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Hybrid Relay-Reflecting Intelligent Surface-AssistedWireless Communication

Nhan Thanh Nguyen, Quang-Doanh Vu, Kyungchun Lee, Senior Member, IEEE, and Markku Juntti, Fellow, IEEE

Abstract—Reconfigurable intelligent surface (RIS) hasemerged as a cost- and energy-efficient solution to enhancethe wireless communication capacity. However, recent studiesshow that a very large surface is required for a RIS-assistedcommunication system; otherwise, they may be outperformedby the conventional relay. Furthermore, the performance gain ofa RIS can be considerably degraded by hardware impairmentssuch as limited-resolution phase shifters. To overcome thosechallenges, we propose a novel concept of hybrid relay-reflectingintelligent surface (HR-RIS), in which a single or few elementsare deployed with power amplifiers (PAs) to serve as activerelays, while the remaining elements only reflect the incidentsignals. Two architectures are proposed, including the fixed anddynamic HR-RIS. Their coefficient matrices are obtained basedon alternating optimization (AO) and power allocation strategies,which enable understanding the fundamental performances ofRIS and relaying-based systems with a trade-off between thetwo. The simulation results show that a significant improvementin both the spectral efficiency (SE) and energy efficiency (EE)with respect to the conventional RIS-aided system can beattained by the proposed schemes, especially, by the dynamicHR-RIS. In particular, the favorable design and deployment ofthe HR-RIS are analytically derived and numerically justified.

I. INTRODUCTION

The requirement of enhanced wireless communication ser-vices has created the demand to use higher frequency bandsincluding the millimeter-wave (mmWave) bands. The highfree-space path loss in those bands can be compensated forby using large antenna arrays at both ends of the link [1]–[3].Therefore, massive multiple-input multiple-output (MIMO)transmission is built-in into the mmWave system design to at-tain a significant improvement in spectral efficiency (SE) fromthe large bandwidth and high beamforming gains. However,the mmWave massive MIMO system generally requires in-creased power consumption, hardware cost, and computationalcomplexity for large-array signal processing [4]. Furthermore,most practical mmWave links require line-of-sight (LoS) prop-agation, because otherwise, the received signal power is toolow for reliable communications. Recently, a new technologycalled reconfigurable intelligent surface (RIS), also knownas intelligent reflecting surface (IRS) [5] or large intelligent

The research has been supported in part by Academy of Finland under6Genesis Flagship (grant 318927) and EERA Project (grant 332362).

N. T. Nguyen and Markku Juntti are with Centre for Wireless Commu-nications, University of Oulu, P.O.Box 4500, FI-90014, Finland, (e-mail:nhan.nguyen@oulu.fi, markku.juntti@oulu.fi)

Quang-Doanh Vu was with the Centre for Wireless Communications,University of Oulu. He is now with the Mobile Networks, Nokia, 90650 Oulu,Finland (e-mail: quang-doanh.vu@nokia.com)

K. Lee are with the Department of Electrical and Information Engineering,Seoul National University of Science and Technology, Seoul 01811, Republicof Korea (e-mail: kclee@seoultech.ac.kr).

surface (LIS) [6] has emerged as a new promising solution toovercome the challenge of mmWave massive MIMO systems[7]–[11]. It may enable LoS-type connectivity on the mmWavebands by steering a narrow beam from the RIS to the receiverassuming that LoS links exist between the transmitter and RISas well as between the RIS and receiver. However, the potentialbenefits of a RIS do not limit to the mmWave bands but areof interest also for the sub-6 GHz frequencies.

A. Related Works

RIS is often assumed to be designed as a planar meta-surface or realized as a planar array of numerous passive re-flecting elements, which are connected to a controller, allowingmodifying the phases of the incident signals independentlyin real-time. As a result, its phase shifts can be optimizedto make the wireless channel between the transmitter andreceiver more favorable for the communication [4], [5], [8],[9]. Several potential technologies for RIS implementationexist. It can be constructed as an array of discrete phaseshifters, which control the impedance values of the elements.Thereby the phase shifts can steer the reflected signal to thedesired direction of the receiver. Such RIS is called the discreteRIS with no baseband processing capability [8], [9], [11]. Thephase shift control is typically assumed to be designed in someactive network element such as a base station (BS) or accesspoint (AP). The phase shift values are then transferred viasome control link from the BS/AP to the RIS. Furthermore,a RIS with active receive elements introduced by Taha et al.in [12], [13] has also been considered to enable the channelestimation at the RIS.

The initial studies on RIS in the literature mainly focuson the performance and design aspects of the RIS-assistedcommunication systems. Specifically, Hu et. al in [14], [15]show that the capacity per RIS area-unit converges to P

2σ2

as the wavelength goes to zero, where P is the transmitpower per area-unit, and σ2 is the additive white Gaussiannoise (AWGN) power spectral density. In [16], an adaptivephase shifter design based on hierarchical codebooks andlimited feedback from the mobile station (MS) is proposedfor a RIS-aided mmWave MIMO system for both accuratepositioning and high data rate transmission. In [17], theasymptotic achievable rate in a RIS-assisted downlink systemis analyzed under practical reflection coefficients, and a passivebeamformer and a modulation scheme that can be used ina RIS without interfering with existing users is proposed toincrease the achievable system sum-rate. Furthermore, in [18],the propagation channel of point-to-point MIMO systems is

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enriched by using the RIS so that more paths with differentspatial angles are added. As a result, the rank of the channelmatrix is enhanced, and the multiplexing gain is achievedeven when the direct path has low rank. The gain of RISis further investigated in [19] and [20] for multiple-input-single-output (MISO) downlink channels operated at sub-6GHz and mmWave frequencies, respectively. It is shown thatthe RIS of N elements can achieve a total beamforminggain of N2 [19] and allows the received signal power toincrease quadratically with N [20]. Moreover, the achievabledata rate of communication systems assisted by a practicalRIS with limited phase shifts and hardware impairments arecharacterized in [21]–[26].

Another important line of studies on RIS focuses on thecapacity/data rate maximization of various RIS-assisted com-munication systems [27]. Specifically, in [28], the problemof maximizing the achievable rate of a RIS-enhanced single-input-single-output (SISO) orthogonal frequency division mul-tiplexing (OFDM) system has been solved by jointly opti-mizing the transmit power allocation and the RIS passivearray reflection coefficients. In contrast, the MISO systemshave been considered in [23], [29]–[32]. In particular, while[29]–[31] focus on the joint transmit and passive beamform-ing problem of RISs with continuous phase shifts to attainsubstantially increased capacity/data rate, Han et al. in [23]consider the discrete phase shifts and justify that only a two-bit quantizer is sufficient to guarantee a high capacity, whichagrees with the finding in [32]. The capacity/rate optimizationof RIS-assisted MIMO systems has been recently consideredin [4], [25], [33]. In [4], an efficient alternating optimization(AO) method has been proposed to find the locally optimalphase shifts of the RIS for both the narrowband frequency-flatand broadband frequency-selective MIMO OFDM systems.In particular, it is shown that with the proposed AO-basedRIS design, the channel total power, rank, and conditionnumber can be significantly improved for capacity enhance-ment. In [25], a RIS-enhanced full-duplex (FD) MIMO two-way communication system is considered. The system sum-rate is maximized through jointly optimizing the transmitbeamforming and the RIS’s coefficient matrix. Specifically,the non-convex optimization problem is decoupled into threesub-problems, which are solved iteratively, and its efficiencyis justified by simulation results.

B. Motivations and ContributionsIn this paper, we propose a novel semi-active RIS-aided

beamforming concept in which active beamforming (relaying)can be applied together with passive beamforming (reflecting).The idea is to activate a few elements of the RIS by connectingthem with radio frequency (RF) chains and power amplifiers(PAs), allowing them to modify not only the phases but also theamplitudes of the incident signals. In this respect, these fewactivated elements become active amplify-and-forward (AF)relaying elements, and the conventional RIS becomes a hybridrelay-reflecting intelligent surface (HR-RIS). We use this termthroughout the paper to refer to the proposed architecture.

The HR-RIS may potentially be realized by recently intro-duced low-power complementary metal-oxide semiconductor

(CMOS)-based technologies, such as the reflection amplifier(RA) [34], [35], which can turn not only the phases but alsothe amplitudes of the incident signals. Thereby a possibleimplementation of the proposed architecture could producea meta-surface with RA elements. However, this methodcan be cost-inefficient because the number of elements inthe HR-RIS is large, while only a few active elements arerequired, as will be proven in the sequel. Therefore, a morepractical solution appears to be upgrading the conventionalRIS by replacing some elements with the RAs. Obviously,compared to the RIS, HR-RIS requires additional complexityfor hardware implementation and signal processing of theactive elements. However, those are only required for a singleor few active elements. Considering that the total number ofelements in the conventional RIS is very large, the hardwareand computational costs increase only moderately. However,the detailed HR-RIS implementation is out of the scope of thispaper. We focus on proposing it and analyzing its potential forsystem performance improvement.

The HR-RIS is motivated by recent comparisons betweenthe conventional relay and RIS, the practical deployment ofRIS with hardware impairments, and the demand for channelestimation at the RIS. First, RIS can be thought in somerespect similar to an FD relay [36], [37], but it consists ofpassive elements without the active power amplifier, enablingpassive beamforming without requiring any active power andintroducing no thermal noise. However, a main limitation ofthe RIS compared to the relays is the fact that the reflectionlimits the degrees of freedom in the beamforming. Therefore,HR-RIS, a hybrid architecture between the RIS and relay, canleverage the advantages while mitigating the disadvantages ofRIS and relay. Second, based on the performance compar-isons of relay and RIS in [19] and [36], we found that theHR-RIS is promising to provide a remarkable performanceimprovement. Specifically, both Bjornson et al. [36] and Wuet al. [19] show that a very large RIS is needed to outperformdecode-and-forward (DF) relaying; otherwise, it can be easilyoutperformed even by a half-duplex (HD) relay with fewelements. Furthermore, as the RIS becomes sufficiently large,the performance improvement is not significant if the numberof elements in the RIS increases. This implies that if a fewpassive elements of the RIS are replaced by active ones, thereduction in the passive beamforming gain is just marginal,while the gain from active relaying can be significant. Third,it is shown in [22]–[25] that RISs with limited-resolutionphase shifts have considerable performance loss with respectto those with sufficiently large resolution. However, as theamplitudes of a few elements are adjustable, they can beoptimized to compensate for the performance loss due to thelimited-resolution phase shifts. The aforementioned aspectsmotivate a new architecture of the intelligent meta-surface,allowing it to leverage the benefits of both the RIS and relay,as the HR-RIS proposed in this paper.

We recognize that there are several practical open problemsrelated to the practical and efficient implementation of HR-RIS. What is more, assume perfect channel state information(CSI) [4], [19], [22], [38]. However, with the proposed HR-RIS architecture, the assumption is less restrictive than with

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a passive RIS, because the active processing chains can bereadily used for channel estimation using their built-in activeelements [12]. Thereby, our paper proposes the concept anddemonstrates its significant system-level potential. This givesmore understanding on the fundamental trade-offs betweenRIS and relaying as part of a MIMO communication link.

The major contributions of the paper are summarized asfollows:

• We propose the novel HR-RIS concept, which enablesa hybrid active-passive beamforming scheme rather thanfully-passive beamforming of the conventional RIS. Itrequires activating only a single or few elements ofthe RIS to serve as active relays to attain remarkableperformance improvement. In particular, HR-RIS exploitsthe benefits of relaying while mitigating the limitation ofpassive reflecting.

• We propose two HR-RIS architectures, namely, fixed anddynamic HR-RIS. In the former, the active elements arefixed in manufacture; by contrast, those in the latter canbe adaptively configured to improve the performance andsave power consumption. The coefficient matrices of theHR-RIS are optimized in the formulated SE maximizationproblem, which is solved by the AO and power allocationschemes. Furthermore, the favorable deployment andperformance gains of the HR-RIS are derived analyticallyand numerically verified by computer simulations.

• The total power consumption and energy efficiency ofthe system with the proposed HR-RIS is analyzed andcompared to that with the conventional RIS. It is shownthat the former requires higher power consumption thanthe latter because additional power is consumed for theactive processing. However, with a small number of activeelements in the proposed HR-RIS, the improvement inboth the SE and energy efficiency (EE) is guaranteed, asnumerically demonstrated by the simulation results.

• The paper bridges the theoretical gap between a passiveRIS and active AF relay aiding a MIMO communicationlink. This gives more insight into the fundamental perfor-mance of each of the approaches and attainable SE andEE.

Structure: The rest of the paper are as follows. In SectionII, we introduce the concept of the HR-RIS and the systemmodel of the HR-RIS-aided MIMO system, and the problemof SE maximization is formulated. Its efficient solution isfound in Section III. Based on that, the coefficient matricesof the fixed and dynamic HR-RIS architectures are derived inSection IV. Their power consumption is investigated in SectionV. Simulation results are shown in Section VI, and, finally,conclusions are drawn in Section VII.

Notations: Throughout this paper, numbers, vectors, andmatrices are denoted by lower-case, bold-face lower-case,and bold-face upper-case letters, respectively. (·)∗ and (·)Hdenote the conjugate of a complex number and the conjugatetranspose of a matrix or vector, respectively. IN denotesthe identity matrix of size N × N , and diag{a1, . . . , aN}represents a diagonal matrix with diagonal entries a1, . . . , aN .Furthermore, |·| denotes either the absolute value of a scalar

or determinant of a matrix, and the expectation operator isdenoted by E {·}.

II. HR-RIS ARCHITECTURE, SYSTEM MODEL, AND SEPROBLEM FORMULATION

A. Proposed HR-RIS Beamforming Architecture

The HR-RIS is equipped with N elements, including Mpassive reflecting and K active relay elements (M + K =N). Here, the passive element can only shift the phase whilethe active one can tune both the phase and amplitude of theincident signal. As such, for K = 0, the HR-RIS serves asthe conventional RIS; by contrast, for K = N , it becomes aconventional relay station. It is noted that an active elementwould consume more power for processing compared to thepassive one due to the operation of the RF chain. Thus, forpractical deployment, we are interested in the case that 1 ≤K � M . Furthermore, similar to the conventional RIS, weassume that each (active/passive) element of the HR-RIS canre-scatter the signal independently (without cross-interference)by an individual coefficient [4].

We now introduce some notations in order to mathemati-cally model the HR-RIS. In particular, let us denote by A,A ⊂ {1, 2, . . . , N}, the index set of the positions of the Kactive elements. Let αn denote the relay/reflection coefficientused at the nth element, which is given as

αn =

{|αn| ejθn , if n ∈ Aejθn , otherwise

, (1)

where θn ∈ [0, 2π) represent the phase shift. We notethat |αn| = 1 for n /∈ A. We also define three diag-onal matrices constructed from {αn}n=1,...,N . Let Υ =diag{α1, . . . , αN} ∈ CN×N be the diagonal matrix of thecoefficients. We define an additive decomposition for it asΥ = Φ + Ψ, where Φ = diag{φ1, . . . , φN} ∈ CN×N ,Ψ = diag{ψ1, . . . , ψN} ∈ CN×N ,

φn =

{0, if n ∈ Aαn = ejθn , otherwise

,

ψn =

{αn = |αn| ejθn , if n ∈ A0, otherwise

.

In other words, Φ and Ψ contain the passive reflecting andactive relaying coefficients, respectively.

There can be two architectures for the HR-RIS, namely, thefixed and dynamic HR-RIS. In the former, the number andpositions of the active elements, i.e., set A, are predefinedand fixed as illustrated in Fig. 1(a) [12], [13]. By contrast, inthe dynamic HR-RIS architecture, the number and positionsof active elements can be dynamically changed according tothe propagation condition. In other words, set A can be adesign parameter in this architecture. More specifically, thereare a number of RF-PA chains each can be turned on/off. Anelement of HR-RIS is active if it connects to an “ON” RF-PAchain; otherwise, it serves as a passive reflecting element [12],[13], [39]. Such connection can be done via a switching (SW)network as illustrated in Fig. 1(b). Furthermore, the signalprocessing in the active elements are controlled by an HR-RIS

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(a) Fixed HR-RIS architecture. All 𝐾 RF-PA chains are ON.

RF PA1

𝑁

𝐾 + 1

⋯⋯

RF PA𝐾

HR-RIS

Controller

baseband unit

RF PA

RF PA

SW network

1

2

𝑁

(b) Dynamic HR-RIS architecture. 𝔸⋆ out of 𝐾 RF-PA chains are ON.

𝑛

on/off

⋯⋯

⋯

⋯⋯

HR-RIS

Controller

baseband unit

Fig. 1. Illustration of the fixed and dynamic HR-RIS architectures.

BS

𝑯𝑡 𝑯𝑟

Active relay element

Passive reflecting element

𝚼 = diag 𝛼1, … , 𝛼𝑁

𝑁 elements

MS

Fig. 2. Illustration of the HR-RIS-aided communication system.

controller and baseband unit [12], [13]. In this paper, both thefixed and dynamic HR-RIS architectures are considered.

B. System Model

We consider a downlink transmission between a BS and anMS. Suppose that there is no direct link between the BS andthe MS due to, e.g., severe pathloss or blockage [40], andthe communications is aided by the HR-RIS as illustrated inFig. 2 [12], [13]. Let us denote by Nt and Nr the numbers ofantennas equipped at the BS and MS, respectively, where Nt ≥1 and Nr ≥ 1. Let Ht ∈ CN×Nt and Hr ∈ CNr×N denotethe channel between the BS and the HR-RIS, and betweenthe HR-RIS and the MS, respectively. Let x ∈ CNt×1 be thetransmitted signal vector, where E

{xxH

}= PBSINt

and PBS

is the transmit power at the BS. With these newly introducednotations, the received signals at the MS can be written as

y = HrΦHtx + HrΨHtx + HrΨnH + nMS

= (HrΦHt + HrΨHt) x + n= HrΥHtx + n, (2)

where nH ∼ CN (0, σ2HIK) and nMS ∼ CN (0, σ2

MSINr ) arethe complex additive white Gaussian noise (AWGN) vectorsat the K active relay elements of the HR-RIS and at the MS,respectively; and n = HrΨnH+nMS represents the total effec-tive noise at the MS. For notational simplicity, we assume thatσ2

H = σ2MS = σ2 and n ∼ CN

(0, σ2

(INr

+ HrΨΨHHHr

)).

C. Problem Formulation

Based on (2), the SE of an HR-RIS-aided MIMO systemcan be expressed as

f0({αn}) = log2

∣∣∣INr+ ρHrΥHtHH

t ΥHHHr R−1

∣∣∣ ,

where {αn} represents the set of all the relay/reflectingcoefficients on the main diagonal of Υ, i.e., {αn} ={α1, α2, . . . , αN}. Furthermore,

R = INr+ HrΨΨHHH

r ∈ CNr×Nr (3)

is the aggregate noise covariance matrix, and ρ = PBS

σ2 . Thetransmit power of the active elements of the HR-RIS is

Pa({αn}) , trace(Ψ(HtE

{xxH

}HHt + σ2IN

)ΨH

)= trace

(Ψ(PBSHtHH

t + σ2IN)ΨH

). (4)

The problem of designing relay/reflection coefficients of theHR-RIS for maximizing the SE can be formulated as

(P0) maximize{αn}

f0({αn}) (5a)

subject to |αn| = 1 for n /∈ A (5b)Pa({αn}) ≤ Pmax

a (5c)

where Pmaxa is the power budget of the active elements.

Function f0({αn}) is nonconvex with respect to {αn}. Inaddition, the feasible set of (P0) is nonconvex due to theunit-modulus constraint (5b). Consequently, problem (P0) isintractable, and it is difficult to find an optimal solution. In thefollowing section, we develop an efficient solution to (P0).

III. EFFICIENT SOLUTION TO (P0)

A. A Tractable Approximation of (P0)

As the first step of developing an efficient solution to(P0), we approximate the problem into a more tractableone. Specifically, f0({αn}) is upper bounded by f({αn}), asfollows:

f0({αn}) = log2

∣∣∣INr + ρHrΥHtHHt ΥHHH

r R−1∣∣∣

= log2

∣∣∣R + ρHrΥHtHHt ΥHHH

r

∣∣∣− log2 |R| (6)

≤ log2

∣∣∣INr+ HrΨΨHHH

r + ρHrΥHtHHt ΥHHH

r

∣∣∣(7)

, f({αn}),

where the inequality in (7) follows by substituting (3) to (6).Clearly, the equality occurs when A = ∅. In addition, thesmaller the term log2 |R| is, the tighter the bound is. We canalso observe that log2 |R| is small when the path loss is largeand the number of active elements is small (since Ψ becomesvery sparse when only a single or few active elements are

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employed at the HR-RIS). These observations motivate us toarrive at the following approximation of (P0)

(Phybrid) maximize{αn}

f({αn})

subject to (5b), (5c).

Although f({αn}) is still nonconvex with respect to {αn}, itsstructure allows us to develop an efficient solution via the AOmethod. The details are presented in the following subsection.

B. Efficient Solution to (Phybrid): an AO Approach

Generally, the proposed solution is a sequential procedure:in each iteration, a specific coefficient of HR-RIS is updatedwhen the others are fixed. We first further expand f({αn})to extract the role of variable αn, for any n ∈ {1, ..., N}.Specifically, let rn ∈ CNr×1 be the nth column of Hr, andtHn ∈ C1×Nt be the nth row of Ht, i.e., Hr = [r1, . . . , rN ] andHt = [t1, . . . , tN ]H . Since Υ and Ψ are diagonal matrices,we have HrΥHt =

∑Nn=1 αnrntHn and HrΨ =

∑n∈A αnrn.

Consequently, we can express f({αn}) as follows

f({αn}) = log2

∣∣∣An + |αn|2 Bn + αnCn + α∗nCHn∣∣∣ , (8)

where matrices An, Bn, and Cn are given by

An , INr+

∑i∈A,i6=n

αiri

∑i∈A,i6=n

α∗i rHi

+ ρ

N∑i=1,i6=n

αiritHi

N∑i=1,i6=n

α∗i tirHi

, (9)

Bn , rnrHn + ρrntHn tnrHn , (10)

Cn , rn

∑i∈A,i6=n

α∗i rHi

+ ρrntHn

N∑i=1,i6=n

α∗i tirHi

(11)

for n ∈ A, and

An , INr + ρ

N∑i=1,i6=n

αiritHi

N∑i=1,i6=n

α∗i tirHi

, (12)

Bn , ρrntHn tnrHn , (13)

Cn , ρrntHn

N∑i=1,i6=n

α∗i tirHi

(14)

for n /∈ A. We note that matrices An,Bn, and Cn donot contain variable αn. This means that if all variables{αi}Ni=1,i6=n are fixed, the three matrices are determined.

Similarly, we also extract the role of variable αn inPa({αn}) by rewriting

Pa({αn}) = trace(PBSΨHtHH

t ΨH + σ2ΨΨH)

= σ2trace(ΨΨH

)+ PBStrace

(ΨHtHH

t ΨH)

= σ2∑n∈A|αn|2 + PBS

∑n∈A|αn|2 ‖tn‖2

=∑n∈A|αn|2 ξn, (15)

where ξn , σ2 +PBS ‖tn‖2. Let Pa ,∑i∈A,i6=n |αi|

2ξi, n ∈

A. We can see that Pa is a constant if all variables {αi}Ni=1,i6=nare fixed. Therefore, the role of αn in Pa({αn}) can be seenin the following expression of Pa({αn}):

Pa({αn}) = |αn|2 ξn + Pa. (16)

1) Problem of Updating αn: In the AO approach, in aniteration of updating αn, {αi}Ni=1,i6=n are fixed. Therefore,the objective function f({αn}) in (8) can be rewritten in theexplicit form

fn(αn) = log2

∣∣∣An + |αn|2 Bn + αnCn + α∗nCHn∣∣∣ , (17)

which has only one variable, i.e., αn. Moreover, because Anis of full-rank (rank(An) = Nr) and invertible, we can rewritefn(αn) as

fn(αn) = log2 |An|+ gn(αn), (18)

where

gn(αn) , log2

∣∣∣INr+ |αn|2 A−1

n Bn + αnA−1n Cn + α∗nA−1

n CHn∣∣∣ .

(19)

In (18), log2 |An| is a constant with αn. Therefore, the problemof updating αn, denoted by (Pupdate), is given bymaximize

αn

gn(αn) subject to |αn| = 1, for n /∈ A

maximizeαn

gn(αn) subject to |αn|2 ≤ Pmaxa −Pa

ξn, for n ∈ A

(20)

2) Solution to (Pupdate): (Pupdate) in (20) admits a closed-form solution, and, thus, it is efficient for practical implemen-tation. To derive it, we rewrite gn(αn) as

gn(αn) = log2

∣∣Dn + αnA−1n Cn + α∗nA−1

n CHn∣∣

= log2 |Dn|+ log2

∣∣INr+ αnE−1

n Cn + α∗nE−1n CHn

∣∣ ,(21)

where Dn = INr + |αn|2 A−1n Bn, and En = AnDn. We start

investigating the objective function gn(αn) by considering thefirst term in (21), i.e., log2 |Dn|. Specifically, we note thatin Dn, rank(A−1

n Bn) ≤ rank(Bn) = 1. In addition, theprobability that rank(A−1

n Bn) = 0 is almost zero (it onlyhappens when A−1

n Bn = 0). Consequently, we generally haverank(A−1

n Bn) = 1. Also, we observe that A−1n Bn is non-

diagonalizable iff trace(A−1n Bn) = 0 [4], which rarely hap-

pens in general. Thus, we almost surely have trace(A−1n Bn) 6=

0 and A−1n Bn is diagonalizable. As a result, we can write as

A−1n Bn = QnΓnQ−1

n based on the eigenvalue decomposition(EVD) where Qn ∈ CNr×Nr , Γn = diag {γn, 0, . . . , 0}, andγn is the sole non-zero eigenvalue of A−1

n Bn. Finally, sinceAn and Bn are both positive semi-definite, γn is non-negativeand real. Thus, we can write

log2 |Dn| = log2

∣∣∣INr+ |αn|2 QnΓnQ−1

n

∣∣∣= log2

∣∣∣Qn (INr+ |αn|2 Γn

)Q−1n

∣∣∣= log2

∣∣∣INr + |αn|2 Γn

∣∣∣= log2

(1 + |αn|2 γn

). (22)

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We now focus on the second term in (21). Followingthe similar arguments for the first term, we also almostsurely have E−1

n Cn diagonalizable. Thus, we have E−1n Cn =

UnΛnU−1n based on the EVD, where Un ∈ CNr×Nr , Λn =

diag {λn, 0, . . . , 0}, and λn is the sole non-zero eigenvalue ofE−1n Cn. Now, let Vn , UH

n EnUn, vn be the first element ofthe first column of V−1

n , and v′n be the first element of thefirst row of Vn. As a result, we can write [4]

log2

∣∣INr + αnE−1n Cn + α∗nE−1

n CHn∣∣

= log2

(1 + |αn|2 |λn|2 + 2R(αnλn)− v′nvn |λn|

2), (23)

where R(·) denotes the real part of a complex number. Thedetailed derivation of (23) can be found in [4]. We note that,compared to [4], the additional coefficient |αn|2 is due to theactive relaying coefficient in the HR-RIS, which does not existfor the conventional RIS [4].

In summary, based on (22) and (23), we have

gn(αn) = log2

(1 + |αn|2 γn

)+ log2

(1 + |αn|2 |λn|2 + 2R(αnλn)− v′nvn |λn|

2). (24)

It is observed that an optimal solution to (Pupdate), denotedby α?n, admits the form

α?n =

{|α?n| e−j arg{λn}, n ∈ Ae−j arg{λn}, otherwise

. (25)

It is observed that the amplitude |α?n| is not available at thisstage because it requires a determined A. To this end, it isnatural to consider two scenarios: A is predetermined andfixed in the manufacture, and A is dynamic and optimizedbased on the propagation condition, associated with the fixedand dynamic HR-RIS architectures illustrated in Figs. 1(a)and 1(b), respectively. The solution {α?n} dedicated to thesearchitectures will be derived in the next section.

IV. BEAMFORMING COEFFICIENT MATRIX OF FIXED ANDDYNAMIC HR-RIS

A. Fixed HR-RIS Architecture

In the fixed HR-RIS, A is available to determine {|α?n|}n∈A.Specifically, |α?n|, n ∈ A, can be determined based on the factthat, given the optimal form in (25), gn(α?n) monotonicallyincreases with |α?n|. Therefore, from (20), we obtain

|α?n| =

√Pmax

a − Pa

ξn, n ∈ A. (26)

As a result, the optimal solution to (Pupdate) is given as

α?n =

{√Pmax

a −Pa

ξne−j arg{λn}, n ∈ A

e−j arg{λn}, otherwise. (27)

The solution (27) and the given A are readily used toobtain the coefficient matrix of the fixed HR-RIS scheme,as outlined in Algorithm 1. Specifically, at the initial stage,coefficients {αn} are randomly generated to have arbitraryphases and amplitudes satisfying |αn| = 1, n /∈ A and∑n∈A |αn|

2ξn = Pmax

a , based on (5c), (15), and (26). During

Algorithm 1 Find coefficients of the fixed HR-RIS

Input: Ht,Hr,A.Output: {α?1, . . . , α?N}.

1: Randomly generate {αn} with |αn| = 1, n /∈ A, and∑n∈A |αn|

2ξn = Pmax

a .2: while objective value does not converge do3: for n = 1→ N do4: if n ∈ A then5: Compute An,Bn, and Cn based on (9)-(11).6: Dn = INr + |αn|2 A−1

n Bn, En = AnDn.7: else8: Compute An,Bn, and Cn based on (12)-(14).9: Dn = INr

+ A−1n Bn, En = AnDn.

10: end if11: Find λn as the sole non-zero eigenvalue of E−1

n Cn.12: Update α?n as (27).13: end for14: end while

steps 2–13, the coefficients are alternatively updated based on(27). It terminates when a convergence criteria on the objectivef({αn}) is reached. The iteration procedure in Algorithm 1guarantees the convergence. To see this, we note that objectivefunction f({αn}) is upper bounded. In addition, (Pupdate)is solved optimally by (27). Thus, the resultant sequence off({αn}) is non-decreasing over iterations [4]. We make tworemarks on the design and deployment of the fixed HR-RIS.

Remark 1: Inserting Pa ,∑i∈A,i6=n |αi|

2ξi, n ∈ A to

(26), it is observed that a larger K results in a smaller |α?n|.Therefore, increasing K does not always guarantee the SEimprovement of the fixed HR-RIS with respect to the conven-tional RIS. In particular, with a limited power budget Pmax

a ,the HR-RIS can have |α?n| < 1, causing signal attenuation, andhence degrades the SE. In this case, the fixed HR-RIS with asmall K is easier to attain SE gains than that with numerousactive elements. Furthermore, in the fixed HR-RIS, we have∑n∈A |αn|

2ξn = Pmax

a , implying that Pmaxa >

∑n∈A ξn is

necessary for |αn| > 1,∀n ∈ A.Remark 2: By inserting ξn , σ2 + PBS ‖tn‖2 to (26), it is

observed that with a fixed Pmaxa , a lower transmit power PBS

and/or smaller channel gain ‖tn‖2 from the transmitter resultsin a higher power amplifier coefficients |α?n|, and thus, a moresignificant performance improvement can be achieved.

Remarks 1 and 2 will be further numerically justified anddiscussed in Section VI. The fixed number and positions ofactive elements in the fixed HR-RIS limit the beamforminggain of the HR-RIS. In the next section, we propose amore robust architecture, named the dynamic HR-RIS, whichovercomes the drawbacks of the fixed HR-RIS to guaranteethe SE improvement.

B. Dynamic HR-RIS Architecture

Unlike the fixed HR-RIS architecture, in the dynamic HR-RIS illustrated in Fig. 1(b), the number and positions of theactive elements can be cast as design parameters. In particular,

7

the problem of SE maximization for the dynamic HR-RIS canbe formulated as

(P1) maximize{αn},A

f0({αn}) (28a)

subject to (5b)− (5c), (28b)|A| ≤ K,A ⊂ {1, 2, . . . , N}, (28c)

where the constraint (28c) means that the number of activeelements in the HR-RIS does not exceed the maximum numberof RF-PA chains. Problem (P1) does not only inherit thenumerical challenge of (P0) but also includes the difficultyfrom cardinality constraint (28c). We develop an efficientsolution to (P1) for practical use below.

Similar to problem (P0), firstly, we employ the upper boundof f0({αn}) (see (7)) to obtain an approximate but moretractable problem of (P1), which is given as

(Pdyn) maximize{αn},A

f({αn}) (29a)

subject to (5b)− (5c), (29b)|A| ≤ K,A ⊂ {1, 2, . . . , N} (29c)

At this point, a naive approach to solve (Pdyn) would bebased on an exhaustive search, i.e., using the result presentedin Section III to determine the coefficients {αn} correspondingto each valid set A, then choosing the one providing the bestperformance. Clearly, such an approach requires high compu-tational cost due to excessively large numbers of combinationsof A.

We now present an efficient solution to (Pdyn). Recallthat we can write αn = |αn| ejθn . Based on this, let usintroduce two matrices as Θ = diag

{ej{θ1}, . . . , ej{θN}

}and

Υ = diag {|α1| , . . . , |αN |} with |αn| = 1, n /∈ A. Then wecan write Υ = ΘΥ. This motivates us develop an alternatingprocedure applied on three sets of variables {Θ, Υ,A}. Next,we describe the approach determining each of {Θ, Υ,A}when the others are fixed, which are the key steps in ourproposed solution.

1) Determine {θn}: For this step, we suppose that |αn| =1,∀n, which corresponds to the conventional RIS where allelements are passive. Then, we determine {θn} via solvingthe following problem extracted from (Pdyn)

(P1dyn) maximize{αn}

f({αn})

subject to |αn| = 1,∀n.

A solution to (P1dyn) can be obtained by using the sameapproach presented in Section III-B (i.e., see (25)) for thecase n /∈ A.

2) Determine A and {|αn|}: We recall that the passivereflecting elements have unit modulus, i.e., |αn| = 1,∀n /∈ A.Thus, we only need to determine |αn| ,∀n ∈ A. Furthermore,an active element only results in performance gain with respectto a passive element if |αn| > 1; otherwise, it causes perfor-mance loss due to signal attenuation. Therefore, to minimizethe loss, in the dynamic HR-RIS architecture, the RF-PAchain connected to the nth element should be turned off if

|αn| < 1. As a result, in the dynamic HR-RIS scheme, wehave |αn| > 1,∀n ∈ A, and the solution in (25) becomes

α?n =

{|α?n| e−j arg{λn}, |α?n| > 1, n ∈ Ae−j arg{λn}, otherwise

,

with {arg{λn}} being determined in the previous step. Withthis solution, it is obvious that R(α?nλn) = |α?n| |λn|, and (24)can be rewritten as

gn(α?n) = log2

[(1 + |α?n|

2γn

)×(

1 + |α?n|2 |λn|2 + 2 |α?n| |λn| − v′nvn |λn|

2)]

= log2

[1 + ζ2

n |α?n|4

+ o(|α?n|

3)], (30)

where ζn , |λn|√γn, and o

(|α?n|

3)

represents the sum ofthe remaining terms associated with |α?n|

c, c ≤ 3. Because

|αn| > 1, n ∈ A, we can ignore o(|α?n|

3)

in (30) and

approximately maximize gn(αn) via maximizing gn(α?n) ,

log2

(1 + ζ2

n |α?n|4)

. Moreover, because gn(α?n) monotoni-cally increases with |α?n|, maximizing it is equivalent tomaximizing ˆgn(α?n) , log2

(1 + ζn |α?n|

2)

. The observationimplies that to find |α?n| that maximizes gn(α?n),∀n ∈ A, wecan consider the following problem

maximizeA,{|αn|2}

∑n∈A

log2

(1 + ζn |αn|2

)(31a)

subject to∑n∈A|αn|2 ξn ≤ Pmax

a and (29c), (31b)

where the first constraint in (31b) is the transmit powerconstraint of the active elements. Let pn , |αn|2 ξn denotethe power allocated to the nth active element, n ∈ A. Theproblem in (31) can be rewritten as

maximizeA,{pn}

∑n∈A

log2

(1 +

ζnξnpn

)(32a)

subject to∑n∈A

pn ≤ Pmaxa and (29c), (32b)

which allows A and {pn} to be obtained. Specifically, itis clear from (32a) that the optimal set of active elements,i.e., A?, is the set of indices of the K largest elements in{ζ1ξ1, . . . , ζNξN

}, where ζn

ξn,∀n are obtained during determining

{θn} in the previous step, i.e., in Section IV-B1.Once A? is determined, {pn} can be solved by performing

power allocation to the active elements in A?. The opti-mal solution is the well-know water-filling solution p?n =

max{

1µ −

ξnζn, 0}

, with µ satisfying (32b). As a result, we

obtain |α?n|2

=p?nξn

= max{

1µξn− 1

ζn, 0}

, n ∈ A?. Conse-quently, the amplitudes of elements in the dynamic HR-RISscheme is given as

|α?n| =

{max

{√1µξn− 1

ζn, 1}, n ∈ A?

1, otherwise. (33)

The proposed procedure finding efficient solution to theproblem of SE maximization in dynamic HR-RIS architecture

8

Algorithm 2 Find coefficients of the dynamic HR-RIS

Input: Ht,Hr.Output: A?, {α?n}.

1: Randomly generate Θ = diag{ejθ1 , . . . , ejθN

}and as-

sign Υ = Θ.2: while objective value does not converge do3: for n = 1→ N do4: Compute An,Bn, and Cn based on (12)-(14).5: Dn = INr

+ A−1n Bn, En = AnDn.

6: Find λn and γn as the sole non-zero eigenvalue ofE−1n Cn and A−1

n Bn, respectively.7: θ?n = − arg{λn}, α?n = e−jθ

?n .

8: end for9: end while

10: Compute ζnξn

, n = 1, . . . , N , where ζn , |λn|√γn and

ξn , σ2 + PBS ‖tn‖2.11: Set A? to the indices of the K largest value in{

ζ1ξ1, . . . , ζNξN

}.

12: Obtain {|α?n|} given in (33).13: Obtain {α?n} based on (25).

is outlined in Algorithm 2. It starts with randomly generating{αn} with unit modules. Then steps 2–9 are executed todetermine {θ?n}. After that, A? is determined in steps 10 and11, allowing {|α?n|} to be obtained in step 12. Finally, thecoefficients of the dynamic HR-RIS are derived in step 13based on (25).

V. POWER CONSUMPTION ANALYSIS

A. Fixed HR-RIS ArchitectureIn the fixed HR-RIS, the number of active and passive

elements are fixed to K and M , respectively. Therefore, thetotal power consumption of the whole MIMO communicationssystem aided by the fixed HR-RIS can be modeled as [36],[41], [42]

P fix.H =

PBS

τBS+Pa

τa+ P fix.

c,H +MPp, (34)

where τBS, τa ∈ (0, 1] are the power amplifier efficiencies ofthe BS and active elements at the HR-RIS, respectively, Pa

is given in (16), and Pp is the power required for a passivereflecting element [36]. Furthermore, P fix.

c,H is the total circuitpower consumption of the fixed HR-RIS-aided MIMO systemand can be computed as [41]

P fix.c,H = NtPBS,dynamic +KPa,dynamic + PBS,static + Pa,static,

with PBS,dynamic denoting the dynamic power consumptionof each RF chain of the BS, and PBS,static denoting thestatic power overhead of the BS, including baseband process-ing, power suply, and cooling power consumption. Similarly,Pa,dynamic and Pa,static are the dynamic and static powerconsumption of the active relay elements in the HR-RIS.

B. Dynamic HR-RIS ArchitectureIn the dynamic HR-RIS architecture, the number of active

and passive elements vary depending on Pmaxa and ξn ac-

cording to (33). Let |A?| denote the number of actual active

BS HR-RIS

MS

𝑥H = 𝑑𝑡

𝑦MS

𝑥MS𝑦

𝑥

𝑑𝑟

Fig. 3. Horizontal locations of the BS, MS, and HR-RIS.

elements whose amplitudes are larger than one, where A? isobtained in Algorithm 2. As a result, the number of passiveelements is given as N−|A?|, and the total power consumptionof a dynamic HR-RIS-assisted MIMO system is given by

P dyn.H =

PBS

τBS+Pa

τa+ P dyn.

c,H + (N − |A?|)Pp +NPSW,

(35)

where the last term accounts for the total power consumed bythe N switches in Fig. 1(b), each requires a power of PSW, andP dyn.

c,H is the total circuit power consumption of the dynamicHR-RIS-aided MIMO system, given as [41]

P dyn.c,H = NtPBS,dynamic + |A?|Pa,dynamic

+ PBS,static + Pa,static.

C. Conventional RIS Architecture

We compare the total power consumption of the proposedHR-RIS-assisted system to that of the conventional RIS-assisted system. We note that in the latter, there is no activerelay element, but N = M + K passive reflecting elementsare used at the RIS. Therefore, the total power consumptionof the RIS-aided system can be expressed as

PRIS =PBS

τBS+ Pc,RIS +NPp, (36)

where Pc,RIS = NtPBS,dynamic +PBS,static is the total circuitpower consumption of the dynamic HR-RIS-aided MIMOsystem

By comparing (34) and (35) to (36), with the note thatPpassive � Pa,dynamic, the increased total power consumptionof the fixed and dynamic HR-RIS-aided systems with respectto the RIS-assisted system can be respectively given as

∆P fix.H =

Pa

τa+K (Pa,dynamic − Pp) + Pa,static,

∆P dyn.H =

Pa

τa+ |A?| (Pa,dynamic − Pp) + Pa,static,

which linearly increases with the number of active elements,i.e., K in the fixed HR-RIS and |A?| in the dynamic HR-RIS. The numerical comparison of these architectures will bepresented in the next section.

VI. SIMULATION RESULTS

In this section, numerical results are provided to validatethe proposed HR-RIS schemes. We assume that uniform lineararrays (ULAs) are deployed at the BS and MS, respectively.By contrast, an uniform planar array (UPA) of N elementsis employed for the HR-RIS. Furthermore, half-wavelengthdistancing between array elements are assumed for the BS,

9

MS, and HR-RIS. As shown in Fig. 3, in a two-dimensionalcoordinate, we assume that the BS is deployed at a fixedlocation (0, 0). By contrast, the HR-RIS and MS are locatedat (xH, 0) and (xMS, yMS), respectively, where xH and xMS

can vary. As a result, the distance between the BS and MSis given by dt = xH, while that between the HR-RIS and theMS is dr =

√(xH − xMS)2 + y2

MS. The path loss of a linkdistance d is given by [4], [19] β(d) = β0

(d

1m

)ε, where β0

is the path loss at the reference distance of 1 meter (m), andε is the path loss exponent.

For small-scale fading, we assume the Rician fading channelmodel [4], [19]. As a result, the small-scale fading channelbetween the BS and the HR-RIS is modeled by [4], [19]

Ht =

(√κt

1 + κtHLoSt +

√1

1 + κtHNLoSt

), (37)

where HLoSt and HNLoS

t represent the deterministic LoS andnon-LoS (NLoS) components, respectively. The NLoS channelis modeled by the Rayleigh fading, with the entry on theith row and jth column of HNLoS

t being given as hNLoSt,ij ∼

CN (0, 1). The LoS component for each channel is modeled asa deterministic component, i.e., HLoS

t = aH(θH, φH)aHBS(θBS),where aBS(θBS) and aH(θH, φH) are the array response vectorsat the BS and HR-RIS, respectively. Here, the nth elementof aBS(θBS) and aH(θH, φH) are given as aBS,n(θBS) =ejπ(n−1) sin θBS , n = 1, . . . , Nt and aH,n(θH, φH) =

ejπ(b nNxc sin θH sinφH+(n−b n

NxcNx) sin θH cosφH), n = 1, . . . , N

[4]. θBS, θH ∈ [0, 2π) denote the angle-of-departure (AoD) atthe BS and the azimuth angle-of-arrival (AoA) at the HR-RIS,respectively, and φH ∈ [−π/2, π/2) denotes the elevation AoAat the HR-RIS. Furthermore, in (37), κt is the Rician factor.With κt → 0, Ht approaches the Rayleigh fading channel,and with κt →∞, Ht becomes the LoS channel. The channelmatrix between the BS and the HR-RIS, i.e., Ht, is obtainedby multiplying Ht by the square root of the correspondingpath loss β(dt), i.e., Ht =

√β(dt)Ht. The channel matrix

between the HR-RIS and MS is modeled similarly.In this work, unless otherwise stated, we set β0 = −30 dB,

{εt, εr} = {2.2, 2.8}, σ2 = −80 dBm, {xMS, yMS, xH} ={40, 2, 51} m, and {κt, κr} = {∞, 0} [19]. Furthermore, theresolution of the phase shifts is set to b = 2 bits, i.e., the phaseshifts are selected from the set F =

{2π(q−1)

Q

}, q = 1, . . . , Q,

where Q = 2b. All the numerical results are averaged over 100independent channel realizations.

A. Improvement in SE of the Proposed HR-RIS

We at first consider in Fig. 4 a 4× 2 MIMO system, wherea small HR-RIS with N = 4 and K = 1 is deployed, toshow that the proposed schemes can perform near-optimal.With K = 1, the HR-RIS has only a single active element,whose transmit power is set to Pmax

a = {−10, 0, 10} dBm.To obtain the optimal performance, we perform the exhaustivesearch over all the possible solutions. Specifically, for the fixedHR-RIS, the index of the active element is set to one, and theoptimal solution is searched over all combinations of N phasevalues, each is in F . For the dynamic HR-RIS, the exhaustivesearch scheme searches for the optimal combination of N

10 15 20 25 30 35 40

4

5

6

7

8

9

10

11

12

13

14

Fig. 4. SEs of the proposed HR-RIS schemes compared to those obtainedby the exhaustive search for Nt = 4, Nr = 2, N = 4, K = 1,

M = N −K = 3, and Pmaxa = {−10, 0, 10} dBm.

10 15 20 25 30 35 40

0

2

4

6

8

10

12

14

16

Fig. 5. SEs of the proposed HR-RIS schemes compared to those of theconventional RIS schemes for Nt = 32, Nr = 2, N = 50, K = 1,

M = N −K = 49, and Pmaxa = {−10, 0, 10} dBm.

phase values and the index of the active element. It is observedfrom Fig. 4 that the proposed schemes perform very closeto the optimal exhaustive search counterparts for all theconsidered values of Pmax

a . In the subsequent simulations, weconsider larger systems and omit the results for exhaustivesearch because they require excessively high computationaland time complexity.

In Fig. 5, we show the performance improvement of theproposed HR-RIS architectures in terms of SE. Specifically,a 32 × 2 MIMO system aided by an intelligent surface ofN = 50 elements is considered. The SEs of the proposedfixed and dynamic HR-RIS schemes are shown for K = 1,M = N − K = 49, and Pmax

a = {−10, 0, 10} dBm. Forcomparison, we also show the SEs of the conventional RISwith N fully passive reflecting elements, whose phases areeither randomly generated or optimized using the AO methodin [4]. Furthermore, in all the simulation results for the fixedHR-RIS scheme, A is fixed to {1, . . . ,K}. It is noted that thecoefficients of the HR-RIS and AO-based RIS [4] are obtainedbased on the same optimization method, i.e., the AO approach.

It can be observed from Fig. 5 that the RIS with randomphases has poor performance compared to the optimizedRIS. By contrast, the HR-RIS schemes achieve significantimprovement in the SE with respect to the conventional RIS,

10

10 20 30 40 50 60 70 80 90 100

0

2

4

6

8

10

12

14

16

18

20

22

Fig. 6. SE improvement for different positions of the HR-RIS withNt = 32, Nr = 2, K = 1, PBS = 30 dBm, and Pmax

a = 0 dBm.

especially at low PBS. This numerically justifies the observa-tion in Remark 2, which states that a higher performance gaincan be obtained by the HR-RIS for low PBS. Furthermore,it is seen that the HR-RIS only requires a small Pmax

a toachieve remarkable SE improvement at low PBS, and withPmax

a = {0, 10} dBm, sustainable SE gains of the HR-RISare seen for the whole considered range of PBS. For example,the fixed and dynamic HR-RIS only requires Pmax

a = 0 dBmto save more than 10 dBm and 15 dBm for the transmit powerat the BS, respectively. However, with a small power budgetat the HR-RIS, but a large transmit power at the BS, e.g., withPmax

a = −10 dBm and PBS ≥ 35 dBm, the performance gainof the HR-RIS decreases, and it performs comparably with theconventional RIS. This observation agrees with the discussionin Remarks 1 and 2. The SE gains of the proposed schemes forlow PBS have been clearly shown in this figure. Therefore, inthe following results on SE, we consider high values of PBS.

In Fig. 6, we investigate the SEs of the HR-RIS for differentgeographical deployments of the RIS/HR-RIS by varying xH.We note that for comparison, the RIS and HR-RIS are set to beplaced at the same position. In this simulation, we set Nt = 32,Nr = 2, K = 4, PBS = 30 dBm, and Pmax

a = 0 dBm.The BS is placed at the coordinate (0, 0), while two positionsare considered for the MS, including {(40, 2), (100, 2)}. TheHR-RIS/RIS is placed at (0, xH), where xH ∈ [10, 100] m,implying that the HR-RIS/RIS moves along the x−axis in Fig.3. An observation from Fig. 6, which agrees with the findingin [36], is that both the RIS and HR-RIS have decreased SEswhen they move far away from either the BS or the MS, andthe highest SEs are attained when they are closest to the MS.However, in this figure, we focus more on the SE gains of theproposed HR-RIS versus xH. Specifically, it is seen that as xH

grows, path loss β(dt) increases, leading to a smaller ξn in(26) and (33). As result, more significant active beamforminggains are attained, as discussed in Remark 2. Furthermore, thedynamic HR-RIS exhibits large gains over the others in theworst scenarios, i.e., when the MS is far from both the HR-RIS/RIS and BS. It can be concluded from Fig. 6 that theproposed HR-RIS is robust over geographical deployment.

Furthermore, we investigate the SE improvement of theproposed schemes for different values of K in Fig. 7. Inthis figure, the proposed schemes are compared to not only

the conventional RIS with random phases and AO-based RIS,which are equipped with N elements, but also to the RISand the fully-active relay equipped with only K independentelements. We consider PBS = 30 dBm, Pmax

a = {−10, 0, 10}dBm, and the other simulation parameters are the same asthose in Fig. 5. The following observations are noted:

• Clearly, the SEs of the conventional fully-passive RISschemes are constant with K. Considering the RIS andrelay with only K elements, their SEs monotonicallyincrease with K. Furthermore, it is seen that as Kincreases, the RIS performs closer to the relay with thesame number of elements. In particular, a sufficientlylarge RIS can outperform the relay with a limited powerbudget, as seen in Fig. 7(a). These observations agreewith the findings in [36] and [19]. However, for a smallK, the relay performs far better than the RIS, even with alimited power budget, which has motivated the proposalof the HR-RIS in this work.

• Comparing the SEs of the proposed HR-RIS and theconventional RIS with N elements, in all the consideredscenarios, the former only requires a single active elementto outperform the latter. However, as we discussed inRemark 1, it is clear for the fixed HR-RIS that increasingK does not guarantee the SE improvement, especiallyfor low Pmax

a in Fig. 7(a). More specifically, as Kincreases, the SEs of both the fixed and dynamic HR-RIS quickly increases for small K, then they graduallyreach their peaks at a small or moderate K beforedegradation. Therefore, we can conclude that a smallnumber of active elements is sufficient for the HR-RISto achieve significant improvement in SE with respect tothe conventional RIS.

• The advantage of the dynamic HR-RIS in terms of SEcompared to the fixed HR-RIS can be clearly seen inFig. 7. Specifically, unlike the fixed HR-RIS, in all theconsidered scenarios, dynamic HR-RIS achieves higheror the same SE compared to the AO-based RIS. A higherPmax

a provides it with a higher gain, but small Pmaxa

does not cause any performance loss because the activeelements can be deactivated to serve as the conventionalreflecting elements if they cannot amplify the signal. Ingeneral, it can be concluded that the dynamic HR-RIS ismore robust than the fixed one.

In Fig. 8, the SEs of the proposed HR-RIS and the con-ventional RIS schemes are shown for Nt = 32, Nr = 2,K = 1, PBS = 30 dBm, Pmax

a = {−10, 0, 10} dBm, andN ∈ [20, 200]. It is seen that the HR-RIS schemes attainthe improvement in SE for all the considered values of Nwith respect to the conventional RIS schemes. In particular,it is observed that the performance improvement becomesless significant as N increases. Nevertheless, the gain of theproposed schemes is still considerable at large N , especiallywith Pmax

a = 10 dBm. With a larger Pmaxa , the dynamic HR-

RIS achieves higher SE gains with respect to the fixed one.

11

1 10 20 30 40 50

0

2

4

6

8

10

12

1 10 20 30 40 50

0

2

4

6

8

10

12

14

1 10 20 30 40 50

0

2

4

6

8

10

12

14

16

Fig. 7. SEs of the proposed HR-RIS schemes compared to those of the conventional RIS schemes and the relay with K active elements for Nt = 32,Nr = 2, N = 50, K = {1, 2, . . . , N}, PBS = 30 dBm, and Pmax

a = {−10, 0, 10} dBm.

20 40 60 80 100 120 140 160 180 200

4

6

8

10

12

14

16

Fig. 8. SEs of the proposed HR-RIS schemes compared to those of theconventional RIS schemes for Nt = 32, Nr = 2, N ∈ [20, 200], K = 1,

PBS = 30 dBm, and Pmaxa = {−10, 0, 10} dBm

B. Power Consumption and EE of the HR-RIS Schemes

In this subsection, the power consumption and EE of theproposed HR-RIS architectures are investigated and comparedto those of the conventional RIS. Specifically, the total powerconsumption of the MIMO system aided by the HR-RISschemes and RIS, i.e., P fix.

H , P dyn.H , and PRIS, are computed

based on (34), (35) and (36), respectively. The componentpower consumption is assumed as follows: PBS,dynamic = 40dBm, Pa,dynamic = 35 dBm, PBS,static = 35 dBm, Pa,static =30 dBm, Pp = PSW = 5 mW, and τa = τBS = 0.5[36], [42], [43]. Then, the EE of a scheme is given asEE = WSE

P [bps/W], where W denotes the system bandwidth,and P ∈ {P fix.

H , P dyn.H , PRIS} is the total power consumption

of the HR-RIS or RIS-assisted MIMO system. We considerthe system bandwidth of 10 MHz [4].

In Fig. 9, the EEs corresponding to the SEs in Fig. 5 areshown. The proposed HR-RIS schemes achieve much higherEEs compared to those of the conventional RIS schemes forlow and moderate PBS. At high PBS, HR-RIS and RIS havecomparable EEs. The EE improvement of the HR-RIS schemesshown in this figure is similar to the SE improvement shownin Fig. 5. In the following results on the EE, we focus oncomparing the power consumption and EEs of the HR-RISand RIS for high PBS because that for low PBS is clear from

10 15 20 25 30 35 40

0

0.5

1

1.5

2

2.5

3

3.5

4

4.510

5

Fig. 9. EEs of the proposed HR-RIS schemes compared to those of theconventional RIS schemes for Nt = 32, Nr = 2, N = 50, K = 1,

M = N −K = 49, and Pmaxa = {−10, 0, 10} dBm.

this figure. In particular, the dynamic HR-RIS will be shownto perform robustly and can save power to ensure high EEswith low Pmax

a .Assuming the same simulation parameters as those in Fig.

7, we show the power consumption of the considered schemesfor K = {1, 2, . . . , N} in Fig. 10. As analyzed in Section V, itis clear that the proposed HR-RIS architectures require higherpower consumption than the RIS. However, for small Pmax

a ,as K increases, the changing properties of P fix.

H and P dyn.H are

of much different. Specifically, for all the considered Pmaxa ,

P fix.H linearly increases with K, which is, however, only correct

for the dynamic HR-RIS with Pmaxa = {0, 10} dBm. In Fig.

10(a), P dyn.H almost linearly increases at first, then quickly

drops to a power value that is as low as PRIS. The reasonis that, with numerous active elements, Pmax

a = −10 dBmis not sufficient to share among all the active elements forsignal amplifying. Therefore, the dynamic HR-RIS deactivatesthem to save power and also to preserve the SE, as justifiedin Fig. 7(a). Focusing on the results for small K, e.g., K ={1, 2, . . . , 4}, it is observed from Figs. 5, 7, and 10 that theHR-RIS requires a reasonably additional power consumptionto attain significant improvement in the SE.

Following the investigation of SEs and power consumptionof the HR-RIS and RIS in Figs. 7 and 10, with the samesimulation parameters, we investigate their EEs in Fig. 11.

12

1 10 20 30 40 50

320

340

360

380

400

420

440

460

480

500

1 10 20 30 40 50

320

340

360

380

400

420

440

460

480

500

1 10 20 30 40 50

320

340

360

380

400

420

440

460

480

500

1 2 3 4325

330

335

1 2 3 4325

330

335

1 2 3 4325

330

335

Fig. 10. Total power consumption of the proposed HR-RIS schemes compared to those of the conventional RIS schemes for Nt = 32, Nr = 2, N = 50,K = {1, 2, . . . , N}, PBS = 30 dBm, and Pmax

a = {−10, 0, 10} dBm.

1 10 20 30 40 50

1.5

2

2.5

3

3.510

5

1 10 20 30 40 50

1.5

2

2.5

3

3.5

410

5

1 10 20 30 40 50

1.5

2

2.5

3

3.5

4

4.510

5

Fig. 11. EEs of the proposed HR-RIS schemes compared to those of the conventional RIS schemes for Nt = 32, Nr = 2, N = 50, K = {1, 2, . . . , N},PBS = 30 dBm, and Pmax

a = {−10, 0, 10} dBm.

It is seen that the EE of the fixed HR-RIS rapidly decreasesas K increases due to its increased power consumption, asshown in Fig. 10. However, the EEs of the proposed HR-RIS architectures are the same or much higher than those ofthe conventional AO-based RIS at small K. At large K, thedynamic HR-RIS still exhibits the comparable EE with respectto the AO-based RIS for Pmax

a = −10 dBm. The results inthis figure and Fig. 7 further show that a spectral- and energy-efficient HR-RIS should be equipped with a small number ofactive elements. However, if the favorable conditions on PBS,Pmax

a , or K are not guaranteed, the dynamic HR-RIS can stillmanage to ensure no or small loss in EE.

VII. CONCLUSION

We proposed a novel HR-RIS to assist MIMO communica-tion systems with significantly improved SE and EE comparedto that assisted by the conventional RIS. The HR-RIS isa semi-passive beamforming architecture, in which a fewelements are capable of adjusting the incident signals’ power.It has been designed and optimized by the AO and powerallocation strategies, resulting in two different architectures,namely, the fixed and dynamic HR-RIS. In the former, theactive elements are fixed; by contrast, those in the latter canbe dynamically selected based on their capability of enhancingthe SE, and are powered based on the water-filling power

allocation method. The analytical results suggest the favorabledesign and deployment of the HR-RIS. Specifically, it shouldbe equipped with a small number of active elements becauseemploying numerous active elements does not guarantee SE orEE improvement, while consuming high power. Furthermore,the HR-RIS can attain significant SE and EE improvementwhen the transmit power at the transmitter is small or moder-ate, and the improvement is more significant when the distancebetween the transmitter and the HR-RIS is large. Finally,intensive simulations have been performed to numericallyjustify the findings. The results reveal that the HR-RIS hasthe potential to outperform both the passive RIS and activeAF relaying in certain communications channels.

Several important open problems for future work remain.The practical implementation of the HR-RIS using efficientRAs needs more attention. In particular, the realization ofthe FD type AF relay assumed has practical challenges.Regardless, this paper provides theoretical insight in bridgingthe gap between the RIS and active relaying concepts. Thesystem optimization from the EE perspective should be furtheraddressed because we did not directly optimize the EE metric.The CSI acquisition and HR-RIS performance under imperfectCSI, hardware imperfections, and practical wireless channelmodels need further investigation. Further benefits of HR-RIS can be expected via channel estimation using the active

13

components compared to the passive RIS case. The activecomponents were assumed to be ideal AF relaying withno processing delays similar to the passive reflection. Thepractical delays and use of DF relaying need to be addressedin future works.

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