1 Robust Optimization of RIS in Terahertz under Extreme Molecular Re-radiation Manifestations

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Robust Optimization of RIS in Terahertz under

Extreme Molecular Re-radiation Manifestations

Anish Pradhan, Mohamed A. Abd-Elmagid, Harpreet S. Dhillon and Andreas F.

Molisch

Abstract

Terahertz (THz) communication signals are susceptible to severe degradation because of the molec-

ular interaction with the atmosphere in the form of subsequent absorption and re-radiation. Recently,

reconﬁgurable intelligent surface (RIS) has emerged as a potential technology to assist in THz com-

munications by boosting signal power or providing virtual line-of-sight (LOS) paths. However, the

re-radiated energy has either been modeled as a non-line-of-sight (NLOS) scattering component or as

additive Gaussian noise in the literature. Since the precise characterization is still a work in progress,

this paper presents the ﬁrst comparative investigation of the performance of an RIS-aided THz system

under these two extreme re-radiation models. In particular, we ﬁrst develop a novel parametric channel

model that encompasses both models of the re-radiation through a simple parameter change, and then

utilize that to design a robust block-coordinate descent (BCD) algorithmic framework which maximizes

a lower bound on channel capacity while accounting for imperfect channel state information (CSI). In

this framework, the original problem is split into two sub-problems: a) receive beamformer optimization,

and b) RIS phase-shift optimization. As the latter sub-problem (unlike the former) has no analytical

solution, we propose three approaches for it: a) semi-deﬁnite relaxation (SDR) (high complexity), b)

signal alignment (SA) (low complexity), and c) gradient descent (GD) (low complexity). The time

complexities associated with the proposed approaches are explicitly derived. We analytically demonstrate

the limited interference suppression capability of a passive RIS by deriving the stationary points of signal-

to-interference and noise ratio (SINR) of a one-element RIS system with one interferer. Our numerical

results also demonstrate that slightly better throughput is achieved when the re-radiation manifests as

scattering.

A. Pradhan, M. A. Abd-Elmagid, and H. S. Dhillon are with Wireless@VT, Department of ECE, Virginia Tech, Blacksburg,

VA, USA (email: {pradhananish1, maelaziz, hdhillon}@vt.edu). A. F. Molisch is with the Wireless Devices and Systems Group,

Ming Hsieh Department of Electrical and Computer Engineering, University of Southern California, Los Angeles, CA, USA

(email: molisch@usc.edu). This work was supported by U.S. National Science Foundation under Grant ECCS-2030215.

This paper was presented in part at the IEEE Globecom 2021, Madrid, Spain [1].

arXiv:2210.00570v1 [eess.SP] 2 Oct 2022

Index Terms

Reconﬁgurable intelligent surface, terahertz, molecular re-radiation, imperfect CSI, robust optimiza-

tion.

I. INTRODUCTION

With the standardization of 5G new radio (NR), it is now well-accepted that the traditional

sub-6 GHz spectrum by itself is not sufﬁcient to meet the ever-expanding network demands in

the near future [2], [3]. This has led to the pursuit of utilizing higher frequency bands, which

ultimately resulted in the recent commercialization of mmWave communication. However, with

the advent of extended reality (xR) technologies, even higher data rates - up to 1 Tbit/s - for

which mmWave bandwidths are not sufﬁcient anymore, are required [4]. The reason is that the

xR ecosystem imposes very stringent requirements on throughput of the wireless communication

technologies sustaining it. Once realized, the xR applications are expected to revolutionize many

industry sectors, including, but not limited to, healthcare, entertainment, and eCommerce. To

support such applications, there has been a recent interest in exploring the possibility of utilizing

the THz (0.1-10 THz) spectrum, which lies above the mmWave band [2]. Recent breakthroughs

in the research of high-power THz sources [5], [6] have further increased the viability of utilizing

this spectrum.

However, THz communication links are highly susceptible to blockages, both by static objects,

and by dynamic objects including the users operating the VR [7]. Static blockages consistently

prevent suitable quality of experience (QoE), while dynamic blockages result in a sudden decrease

in throughput and are detrimental to the immersion of xR. A further impairment for THz signals

is the molecular re-radiation that can manifest as either noise or NLOS component of the signal.

Inspired by the recent standardization efforts by various organizations, a potential solution is

to deploy the emerging RIS technology that can create virtual LOS links to enhance throughput

in situations where direct LOS links are blocked. Yet, the integration of RIS with THz commu-

nication links presents the following challenges: a) accurate characterization of the molecular

re-radiation, b) channel estimation issues due to RIS, and c) subsequent operation of RIS under

imperfect CSI. To overcome these challenges, we ﬁrst develop a parametric THz channel model

in this paper that captures both manifestations of re-radiation, and then use that model to present a

novel alternating RIS optimization framework, where the RIS phase shift and receive beamformer

are jointly optimized.

A. Background and Prior Art

We will now discuss in more detail each of the aforementioned challenges associated with

integrating RIS with THz communication links. We start this discussion with the challenge of an

accurate characterization of the molecular re-radiation, which is the less understood and is also

the prime motivation behind this work. In most of our typical communication scenarios, water

vapor is one of the primary constituents in the molecular makeup of the wireless medium. Since

water molecule, like many other atmospheric molecules [8], has many rotational absorption lines

through the THz band [9], these molecules are highly susceptible to being excited by the THz

communication signals. In particular, the transmitted EM wave causes molecular absorption by

exciting the molecules from lower to higher energy states. These higher energy molecules re-

radiate absorbed energy in a similar frequency range while returning to the ground state. For

many decades, the process of such atomic and molecule re-radiation has been referred to as

radiation trapping in the physics literature [10]. In the existing THz literature, this re-radiation

often manifests as additive Gaussian noise based on sky-noise models [8], [11]. This is an

approximation that results from the fundamental difference of the physical phenomena dictating

the two [8]. To our knowledge, no measurement studies have adequately supported this model

until now. Furthermore, there is some support in the literature [8], [12], [13] for describing this

phenomenon as scattering, with the presence of multiple scattered copies of the signal due to re-

radiation. Note that this scattering could potentially result in delay dispersion [8] and frequency

dispersion [10], which is beyond the scope of this paper. Following the scattering assumption,

[14] recently characterized the THz channel as a Rician channel, with the Rician factor computed

from the molecule absorption coefﬁcient. In the literature, both manifestations (i.e., as noise and

as scattering) have been employed separately, and determining the prevalence of each effect is

difﬁcult. To be more speciﬁc, the exact effect will most likely exist as a combination of these two

extreme situations. There is no way to accurately deﬁne the exact effect without comprehensive

measurement studies. Given this, one of the reasonable things to do with the current information

is to explore the two extreme circumstances and quantify their inﬂuence on the RIS performance

which is what we do in this paper.

Due to the peculiarities of the THz links, the techniques developed in the beamforming

literature on RIS [15]–[20] in the sub-6 GHz spectrum (that deal with the joint optimization of

RIS phase-shifts and receiver beamformer) cannot be trivially extended. The interplay between

RIS and THz band has been recently studied in [7], [21]–[26]. The authors of [21] proposed a

sub-optimal search method to optimize RIS discrete phase-shifts while the authors of [22] jointly

optimized the RIS location, phase-shift and THz sub-bands to improve system performance. A

deep reinforcement learning-based algorithm has been used to optimize the reliability and rate

for RIS-operated virtual reality systems in the THz band [7]. A physically consistent near-ﬁeld

channel model for RIS-THz systems was developed in [23] while the secrecy rate for an RIS-

aided THz system was optimized in [24]. A particle swarm optimization-based method with

limited channel estimation was used to optimize the RIS phase shifts in a THz band [25]. The

error performance of an RIS-assisted low earth orbit satellite network has been analyzed in [26].

However, the above prior works studying RIS in the THz band neglected either the two possible

manifestations of the molecular re-radiation, or the cumulative effect of this re-radiation along

with the RIS conﬁguration on the receiver noise [7], [21]–[26]. Further, they did not account for

the natural challenge of imperfect CSI resulting from the passive nature of RIS elements, and

non-cooperation from the interfering nodes.

The robust optimization of RIS-aided THz systems (against the imperfect CSI) has only

been considered in a handful of recent works [27]–[29]. These works used semideﬁnite pro-

gramming (SDP) techniques while ignoring the peculiar characteristics of RIS-THz integration.

Such techniques suffer from high computational cost that decreases the energy efﬁciency of the

network, and hence defeat the purpose of low-cost RISs [30]. We bridge this gap by developing

a parametric THz channel model that accounts for both assumptions of re-radiation, as well as

three BCD-based joint optimization approaches of varying complexity for the proposed channel

model under imperfect CSI. We use a lower bound on the channel capacity as our objective as

the exact channel capacity is unknown for considering interference in our system model [31].

Different from [27], [28], the achievable throughput expression in our objective function assumes

that the receiver only has access to imperfect CSI, which reﬂects the reality more precisely. Our

objective function is also consistent with the discussion of the uplink spectral efﬁciency under

imperfect CSI in [31, eq. (4.1)]. To the best of our knowledge, no comparative study exists

that analyzes a jointly-optimized multi-antenna system in a THz environment with two extreme

assumptions regarding molecule re-radiation.

B. Contributions

We study an RIS-aided THz system setting that consists of a single-stream transmitter (Tx)

communicating with an RIS-aided multi-antenna receiver (Rx) in the THz band in the presence of

potentially multiple single-stream interferers. For this setup, our objective is to jointly optimize

the RIS’s phase shift and receive beamformer while assuming imperfect CSI knowledge. Our

key contributions in this paper are listed next.

A novel parametric THz channel. We propose a new parametric THz channel model that

accounts for the following two extreme manifestations of re-radiation in the THz spectrum

through a single parameter change: a) re-radiation is assumed as Gaussian noise, and b) re-

radiation is assumed as an NLOS component of the signal. We also characterize the cumulative

effect of molecular re-radiation and the RIS conﬁguration utilizing this parameter.

Three robust BCD algorithms. We formulate an optimization problem in which we jointly

optimize the RIS’s phase shift vector and receive beamformer vector with the objective of

maximizing a lower bound on the channel capacity. Due to the coupling between the two sets

of optimization variables (i.e., the RIS’s phase shift vector and receive beamformer vector) in

its objective function, the formulated problem turns out to be non-convex, and hence its global

optimal solution cannot be obtained using standard convex optimization techniques. Because of

that, we aim to obtain an efﬁcient solution through the BCD algorithm. In this algorithm, we split

the original problem of two sets of optimization variables into the following two sub-problems

of one set of variables each: a) receive beamforming vector optimization problem, and b) RIS’s

phase shift optimization problem. These sub-problems are then solved in an alternative manner

until they converge to an efﬁcient solution of the original problem. As the latter sub-problem

does not have a closed-form solution unlike the former, we propose three algorithms of varying

complexity to solve the RIS sub-problem. First, we propose a conventional SDR approach as

a baseline. Due to the high time complexity of the SDR approach, we then present the SA

approach for its speed, where the expected receive signal strength is maximized rather than

the original objective function. This approach provides a good sub-optimal solution when the

interference power in the network is low. However, in a network with a moderate amount of

interference, we can achieve better performance without sacriﬁcing any speed by utilizing the

gradient descent algorithm, which is our third proposed approach. These approaches consider

the direct links of both users and interferers under imperfect CSI. Our objective function also

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1RobustOptimizationofRISinTerahertzunderExtremeMolecularRe-radiationManifestationsAnishPradhan,MohamedA.Abd-Elmagid,HarpreetS.DhillonandAndreasF.MolischAbstractTerahertz(THz)communicationsignalsaresusceptibletoseveredegradationbecauseofthemolec-ularinteractionwiththeatmosphereintheformofsubsequentab...

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