Hybrid Beamforming for Reconfigurable Intelligent Surface based Multi-user Communications: Achievable Rates with Limited Discrete Phase Shifts

1 Hybrid Beamforming for Reconfigurable Intelligent Surf ace base d Multi-user C ommunications: Achiev able Rates with Limited Discrete Phas e Shifts Boya Di, Member , IEEE , Hongliang Zhang , Member , IEEE , Ling yang Song, F ellow , IEEE , Y ongh u i Li, F ellow , IEEE , Zhu Han, F ellow , IEE E , and H. V incent Poor, F ellow , IEEE Abstract Reconfigura ble intelligent sur face (RIS) has drawn con sid e rable atten tio n fro m the research society recently , which creates fav orab le propag ation con ditions by contro llin g th e pha se shifts of the r eflected wa ves at the surface, thereby enhancing wireless transmissions. In this paper, we study a downlink multi-user system where the transm ission fr om a mu lti-antenna base station (BS) to various users is achieved by the RIS reflecting the inciden t signals of t he BS towards th e users. Unlike mo st existing works, we consider the prac tica l case wher e only a limited numb er o f discrete phase shifts can be realized b y th e finite-sized RIS. Based on the reflection-do minated one-hop pr opagation model between the BS and users via the RIS, a hy brid beamforming scheme is propo sed and th e su m -rate m aximization problem is for mulated. Specifically , the contin uous digital beamfor ming an d discr e te RIS-based an alog beamfor ming are perform ed at the BS an d the RIS, resp ectiv ely , and an iterativ e algor ith m is designed to solve this problem. Both th eoretical analysis and numerical validations show th a t the RIS-based system can achieve a go od sum-rate perfo r mance by setting a reasonab le size of RIS and a sm a ll n umber of discrete pha se shifts. Index T erms Reconfigura ble intellige n t su r face, Hy brid beamf orming, Multi-user commu nications, Limited dis- crete phase shifts, Non- c on vex op timization Boya Di is with Depart ment of Electronics Engineering, Peking Uni versi ty , Beijing, China and with Department of Computing, Imperial Colle ge L ondon, London, UK (email: diboya 92@gmail.com). Honglian g Zhang is with Department of Electroni cs Engineering, Peking Univ ersity , Beiji ng, China and with Departmen t of Electroni cal and Computer Engineeri ng, Uni ver sity of Houston, Houston, T X, USA (email: diboya92@g mail.com). Lingyang Song is with Department of Electronics Engineering, Peking Uni versity , Beijing, China (email: lingyang.song@pk u.edu.cn). Y onghui Li is with School of Electrical and Information Engineering, the Uni versity of Sydney , Australia (yonghui.li@ sydney .edu.au). Zhu Han is with Department of Electronic al and Computer Engineeri ng, Univ ersity of Houston, Houston, T X, USA (zhan2@uh.edu). H. V incent Poor is with School of Engineering and Applied Science, Prince ton Univ ersity , NY , USA (poor@princet on.edu). 2 I . I N T R O D U C T I O N The past decade has wi tnessed an enormous increase i n the number of m obile de vices [1], triggering urgent needs for high-speed and seamless data services in future wireless systems. T o meet such dem ands, one fundamental issue is how to improve th e li n k quality in th e comp l icated time-varying wireless en vironments in volving unpredictabl e fading and st rong shadowing effects. V ar ious technologies ha ve been de veloped such as relay [2] and massive mul t iple input and multiple output (MIMO) [3], aimin g to active ly strengthen the target s i gnals by forwarding and taking advantage of mult i-path ef fects, respectively . Howe ver , these t echni ques require extra hardware impl em entation with in e vitable power consum ption and high compl exity for sig nal processing, and the quali t y of services is also not always guaranteed in harsh propagation en v i ronments. Recently , th e development of meta-surfaces [4] has g iv en rise to a new transmi ssion t echnique named reconfigurable intellig ent surface (RIS), whi ch sh apes the p ropagation en v i ronment i n to a desirable form by con trolling the electromagnetic respons e of m ultiple scatters [5]. Specifically , the RIS is an ultra-thin surface inl aid with mul tiple su b -wa velength scatters, i.e., RIS elements, whose el ectrom agnetic response (such as phas e shifts) can be controlled b y sim p l e programmable PIN diodes [6]. Based on the ON/OFF functions of PIN diodes, only a limited number of discrete phases shifts can be achiev ed by the RIS [7]. Instead o f scattered wav es emanated from t raditional antennas, t he sub -wa velength separation between adjacent RIS elements enables the refracted and reflected wa ves to be generated via su p erposition of incident wav es at the surface [8]. Benefited from such a programmable characteristic of mo lding the wa vefronts i nto desired shapes, the RIS serves as a part of reconfigurable propagati on en vironment such that the recei ved signals are directly reflected towards the receiv ers wi thout any extra cost of power sources or hardwares [9], thereby im proving the link quality and coverage. T o exploit the potential of RIS techniques, m any existing works have considered the RIS as a reflection-type surface deployed between sources and destinations i n either point-to-point communication s [5], [10]–[12] or multi -user s ystems [13]–[16] for higher data rates or energy saving. In [11], a point-to-point RIS-assisted mu lti-input single-out s y stem has been inv estigated where the beamformer at the transmi t ter and continuous phase shifts of the RIS are jo intly optimized to m aximize t h e s u m rate. In [15], a channel estimation p rotocol h as been proposed 3 for a multi-us er RIS-assisted system and the cont inuous phase shifts ha ve b een d esigned t o maximize the minimu m user date. In [16], the autho rs have mini mized the transmi t power of the access point by opt imizing the continuous d i gital beamformin g and di screte phase shifts. An algorithm has been designed for t he singl e-user case and e xtended to th e multi-user case. Howe ver , two maj o r issues still remain to be further discussed in the op en literature. • F or th e m u lti-user cas e, how to determine the limited discr ete phase shifts dir ectly su ch that the i nter-user interfer ence can be elimina t ed? How does th e quantizati on level i nfluence t h e sum rate of the system? • Consideri n g the s t r engthened coupling between propa gation and discr ete phase shifts br ought by the dominant r eflection ray via th e RIS, how do we design the s ize of RIS and perf orm beamforming in a multi-antenna system to achie ve the maximum sum rate? T o address the abov e issues, in thi s paper , we consider a do wnlink multi-user multi-antenna system where the direct links between th e multi -antenna base stati on (BS) and users s u f fer from deep shadowing. T o pro vide high-quality data services, a RIS wit h li m ited di s crete phase shifts is deployed between the BS and users such t h at the si gnals sent by the BS are reflected by t he RIS towa rds the users. Since t h e incident wav es are reflected rather than scattered at th e RIS, it is t he reflection-based one-hop ray [17] via the RIS that dominates the propagation between the BS and users. Therefore, the propagation m o del in the RIS-based system differ s from those for tradi t ional two-hop relays and one-hop direct links in M IMO syst em s, reve aling an inner connection between the ph ase shifts and the propagation pat h s . T o achieve better directional reflection rays towards the desired users, it is vitally im portant to determine t he phase shifts of all RIS elements, the process of which is al s o known as the RIS configuration . Such built-in programmabl e configuration [5] is actually equiv alent to analog beamforming, realized by the RIS inherently . Since the RIS elements do not have any digi t al processing capability , we consid er the hybrid beamforming (HBF) [18] consist ing of the digital beamforming at the BS and RIS configuration based analog b eamforming. A novel HBF scheme for RIS-based communicatio ns with dis crete phas e shifts is thus required for bett er shapi n g t he propagation en vironment and sum rate maximization. Designing an HBF schem e presents se vera l majo r challenges. F irst , the refle ction-dom inated one-hop p rop agation and the RIS configuration based analog b eamform ing are coupled with each ot her , rendering the optimal scheme very hard to be obtained. The traditio nal b eamforming 4 schemes with separate channel matrix and analog beamform er do no t work any more. Second , discrete phase shifts required by the RIS renders the sum rate maximization to be a mixed integer programming problem, w h ich is non-trivial to be solved esp ecially i n th e comp lex domain. Thir d , giv en the dense placing of the RIS elements, t he correlatio n between elements may degrade the data rate performance. Thus, it is necessary to explore how the achiev able rate is influenced by the size of RIS, which is challengi n g due to the complicated propagation en vironments especially in a m ulti-user case. Through s o lving the above challenges, we aim t o design an HBF s cheme for the RIS- based mu l ti-user system with limited discrete phase shi ft s to maximi ze the sum rate. Our main contributions can be summarized below . • W e consider a downlink RIS-based multi-us er sys tem where a RIS with limited dis crete phase s hifts reflects signals from the BS tow ards various users. Giv en a reflection-based one-hop propagation model, we design an HBF scheme where th e digital beamforming is performed at th e BS, and the RIS- based analog beamforming is conducted at the RIS . • A mixed-integer sum rate maximization problem for RIS-based HBF is formulated and decomposed into two subproblems. W e propose an iterativ e algo rithm in which the di g ital beamforming subprobl em i s solved by zero-forcing (ZF) beamforming with power allocation and the RIS-based analog beamforming is solved by t he outer approximation. • W e prove th at t he proposed RIS-based HBF scheme can save as much as half o f t he radio frequency (RF) chains compared to traditional HBF schemes. Extending from our theoreti cal analysis on the pure Line-of-Sight (LoS) case, we revea l the influence of the size of RIS and the number of discrete p hase shifts on the sum rate both t heoretically and num erically . The rest of this paper is or ganized as follows. In Section II, we introduce the system model of the downlink RIS-based MU multi-antenna syst em. The frequency-response mo d el of the RIS and the channel m odel are deri ved. In Seciton III, the HBF scheme for t h e RIS-based system is propos ed. A sum rate m aximization problem is form ulated and decompos ed into two subproblems: digital beamform i ng and RIS configuration b ased analog beamforming. An iterative algorithm is developed in Section IV to solve the above two subproblems and a sub-optimal solution is obtained. In Section V , we com pare the RIS-based HBF scheme with the traditional one t heoretically , and discuss ho w to achiev e the maximum sum rate in t h e pure LoS case. The 5 complexity and con vergence of the proposed alg o rithm are also analyzed. Numerical result s in Section VI ev aluate the performance o f the proposed algo rithm and validate o u r analysis. Finally , conclusions are dra wn in Section VII. Notations: Scalars are denoted by itali c letters, vectors and matrices are denoted by b old-face lower -case and uppercase letters, respecti vely . For a complex-v alued vector x , k x k denotes its Euclidean norm, and diag ( x ) denotes a diagonal matrix whose diagonal element is the corresponding element in x . For a square m atrix S , Tr ( S ) denotes its trace. For any general matrix M , M H and M T denote it s conju g at e transpose and transpose, respectively . I , 0 , and 1 denote an identity matrix , all-zero and all-one vectors, respectively . I I . S Y S T E M M O D E L In t his section, we first introduce the RIS-based downlink multi-user multi-antenna syst em in which t h e BS with mu l tiple antennas serves var ious si ngle-antenna users via th e RIS such that the propagation en vironment can be pre-designed and configured to optimize the sys tem performance. The discrete phase shift m odel of RIS and the channel model are th en constructed, respectiv ely . A. Scenario Description Consider a d ownlink multi-user commun i cation system as sh own in Fig. 1 where a BS equipped with N t antennas transmits to K single-antenna users. Due to t he complicated and dynamic wireless en vironm ent in v olving unexpected fading and potential o b stacles, the BS - user l ink may not be s table enough or ev en be in out age. T o all e viate t h is iss u e, we consider to deploy an RIS between the BS and users, which reflects the signals from the BS and directly projects to the u sers by actively shaping the propagati on en viron ment into a desirable form. An RIS consists of N R × N R electrically controlled RIS elements as shown in Fig. 1, each of which i s a sub -wav elength meta-material particle with very small features. An RIS controller , equipped with se veral PIN diodes, can control the ON/ OFF state of the connection b etw een adjacent metal plates where the RIS element s are lai d , thereby m anipulating the electromagnetic response of RIS elements t ow ards incident wa ves. Due to t hese built-in programmable elements , the RIS requi res no extra active power sources nor do not hav e any signal processi ng capability such as decodi ng [5]. In ot h er words, it serves as a l ow-c ost reconfigurable phased array t hat 6 Base St ation User 1 User 2 User K RIS controlle r RIS element Incide nce ray c Reflec tion ra y Mult i-path scatte ring Fig. 1. System model of the RIS-based do wnlink multi-user communication system only relies on the combi nation of m ultiple prog ram m able radiating elements to realize a desired transformation on the transmitted, receiv ed, or refl ected wa ves [9]. B. Reconfigurable Intelligent Surface with Limited Discr ete Phase Shi fts As sho wn i n Fig. 2, t he RI S is achie ved by the b -bit re-programmable meta-material, which has been implemented as a set of radiative elements layered on a guidin g structure foll owing the wav e gu ide techni q ues, forming a 2-dim ens ional (2D) planar ant enn a array [7]. Being a miniature radiative element, t he field radiated from t h e RIS element has a phase and amplitu de determined by this element’ s pol arizability , whi ch can be tuned by the RIS control l er via mul tiple PIN diodes (ON/OFF) [5]. Howe ver , the phase and amplitu d e introdu ced by an RIS element are not generated randomly; instead they are constrained by t h e Lorentzian resonance response [4], which greatly li mits the range of phase v alues. Based on such constraint s, one common manner of implement ation is to constrain the amplitude and sample the ph ase v alues from the finite feasible set [4] su ch t hat th e v oltage-controll ed diodes can easil y m anipulate a d i scr ete set of phase values at a ve ry low cost. 7 RIS el ement Design diagram Fig. 2. Schematic structure of b -bit encoded RIS Specifically , we assume t hat each RIS element is encoded by the controller (e.g., v ia PIN diodes) to conduct 2 b possible ph ase shifts to reflect t he radio wa ve. Due to the frequency- selectiv e nature of the meta-materials, these elements only vibrate i n resonance with the in coming wa ves over a narro w band centerin g at the resonance frequency . W itho ut loss o f generality , we denote the frequency response of each element ( l 1 , l 2 ) at the l 1 -th row and l 2 -th column of the 2D RIS within the considered frequency range as q l 1 ,l 2 , 0 ≤ l 1 , l 2 ≤ N R − 1 . Since the RIS is b -bit controllable, 2 b possible configuration modes (i.e., ph ases) of each q l 1 ,l 2 can be defined according to the Lorentzian resonance respons e [4]. q l 1 ,l 2 = j + e j θ l 1 ,l 2 2 , θ l 1 ,l 2 = m l 1 ,l 2 π 2 b − 1 , m l 1 ,l 2 ∈  0 , 1 , . . . , 2 b − 1  , 0 ≤ l 1 , l 2 ≤ N R − 1 , (1) where θ l 1 ,l 2 denotes the p hase shift of RIS element ( l 1 , l 2 ) . For con venience, we refer to b as the number of quantization bits . Since the 2D RIS is constructed based on the u ltra-thin wa ve guides, the propagation ins ide th e 2D RIS is influenced b y both the locatio n and the wav e-number of each meta-surface element. The latter one is also a function of the frequency , which reflec ts the frequency-selectiv e nature of RIS. Denote t he posi t ion of the element ( l 1 , l 2 ) as p l 1 ,l 2 . The propagati on in troduced by this element can then be give n by e − j β l 1 ,l 2 ( λ ) 2 π λ p l 1 ,l 2 , where β l 1 ,l 2 ( λ ) is the wa ve-number of RIS element ( l 1 , l 2 ) , and λ is the corresponding wave length. F or simplicity , we assume that 8 the wa ve-number β l 1 ,l 2 ( λ ) remains the sam e for all RIS element within t h e considered n arrow band 1 . C. Reflection-domin a ted Chan n el Model In this sub s ection, we m odel t h e chann el between antenna 0 ≤ n ≤ N t − 1 of t he BS and user k . Specifically , instead of the tradition al two-hop channel for relays [10], we us e the one-hop reflection ray to model the dominant channel between the BS and users via th e RIS which only passiv ely reflects the receiv ed sign als . The key reason can be detailed below . Due to s mall spacing between adjacent RIS elements (usu ally much less than the wa velength), the signals projected onto the surface are no longer just scattered rando mly into the open space li ke those signals sp read by the t radi t ional antennas. Instead, the sup erpos ition of spherical wa ves facilitated by a number of miniature scatters enables refracted and reflected wa ves [8] witho ut any extra decoding or signal forwarding procedures. Therefore, unlike the tradit ional s cattering- based propagation model where signals trav el independently along the BS - RIS and RIS - user paths, in the considered scenario signals are only passiv ely reflected by the RIS along the reflection-based path due t o the coup ling effect of RIS elements. Moreover , benefited from the directio n al reflections of t he RIS, the BS - RIS - user link is usually stronger than other multi paths as well as the degraded direct link between the BS and the user [17]. Therefore, w e model th e channel between the BS and each u s er k as a Ricean model such that the BS - RIS - user l ink acts as the do m inant “LoS” compon ent and all the other paths together form the “non-Lo S (NLOS)” component. Specifically , let D ( n ) l 1 ,l 2 and d ( k ) l 1 ,l 2 denote t he dis tance between antenna n and RIS element ( l 1 , l 2 ) , and that between user k and RIS element ( l 1 , l 2 ) , respectively . The “LoS” chann el between the signal transmitt ed by the BS at antenna 1 ≤ n ≤ N t to user k vi a RIS element ( l 1 , l 2 ) can be giv en by h ( k ,n ) l 1 ,l 2 = h D ( n ) l 1 ,l 2 + d ( k ) l 1 ,l 2 i − α · e − j β l 1 ,l 2 ( λ ) 2 π λ h D ( n ) l 1 ,l 2 + d ( k ) l 1 ,l 2 i , (2) where α is the path loss p arameter . T h erefore, the channel m odel b et w een each antenna n of 1 This model can be easily extended to a frequency-selecti ve case where β l 1 ,l 2 ( λ ) varies with the working frequency . The propagation can then be modelled by a filter with finite impulse r esponse [9]. 9 Fig. 3. Placement of the antenna arrays at the BS and the RIS the BS and user k vi a RIS element ( l 1 , l 2 ) can be written by ˜ h ( k ,n ) l 1 ,l 2 = r κ 1 + κ h ( k ,n ) l 1 ,l 2 + r 1 1 + κ P L  D ( n ) l 1 ,l 2 + d ( k ) l 1 ,l 2  h ( k, n ) N LO S , ( l 1 ,l 2 ) , (3) where κ is t he Rician factor , P L ( · ) is the path lo ss model for NLOS t ransmissions , and h ( k ,n ) N LO S , ( l 1 ,l 2 ) ∼ C N (0 , 1) is the small -scale NLO S component. Here we assum e that the perfect channel state information is known to the BS via comm unicating with the RIS controller o ver a dedicated wireless link. A num b er of channel estimation methods can be found in [15], [16 ], which is o u t of the scop e of this paper . For the case where chann el information is partiall y known to the BS, we wi ll consider th e pure LoS transmission and di scuss it in detail in Section V . Geometric Model of the Pr opagation : W e now deriv e t he above distances D ( n ) l 1 ,l 2 and d ( k ) l 1 ,l 2 by using t he g eom etry informati on [19]. W ithout loss of generality , we assume that the uni form planar arrays (UP A) and uniform linear arrays (ULA) are deployed at the RIS and the BS, respectiv ely . The UP A is placed in a way that the surface is perpendicular to the ground . T o describe the geometry of the above ment i oned UP A and ULA, we employ t he spherical coordinates as shown in Fig. 3. For the RIS, we define t h e l ocal origin as the higher corner of the surface; while for th e BS, its local orig in is set as one end of t h e linear array . The y axis is set along the direction of BS antenn a 0 - elem ent (0 , 0) link, and the z axis is p erpendi cular to the ground. For con venience, the di rectio n s of t he ULA and RIS on the x - y plane can be depicted by the angles θ B ∈ [0 , 2 π ] and θ R ∈ [0 , 2 π ] , respectiv ely , as sho wn in Fig. 3. W e can 10 then define the principle directions of t he linear antenna arrays at the BS and the RIS, d eno t ed by n B and n n (1) R , n (2) R o , as n B = n x cos θ B + n y sin θ B , (4a) n (1) R = n x cos θ R + n y sin θ R , (4b) n (2) R = n z , (4c) where n x , n y , and n z are directions o f the x , y , and z axis , respecti vely . W e denote the u n i form s eparation between any two adjacent elements in the above defined ULA and UP A as d B and n d (1) R , d (2) R o , respectively . The position o f any antenna n at the BS can be represented by c ( n ) B = nd B n B , (5) and the p o sition of any RIS element ( l 1 , l 2 ) can be giv en by c ( l 1 ,l 2 ) R = l 1 d (1) R n (1) R + l 2 d (2) R n (2) R + D (0) 0 , 0 n y . (6) Based on (4) - (6) , the distance betw een ant enna n and RIS element ( l 1 , l 2 ) , i.e., D ( n ) l 1 ,l 2 , can be calculated as D ( n ) l 1 ,l 2 =    c ( l 1 ,l 2 ) R − c ( n ) B    2 (7a) =   l 1 d (1) R cos θ R − nd B cos θ B  2 +  l 1 d (1) R sin θ R + D (0) 0 , 0 − nd B sin θ B  2 +  l 2 d (2) R  2  1 2 (7b) ≈  l 1 d (1) R sin θ R + D (0) 0 , 0 − nd B sin θ B  +  l 1 d (1) R cos θ R − nd B cos θ B  2 +  l 2 d (2) R  2 2 D 0 0 , 0 , (7c) where (7 c ) is ob tained by ado p ting √ 1 + a ≈ 1 + a/ 2 when a ≪ 1 . Denote the positi on of each user k as c k = ( d x,k , d y , k , d z ,k ) . Th e distance between the RIS element and any user k can be expressed b y d ( k ) l 1 ,l 2 =    c ( l 1 ,l 2 ) R − c k    2 . (8) Since the d istance b etween any two antennas or RIS elements i s much s m aller t han the distance between the BS and the user , i.e., D ( n ) l 1 ,l 2 + d ( k ) l 1 ,l 2 ≫ d (1) R , d (2) R , d B , we assume th at the path loss 11 of each BS-user link i s the sam e, ignoring th e influence broug h t by di fferent antennas or RIS elements. Therefore, t h e channel propagation h ( k ,n ) l 1 ,l 2 can be re w ri t ten as h ( k, n ) l 1 ,l 2 = h D (0) 0 , 0 + d ( k ) 0 , 0 i − α e − j β l 1 ,l 2 ( λ ) 2 π λ h D ( n ) l 1 ,l 2 + d ( k ) l 1 ,l 2 i , (9) where D ( n ) l 1 ,l 2 and d ( k ) l 1 ,l 2 are given above in (7 ) and (8) . Based on the propagation characteristics introduced above, we will in vestigate how RIS can be util ized to assist multi-user t ransmissions in the following section. I I I . R I S - B A S E D H Y B R I D B E A M F O R M I N G A N D P RO B L E M F O R M U L A T I O N F O R M U LT I - U S E R C O M M U N I C A T I O N S Note that the RIS usually consis t s of a large numb er of RIS elements, which can be viewed as antenna elem ents fay aw ay from t h e BS, inherently capable of realizing analog beamforming via RIS configuration. Howe ver , these RIS elements d o not have any dig i tal processing capacity , requiring signal processing to be carried out at t he BS. In thi s section, to realize reflected wa ves towa rds preferable directions, we present an HBF scheme for RIS-based multi-us er communications gi ven the phase sh i ft mo d el and the channel model of RIS-based transmis sions in Section II. As shown in Fig. 4, the digital beamformin g is performed at th e BS while the analog beamform ing is achiev ed by the RIS with discrete phase shifts. Based on the considered HBF scheme, we form ulate a s u m rate maximi zation prob lem, and t hen decompose it into th e digit al beamforming s u bproblem and the RIS configuratio n based analog beamforming subproblem. A. Hybrid Beamforming Scheme 1) Digital Beamformi ng at the BS: The BS first encodes K diffe rent data steams via a di g i tal beamformer , V D , of size N t × K , satisfying N t ≥ K . After up -conv erting the encod ed signal over th e carrier frequency and allo cating t he transmit powers, the BS sends users’ sign als di rectly through N t antennas. D eno t e the intended signal vector for K users as s ∈ C K × 1 . The transm itted signals of th e BS can be given by x = V D s . (10) 12 Fig. 4. Block diagram of the RIS-based transmission between the B S and user k . 2) RIS Configuration based Analog Beamforming: After travelling through the reflection- dominated channel introduced in Section II-C, the recei ved signal at t he antenna of user k can then be expressed as z k = X n X l 1 ,l 2 φ ( k, n ) l 1 ,l 2 h ( k, n ) l 1 ,l 2 q l 1 ,l 2 V D k,n s k + X k 6 = k ′ X n X l 1 ,l 2 φ ( k, n ) l 1 ,l 2 h ( k, n ) l 1 ,l 2 q l 1 ,l 2 V D k ′ ,n s k ′ + w k , (11) where w k ∼ C N ( 0 , σ 2 ) is the additive whit e G aus sian noise and V D k,n denotes the k -th elem ent in row n of matrix V D . In (11) , φ ( k, n ) l 1 ,l 2 denotes the reflection co-ef ficient of the RIS element ( l 1 , l 2 ) with respect to the transmitt ing antenna n and user k . In practice, it is a function of the i ncidence and reflection angels, but here witho u t los s of generality we assume th at φ ( k ,n ) l 1 ,l 2 = φ ( k ) , ∀ n, l 1 , l 2 . W e ignore the couplin g between any two RIS elements here for simpli city , and t h us the receiv ed signal of each user k comes from the accumulated radiations of all RIS elements, as shown in (11) . This is a comm o n assum ption widely used i n the literature on both meta-surfaces [4] and traditional antenna arr ays [20]. 3) Received Signal at the User: For each user k , after it recei ves the signal z k , it do wn con ver ts the signal to the baseband and then recover s the final signal. The whole transmissio n model of K users can be formulated by ˜ z k = F V D s + w , (12) where w = [ w 1 , · · · , w K ] T is the noise vector . The transmi ssion matrix F i n (12) is defined as F = X l 1 ,l 2 q l 1 ,l 2 ( H l 1 ,l 2 ◦ Φ ) , (13) 13 where H l 1 ,l 2 and Φ are both K × N t matrices consisting of elements n h ( k ,n ) l 1 ,l 2 o and  φ ( k )  , respectiv ely . The notion ◦ implies the elem ent-by-element multipl i cation of two matri ces. B. Sum R a te Maximiz a tion Problem F ormulat ion T o explore how the H BF design influences the sum-rate performance, we e valuate the achiev- able data rates of all users in the RIS-based system. Based on (11) and (12 ) , we first rewrite the receive d sign al of user k in matrix form as z k = F H k V D ,k s k + X k ′ 6 = k F H k V D ,k ′ s k ′ | {z } inter-user interference + w k , (14) where F k and V D ,k denote th e k -th columns of matrices F and V D , respectiv ely . The achiev able rate of user k can then be giv en by R k = lo g 2    1 +   F H k V D ,k   2 P k ′ 6 = k   F H k V D ,k ′   2 + σ 2    . (15) W e aim to maxim i ze the achie vable rates of all users by opt i mizing the digital beamformer V D and the RIS configuration { q l 1 ,l 2 } , as formulated below: maximize V D , { q l 1 ,l 2 } X 1 ≤ k ≤ K R k (16a) sub j ect to T r  V H D V D  ≤ P T , (16b) q l 1 ,l 2 = j + e j θ l 1 ,l 2 2 , 0 ≤ l 1 , l 2 ≤ N R − 1 , (16c) θ l 1 ,l 2 = m l 1 ,l 2 π 2 b − 1 , m l 1 ,l 2 ∈  0 , 1 , . . . , 2 b − 1  , (16d) where P T is the total transmit power of the BS. C. Pr oblem Decompositi on Note that prob l em (16) is a mixed integer non-con vex opt i mization p rob lem which i s v ery challenging d u e to the large number of discrete variables { q l 1 ,l 2 } as well as the coupling bet w een propagation and RIS configuration based analog beamformi n g. Tra dition al analo g b eam form ing design methods with finite resolutio n phase shifters [18] may not fit well since it is non-trivial to decouple the transmis s ion m atrix F into the product of a channel matrix and a beamformer 14 matrix in ou r case. T o solve t h is p roblem ef ficiently , we decoup l e it i nto two subproblem s as shown below . 1) Digital Beamforming: Given RIS configuration { q l 1 ,l 2 } , the digital beamforming subprob- lem can be written by maximize V D X 1 ≤ k ≤ K R k , (17a) sub j ect to T r  V H D V D  ≤ P T , (17b) where F is fix ed. 2) RIS Config u ration based Analog Beamformin g : Based on constraint (16 c ) , th e RIS con- figuration subprobl em with fixed beamform er V D is equivalent to maximize { θ l 1 ,l 2 } X 1 ≤ k ≤ K R k , (18a) sub j ect to θ l 1 ,l 2 = m l 1 ,l 2 π 2 b − 1 , m l 1 ,l 2 ∈  0 , 1 , . . . , 2 b − 1  . (18b) In the next section , we will design two algorithms to solve these s ubproblems, respectively . I V . S U M R A T E M A X I M I Z A T I O N A L G O R I T H M D E S I G N In t his section, we will develop a sum rate m aximization (SRM) alg orithm to obt ain a suboptim al sol ution of problem (16) in Section III. Specially , we iterative ly so l ve subproblem (17) giv en RIS configuration { q l 1 ,l 2 } , and solve subproblem (18) giv en b eamform er V D . Finally , we will summarize the overall algorit hm and provide con ver gence and complexity analysis. A. Digi tal Beamforming Algori thm Subproblem (17) is a well-known digital b eamforming problem. According to th e results in [21], the ZF digital beamformer can obt ain a near optimal solution . Therefore, we consider ZF beamforming together with power allocation as the beamformer at the BS to alleviate the interference amon g users. Based on the results in [22], t h e beamformer can be giv en by V D = F H ( F F H ) − 1 P 1 2 = ˜ V D P 1 2 , (19) where ˜ V D = F H ( F F H ) − 1 and P is a diagon al matrix whose k -th diagonal element is the recei ved power at t he k -th user , i.e., p k . 15 Algorithm 1 Digi t al Beamformi n g Algo ri t hm 1: Solve power allocation problem (21); 2: Obtain t he opti mal power allo cation result (22); 3: Derive the beamformer matrix from th e optim al p ower allocation b ased on (19); In the ZF beamforming, we have the following constraints: | F H k ( V D ) k | = √ p k , | F H k ( V D ) k ′ | = 0 , ∀ k ′ 6 = k . (20) W i th these const raints, su b problem (17) can be reduced t o the following p ower allocation problem: max { p k ≥ 0 } X 1 ≤ k ≤ K log 2  1 + p k σ 2  , (21a) sub j ect to T r  P 1 2 ˜ V H D ˜ V D P 1 2  ≤ P T . (21b) The opti mal solution of this problem can b e obtained by water-fi lling [23] as p k = 1 ν k max  1 µ − ν k σ 2 , 0  , (22) where ν k is the k -th diagonal element of ˜ V H D ˜ V D and µ is a normali zed factor which i s selected such that P 1 ≤ k ≤ K max { 1 µ − ν k σ 2 , 0 } = P T . The algorithm can be s ummarized in Algorithm 1. B. RIS Configu ration based Analog Beamformi n g A l gorithm Since we iterate between t h e dig ital beamformi ng and RIS configuration based analog beam- forming, the l atter can be opt imized assuming ZF precoding as shown in (19). Since the data rate with ZF precoding i n (21) onl y depends on the RIS configuration through the power const raint (21b), the RIS configurati on based analog beamformin g problem can be reformulated as a power minimizati on problem : min θ l 1 ,l 2 f ( F ) , (23a) sub j ect to θ l 1 ,l 2 = m l 1 ,l 2 π 2 b − 1 , m l 1 ,l 2 ∈ { 0 , 1 , . . . , 2 b − 1 } , (23b) where f ( F ) = Tr ( ˜ V D P ˜ V H D ) = T r ( P 1 2 ˜ V H D ˜ V D P 1 2 ) = Tr (( ˜ F ˜ F H ) − 1 ) . (24) 16 Here, ˜ F = P − 1 2 F . Since ˜ F ˜ F H is a symmetric, posit ive semi-definite m atrix, w e can transform this problem into a s em i-definite programming (SDP) problem. Let Tr (( ˜ F ˜ F H ) − 1 ) = T r ( w K I K ) . According t o Schur complement [25], t h e problem can b e rewritten by min θ l 1 ,l 2 ,w w , (25a) sub j ect to Z =   w K I K I K I K ˜ F ˜ F H    0 , (25b) θ l 1 ,l 2 = m l 1 ,l 2 π 2 b − 1 , m l 1 ,l 2 ∈ { 0 , 1 , . . . , 2 b − 1 } , (25c) where X  0 m eans that matrix X is a symmetri c and positive s emi-definite matrix. Remark o n the tractabi lity of the formulated problem : This problem is a mix-integer SDP , which is generally N P-hard. Moreover , any two discrete variables { θ l 1 ,l 2 } are coupled wi t h each other via constraint (25 b ) , which makes the prob l em ev en more complicated. One com m only used soluti on is to first relax the discrete variables into continuous ones and then round the obtained solution to s atisfy the discrete constraints . Howe ver , for the RIS-based systems, the typical value of the number of quantization bit s is usu ally very sm all (e.g., 2 or 3) such that the round-off method s will lead t o inevitable performance degrade. T o a v oid th e above i ssue, we consider to sol ve t h e SDP discretely . In t he following, we first present the following Proposition 1 to transform the n o n linear fun ctions in (25) with respect to θ l 1 ,l 2 into linear ones, as proved in Appendix A . W e then use the outer approximation method [26] to s olve this problem. Pr oposition 1. Let a =  − (2 b − 1) π 2 b − 1 , . . . , − mπ 2 b − 1 , . . . , mπ 2 B − 1 , . . . , (2 b − 1) π 2 b − 1  , c =  cos  − (2 b − 1) π 2 b − 1  , . . . , cos  − mπ 2 b − 1  , . . . , cos  mπ 2 b − 1  , . . . , cos  (2 b − 1) π 2 b − 1  , s =  sin  − (2 b − 1) π 2 b − 1  , . . . , sin  − mπ 2 b − 1  , . . . , sin  mπ 2 b − 1  , . . . , sin  (2 b − 1) π 2 b − 1  . W e intr oduce a bina ry vector x l 1 ,l 2 , wher e x l 1 ,l 2 i indicates whether θ l 1 ,l 2 = a i , and a binary vector y l 1 ,l 2 ,l 1 ′ ,l 2 ′ for p h ase differ ence ∆ θ l 1 ,l 2 ,l 1 ′ ,l 2 ′ = θ l 1 ,l 2 − θ l 1 ′ ,l 2 ′ . Ther efor e, p roblem (25) can 17 be r ewritten by min x l 1 ,l 2 , y l 1 ,l 2 ,l 1 ′ ,l 2 ′ ,w w , (26a) sub j ect to Z =   w K I K I K I K ˜ F ˜ F H    0 , (26b) k x l 1 ,l 2 k 1 = 1 , e T x l 1 ,l 2 = 0 , (26c) a T ( x l 1 ,l 2 − x l 1 ′ ,l 2 ′ ) = a T y l 1 ,l 2 ,l 1 ′ ,l 2 ′ . (26d) Her e, e is a cons tant vector whose first 2 B − 1 element s ar e 1 and others are 0. Problem (26) is a mix-integer SDP with li near constraints, which can b e solved by the outer approximation method. Th e basic idea of the outer approxim ation method is to enforce t he SDP const raint vi a linear cuts and transform the origi nal problem into a m ix-integer linear programming one, which can be s o lved by the b ranch-and-boun d alg orithm [27 ]. In the following, we will elaborate on how to enforce the SDP constraints via li near cuts. Assume that a solutio n is ¯ w , ¯ x l 1 ,l 2 , ¯ y l 1 ,l 2 ,l 1 ′ ,l 2 ′ . In most mix-integer programming probl ems, it is very com mon to u se the gradient cuts to approach th e feasible set. Howe ver , the fun ct i on of smallest eigen va lues is not always differentiable. Therefore, we use the characterization i n stead [26]. No te that Z  0 is equiv alent t o u T Z u ≥ 0 for arbitrary u . If Z with ¯ w , ¯ x l 1 ,l 2 , ¯ y l 1 ,l 2 ,l 1 ′ ,l 2 ′ is not posit iv e semi-definite, we compute eigen vector u ass ociated with the small est eigen value. Then u T Z u ≥ 0 (27) is a valid cut that cuts off ¯ w , ¯ x l 1 ,l 2 , ¯ y l 1 ,l 2 ,l 1 ′ ,l 2 ′ . The RIS configuration based analog beamformi ng algorithm can be summarized in Algorithm 2. C. Overall Algorit hm Description Based on t he result s presented in the previous two s ubsections, we propose an overall it erative algorithm, i.e., the SRM algorit h m , for solvin g the origin al probl em in an i t erativ e manner . Specially , the beamformer V D is solved by Alg o rithm 1 w h ile keeping the RIS configuration fixed. After obtaini ng the results, we will optimize the RIS configuration θ l 1 ,l 2 by Algorithm 2. Those obtained results are set as t he initi al sol ution for subsequent iterations . Define R as 18 Algorithm 2 RIS Configuration based Analog Beamformi n g Algorithm 1: Remove t h e semi-definite cons traint of problem (26) and sol ve an initial solution; 2: repe at 3: Use the branch-and-bound m ethod to solve problem (26) and obtain the opti mal solution x l 1 ,l 2 for each RIS element; 4: Check the fea sibil i ty; 5: If the obtained solution is no t feasible, add a cut according to (27); 6: until The obtained soluti on is feasible; 7: Derive the phase shits according t o the o b tained soluti on. the value of t h e objectiv e funct i on. T he two subproblems will be solved alternatively until i n iteration t the value d i f ference of the objective functions between two adjacent it eratio n s is less than a predefined threshold π , i.e., R ( t +1) − R ( t ) ≤ π . D. Con ver gence an d Comple xity Analysi s W e no w analyze the con ver gence and complexity of our proposed SRM algorithm. 1) Con ver gence: First, according to Algorith m 1, in the digital beaforming su b p roblem, we can obtain a better resul t given RIS configuration θ ( t ) in the ( t + 1) -th iteration. Therefore, we hav e R ( V ( t +1) D , θ ( t ) ) ≥ R ( V ( t ) D , θ ( t ) ) . (28) Second, given the beamforming resul t V ( t +1) D , we m aximize su m rate of all users, and thus , the following inequal i ty holds: R ( V ( t +1) D , Q ( t +1) ) ≥ R ( V ( t +1) D , θ ( t ) ) . (29) Based the abo ve inequalities, we can obtain R ( V ( t +1) D , θ ( t +1) ) ≥ R ( V ( t ) D , Q ( t ) ) . (30) which i m plies that the objective value of the o ri g inal prob lem is non-decreasing after each iteration of the SRM algorithm. Sin ce t h e objective value is upper b ounded, th e proposed SRM algorithm is guaranteed to con ver ge. 19 2) Complexity: W e consider the complexity of the p rop osed algorith ms for two subproblems separately . • In the digital beamforming subproblem, we need to o p timize the recei ved p ower for each user according to (22). Therefore, it s computatio nal com plexity is O ( K ) . • In the RIS configuration based analog beamforming s u bproblem, we solve a series o f linear programs by the branch-and-bound m ethod. Since only one element in x l 1 ,l 2 can be 1, x l 1 ,l 2 can hav e at m ost 2 b possible soluti ons. Thus, the scale of the comput at i onal complexity o f each linear program i s O (2 bN 2 T ) . V . P E R F O R M A N C E A NA L Y S I S O F R I S - B A S E D M U L T I - U S E R C O M M U N I C A T I O N S In t his section, we compare t he RIS-based H BF scheme with the traditional ones in terms of the minimum number of required RF chains. A special case, i.e., the pure Lo S transmission, is also cons idered to e xplore theoretically ho w the size of RIS and its placement influence the achie vable rates. A. Comparis on with T raditi onal Hybri d Beamforming W e adopt the fully digi tal beamforming scheme as a benchmark to com pare the traditional and RIS-based HBF schemes. In the traditional HBF , it has already been proved that when the number of RF chains is n o t smaller than twice the numb er of target data streams, any fully digital beamform i ng matrix can be realized [18]. Howe ver , in th e RIS-based HBF , the inherent analog beamforming (i.e., the RIS configuration) is clo sely coupled wi th propagation, which off ers more freedom for shaping the propagation en vironment than t h e traditional scheme. T o capture this characteristic, we explore a new condi tion for the RIS-based system to achieve full y digital beamform i ng. W e start by describing th e full y di gital beamformi n g schem e in an RIS-based system. Consider an ideal ca se where each RIS element d i rectly connects with an RF chain and ADC as i f i t is part of the BS 2 . The fully digital beamformer can then be denoted by V F D ∈ N 2 R × K , based on which we present the following propo sition, which wil l be proved in Appendix B. 2 In practice, the RIS does not connect to t he RF chain directly unless it is i nstall ed at the BS, which is not the truth i n our case where RIS actually only r efl ects signals. T herefore, we only consider such an ideal scheme as a benchmark to ev aluate the effe ctiv eness of RIS-based HBF 20 Pr oposition 2. F or the RIS-based HBF scheme with N 2 R ≥ K N t , to achieve a ny fully digital beamforming s cheme , the number of transmi t antennas at the BS sh ould n ot be smal ler th a n the number of single-antenna users, i.e., N t ≥ K . This impli es two conditions for the proposed RIS -based scheme to achiev e the fully digial beamforming s cheme. First, the s ize of RIS should be n o s maller than t h e product of th e number of users and the si ze of the antenna array at the BS. Second, the number of t ransmit antennas at the BS should be no s m aller than the num ber of sin g le-antenna users. Remark on dedicated har dwar e r eduction for anal og beamf o rming : In the tradit ional HBF , to offl oad part of the digital baseband processi ng to the analog domain, a number of RF chain s are required at the BS to feed t he analog beamformer , equipped with necessary hardwares such as m ixers, filters, and ph ase shi fters [28]. In contrast, as shown in Propositio n 2, the numb er of minimum RF chains requi red by the RIS-based HBF to achie ve the fully digi al beamforming has already b een reduced by half compared to the traditional schem e. Moreover , t he phase shifters can be saved s ince the RIS inherently realizes analog beamforming owning to its flexible phy s ical structure [29]. B. Special Case: P u r e Line-of-Sight T ransmissions W e consider th e data rate obtained by the pu re Lo S case as a lower bound of the achiev able rate and analy ze optimal RIS p lacement to provide o rthogonal communi cation links. The LoS case also reveals i nsights on how the achiev abl e data rate is influenced by RIS design and placement. Since a multit ude of RIS elements are placed in a s ub-wa velength order , spacial correlation between these elements are inevitable. In th is case, the chann el matrix F approaches a low-rank matrix, leading to a degraded performance in t erm s of the achiev able data rate. Traditionally , we can u tilize the mu l ti-path effect to decorrelate di f ferent channel l inks between the t ranscei ver antennas. Howe ver , i t is also important to und erst and ho w the system works in the pure LoS case, especially when it comes to th e RIS-based systems where the reflection-based one-hop link between the BS and users acts as the do m inated “LoS” component and is us u ally much stronger than other multi-paths as well as the degraded direct lin ks. In the pu re LoS case, we aim to achie ve a high-rank LoS channel matrix by desi g ning the 21 size of RIS and t he antenna array at the BS. Different from those existing works on antenna array design for LoS MIMO system s , the discrete phase shift s depicted by { q l 1 ,l 2 } needs to be considered in the RIS-based propagation. W e thus present the results in Proposition 3 below , which is prov ed i n Appendi x C. Pr oposition 3. In the RIS-based system, t o make differ ent links between the BS a n d user k via the R IS orthogonal to each other , the RIS d esign should sat i sfy the following cond i tions: d (1) R d B = λD (0) 0 , 0 N R cos θ R cos θ B , (31a) N R − 1 X l 2 =0 (1 + sin θ l 1 ,l 2 ) = N R − 1 X l 2 =0  1 + sin θ l ′ 1 ,l 2  , ∀ 0 ≤ l 1 , l ′ 1 ≤ N R − 1 . (31b) This proposit i on s hows that the achiev able data rate is h i ghly related to the size of the RIS and its placement. For con venience, when all other parameters are fixed, we refer to the t hr eshold of the RIS size as N th R = λD (0) 0 , 0 d (1) R d B cos θ R cos θ B . (32) For the pure LoS case, the sum rate maximization problem can still be formul at ed as (16) wit h one extra constraint (31 b ) . Our proposed SR M algorithm can be u tilized to solve t his problem after we con vert the extra constraint (31 b ) into a linear one sh own below by following the transformations in Proposition 1, N r − 1 X l 2 =0 ( x l 1 ,l 2 − x l 1 ′ ,l 2 ) s T = 0 . (33) V I . S I M U L A T I O N R E S U L T S In this section, we e valuate the performance of our prop osed algorithm for RIS-based HBF in t erms of the su m rate. W e sho w how the sy stem performance is influenced by the SNR, number of users, the si ze of RIS, and the number of quantization bits for discrete phase shi fts. For comparison, the fol lowing algorith m s are performed as well. • Simul ated an nealing : W e utilize the simulated annealing m ethod [30] to approach the gl obal optimal solut i on of the sum rate maximi zati o n problem with discrete phase shifts. The maximum numb er of iterations is set as 10 7 . 22 0 5 10 15 20 Antenna separation at the BS d B (m) 0 5 10 15 20 25 30 35 40 Sum rate (bits/s/Hz) b = 2 b = 3 b = 4 b = 8 Fig. 5. Sum rate v .s. antenna separation at the BS (SNR = 2 dB, N t = K = 5 , N R = 6 ) • Pur e LoS case : W e cons i der the pure LoS case as a lower bound to ev aluate the performance of the HBF s chem e. The proposed i terativ e SRM algorit hm can s till be utilized to solve the corresponding opti mization problem . • Random phase s hift : Algorit h m 1 is first performed, fol lowed b y a random algorit hm to solve the RIS-based analog beamformi n g subproblem (18) . • RIS-based HBF with continuous pha s e shif ts : This scheme only serves as a benchmark when we in vestigate ho w t he discreteness l evel (i.e., the number of quantization bits, b ) influences the sum rate. The discrete constraint in the original subp roblem (1 8) is relaxed to θ l 1 ,l 2 ∈ [0 , 2 π ] . The HBF sol ution is obtained by iteratively performing Algorit hm 1 and the gradient descent m ethod. In our simulatio n , we set t h e d i stance between the BS and the RIS, D (0) 0 , 0 , as 20m, and users are randomly depl oyed within a half circle of radius 60 m centering at the RIS . The antenna array at the BS and the RIS are placed at ang l es o f 15 ◦ and 3 0 ◦ to the x axis, respectiv ely . The transmit power of t h e BS P T is 20 W , t he carrier frequency i s 5.9 GHz, the ant enna separation at t he BS d B is 1 m, the RIS element separation d R is 0.03 m, and the Rician fading parameter 23 0 10 20 30 40 50 60 70 Size of the RIS N R 0 20 40 60 80 100 120 Sum rate (bits/s/Hz Proposed algorithm Pure LoS case Random algorithm Fig. 6. Sum rate v .s. size of RIS N R (SNR = 2dB, K = N t = 5 , b = 2 ) κ is 4 [31 ]. W e set the si ze of the RIS N 2 R ranging b etw een 5 2 ∼ 65 2 , the number of antennas at the BS N t and the number of u s ers K between 5 ∼ 15, t h e discreteness le vel o f RIS b between 1 ∼ 5 , and t he SNR (defined as P T /σ 2 ) between -2 dB ∼ 10 dB. Specifically , for the LoS case, giv en the above parameters the designi n g rule in (31 a ) is onl y satisfied when the size of RIS is set as 40 or the antenna separation at the BS is set as 6.75 m. W e present Fig. 5 t o verify Proposition 3 in t he pure LoS case. The figure shows the sum rate of all users versus the antenna separation at the BS, d B , with different numbers of quantization bits in the pure LoS case. Given the parameters set above, according to Proposition 3, the optimal value of d B should be 6.75 m so as to orth o gonalize the channel links between each antenna of the BS and a user k via any RIS elem ent . W e observe that the op t imal sum rate can be achiev ed when d B is around 5 ∼ 6 . 75 m. Th e num erically op t imal va lue of d B approaches the theoretical result as the number of quant ization bits, b , grows, impl ying that such fluctuatio n around the optimal value comes from the dis crete phase shi ft s of RIS. When b i s large enough, i.e., b = 8 in Fig. 5, the optimal d B equals 6.75 m , which justifies Proposi tion 3. 24 1 2 3 4 5 Number of quantization bits b 50 55 60 65 70 75 80 85 Sum rate (bits/s/Hz) Discrete phase shifts (N R = 6) Continuous phase shifts (N R = 6) Discrete phase shifts (N R = 10) Continuous phase shifts (N R = 10 Fig. 7. Sum rate v .s. number of quantization bits b (SNR = 2dB, K = N t = 5 , N R = 6 ) Fig. 6 shows the sum rate of all users versus the size of RIS 3 with b = 2 , N t = K = 5 . W e observe that the sum rate grows rapidly with a s m all size of RIS and gradually flattens as the size o f RIS continues to i ncrease 4 . The inflection poi n t of each curve shows up around N R = 40 , wh i ch verifies t h e threshold (32) given by Proposi t ion 3 very well. When the RIS size N R exceeds 40, though Proposition 3 does not hold, the sum rate does not drop since the RIS can always t u rn off those extra RIS elements to maintain the sum -rate performance. This figure also implies t hat though Proposit i on 3 is obtained in the pure LoS case, it also sheds insight into the RIS placement and array design in a more general case wi th small-scale fading. Moreover , Fig. 6 also shows that t h e performance of RIS-based beamforming wi th small-scale fading is much better than that of the pure LoS case when the size of RIS is small. Such a gain comes from the reduced correlation between differe nt channel links owning to m ulti-path effects. As the size o f RIS grows, Proposition 3 is satisfied such that the channel links in the pure LoS case are orthogonalized, m aking the gap b et ween these two cases sm aller . 3 For con venience, here we adopt N R to represent the size of RIS to better display t he curves. 4 W e do not sho w the simulated annealing algorithm in this figure due to its high complexity wi th a l arge size of RIS. 25 -2 0 2 4 6 8 10 SNR (dB) 40 50 60 70 80 90 Sum rate (bits/s/Hz) Simulated annealing Proposed algorithm Random algorithm (a) 4 6 8 10 12 14 16 18 Number of users K 40 60 80 100 120 140 160 180 Sum rate (bits/s/Hz) Simulated annealing Proposed algorithm Random algorithm (b) Fig. 8. a) Sum rate v .s. S NR ( K = N t = 5 , b = 2 , N R = 6 ); b) Sum r ate v .s. number of users (SNR = 2 dB, N t = K , b = 2 , N R = 6 ) Fig. 7 depicts the sum rate of all users versus the number of quantization bits b for discrete phase shifts i n RIS configuration with SNR = 2 dB, N t = K = 5 , and N R = 6 . As the nu mber of quantization bi ts increases, the s um rate obtained by our proposed alg orithm wit h discrete ph ase shifts app roaches that in the conti n u ous case. When the size of RIS gro ws, the gap between the discrete and continuous cases shrinks since a larger RIS usually provides more freedom of generating directional beams. Not e that the implem ent ation d iffi culty increases dramatically i n practice with t he number of quantization bits. A trade-of f can then be achie ved between the s u m rate and number of quantization bit s. Fig. 8(a) and Fig. 8(b) show the sum rate of all users versus SNR and the number of users, respectiv ely , obtained by different algorithms with a RIS of size 6 × 6 (i.e., N R = 6 ), b = 2 quantization bits for phase shifts, equal num ber of transm it antennas at t he BS and the downlink users. In Fig. 8(a), the s um rate increases with SNR since more po wer resources are allocated by the BS. In Fig . 8(b), the sum rate grows with the number of users s ince a higher diversity gain is achieve d. From both figures, we observe that the performance of our proposed algorithm is close to t hat of t h e simulated annealing method and much better than the random algorithm . This indi cates the ef ficiency o f our proposed alg o rithm to s olve the RIS- based HBF problem. 26 V I I . C O N C L U S I O N S A N D D I S C U S S I O N In this paper , we have studied a RIS-based downlink multi-us er multi -antenna system in the absence of di rect links between the BS and users. Th e BS transmit s signals to users via the reflection-based RIS with limited discrete phase shifts. T o better d epi ct the close coupling between channel propagation and the RIS configuration p attern selection, we haved considered a reflection-domin ated one-hop propagation m odel between the BS and users. Based on this model, we have carried ou t an HBF scheme for su m rate maximi zati o n where the con t inuous digital b eamform ing has been performed at the BS and the discrete analog beamform i ng has been achie ved inherently at t he RIS via configuration pattern selection. The sum rate maxi mization problem has b een decomposed into two subprobl em s and solved iterativ ely by our proposed SRM algorithm . Three remarks can be drawn from the theoretical analy s is and numerical results, providing insights for RIS -based system design. • The sum rate of th e RIS-based system with discr ete pha se shift s incr eases rapidly when the number of quant ization bi t s b i s small , an d g radually appr oaches the s um rate achiev ed in the continuous case if b is lar ge enoug h. • The s um rate incr eases with the si ze of R IS and con ver ges to a stabl e value as the s ize of RIS gr ows to re ach the th reshold determined by Pr op osition 3. • The minimum num b er of transmit antennas at the BS r equir ed to achieve any f ully digital beamforming scheme is only half of tha t in traditional HBF schemes, implying t h at the RIS-based HBF sc heme can g r eatly r educe the cost of d edicated har dwar e. The above remarks h a ve indicated that when designing the RIS-based systems, a moderate size of RIS and a very small number of quanti zation bits are enough to achiev e the satisfying sum rate at lo w cost. A P P E N D I X A P R O O F O F P RO P O S I T I O N 1 Note that ˜ F ˜ F H = P − 1 2 X l 1 ,l 2 q l 1 ,l 2 ( H l 1 ,l 2 ◦ Φ ) X l 1 ′ ,l 2 ′ q H l 1 ′ ,l 2 ′ ( H l 1 ′ ,l 2 ′ ◦ Φ ) H P − 1 2 = X l 1 ,l 2 q l 1 ,l 2 X l 1 ′ ,l 2 ′ q H l 1 ′ ,l 2 ′ P − 1 2 ( H l 1 ,l 2 ◦ Φ )( H l 1 ′ ,l 2 ′ ◦ Φ ) H P − 1 2 27 = X l 1 ,l 2 X l 1 ′ ,l 2 ′ ( j + e j θ l 1 ,l 2 )( − j + e − j θ l ′ 1 ,l ′ 2 ) 4 A l 1 ,l 2 ,l 1 ′ ,l 2 ′ = X l 1 ,l 2 X l 1 ′ ,l 2 ′ A l 1 ,l 2 ,l 1 ′ ,l 2 ′ 4  cos( θ l 1 ,l 2 − θ l 1 ′ ,l 2 ′ ) + j sin( θ l 1 ,l 2 − θ l 1 ′ ,l 2 ′ )  + X l 1 ,l 2 X l 1 ′ ,l 2 ′ A l 1 ,l 2 ,l 1 ′ ,l 2 ′ 4  cos( θ l 1 ′ ,l 2 ′ ) + cos ( θ l 1 ,l 2 ) + 1  = X l 1 ,l 2 X l 1 ′ ,l 2 ′ A l 1 ,l 2 ,l 1 ′ ,l 2 ′ 4  cos(∆ θ l 1 ,l 2 ,l 1 ′ ,l 2 ′ ) + j sin(∆ θ l 1 ,l 2 ,l 1 ′ ,l 2 ′ )  + X l 1 ,l 2 X l 1 ′ ,l 2 ′ A l 1 ,l 2 ,l 1 ′ ,l 2 ′ 4  cos( θ l 1 ′ ,l 2 ′ ) + cos ( θ l 1 ,l 2 ) + 1  , (34) is not l inear with respect to θ l 1 ,l 2 . T aking t he advantage of the di screte p rop erty of θ l 1 ,l 2 , we can further transform the non-linear functions into linear ones. W i th the definition s of x l 1 ,l 2 , we have cos( θ l 1 ,l 2 ) = x l 1 ,l 2 c T , sin( θ l 1 ,l 2 ) = x l 1 ,l 2 s T , (35) with k x l 1 ,l 2 k 1 = 1 . It is also worthwhile to p o i nt out t hat the v alue of θ l 1 ,l 2 only falls in the range [0 , 2 π ) , and thus, we have e T x l 1 ,l 2 = 0 . (36) Similarly , according t o the definitions of y l 1 ,l 2 ,l 1 ′ ,l 2 ′ , we ha ve cos(∆ θ l 1 ,l 2 ,l 1 ′ ,l 2 ′ ) = y l 1 ,l 2 ,l 1 ′ ,l 2 ′ c T , sin(∆ θ l 1 ,l 2 ,l 1 ′ ,l 2 ′ ) = y l 1 ,l 2 ,l 1 ′ ,l 2 ′ s T , (37) with a T ( x l 1 ,l 2 − x l 1 ′ ,l 2 ′ ) = a T y l 1 ,l 2 ,l 1 ′ ,l 2 ′ . (38) W i th these transformati ons, ˜ F ˜ F H is linear with respecti ve to x l 1 ,l 2 and y l 1 ,l 2 ,l 1 ′ ,l 2 ′ . A P P E N D I X B P R O O F O F P RO P O S I T I O N 2 W e first prove th at wh en N t = K , any ful ly digital b eamform ing schem e can be achieved by the RIS-based HBF scheme if N 2 R ≥ K N t holds. W e t hen state that it is not possi ble to achieve digital beamform i ng when N t < K . Therefore, N t ≥ 2 K is a s uffi cient conditi on. 28 i) Denote the channel matrix between the RIS and t h e users as H F D ∈ K × N 2 R . Since N t = K , the k th colum n o f th e digital beamformer can be expressed by V D ,k =  0 T , v k ,k , 0 T  T , where v k ,k is the k th element of the the k th column in V D ∈ R N t × K . T o satisfy H F D V F D = F V D , we hav e [ . . . , f m,k , . . . ]           0 . . . v k ,k . . . 0           = ( H F D V F D ) m,k , (39) i.e., X l 1 ,l 2 1 2 φ ( k ) h  j + e j θ l 1 ,l 2  g m,k l 1 ,l 2 i v k ,k = ( H F D V F D ) m,k , (40) for all 0 ≤ m, k ≤ K , where f m,k is the element of m atri x F . Note that the term m ultiplied by v k ,k in ( 4 0) can achiev e di f ferent magnit udes owning to th e linear combinati on of channel coef ficients, which is different from the tradi t ional HBF [20]. At least on e soluti on can be foun d for this set of equations if the nu m ber of equatio ns is no s maller than the nu mber of variables, i.e., N 2 R ≥ K N t . This also holds for N t > K s i nce we can always use the solu tion for N t = K and set t hose extra variables t o be zero. ii) W e ob serve that r ank ( H F D V F D ) = K and r ank ( FV D ) = min { K , N t } . If N t < K , then r ank ( H F D V F D ) > r ank ( FV D ) , implyi ng that the RIS-based HBF cannot im plement the ful l y digital beamform i ng scheme. This completes the proof. A P P E N D I X C P R O O F O F P RO P O S I T I O N 3 Note that each user k goes throu gh different path loss es due to various posi tions, which natu- rally varies the corresponding chann el coefficients. Therefore, we only focus on orthogo nalizing diffe rent link s from t he BS to the same us er . Since th e path lo s s is the sam e for d iffe rent links with respect to user k , we consider the channel response between one BS antenna n and all RIS elements as f ( n,k ) =  q 0 , 0 e − j 2 π λ  D ( n ) 0 , 0 + d ( k ) 0 , 0  , q 0 , 1 e − j 2 π λ  D ( n ) 0 , 1 + d ( k ) 0 , 1  , · · · , q N R − 1 ,N R − 1 e − j 2 π λ  D ( n ) N R − 1 ,N R − 1 + d ( k ) N R − 1 ,N R − 1   . (41) 29 T o keep any two dif ferent links orth ogonal to each o ther , the following condition shoul d be satisfied, h f ( n a ,k ) i H · f ( n b ,k ) = 0 , ∀ n a 6 = n b , (42) where n a and n b denote two different transmit antennas. By subst i tuting (41) , (42) can b e rewritten by N R − 1 X l 2 =0 N R − 1 X l 1 =0 q l 1 ,l 2 · ( q l 1 ,l 2 ) ∗ e j 2 π λ  D ( n a ) l 1 ,l 2 − D ( n b ) l 1 ,l 2  = 0 . 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