}_{\text{r},k}^{H}}{{\bm{\Psi}}_{k}}(\bm{\theta}){{\textbf{H}}}_{\text{i},k}{{\textbf{x}}_{k}}}}_{\text{Reflected
auxiliary
signal}:\,{\overline{\textbf{y}}}_{{\text{r}},k}}+{\textbf{n}}_{k}},$ (3a)
$\displaystyle{\widetilde{\textbf{y}}}_{k}={\underbrace{{{\varepsilon_{k}\widetilde{\upsilon}_{0}}}{{{\textbf{W}}^{H}_{k}}{{\textbf{H}}}_{{\text{d}},k}}{{\textbf{x}}_{k}}}_{\text{Auxiliary
signal}:\,{\widetilde{\textbf{y}}}_{{\text{d}},k}}}+{\underbrace{\widetilde{\upsilon}_{1}{{{\textbf{W}}^{H}_{k}}{{\textbf{H}}_{\text{r},k}^{H}}{{\bm{\Psi}}_{k}}(\bm{\theta}){{\textbf{H}}}_{\text{i},k}{{\textbf{x}}_{k}}}}_{\text{Reflected
dominant
signal}:\,{\widetilde{\textbf{y}}}_{{\text{r}},k}}+{\textbf{n}}_{k}},$ (3b)

where
${{\textbf{W}}^{H}_{k}}={{\textbf{W}}^{H}_{{\text{BB}}_{k}}}{{\textbf{W}}_{{\text{RF}}_{k}}}$;
${\textbf{n}}_{k}\sim\mathcal{CN}\left\\{0,{\sigma}_{k}^{2}\textbf{I}\right\\}$
is the circularly symmetric complex additive white Gaussian noise (AWGN) at
the $k$th MS; ${\sigma}_{k}^{2}$ represents the noise variance associated with
the $k$th MS; ${\textbf{I}}\in\mathbb{C}^{N_{r}\times N_{r}}$ is the identity
matrix. For ease of presentation, we also denote ${\textbf{x}}_{\text{d},k}$
and ${\textbf{x}}_{\text{r},k}$ as the direct and indirect signal impinging on
the antenna array, respectively. For convenience, we denote
${\textbf{U}}_{k}={{\textbf{F}}}^{H}_{k}{\textbf{F}}_{k}$ as the transmit
beamforming gain, ${\textbf{V}}_{k}={{\textbf{W}}_{k}^{H}}{{\textbf{W}}_{k}}$
as the receive beamforming gain, and
${\textbf{Z}}_{k}(\bm{\theta})={{{\textbf{H}}_{\text{r},k}^{H}}{{\bm{\Psi}}_{k}}(\bm{\theta}){\textbf{H}}_{\text{i},k}}$.
The total beamforming gain is thus given by
${\textbf{G}}={\textbf{V}}_{k}{\textbf{U}}_{k}$. The signal-to-interference-
plus-noise ratio (SINR) on the $k$th MS can now be defined in a closed form as

$\displaystyle\overline{\Gamma}_{k}(\bm{\theta},{\textbf{G}})=$
$\displaystyle\frac{\varepsilon_{k}^{2}\overline{\upsilon}_{0}^{2}}{J_{0}}\left|{\textbf{H}}_{\text{d},k}{\textbf{G}}{\textbf{H}}_{\text{d},k}^{H}\right|+\frac{\overline{\upsilon}_{1}^{2}}{J_{0}}\left|{\textbf{Z}}_{k}(\bm{\theta}){\textbf{G}}{\textbf{Z}}_{k}^{H}(\bm{\theta})\right|,\;\;$
(4a) $\displaystyle\widetilde{\Gamma}_{k}(\bm{\theta},{\textbf{G}})=$
$\displaystyle\frac{\varepsilon_{k}^{2}\widetilde{\upsilon}_{0}^{2}}{J_{0}}\left|{\textbf{H}}_{\text{d},k}{\textbf{G}}{\textbf{H}}_{\text{d},k}^{H}\right|+\frac{\widetilde{\upsilon}_{1}^{2}}{J_{0}}\left|{\textbf{Z}}_{k}(\bm{\theta}){\textbf{G}}{\textbf{Z}}_{k}^{H}(\bm{\theta})\right|,$
(4b)

where $J_{0}=N_{t,k}\sigma_{k}^{2}+\sum_{j\neq k}g_{j}$ is the total signal
power received from all the other MSs, and $\sigma_{k}^{2}$ is a measure of
the noise at the $k$th MS. Under the assumption that the perfect CSI is
available at the transceiver, the achievable rate of the $k$th user, denoted
by $\mathcal{R}_{k}$ and measured in bit/s/Hz, can be calculated as

$\displaystyle\overline{\mathcal{R}}_{k}(\bm{\theta},\textbf{G})$
$\displaystyle={\log_{2}}\det\left({\textbf{I}}_{k}+\frac{1}{J_{0}}\overline{\textbf{H}}_{k}({\bm{\theta}}){\textbf{G}}{\overline{\textbf{H}}}_{k}^{H}({\bm{\theta}})\right),$
(5a) $\displaystyle\widetilde{\mathcal{R}}_{k}(\bm{\theta},\textbf{G})$
$\displaystyle={\log_{2}}\det\left({\textbf{I}}_{k}+\frac{1}{J_{0}}\widetilde{\textbf{H}}_{k}({\bm{\theta}}){\textbf{G}}{\widetilde{\textbf{H}}}_{k}^{H}({\bm{\theta}})\right).$
(5b)

In such a case, the downlink sum rate can be determined as

$\displaystyle{\mathcal{R}}_{\text{sys}}={\sum\limits}_{k\in\mathcal{K}^{\text{LoS}}}\overline{\mathcal{R}}_{k}({\bm{\theta}},{\textbf{G}})+{\sum\limits}_{k\in\mathcal{K}^{\text{NLoS}}}\widetilde{\mathcal{R}}_{k}(\bm{\theta},\textbf{G}).$
(6)

###  II-B Channel Model

Due to the inherent sparsity of mmWave channels, we adopt a geometric channel
model as proposed in[22] to describe the characteristics of path clusters,
i.e., taps, in the angular domain between transceiver pairs. At the $k$th MS,
suppose that the number of effective channel clusters for the BS-MS links, the
BS-RIS links, and the RIS-MS links are $L_{\text{d},k}$, $L_{\text{r},k}$, and
$L_{\text{i},k}$, respectively. With this clustered channel model, the channel
matrices in (3) can be written as

$\displaystyle{\textbf{H}}_{\text{d},k}={\frac{1}{\sqrt{L_{\text{d},k}}}}\sum\limits_{\ell=1}^{{L}_{\text{d},k}}\textsl{g}^{\ell}_{\text{d},{k}}{\textbf{a}}_{{\text{MS}}}\left(\psi^{\text{AoA}_{\ell}}_{{\text{d},{k}}}\right){\textbf{a}}_{{\text{BS}}}\left(\phi^{{\text{AoD}_{\ell}}}_{{\text{d},{k}}}\right)^{H},$
(7)
$\displaystyle{\textbf{H}}_{\text{r},k}={\frac{1}{\sqrt{L_{\text{r},k}}}}\sum\limits_{\ell=1}^{L_{\text{r},k}}\textsl{g}^{\ell}_{\text{r},{k}}{\textbf{a}}_{{\text{MS}}}\left(\psi^{\text{AoA}_{\ell}}_{{\text{r},{k}}}\right){\textbf{a}}_{{\text{RIS}}}\left(\phi^{\text{AoD}_{\ell}}_{{\text{r},{k}}},\varphi^{\text{AoD}_{\ell}}_{{\text{r},{k}}}\right)^{H},$
(8)
$\displaystyle{\textbf{H}}_{{\text{i},k}}={\frac{1}{\sqrt{L_{\text{i},k}}}}\sum\limits_{\ell=1}^{L_{\text{i},k}}\textsl{g}^{\ell}_{{\text{i},k}}{\textbf{a}}_{{\text{RIS}}}\left(\phi^{\text{AoA}_{\ell}}_{{\text{i},k}},\varphi^{\text{AoA}_{\ell}}_{{\text{i},k}}\right){\textbf{a}}_{{\text{BS}}}\left(\phi^{\text{AoD}_{\ell}}_{{\text{d},k}}\right)^{H},$
(9)

where $\textsl{g}^{\ell}_{\text{d},{k}}$, $\textsl{g}^{\ell}_{\text{r},{k}}$
and $\textsl{g}^{\ell}_{{\text{i},k}}$ are the complex gains of the $\ell$th
tap of the corresponding links, respectively;
$\left\\{\psi^{\text{AoA}_{\ell}}_{{\text{d},{k}}},\phi^{{\text{AoD}_{\ell}}}_{{\text{d},{k}}}\right\\}$,
$\left\\{\psi^{\text{AoA}_{\ell}}_{{\text{r},{k}}},\left\\{\phi^{\text{AoD}_{\ell}}_{{\text{r},{k}}},\varphi^{\text{AoD}_{\ell}}_{{\text{r},{k}}}\right\\}\right\\}$
and
$\left\\{\left\\{\phi^{\text{AoA}_{\ell}}_{{\text{i},k}},\varphi^{\text{AoA}_{\ell}}_{{\text{i},k}}\right\\},\phi^{\text{AoD}_{\ell}}_{{\text{d},k}}\right\\}$
are the azimuth (horizontal) AoA and AoD pairs of the $\ell$th tap for the BS-
MS links, the RIS-MS links, and the BS-RIS links, respectively. Moreover,
${\textbf{a}}_{{\text{BS}}}(\phi)$, ${\textbf{a}}_{{\text{MS}}}(\phi)$ and
${\textbf{a}}_{{\text{RIS}}}(\phi,\varphi)$ represent the transmit (BS),
receive (MS) and reflect (RIS) array responses, i.e., steering vectors,
respectively, given by

$\displaystyle{\textbf{a}}_{\text{BS}}({\phi})=\frac{1}{{\sqrt{{N_{t,k}}}}}{\left[{1,{e^{j{{\cos}}({{\phi}})}}},\dots,{e^{j({N_{t,k}}-1){{\cos}}({{\phi}})}}\right]^{T},}$
(10)
$\displaystyle{\textbf{a}}_{\text{MS}}({\psi})=\frac{1}{{\sqrt{{N_{r}}}}}\left[{1,{e^{j{{\cos}}({\psi})}},\dots,{e^{j({N_{r}}-1){{\cos}}({\psi})}}}\right]^{T},$
$\displaystyle{\textbf{a}}_{\text{RIS}}({\phi},\varphi)=\frac{1}{{\sqrt{{\mathcal{N}_{k}}}}}\left[1,{e^{j\xi_{1}(\phi,\varphi)}},\dots,{e^{j\xi_{\imath\jmath}(\phi,\varphi)}}\right]^{T},$

where
${\xi}_{\imath\jmath}(\phi,\varphi)=(\imath-1){{\sin}}\,{\varphi}{{\sin}}{\phi}+(\jmath-1){{\cos}}{\phi}$
is the phase of an incoming plane wave at the $\imath\jmath$th RIS element;
$\lambda$ is the wavelength, and $d$ is the inter-element distance of the RIS.
For simplicity, we adopt a uniformly normalized geometric array factor to
describe the layout of the radiating elements of BS, RIS, and MS.
In general, the CSI acquisition of the cascaded channel is challenging due to
the passive nature of RIS. However, from the perspective of the angular
domain, the uncertainty of $\widetilde{\textbf{H}}_{\text{r},k}$ is
significantly greater than that of $\widetilde{\textbf{H}}_{\text{d},k}$, and
the channel estimation errors of the cascaded BS-RIS-MS links are mainly
determined by the reflection links. A important reason is that the BS-RIS link
is actually a fixed point-to-point cluster channel, which means that
$\widetilde{\textbf{H}}_{\text{i},k}$ could be completely known in the angular
domain.
Hence, the uncertainty of CSI in RIS-aided mmWave communication systems can be
simplified to be solely described by the estimated error of the RIS-MS
channel. As a result, estimated $\hat{\widetilde{\textbf{H}}}_{\text{r},k}$
can be calculated as [23]

${\widetilde{\textbf{H}}}_{\text{r},k}(\hat{\bm{\theta}})={{\varsigma}_{k}}{{\widetilde{\textbf{H}}}}_{\text{r},k}({\bm{\theta}})+\sqrt{1-\varsigma_{k}^{2}}\textbf{E},~{}\forall\,k\in\mathcal{K}^{\text{NLoS}},$
(11)

where $\varsigma_{k}\in[0,1]$ represents the reliability of the estimate, and
E is an error matrix with i.i.d. $\mathcal{CN}(0,1)$ distributed entries.
In this way, the uncertainty of the cascaded BS-IRS-MS channel can ultimately
be reflected as inaccurate estimates of PSs. The mean squared error (MSE) of
the channel uncertainty associated with the inaccurate estimates of PSs can be
written as

$\text{MSE}_{{\bm{\theta}}}={\mathbb{E}}\left\\{\left|\left|{\bm{\theta}}-{\hat{\bm{\theta}}}\right|\right|^{2}\right\\},~{}\forall\,k\in\mathcal{K}^{\text{NLoS}}.$
(12)

###  II-C Problem Formulation

For potentially blocked user $k$, our goal here is to maximize the achievable
rate by jointly adapting transmit beamformer ${\textbf{U}}_{k}$, receive
beamformer ${\textbf{V}}_{k}$, and PSM $\bm{\Psi}_{k}(\bm{\theta})$, while
striving to minimize the outage probability. To find a feasible solution, we
propose a pragmatic approach that sequentially tackles the following two-stage
optimization problems: the minimization of outage probability and the
maximization of achievable rate.
We first consider the minimization of outage probability for arbitrary MS $k$
under pure NLoS conditions, neglecting the case of any LoS. Let
$\Pr\left\\{\widetilde{\Gamma}_{k}(\bm{\theta},{\textbf{G}})\leq{\gamma_{\text{th}}}\right\\}$
be the cumulative outage probability of MS $k$. Mathematically, this
minimization problem can be formulated as

$\displaystyle(\mathcal{P}_{1}):$
$\displaystyle{\mathop{\mathrm{\min}}\limits_{{\bm{\theta}},\,{\textbf{G}}}}~{}\Pr\left(\widetilde{\Gamma}_{k}(\bm{\theta},{\textbf{G}})\leq{\gamma_{\text{th}}}\right),\quad
k\in\mathcal{K}^{\text{NLoS}},$ (13)
$\displaystyle\;\;\mathrm{s.t.~{}}~{}\mathrm{Tr}(\textbf{G})\leq\Omega_{\text{tot}},$

where $\gamma_{\text{th}}$ is the required SINR threshold, corresponding to
the minimum acceptable rate. By the definition of the outage probability, an
outage event will take place when the SINR falls below threshold
$\gamma_{\mathrm{th}}$.
In the second stage, we aim to find a set of optimum ${\textbf{U}}_{k}$,
${\textbf{V}}_{k}$, and ${\bm{\theta}}_{k}$ that maximize the achievable rate
under the total power constraint $\Omega_{\text{tot}}$. This can be cast as
the following optimization problem:

$\displaystyle(\mathcal{P}_{2}):\,{\mathop{\mathrm{\max}}\limits_{\bm{\theta},\,{\textbf{G}}}}$
$\displaystyle\widetilde{\mathcal{R}}_{k}(\bm{\theta},{\textbf{G}}),\quad
k\in\mathcal{K}^{\text{NLoS}}$ (14) $\displaystyle{\mathrm{s.t.}}$
$\displaystyle\mathrm{Tr}(\textbf{G})\leq\Omega_{\text{tot}},\;\widetilde{\Gamma}_{k}(\bm{\theta},{\textbf{G}})>{\gamma_{\text{th}}},$

where $\Omega_{\text{tot}}$ is the total transmit power constraint. Unless
stated otherwise, we assume that objective function
$\widetilde{\mathcal{R}}_{k}(\bm{\theta},{\textbf{G}})$ per se is convex and
smooth. Because the objective function
$\widetilde{\mathcal{R}}_{k}(\bm{\theta},{\textbf{G}})$ in the involved
variables is nonconvex, the global optimum of this constrained optimization
problem is hard, in general, to be found. Since the number of RIS elements is
large, the AO techniques holding one variable and the other variables fixed in
a cyclical fashion may require more iterations, resulting in a practically
prohibitive duration for solving the optimization problem. Furthermore, it is
decisive for $\mathcal{P}_{1}$ to efficiently detect the availability of
dominant links, especially when a sudden blockage occurs. To facilitate the
following analysis we first need to devise a statistical inference algorithm
based on hypothesis testing for random blockage detection. On the basis of
trustworthy blockage detection, we provide a robust beamforming design and
propose an accelerated PGD algorithm to address the above non-convex
optimization problem $\mathcal{P}_{2}$. More details will be given in the
sequel.

##  III Blockage Detection and Indication

In this section, we provide a general statistical decision framework to solve
the blockage detection problem. By exploiting the reciprocity of angle-domain
spatial channels, we further propose a mechanism of link availability
indication.

###  III-A Binary Hypothesis Testing for Random Blockage Detection

From the perspective of statistical decision theory, we formulate the blockage
detection as a binary hypothesis testing problem, and its process can be
described by a three-tuple
$(\mathcal{H},\mathcal{Y}_{\varepsilon},\mathscr{P}_{\varepsilon})$, where
$\mathcal{H}=\left\\{{\mathcal{H}}_{0}^{\text{LoS}},{\mathcal{H}}_{1}^{\text{NLoS}}\right\\}$
is a set that consists of two hypotheses: ${\mathcal{H}}_{0}^{\text{LoS}}$ is
the null hypothesis of no blockage, and ${\mathcal{H}}_{1}^{\text{NLoS}}$ is
the alternative hypothesis of blockage;
$\mathcal{Y}_{\varepsilon}=\left\\{\mathcal{Y}_{0}^{\text{LoS}},\mathcal{Y}_{1}^{\text{NLoS}}\right\\}$
is the decision region with parameter $\varepsilon$, representing the disjoint
sample sets $\mathcal{Y}_{0}^{\text{LoS}}$ and $\mathcal{Y}_{1}^{\text{NLoS}}$
that decide LoS or NLoS channel condition between the BS and the MS, given
$\mathcal{Y}_{0}^{\text{LoS}}\cap\mathcal{Y}_{1}^{\text{NLoS}}=\varnothing$;
$\mathscr{P}_{\varepsilon}$ is the probability distribution of the random
variable $\varepsilon$ corresponding to blockages.
For the brevity of mathematical representation, we temporarily omit
${\textbf{W}}_{k}$ in the following derivation. Note that, without loss of
generality, we can rewrite (3a) as a time-domain noisy observation with
unknown parameter $\varepsilon$, which is given by

$\displaystyle{{\textbf{y}}}_{q}(\varepsilon)={{\textbf{y}}}_{k}(t_{q})$
$\displaystyle=\overline{\textbf{y}}_{\text{d},k}(t_{q})+\overline{\textbf{y}}_{\text{r},k}(t_{q}-{\tau})+{\textbf{n}}_{k}(t_{q}).$
(15)

Notice that $\overline{\textbf{y}}_{\text{d},k}(t_{q})$ and
$\overline{\textbf{y}}_{\text{r},k}(t_{q}-{\tau})$ represent the signals over
different propagation paths. They are resolvable in terms of path delay
component $\tau$, especially when considering the spatial sparsity of mmWave
channels222The estimation of path delay has been well investigated in previous
literature, and a detailed treatment for this issue is available in [24].
Generally, the time-of-arrival (ToA) detection and received signal power
measurement with a high degree of accuracy can be used to efficiently
distinguish different propagation paths.. This suggests that the presence or
absence of a blockage can be identified through the detection of the direct
signal.
Therefore, our goal here can be equivalently transferred to preemptively
detect whether a strong LoS signal $\overline{\textbf{y}}_{\text{d},k}$ of
interest is present. This can be done by distinguishing between the following
two simple hypotheses [25]:

$\displaystyle\mathcal{H}_{0}^{\text{LoS}}:$
$\displaystyle\;{{\textbf{y}}_{q}}(\varepsilon)=\overline{\textbf{y}}_{\text{d},k}+\overline{\textbf{y}}_{\text{r},k}+{\textbf{n}}_{k},\;$
$\displaystyle\textit{if}\;\;\varepsilon>\varepsilon_{\text{th}},$ (16)
$\displaystyle\mathcal{H}_{1}^{\text{NLoS}}:$
$\displaystyle\;{{\textbf{y}}_{q}}(\varepsilon)=\overline{\textbf{y}}_{\text{r},k}+{\textbf{n}}_{k},\;$
$\displaystyle\textit{if}\;\;\varepsilon<\varepsilon_{\text{th}},$

where $\varepsilon_{\text{th}}$ is a pre-defined threshold, identifying the
absence or presence of the blockage. From (3a) and (3b), we can see that
$\varepsilon$ is only related to the effects of the blockages but independent
of the Rician factor and the fading coefficients, and thus it can be
calculated as

$\displaystyle\varepsilon=\frac{1}{\upsilon_{0}}\sqrt{\frac{|{\overline{\textbf{y}}_{\text{d},k}^{H}}(t_{q}){\textbf{V}}_{k}{\overline{\textbf{y}}_{\text{d},k}(t_{q})}|}{|{\textbf{H}}_{\text{d},k}{\textbf{G}}{\textbf{H}}_{\text{d},k}^{H}|}}.$
(17)

Consequently, the probability mass function (PMF) of this binary hypothesis
testing problem under each hypothesis can be obtained by the following form:

$\displaystyle\mathcal{H}_{0}^{\text{LoS}}:$
$\displaystyle\;{{\mathcal{Y}}_{\varepsilon}}\sim
f_{{\textbf{Y}}_{q}}\left(\varepsilon>\varepsilon_{\text{th}}|\mathcal{H}_{0}^{\text{LoS}}\right),$
(18) $\displaystyle\mathcal{H}_{1}^{\text{NLoS}}:$
$\displaystyle\;{{\mathcal{Y}}_{\varepsilon}}\sim
f_{{\textbf{Y}}_{q}}\left(\varepsilon<\varepsilon_{\text{th}}|\mathcal{H}_{1}^{\text{NLoS}}\right),$

where $p_{{\textbf{Y}}_{k}}(\cdot)$ is the corresponding PMF of different
observations ${\textbf{Y}}_{k}$ associated with $\mathcal{H}_{0}$ or
$\mathcal{H}_{1}$.
Given a fixed tolerable probability of false alarm $\alpha_{k}$, an
advantageous option in practice is to maximize the probability of correct
blockage detection. Upon using the NP lemma, an alternative decision rule is
to leverage the “significance level” of the test for null hypothesis
$\mathcal{H}_{0}$ against alternative hypothesis $\mathcal{H}_{1}$, which is
equivalent to solving the following optimization problem:

$\displaystyle\delta_{\text{NP}}({\textbf{y}}_{q}(\varepsilon))=$
$\displaystyle\mathop{\arg\max}_{\varepsilon}\quad
P_{\text{D}}^{\text{NLoS}}(\varepsilon_{\text{th}}),$ (19)
$\displaystyle\quad\;\text{s.t.}\quad\quad
P_{\text{FA}}^{\text{LoS}}(\varepsilon_{\text{th}})\leq\alpha,$

where $P_{\text{D}}^{\text{NLoS}}(\varepsilon_{\text{th}})$ is the probability
of correct blockage detection;
$P_{\text{FA}}^{\text{LoS}}(\varepsilon_{\text{th}})$ is the probability of
false alarm, corresponding to detection threshold $\varepsilon_{\text{th}}$.
During each decision epoch, we suppose that the MS directly collects $M$
samples on the basis of observation ${\textbf{y}}_{q}(\varepsilon)$ indexed on
$m\in\\{0,1,\dots,M-1\\}$ in time. In this way, we can model the observed data
by a discrete observation set
${\textbf{Y}}=\left\\{{\textbf{Y}}_{1},\dots,{\textbf{Y}}_{q},\dots\right\\}$,
where
${\textbf{Y}}_{q}=\left[{\textbf{y}}_{q}^{(0)},\dots,{\textbf{y}}_{q}^{(m)},\dots\right]^{T}\in\mathbb{C}^{N_{t,k}\times
M}$, and ${\textbf{y}}_{q}^{(m)}$ represents each possible observation [26].
From the basics of probability theory, probability of correct detection
$P_{\text{D}}^{\text{NLoS}}(\varepsilon_{\text{th}})$ and the probability of
false alarm $P_{\text{FA}}^{\text{LoS}}(\varepsilon_{\text{th}})$ can be
respectively calculated as

$\displaystyle P_{\text{D}}^{\text{NLoS}}(\varepsilon_{\text{th}})$
$\displaystyle=P\left(\mathcal{H}_{1}^{\text{NLoS}}|\mathcal{H}_{1}^{\text{NLoS}}\right)$
(20)
$\displaystyle=\prod\limits_{l:\;\mathcal{L}_{l}(\varepsilon_{\text{th}})>\Lambda}\int_{0}^{\varepsilon_{\text{th}}}f_{{\textbf{y}}_{q}^{(m)}}(\varepsilon)\,d\varepsilon=\beta,$

and

$\displaystyle P_{\text{FA}}^{\text{LoS}}(\varepsilon_{\text{th}})$
$\displaystyle=P\left(\mathcal{H}_{1}^{\text{NLoS}}|\mathcal{H}_{0}^{\text{LoS}}\right)$
(21)
$\displaystyle=\prod\limits_{l:\;\mathcal{L}_{l}(\varepsilon_{\text{th}})<\Lambda}\int_{\varepsilon_{\text{th}}}^{1}f_{{\textbf{y}}_{q}^{(m)}}(\varepsilon)d\varepsilon\leq\alpha,$

where $\alpha\in[0,1]$ is the significance level of the test of user $k$,
which can may be viewed as a transition probability of going from the LoS
state to the NLoS state.
Following the above formulation and derivation, the challenge now becomes to
compare the best model in class $\mathcal{H}_{1}$ to the best model in
$\mathcal{H}_{0}$, which can be formalized as follows. For given
$P_{\text{FA}}^{\text{LoS}}=\alpha$, NP’s optimal solution achieving maximum
$P_{\text{D}}$ can be determined by the likelihood rate testing (LRT) as [25,
26]

$\displaystyle\mathcal{L}(\varepsilon_{\text{th}})$
$\displaystyle=\frac{\displaystyle\int_{0}^{\varepsilon_{\text{th}}}f_{{\textbf{Y}}_{q}}\left(\varepsilon|\mathcal{H}_{1}^{\text{NLoS}}\right)\,d\varepsilon}{\displaystyle\int_{\varepsilon_{\text{th}}}^{1}f_{{\textbf{Y}}_{q}}\left(\varepsilon|\mathcal{H}_{0}^{\text{LoS}}\right)\,d\varepsilon}$
(22)
$\displaystyle=\prod\limits_{l=1}^{\mathcal{M}}{\frac{\displaystyle\int_{0}^{\varepsilon_{\text{th}}}{f_{{\textbf{y}}_{q}^{(m)}}\left(\varepsilon|\mathcal{H}_{1}^{\text{NLoS}}\right)\,d\varepsilon}}{\displaystyle\int_{\varepsilon_{\text{th}}}^{1}{f_{{\textbf{y}}_{q}^{(m)}}\left(\varepsilon|\mathcal{H}_{0}^{\text{LoS}}\right)\,d\varepsilon}}\mathop{\frac{>}{<}}^{\mathcal{H}_{1}^{\text{NLoS}}}_{\mathcal{H}_{0}^{\text{LoS}}}\Lambda}.$

In this paper, we construct a deterministic decision rule
$\delta_{\text{NP}}:{\textbf{y}}_{q}\to\\{1,0\\}$ with a LRT threshold
$\Lambda$. Mathematically, it can be reformulated as

$\displaystyle\delta_{\text{NP}}({\textbf{y}}_{q}(\varepsilon))=\begin{cases}1;\quad&\mathcal{L}(\varepsilon_{\text{th}})>\Lambda\\\
\varepsilon_{\text{th}};\quad&\mathcal{L}(\varepsilon_{\text{th}})=\Lambda\\\
0;\quad\quad&\mathcal{L}(\varepsilon_{\text{th}})<\Lambda\end{cases},$ (23)

where $\delta_{\text{NP}}(\cdot)$ is the NP criterion-based statistic test
operator. At this point, we have solved the statistical decision problem
raised at the beginning, which leads to

$\displaystyle{\mathcal{Y}}_{0}^{\text{LoS}}$
$\displaystyle\triangleq\\{{\textbf{y}}_{q}|\delta_{\text{NP}}({\textbf{y}}_{q}(\varepsilon))=0\\},$
(24) $\displaystyle{\mathcal{Y}}_{1}^{\text{NLoS}}$
$\displaystyle\triangleq\\{{\textbf{y}}_{q}|\delta_{\text{NP}}({\textbf{y}}_{q}(\varepsilon))=1\\}.$

Subsequent experimental results show that the NP hypothesis testing is an
efficient method in practice, which facilitates sensing with low complexity to
determine whether a blockage is present or not. The most significant benefit
of this method is that the BS can immediately steer the dominant beam toward
the RIS when a severe blockage is present.

###  III-B Link Availability Indication with Angle Reciprocity

Figure 2: Pictorial illustration of geometrical representation of the channel
reciprocity on the azimuthal plane: Reflecting geometry satisfies certain
constraints imposed by the array geometry between BS-MS and BS-RIS-MS
channels, where $\vartheta$ and $\theta$ denote the spatial AoA and AoD,
respectively.

For the studied system in this paper, it is worth noting that the angle-
dependent characteristic of RIS has been verified through full-wave
simulations[10]. Although the artificial reflection properties of RIS do not
satisfy the reflection theorem, the angle reciprocity still holds in RIS-aided
mmWave systems since the BS and RIS are generally fixed and mmWave is capable
of providing the requisite for the spatial discrimination in the angular
domain [24]. Based on these facts, one important finding in this paper is that
the angle reciprocity is retained in reverse links, e.g., BS-MS links, RIS-MS
links, and BS-RIS links. Geometrically, this suggests that any pair of angles
of incidence and reflection is interdependent, such that the angle reciprocity
uniformly holds on both azimuth (horizontal) and elevation (vertical)
projection planes without explicitly calculating CSI in the opposite
direction. Fig. 2 graphically depicts the angle-dependent relations among the
BS, MS, and RIS.
With this far-field aerial view, denoting ${\mathcal{A}}^{-1}$ as an inverse
operator of the angle transformation and exploiting the aforementioned channel
reciprocity, the mapping relation of the angle reciprocity from AoA to AoD can
be explicitly expressed as

$\displaystyle{\psi}^{\text{AoD}}_{k}\sim{{\mathcal{A}}^{-1}}(\measuredangle{{\psi}^{\text{AoA}}_{k}};\measuredangle{{(\phi,\varphi)}^{\text{AoA}}_{k}}),~{}\forall\,k.$
(25)

To fully exploit the advantages of the proposed blockage detection approach,
we further propose a mechanism of link availability indication. Specifically,
the BS transmits pilot symbols to the MS for channel estimation, and the MS
simultaneously performs blockage detection and sends feedback on the link
availability through dedicated feedback channels. Leveraging the angle
reciprocity revealed in Fig. 2, a complex channel estimation procedure can be
avoided since the location of the MS is prone to be determined by appropriate
positioning techniques[24]. In this case, even if the acquisition of CSI at
the BS is partial, the BS is capable of reconstructing the requisite AoD
through the AoA measurement, and vice versa. Based on the blockage detection
results provided by the MS, the BS can easily decide whether or not to
establish an alternative reflection link. For uncertain BS-RIS-MS links, the
BS is capable of proactively restoring unknown or inaccurate angle information
by means of angle reciprocity, thereby effectively mitigating the impact of
insufficient signal processing capabilities, due to the passive nature of the
RIS.
Through the aforementioned methods, we can solve the blockage detection
problem on an accurate basis. From the reliability perspective, the proposed
blockage-aware approach focuses on preemptively excluding potential outages
caused by random blockages. As a consequence, the system outage performance
and transmission reliability can thus be improved.

##  IV Outage Minimization and Rate Maximization

In this section, we focus on solving previously formulated optimization
problems $\mathcal{P}_{1}$ and $\mathcal{P}_{2}$. It should be emphasized that
both optimization problems are resolved on the premise that the potential link
blockages can be accurately distinguished by means of the methodology
presented in Section III. In this case, since the RIS-MS reflection link can
take effect immediately, the BS has the opportunity to preventively steer the
beam toward the RIS, thereby avoiding the worst-case system outage.

###  IV-A Outage Minimization via Robust Beamforming

It is worth pointing out that if the blockages can be efficiently detected,
the worst-case system outage can be substantially circumvented. As a result,
the system outage probability will be considerably reduced. It suggests that
originally formulated problem $\mathcal{P}_{1}$ has been roughly solved. In
this subsection, we focus on solving the outage-constrained robust beamforming
optimization problem, including jointly tailoring transmit precoders at the BS
and the reflective PSs at the RIS. To combat the uncertainty of the cascaded
BS-RIS-MS channel, we further devise a beamwidth-variant control scheme, which
is capable of further reducing the outage probability and providing more
resilient connectivity when the dominant LoS link is blocked.
In essence, for many RIS aided communication applications, such passive
reflecting surfaces are required to intercept as much radiated energy as
possible from a far-field source, and to generate the reflected EMs with a
sufficiently large directional gain. Also, it is proven that the received
signal power is proportional to the size of the RIS surface [27]. For mmWave
communications, it suggests that if more energy is to be harvested through the
RIS, it is more likely for the BS to choose a narrower redirect beam toward
the RIS. Otherwise, according to the law of conservation of energy, a wider
beam necessarily results in a more dispersed radiation distribution, such that
most of the energy from the far-field BS misses the receiving aperture of the
RIS. An in-depth discussion of the far-field properties of RISs can be found
in [27].
Motivated by the above elaboration, we propose a beamwidth-variant beamforming
solution enabling flexible beamwidth adaptation. Normally, the BS-RIS beam is
designed to be with the narrowest beamwidth, such that the optimization of
transmit precoder ${\textbf{F}}_{k}$ and PSM $\bm{\Psi}(\bm{\theta})$ can be
decoupled. As a consequence, the solution to the outage-constrained problem
$\mathcal{P}_{1}$ can be divided into the following two steps. The narrowest
BS-MIS beam is designed first. Then, assuming a local optimum
${\textbf{F}}_{k}^{*}$ has been obtained, an adaptive beamwidth control method
can be applied to further decrease the outage probability.
The first step of the algorithm considers the design of hybrid precoder
${\textbf{F}}_{\text{BB}_{k}}$ and ${\textbf{F}}_{\text{RF}_{k}}$, with the
purpose to minimize the beamwidth of the main lobe. Most commonly, the
beamwidth can be defined by the half-power beamwidth (HPBW), equivalent to the
angular width of the radiation pattern that is 3-dB lower than the maximum
gain of the beam, i.e., the beam peak [28]. On the azimuthal plane, assume
that the boresight beam peak is at steering angle $\upphi$. The beamwidth can
be defined as difference
$|\upphi^{+}_{3\text{-}\text{dB}}-\upphi^{-}_{3\text{-}\text{dB}}|$, where
$\upphi^{+}_{3\text{-}\text{dB}}$ and $\upphi^{-}_{3\text{-}\text{dB}}$
correspond to the 3-dB angles at the ambilateral direction of the beam peak,
respectively. Thus, the design problem of the narrowest transmit beam can be
written as

$\displaystyle\mathop{\arg\min}\limits_{{\textbf{F}}_{\text{BB}_{k}},\,{\textbf{F}}_{\text{RF}_{k}}}{\Phi}_{\text{BS}}({\textbf{F}}_{\text{BB}_{k}},{\textbf{F}}_{\text{RF}_{k}}):=|\upphi^{+}_{3\text{-}\text{dB}}-\upphi^{-}_{3\text{-}\text{dB}}|.$
(26)

In general, narrowest possible beamwidth
${\Phi}^{\text{min}}_{\text{BS}}({\textbf{F}}_{\text{BB}_{k}},{\textbf{F}}_{\text{RF}_{k}})$
can be determined by optimal beamforming vector ${\textbf{F}}_{\text{BB}_{k}}$
and ${\textbf{F}}_{\text{RF}_{k}}$ in a hierarchical multi-resolution
codebook, where the narrowest beamwidth is associated with the highest level
of codebook. In terms of controlling variable beamwidth, a pioneer
contribution to the design of HAD codebooks can be found in [4].
Meanwhile, it is confirmed that a proper beamwidth plays a vital role in
tracking mobile receivers [29]. Specifically, the beamwidth required to
support the mobile access scenarios is slightly different. It is noteworthy
that the narrower beamwidth demands more precise beam alignment due to the
movement of receivers, while frequent beam alignment may entail considerable
overhead. Furthermore, the narrow beams are more sensitive to angular changes,
because of which the effects of imperfect CSI will be prominent, since the
misaligned beams significantly reduce the signal gain at the receiver,
incurring an increased outage probability. Obviously, the aforementioned facts
still hold for the RIS-aided mmWave access. With regard to RIS reflective
surfaces, it is known that the beamwidth of RIS reduces as the number of
reflective elements increases.
Thus, in the followoing passive beamforming design, we formulate an adjustable
beamwidth control strategy by scaling the number of the RIS reflective
elements. In a more compact form, (13) can be rewritten as

$\displaystyle{\mathop{\mathrm{\min}}\limits_{{\Phi_{\text{RIS}}({\bm{\theta}},\mathcal{N}_{k})},\,{\textbf{G}}}}{}$
$\displaystyle\Pr\left(\widetilde{\Gamma}_{k}(\bm{\theta},{\textbf{G}})\leq{\gamma_{\text{th}}}{|_{{\Phi_{\text{RIS}}({\bm{\theta}}},\mathcal{N}_{k})}}\right),\,k\in\mathcal{K}^{\text{NLoS}},$
(27) $\displaystyle\mathrm{s.t.}$
$\displaystyle\mathrm{Tr}(\textbf{G})\leq\Omega_{\text{tot}},\quad{\mathrm{MSE}}_{\bm{\theta}}<\zeta_{\bm{\theta}}^{\text{max}},$

where $\Phi_{\text{RIS}}({\bm{\theta}},\mathcal{N}_{k})$ is a function of
${\bm{\theta}}$, and $\mathcal{N}_{k}$, controlling the beamwidth of the RIS;
$\zeta_{\bm{\theta}}^{\text{max}}$ is the maximum tolerable level of
inaccurate PS estimates. $\Phi_{\text{RIS}}({\bm{\theta}},\mathcal{N}_{k})$ is
inversely proportional to the distance between the RIS and the MS.
Compared to a constant-beamwidth strategy, the adjustable beamwidth allows for
more flexibility by scaling $\mathcal{N}_{k}$, and there exists an optimal
beamwidth that significantly suppresses the outage probability. It is shown in
the simulation part of this paper that even though a broadened beam may
temporarily reduce the achievable rate of the blocked user, the downlink user
sum rate roughly remains constant. The intrinsic reason for this is that a
wider beam requires less training overhead, especially with a significant
reduction in outage probability. Because channel estimation errors are
inevitable in practice, the stringent accuracy requirements can be relaxed
since the BS can readily reconstruct the indirect RIS-MS channel matrix by
leveraging the angle reciprocity revealed in Fig. 2. In this case, the CSI
estimation error at the BS can be regarded as acceptable as long as the
constraint of the maximum tolerance level $\zeta_{\bm{\theta}}^{\text{max}}$
is satisfied.

###  IV-B Rate Maximization via Projected Gradient Descent

We now consider the globally optimal solution to problem $\mathcal{P}_{2}$. In
general, a multivariate optimization problem similar to problem
$\mathcal{P}_{2}$ can be straightforwardly tackled by the conventional AO
technique, the main idea of which is that each variable is optimized in an
alternating manner while the others are fixed. However, the AO-based algorithm
might not be applicable to RIS-aided mmWave communication systems. An
underlying reason is that an RIS consists of massive passive elements, whereby
the optimization of problem $\mathcal{P}_{2}$ requires a large number of
adjustments of the PSs. This would, of necessity, involve a large number of
optimization variables, and thus the computational complexity becomes
prohibitive. Besides, the objective function ${\widetilde{\mathcal{R}}}_{k}$
in (14) is still nonconvex due to the inequality constraints, which also makes
problem $\mathcal{P}_{2}$ difficult to solve. Among optimization algorithms,
the gradient descent method is a vital tool, and provably better for finding
the local minima of an objective function in the steepest descent direction.
To this end, we devise an accelerated PGD algorithm with Nesterov’s momentum.
From the extreme value theorem, even though objective function
$\widetilde{\mathcal{R}}$ is nonconvex, its constituents can be supposed to be
a series of locally convex and continuously differentiable functions, which
can be regarded as a manifold with the Lipschitz continuous gradient that can
be efficiently solved by numerical schemes. By applying the PGD algorithm, a
local extremum, i.e.,
$\widetilde{\mathcal{R}}^{*}(\bm{\theta}_{k},{\textbf{G}}_{k}):{\mathbb{R}}^{n}\to\mathbb{R}$,
as well as an absolutely global maximum
$\widetilde{\mathcal{R}}^{\star}(\bm{\theta}_{k},{\textbf{G}}_{k})$ can be
found recursively. In what follows, we propose a three-stage approach to
implement this accelerated version of PGD.
1) Solve constrained PGD with projection operation: By applying the
constrained optimality theorem, the optimization of $\mathcal{P}_{2}$ turns
out to find the closest ${\bm{\theta}}^{\star}$ and ${\textbf{G}}^{\star}$ in
a feasible set
$\mathcal{F}=\\{\Theta,\mathcal{G}\\},\mathcal{F}\subset\mathbb{R}^{n}$ to
maximize objective function $\widetilde{\mathcal{R}}$ [30]. For proposed
problem $\mathcal{P}_{2}$, $\Theta$ and G are respectively given by
$\Theta:=\\{\bm{\theta}\in{\mathbb{R}}^{n}|\widetilde{\Gamma}_{k}(\bm{\theta},{\textbf{G}})>{\gamma_{\text{th}}}\\}$
and
${\mathcal{G}}:=\\{\textbf{G}\in{\mathbb{R}}^{n}|\mathrm{Tr}(\textbf{G})\leq\Omega_{t}\\}$
that both of the set of points satisfy the inequality constraints.
To solve problem $\mathcal{P}_{2}$ in a standard gradient descent fashion, we
can reformulate, in terms of the feasible set $\mathcal{F}$, the optimization
problem in a more compact form as follows:

$\displaystyle{\mathop{\arg\min}\limits_{{\bm{\theta}}^{\star}\in\Theta,\;{\textbf{G}}^{\star}\in\mathcal{G}}}$
$\displaystyle{\widetilde{\mathcal{R}}}_{k}^{-1}(\bm{\theta},{\textbf{G}}),\quad
k\in{\mathcal{K}}^{\text{NLoS}}.$ (28)

Similarly, objective function ${\widetilde{\mathcal{R}}}_{k}^{-1}$ holds
continuously differentiable and has a known gradient Lipschitz constant e.g.,
$L$-Lipschitz continuous. Our main focus of this subsection is to find the
global minima of function $\mathcal{R}_{k}^{-1}$, which is equivalent to
excluding all local minima.
In the case of RIS-aided mobile mmWave communications, a key fact is that the
spatial channel can be equivalently represented in the beamspace domain, i.e.,
the angular domain. For the beamspace in $\mathbb{R}^{2}$, we just need to
consider the azimuth and elevation angles during the beam search. In addition,
this fact also suggests that to retain stable connectivity, the RIS reflected
beam corresponding to feasible set $\mathcal{F}$ is necessarily associated
with the motion trajectory of the MS under the NLoS condition [31]. By
leveraging these facts, we can represent whole set ${\mathcal{F}}$ as a set of
mutually independent subsets whose collections are used to represent the
continuous-time approximation of motion trajectory. Under this reasonable
assumption, for each motion trajectory $w$, where $w=1,2,\dots,W$, there
exists
${\mathcal{F}}={\\{{\mathcal{F}}_{1},{\mathcal{F}}_{2},\dots,{\mathcal{F}}_{W}\\}}$.
That is to say, for any subset $\mathcal{F}_{w}\subset\mathcal{F}$,
$\\{{\mathcal{F}}_{w}\\}_{w=1}^{W}$ is a spanning set for $\mathcal{F}$. By
analogy, for given feasible sets
$\Theta=\\{\Theta_{1},\Theta_{2},\dots,\Theta_{W}\\}$ and
$\mathcal{G}=\\{\mathcal{G}_{1},\mathcal{G}_{2},\cdots,\mathcal{G}_{W}\\}$,
the projection operation is correspondingly converted to the subspace
projection along the motion trajectory, with the goal of finding the “closest“
subset to ${\mathcal{F}}_{w}$ in the sense of Euclidean norm. From this the
above reasoning, feasible subset $\mathcal{F}_{w}$ is spanned by

$\displaystyle{{\Theta}_{w}}$
$\displaystyle={\textrm{Span}({\bm{\theta}}_{w})}\,={\textrm{Span}({\theta}_{w,1},{\theta}_{w,2},\dots,{\theta}_{w,\imath\jmath})},$
(29) $\displaystyle{{\mathcal{G}}_{w}}$
$\displaystyle={\textrm{Span}({\textbf{G}}_{w})}={\textrm{Span}({\textbf{g}}_{w,1},{\textbf{g}}_{w,2},\dots,{\textbf{g}}_{w,N_{s}})},$

where ${\textbf{g}}_{w,N_{s}}\in\mathbb{C}^{N_{s}}$ is a row vector of
${\textbf{G}}_{w}$.
We can define
${\textbf{P}}_{\mathcal{F}_{w}}(\textbf{u})=\frac{\textbf{u}\cdot\textbf{v}}{||\textbf{v}||^{2}}\textbf{v}$
as the Euclidean projection from vector u onto vector v in $\mathcal{F}_{w}$.
Exactly speaking, it refers to a mapping relation from $\mathbb{R}^{n}$ to
$\mathbb{R}^{n}$, which in our case is to simply find ${\bm{\theta}}_{w}$ and
${\textbf{G}}_{w}$ that are closest to $\Theta_{w}$ and $\mathcal{G}_{w}$,
respectively. Mathematically, the projection operation onto constraint set
$\Theta_{w}$ and $\mathcal{G}_{w}$ can be cast as the following optimization
problems:

$\displaystyle{\textbf{P}}_{\Theta_{w}}({\bm{\theta}}_{w})$
$\displaystyle=\arg\min_{\bm{\vartheta}\in\Theta_{w}}~{}||\bm{\vartheta}-{\bm{\theta}}_{w}||_{2},$
(30a) $\displaystyle{\textbf{P}}_{\mathcal{G}_{w}}({\textbf{G}}_{w})$
$\displaystyle=\arg\min_{{\textbf{Q}}\in\mathcal{G}_{w}}~{}||{\textbf{Q}}-{\textbf{G}}_{w}||_{2}.$
(30b)

The key idea of the above method is to find suitable vectors
${\bm{\theta}}_{w}$ and ${\textbf{G}}_{w}$ within a smaller subspace
$\mathcal{F}_{w}$. By projecting onto the constraint set, constrained
optimization problem $\mathcal{P}_{2}$ can be fixed through a simple
modification of standard gradient descent. Furthermore, since the PGD
algorithm does not need to search the whole space, the computational
complexity is thereby reduced.
Following the above reasoning and manipulation, we devise an accelerated PGD
algorithm in the next stage. The prestigious Nesterov’s momentum method is
applied to circumvent local minima, especially to prevent the PGD algorithm
from getting stuck around saddle points of $\widetilde{\mathcal{R}}_{k}$.
2) Accelerate PGD with Nesterov’s momentum: In standard Nesterov’s accelerated
gradient method, an extra momentum term is first introduced to the gradient
descent on $\widetilde{\mathcal{R}}_{k}^{-1}$ at each iteration. After that,
parameters ${\bm{\theta}}_{w}$ and ${\textbf{G}}_{w}$ are iteratively updated
by re-projecting each of them onto constraint set $\Theta_{w}$ and
$\mathcal{G}_{w}$. The updating process of Nesterov’s momentum and parameters
over two successive steps is summarized through the following recursive
formulas [32]:

$\displaystyle{\bm{\eta}}_{w}^{(n+1)}$
$\displaystyle={\bm{\nu}}{\bm{\eta}}_{w}^{(n)}-{\mu}_{{\bm{\theta}}_{w}}^{(n)}\nabla_{{\bm{\theta}}_{w}}\widetilde{\mathcal{R}}_{k}^{-1}\left(\bm{\theta}_{w}^{(n)},{\textbf{G}}_{w}^{(n)}\right),$
(31) $\displaystyle{{\bm{\theta}}}^{(n+1)}_{w}$
$\displaystyle={\textbf{P}}_{\Theta_{w}}\left({\bm{\theta}}_{w}^{(n)}+{\bm{\eta}}_{w}^{(n+1)}\right),$

and

$\displaystyle{\bm{\chi}}_{w}^{(n+1)}$
$\displaystyle={\bm{\Delta}}{\bm{\chi}}_{w}^{(n)}-{{\mu}}_{{\textbf{G}}_{w}}^{(n)}\nabla_{{\textbf{G}}_{w}}\widetilde{\mathcal{R}}^{-1}_{k}\left(\bm{\theta}_{w}^{(n)},{\textbf{G}}_{w}^{(n)}\right),$
(32) $\displaystyle{{\textbf{G}}}^{(n+1)}_{w}$
$\displaystyle={\textbf{P}}_{{\mathcal{G}_{w}}}\left({\textbf{G}}_{w}^{(n)}+{\bm{\chi}}_{w}^{(n+1)}\right),$

where ${\bm{\eta}}_{w}^{(n+1)}$ and ${\bm{\chi}}_{w}^{(n+1)}$ are the
accumulated momentum terms at the $(n+1)$th iteration; hyperparameter
$\bm{\nu}\in[0,1]$ and $\bm{\Delta}\in[0,1]$ represent the level of inertia in
the descent direction (a.k.a the momentum coefficients), which can be usually
chosen by trial and/or with a model selection criterion [33];
$\mu_{{\bm{\theta}}_{w}}^{(n)}\in(0,1)$ and
$\mu_{{\textbf{G}}_{w}}^{(n)}\in(0,1)$ are the step sizes used in $n$th
iteration.
Directly following from the derivation in [21], the complex-value gradient of
$\widetilde{\mathcal{R}}^{-1}_{k}$ with respect to ${\bm{\theta}}_{w}$ and
${\textbf{G}}_{w}$ can be explicitly computed as follows:

$\displaystyle\nabla_{{\bm{\theta}}_{w}}\widetilde{\mathcal{R}}^{-1}_{k}({\bm{\theta}}_{w},{\textbf{G}}_{w})$
$\displaystyle=\mathcal{V}_{d}\left({\textbf{H}}_{\text{RU},k}^{H}{\bm{\Sigma}}^{-1}{\textbf{G}}({\bm{\theta}}_{w}){{\textbf{H}}}_{\text{BR},k}\right),$
(33a)
$\displaystyle\nabla_{{\textbf{G}}_{w}}\widetilde{\mathcal{R}}^{-1}_{k}({\bm{\theta}}_{w},{\textbf{G}}_{w})$
$\displaystyle={\textbf{G}}^{H}({\bm{\theta}}_{w}){\bm{\Sigma}}^{-1}{\textbf{G}}({\bm{\theta}}_{w}),$
(33b)

where $\mathcal{V}(\cdot)$ denotes the vectorization operator that stacks the
columns to create a single long column vector. The detailed calculation
procedure of the complex-value gradient is rigorously derived in [21]. Without
repeating the procedure, we choose to omit these derivation details for
brevity, and interested readers are referred to [21] for a complete
exposition.
From a physics perspective, momentum terms ${\bm{\eta}}_{w}^{(n+1)}$ and
${\bm{\chi}}_{w}^{(n+1)}$ are equivalent to the momentum of inertia the
“average” of the sequence ${\bm{\eta}}_{w}^{(n)}$ and ${\bm{\chi}}_{w}^{(n)}$
after $n$ iterations. Particularly, since this momentum-aided PGD method makes
full use of stored knowledge about the gradient of objective function
$\widetilde{\mathcal{R}}_{k}^{-1}$ at previous descent directions. It has been
proven that this learning-based method is capable of better estimating the
next descent step [33], thereby helping to eventually escape the potential
saddle points of the objective function.
3) Determine an appropriate step size: To guarantee a higher convergence rate,
the choice of step size is critical to the PGD algorithm. However, a drawback
of Nesterov’s method is that the convexity parameter of the objective function
is required to compute an appropriate step size. Basically, the strict
convexity of $\widetilde{\mathcal{R}}_{k}$ is hard to guarantee in each step
of the descent process of the gradient since there are different degrees of
convexity. To solve this dilemma, our focus in the sequel is to apply a
backtracking line search in accordance with the Armijo rule to determine an
appropriate step size.
In this approach, the step size is determined in terms of the criteria of
sufficient decrease and does not require prior knowledge of convexity
parameters. We first compute a search direction and then find an acceptable
step size with no additional gradient evaluation. Then, a suitable step size
can be chosen in each iteration $n$ such that the dynamics of the gradient can
be calculated. For ease of notation, we define
${\bm{\xi}}^{(n)}_{{\bm{\theta}}_{w}}=-\nabla_{{\bm{\theta}}_{w}}\widetilde{\mathcal{R}}_{k}^{-1}\left(\bm{\theta}_{w}^{(n)},{\textbf{G}}_{w}^{(n)}\right)$
and
${\bm{\Xi}}^{(n)}_{{\textbf{G}}_{w}}=-\nabla_{{\textbf{G}}_{w}}\widetilde{\mathcal{R}}^{-1}_{k}\left(\bm{\theta}_{w}^{(n)},{\textbf{G}}_{w}^{(n)}\right)$
to be the search direction with respect to $\bm{\theta}_{w}^{(n)}$ and
${\textbf{G}}_{w}^{(n)}$ at the $n$th iteration, respectively.
Letting $\beta_{k}\in(0,1)$ denote the constant discount factor, the
backtracking line search for choosing a proper step size at the $n$th
iteration is to perform pullbacks by iteratively updating
$\mu_{{\bm{\theta}}_{w}}^{(n)}=\beta_{{\bm{\theta}}_{w}}^{m_{k}}$ and
$\mu_{{\textbf{G}}_{w}}^{(n)}=\beta_{{\textbf{G}}_{w}}^{m_{k}}$ with the
smallest integer $m_{k}$. More explicitly, it is equivalent to solving the
following minimization problems:

$\displaystyle\mathop{\arg\min}\limits_{m_{k}}\widetilde{\mathcal{R}}^{-1}_{k}\left({\textbf{P}}_{\Theta_{w}}\left(\bm{\theta}_{w}^{(n)}+\beta_{{\bm{\theta}}_{w}}^{m_{k}}{\bm{\xi}}^{(n)}_{{\bm{\theta}}_{w}}+{\bm{\nu}}{\bm{\eta}}_{w}^{(n)}\right)\right),$
(34)
$\displaystyle\mathop{\arg\min}\limits_{m_{k}}\widetilde{\mathcal{R}}^{-1}_{k}\left({\textbf{P}}_{{\mathcal{G}}_{w}}\left({\textbf{G}}_{w}^{(n)}+\beta_{{\textbf{G}}_{w}}^{m_{k}}{\bm{\Xi}}^{(n)}_{{\textbf{G}}_{w}}+{\bm{\Delta}}{\bm{\chi}}_{w}^{(n)}\right)\right).$

Starting with $m_{k}=0$ and then incrementing $m_{k}$ until the following
optimality conditions (sufficient decrease) are satisfied:

$\displaystyle\widetilde{\mathcal{R}}^{-1}_{k}\left({\bm{\theta}}_{w}^{(n)}\right)-\widetilde{\mathcal{R}}_{k}^{-1}\left({\bm{\theta}}_{w}^{(n+1)}\right)$
$\displaystyle\geq\uppi_{{\bm{\theta}}_{w}}\beta_{{\bm{\theta}}_{w}}^{m_{k}}||{\bm{\xi}}^{(n)}_{{\bm{\theta}}_{w}}||^{2},$
(35)
$\displaystyle\widetilde{\mathcal{R}}^{-1}_{k}\left({\textbf{G}}_{w}^{(n)}\right)-\widetilde{\mathcal{R}}_{k}^{-1}\left({\textbf{G}}_{w}^{(n+1)}\right)$
$\displaystyle\geq\uppi_{{\textbf{G}}_{w}}\beta_{{\textbf{G}}_{w}}^{m_{k}}||{\bm{\xi}}^{(n)}_{{\textbf{G}}_{w}}||^{2}.$

Here, parameters $\uppi_{{\bm{\theta}}_{w}}\in(0,1)$ and
$\uppi_{{\textbf{G}}_{w}}\in(0,1)$ control the acceptable error tolerance of
the line search procedure. To achieve local minima rapidly, larger step size
$\mu_{k}$ is generally preferable for fast descent at current point
$\widetilde{\mathcal{R}}^{-1}_{k}(\bm{\theta},{\textbf{G}})$. In practice,
there is a balance that must be maintained between taking $\mu_{k}$ as large
as possible and not having to evaluate the function at many points. Such a
balance can be obtained with an appropriate selection of parameters $\beta$
and $\upsilon_{k}$, such that saddles point can be avoided.
Now we are ready to directly solve problem ${\mathcal{P}}_{2}$. Specifically,
we devise an iterative algorithm, of which the pseudo codes are summarized in
Algorithm 1. In general, the initial point
$\\{{\bm{\theta}}_{w}^{(0)},{\textbf{G}}_{w}^{(0)}\\}$ can be randomly
initialized or artificially set by guessing the location of a local minimum.
In the standard Nesterov momentum updating step (line 10), hyperparameters
$\bm{\nu}$ and $\bm{\Delta}$ are applied to preserve the historical gradient
information that can produce new gradient values by an arbitrary number of
training steps. With the Nestorov momentum, the size of the next-step gradient
can be foreseeingly adjusted, thereby enabling the PGD algorithm to achieve
faster convergence in a more responsive way. To simplify notation, we use
${\textbf{grad}}_{{\bm{\theta}}_{w}^{(n)}}$ and
${\textbf{grad}}_{{\textbf{G}}_{w}^{(n)}}$ to represent the corresponding
gradient vectors. Through both, we can stipulate a termination criterion: the
average gradient is sufficiently small after a certain number of iterations in
comparison with a prescribed threshold, i.e.,
$||{\textbf{grad}}_{{\bm{\theta}}_{w}^{(n)}}-{\textbf{grad}}_{{\bm{\theta}}_{w}^{(n-1)}}||<\epsilon$.
Therefore, once a stationary point is reached, i.e., the algorithm meets the
preset optimality conditions, this optimization procedure will be terminated,
and the stationary point will be regarded as the global minimum.

Algorithm 1 Solution of projected gradient descent (PGD).

Input: $\\{{\bm{\Theta}}_{w}\\}_{w=1}^{W},\\{{\mathcal{G}}_{w}\\}_{w=1}^{W}$
$\triangleright$ non-empty feasible sets

1:Initialization:
${\bm{\theta}}_{w}^{(0)},{\textbf{G}}_{w}^{(0)},{\bm{\nu}}\leftarrow
0,{\bm{\Delta}}\leftarrow
0,\mu_{{\bm{\theta}}_{w}}^{(0)},\mu_{{\bm{\theta}}_{w}}^{(0)}$

2:for $n=0,1,\dots$ do

3: Compute gradient ${\textbf{grad}}_{{\bm{\theta}}_{w}^{(n)}}$ and
${\textbf{grad}}_{{\textbf{G}}_{w}^{(n)}}$ by (34)

4: Accelerate
${\bm{\eta}}_{w}^{(n+1)}={\bm{\nu}}{\bm{\eta}}_{w}^{(n)}-{\mu}_{{\bm{\theta}}_{w}}^{(n)}{\textbf{grad}}_{{\bm{\theta}}_{w}^{(n)}}$

5:
${\bm{\chi}}_{w}^{(n+1)}={\bm{\Delta}}{\bm{\chi}}_{w}^{(n)}-{{\mu}}_{{\textbf{G}}_{w}}^{(n)}{\textbf{grad}}_{{\textbf{G}}_{w}^{(n)}}$

6: Update
${\bm{\theta}}_{w}^{(n+1)}\leftarrow{\textbf{P}}_{\Theta_{w}}\left({\bm{\theta}}_{w}^{(n)}+{\bm{\eta}}_{w}^{(n+1)}\right)$

7:
${{\textbf{G}}}^{(n+1)}_{w}\leftarrow{\textbf{P}}_{{\mathcal{G}_{w}}}\left({\textbf{G}}_{w}^{(n)}+{\bm{\chi}}_{w}^{(n+1)}\right)$$\triangleright$
projection

8: Try step sizes
$\mu_{{\bm{\theta}}_{w}}^{(n)}:=1,\beta_{{\bm{\theta}}_{w}},\beta_{{\bm{\theta}}_{w}}^{2},\dots,$

9:
$\mu_{{\textbf{G}}_{w}}^{(n)}:=1,\beta_{{\textbf{G}}_{w}},\beta_{{\textbf{G}}_{w}}^{2},\dots,$$\triangleright$
line searching

10: Update current Nesterov’s momentum ${\bm{\nu}},{\bm{\Delta}}$ [33]

11: Repeat until
$||{\textbf{grad}}_{{\bm{\theta}}_{w}^{(n)}}-{\textbf{grad}}_{{\bm{\theta}}_{w}^{(n-1)}}||^{2}<\epsilon$

12:Output: $\bm{\theta}^{*},{\textbf{G}}^{*}$

13:end for

###  IV-C Analysis of Computational Complexity and Convergence

For the proposed PGD optimization algorithm, the overall computational
complexity mainly depends upon the product of the number of required
iterations $I_{\text{PGD}}$ and the amount of computation required for the
projection operations $N_{\text{proj}}$ as well as the gradient calculations
$N_{\text{grad}}$ in each iteration. For any closed convex set
$\mathcal{F}\subset\mathbb{R}^{n}$, the computation of projection operations
is dominated by the computational overhead of a Euclidean projection on
simplex convex set $\\{\Theta,\mathcal{G}\\}$. Correspondingly, the standard
Euclidean norm (cf. (35a) and (35b)) that can be solved efficiently is
required to take
$N_{\text{proj}}={\mathcal{O}}(({\mathcal{N}}_{k}+N_{t,k}^{2}N_{r}^{2})\log({\mathcal{N}}_{k}+N_{t,k}^{2}N_{r}^{2}))$
floating-point operations in general. The computational complexity of the
complex-value gradient is approximated by
$N_{\text{grad}}=\mathcal{O}(\mathcal{N}_{k}N_{t,k}N_{r})$ on condition that
$\mathcal{N}_{k}\gg N_{t,k}$. It is evident that the overall computational
cost of Algorithm 1 is approximate as
$I_{\text{PGD}}\times(N_{\text{proj}}+N_{\text{grad}})$.
The convergence rate of Algorithm 1 can be improved by decoupling the
objective function. According to the inference in Section III-A, when the BS-
MS links are unavailable, the BS will generate the narrowest BS-RIS beam, in
the sense that hybrid precoders ${\textbf{F}}_{{\text{BB}}_{k}}$ and
${\textbf{F}}_{{\text{RF}}_{k}}$ can be viewed as relatively constant or vary
only within a small range. Combined with this mechanism, it is feasible to
substantially decouple the optimization for objective function
$\mathcal{R}_{k}$. To be specific, we can first complete the optimization of
${\textbf{G}}_{w}$, which approximately stays invariant throughout the
algorithm. In this way, the optimization of ${\bm{\theta}}_{w}$ can be
independently proceeded. The advantage of this approach is that even though
this strategy is suboptimal, the optimization of problem $\mathcal{P}_{2}$ can
be configured as a simplex-constrained one-dimensional function, such that
each iteration will come with a much lower computational burden. In the
subsequent simulation, we examine the convergence of the overall algorithm and
provide evidence that the accelerated PGD algorithm converges faster than the
baseline algorithms.

##  V  Numerical Results

TABLE I: System parameters and simulation settings. System Parameter | Simualtion setting
---|---
Carrier frequency | 28 GHz
Bandwidth | 100 MHz
Cell radius | 400 m
Channel model | Rician fading channel model
Path loss | FSPL ratio [21]
Modulation format | QPSK
Receiver noise figure | 5 dB
$N_{t}$ | 32
$N_{r}$ | 4
$\mathcal{N}(\mathcal{I}\times\mathcal{J})$ | 2048 (32$\times$64)

In this section, we evaluate the outage probability and achievable rate with
extensive Monte Carlo simulations. We consider a rectangle RIS comprising
2,048 elements arranged in a $32\times 64$ matrix. The center of ULA of the BS
and the center of the RIS panel are assumed to be in the same horizontal plane
and have the same height, and the position of the RIS is fixed. Without loss
of generality, the distance between BS and RIS $d_{\text{BS-RIS}}$ is set to
100 meters, and the corresponding angle between BS and RIS is fixed at 45
degrees. To evaluate the path loss of direct BS-MS link and cascaded BS-RIS-MS
link, we adopt the total free space path loss (FSPL) ratio model proposed by
[21]. The system is assumed to operate at the 28 GHz carrier frequency, and
the other key simulation setups are given in Table 1.

###  V-A Performance of Blockage Detection

In what follows, our first goal is to investigate the impact of link blockage
$\varepsilon_{k}$. To simulate random blockage events for the downlink, we
adopt a probabilistic model [15], [16], by which each path undergoes a random
and independent blockage. Specifically, we utilize the homogeneous Poisson
process to emulate the statistical properties of blockage behavior of mmWave
communication links between the BS and the MS. To be more specific, random
$\varepsilon_{k}$ signifies the effect of different blockages, and the sudden
blockage event is modeled using a two-dimensional homogeneous Poisson process.

Figure 3: Distribution of the blockages: An example for illustration.

Figure 4: Detection probability vs. false alarm probability given different
SNRs.

For simplicity, the simulated blockage events are randomly generated from a
homogeneous Poisson process with constant $\lambda=10$ and $n=5$ without loss
generality. For each experiment run, the distribution of the number of blocked
channel realizations is independent of the time intervals, in which a fraction
of trials are uniformly distributed. The results are shown in Fig. 3, where we
generate 300 samples, where the strength of $\varepsilon$ is distinguished by
different colors.
According to the basic framework for binary hypothesis testing established in
Section III-A, we further evaluate the performance of the proposed blockage
detection method under different SNR conditions. Fig. 4 plots the detection
probability versus false alarm probability given different SNRs, and
demonstrates the effectiveness of the proposed blockage detection algorithm.
It can be observed from Fig. 4 that with the SNR progressively increasing, the
false alarm probability $P_{\text{FA}}^{\text{LoS}}$ is significantly reduced.
For instance, assuming the optimal threshold to achieve a probability of
detection $P_{\text{FA}}^{\text{LoS}}$ of 0.9, the probability of false alarm
$P_{\text{FA}}^{\text{LoS}}$ drops from 0.1 (SNR=2 dB) to less 0.001 (SNR=10
dB). Moreover, we observe that a higher SNR can substantially improve the
detection threshold, and SNR=10 dB is generally acceptable for blockage
detection. In contrast, blockage detection becomes impossible in the low SNR
regime due to the uncertainty caused by noise. These observations provide
sufficient evidence that when the significance level of a specific alternative
hypothesis is guaranteed, the proposed binary hypothesis test can efficiently
reject the null hypothesis, and thereby is feasible to sense the random
blockages while minimizing the false alarm risk.

###  V-B Outage Performance

Figure 5: Outage probability vs. average achievable rate, where
$\mathcal{N}_{k}$ is set to be 128.

Figure 6: Outage performance vs. $\Omega_{t}$.

Figure 7: Outage performance vs. CSI uncertainty, where different downlink
power is allocated to the edge-user and the Centric-user.

In Fig. 5, we further examine the role of the beamwidth in combating the CSI
uncertainty and its effect on the average achievable rate of the downlink
transmission. To visualize the relation between the beamwidth and CSI
uncertainty, we plot outage probability $\Gamma_{k}$ versus achievable rate
$\mathcal{R}_{k}$ and evaluate the outage performance of the proposed method
under different levels of CSI uncertainty. It can be shown from Fig. 5 that
the outage performance can be substantially improved by increasing the width
of the reflecting beam. In fact, choosing a wider beam may be more
advantageous, albeit at the expense of the achievable rate. In user mobility
scenarios, when the LoS link is blocked, the wider reflecting beam is more
beneficial for minimizing the outage probability, since it provides more
stable and reliable connectivity. A wider beam implies that even though the
estimates of the CSI and/or PSs are relatively coarse, passive beamforming at
the RIS can retain highly reliable connectivity and achieve comparable
performance. It is worth emphasizing that there is a trade-off between outage
performance and achievable rate to overcome the limitation of imperfect CSI
and allow for partial CSI. To be brief, if minimizing the outage probability
is preferential and a bounded rate loss is acceptable, a wider beam is clearly
a better choice, and vice versa.
Next, Fig. 6 shows the outage probability as a function of the transmit power
at the BS. We compare the outage performance of the proposed beamwidth control
scheme with the fixed beamwidth solution given in [9]. In our experiment, the
reliability of estimate $s$ is set to be 0.95, 0.85, and 0.5, so as to
evaluate the impact of the channel estimation error. From Fig. 6, one can
observe that the outage performance of the proposed approach is better than
that of the method in [9] under various CSI uncertainty settings. With the CSI
uncertainty merely exacerbating the channel estimation errors, it leads to the
deterioration of the outage performance. Consequently, the beamwidth is
critical in alleviating the impacts of the outage. The proposed adaptive
beamwidth control method provides a trustworthy solution for tracking a mobile
receiver. This is because a wider reflecting beam is capable of providing
wider coverage, thereby providing resilient connectivity over the indirect
RIS-MS link.
In Fig. 7, we proceed to evaluate the outage probability versus CSI
uncertainty in our proposed framework by taking into account the effects of
the beamwidth and compare the performance against the methods in [9] and [16].
In our experiment, we mainly consider two typical user access scenarios,
namely the edge user and centric user. For a fair comparison, the distances
between the edge- and centric-users to the RIS are respectively set to be 15 m
and 30 m. From the presented results, we can see that the proposed technique
can achieve better outage performance than those of the methods of [9] and
[16]. It is interesting through scrutinizing this case that the centric-user
with a wider and synthesizable beamwidth has a significant outage performance
advantage over the edge-user. This observation suggests that even with
inaccurate angle estimates, the proposed method can efficiently mitigate the
impacts of the beam squint on the MS. It demonstrates the possibility that the
proposed technique has the ability to provide a more resilient connection
through RIS passive beamforming in the presence of blockages, which could be
applied with imperfect CSI. It is important to realize that the wider and
synthesizable beamwidth with a small amount of allocated power inevitably
sacrifices the achievable rate during the transmitting procedure, and
therefore an intrinsic trade-off needs to be considered in this case,
examining such a trade-off is left for future work.

###  V-C Achievable Rate

Figure 8: Achievable rate of the accelerated PGD algorithm versus the CSSCA-
and AO-based schemes.

Figure 9: Downlink sum-rate performance in an RIS-aided mmWave MIMO
communication system, considering four active users and two of them being
blocked.

Figure 10: Convergence rates of PGD yielded by different methods.

For the following analysis, we further investigate the effectiveness of the
proposed PGD-based optimization method for the RIS-aided transmission when the
dominant LoS path is blocked. We analyze two cases for achievable rate
${\widetilde{\mathcal{R}}}_{k}$ and for downlink sum rate
${\mathcal{R}}_{\text{sys}}$. In the first case, the performance metric we
consider is achievable rate $\mathcal{R}_{k}$, a function of the transmit
power as formulated in (10). The performance is compared with references [9]
and [20], with consideration of the impact of the number of RIS elements
$\mathcal{N}_{k}$ on the achievable rate. Accordingly, the PGD optimization is
performed in the scaled-variable form. Unless otherwise specified, we set
parameters $\chi$=0.95 and the RIS-MS distance $D$=30 m, and therefore we can
directly apply either the optimal transmission beam or the optimal reflecting
beams toward an individual user. By examining the results presented in Fig. 8,
we clearly see that the proposed PGD approach significantly boosts the
achievable rate compared to the stochastic successive convex approximation
(CSSCA) algorithm in [9] and AO-based schemes in [20]. We also observe that
scaling the number of RIS elements has a significant effect on the achievable
rate. Intuitively, by increasing the number of PSs $\mathcal{N}_{k}$, the
achievable rate increases. This is because the received signal power is
proportional to the size of the RIS surface, such that not only the indirect
link can be compensated by the increased number of PSs $\mathcal{N}_{k}$, but
also the introduced RIS can create a narrower beam to provide a large
directional gain.
In the second case, we focus on evaluating the achievable sum rate in a
multiuser scenario for the proposed PGD approach and the counterparts under
the same consideration. Specifically, we consider a scenario where there are
four active users, two of whom are completely blocked, whereby each user tries
to receive data from the BS. For the sake of comparison, the communication
mode of each user is still chosen on the basis of the settings in Fig. 8. We
assume in further simulations that the transmission over the direct link is
capable of providing the optimal achievable rate and maintains an
approximately constant rate. We can see from Fig. 9 that without the aid of
the RIS, or if the blockages cannot be effectively sensed, the sum rate of the
system deteriorates severely. In contrast, we observe that with the aid of the
RIS, all algorithms can achieve higher achievable sum rates, whereas the
proposed PGD-based algorithm performs better than the CSSCA-based and AO-based
algorithms. In addition, we study the impact of the number of PSs
$\mathcal{N}_{k}$ on RIS-aided downlink transmission. As expected, the sum
rate ${\mathcal{R}}^{*}_{\text{sys}}$ improves with increased
$\mathcal{N}_{k}$ for each algorithm. It is evident that with a sufficiently
large RIS, the proposed PGD-based algorithm can provide a near-optimal sum
rate in the presence of blockages, similar to the case where the LoS link is
available for all active users.
As a final numerical experiment, we investigate the convergence behavior of
the proposed PGD-based algorithm compared with the aforementioned baseline
algorithms. To further accelerate the convergence of the PGD-based algorithm,
we incorporate the subspace constraint and the momentum term, as described in
Sections IV-A and IV-B, respectively. The results are presented in Fig. 10,
where the convergence rate of the objective gradient versus the number of
iterations is plotted to depict the behavior of convergence to reach the
optimal achievable rate. Our numerical results demonstrate that the proposed
PGD-based algorithm with momentum acceleration outperforms other baseline
algorithms, i.e., CSSCA-based and AO-based algorithms. It is evident from Fig.
10 that the subspace-constrained momentum-accelerated PGD algorithm is the
fastest algorithm. For the case with only the subspace constraint, the
proposed PGD-based method without momentum aid evidently achieves faster
convergence and performs better than the AO-based strategies developed in [9]
and [20], which run a gradient descent algorithm with a fixed step size to
search the local optimum. After about 1,000 iterations, all algorithms get the
stable minimum, corresponding to 70$\%$, 75$\%$, 90$\%$, and 90$\%$ of the
desired optimal achievable rate $\widetilde{\mathcal{R}}_{k}^{*}$,
respectively. Both momentum and subspace suffice convergence to the minimum in
fewer iterations with higher achievable rates. Particularly, by introducing
the momentum term, the perturbations are significantly reduced. This agrees
with the intuition that reducing the momentum coefficients allows for faster
convergence to be achieved. The reason for this phenomenon is that a constant
step size is set to 0.05, while the convergence could readily get stuck due to
a saddle point of the objective function. For conventional gradient descent
methods, the larger step size enables faster convergence, but the risk of
divergence is correspondingly higher. Experimental results illustrate that our
proposed method can efficiently circumvent saddle points by taking an
increasing momentum term, which is helpful in finding the global minimum.

##  VI Conclusion

In this paper, we studied the NP hypothesis testing criterion and the
momentum-based gradient optimization algorithm for solving the outage
probability minimization and achievable rate maximization problems,
respectively. Among them, the NP hypothesis testing provides a trustworthy
means for random blockage detection. Our analysis highlighted the importance
of the NP criterion-based detector in substantially alleviating the impact of
blockages. Preliminary results also verified that with this integrated
blockage sensing capability, the mmWave link availability becomes tractable.
We further confirmed through numerical results that by leveraging the
subspace-constrained and momentum-based optimization algorithms, the proposed
PGD algorithm significantly outperforms the existing baseline algorithms and
provably converges to the global optimum. The corresponding robust beamforming
design can thus guarantee stable and reliable connectivity and has the
possibility to unlock the full potential of mmWave communications. For future
studies, we intend to take a multi-RIS setting into account to combat
distributed blockages and to quantitatively investigate the intrinsic trade-
off between the outage performance and the achievable rate.

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