Multi-Beam Multi-Hop Routing for Intelligent Reflecting Surfaces Aided Massive MIMO
11 Multi-Beam Multi-Hop Routing for IntelligentReflecting Surfaces Aided Massive MIMO
Weidong Mei and Rui Zhang,
Fellow, IEEE
Abstract
Intelligent reflecting surface (IRS) is envisioned to play a significant role in future wireless com-munication systems as an effective means of reconfiguring the radio signal propagation environment.In this paper, we study a new multi-IRS aided massive multiple-input multiple-output (MIMO) system,where a multi-antenna BS transmits independent messages to a set of remote single-antenna users usingorthogonal beams that are subsequently reflected by different groups of IRSs via their respective multi-hop passive beamforming over pairwise line-of-sight (LoS) links. We aim to select optimal IRSs andtheir beam routing path for each of the users, along with the active/passive beamforming at the BS/IRSs,such that the minimum received signal power among all users is maximized. This problem is particularlydifficult to solve due to a new type of path separation constraints for avoiding the IRS-reflected signalinduced interference among different users. To tackle this difficulty, we first derive the optimal BS/IRSactive/passive beamforming solutions based on their practical codebooks given the reflection paths.Then we show that the resultant multi-beam multi-hop routing problem can be recast as an equivalentgraph-optimization problem, which is however NP-complete. To solve this challenging problem, wepropose an efficient recursive algorithm to partially enumerate the feasible routing solutions, which isable to effectively balance the performance-complexity trade-off. Numerical results demonstrate thatthe proposed algorithm achieves near-optimal performance with low complexity and outperforms otherbenchmark schemes. Useful insights into the optimal multi-beam multi-hop routing design are alsodrawn under different setups of the multi-IRS aided massive MIMO network.
Index Terms
Intelligent reflecting surface, massive MIMO, passive beamforming, multi-beam multi-hop routing,graph theory.
Part of this work has been submitted to IEEE International Conference on Communications, Montreal, Canada, 2021 [1].The authors are with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore117583 (e-mails: [email protected], [email protected]). W. Mei is also with the NUS Graduate School, National Universityof Singapore, Singapore 119077. a r X i v : . [ c s . I T ] J a n I. I
NTRODUCTION
Wireless communication systems in the last decade have undergone a remarkable progresswith various advanced technologies successfully implemented, such as adaptive modulationand coding, dynamic resource allocation, hybrid digital and analog beamforming, etc., whichsignificantly enhanced their throughput and efficiency. However, existing wireless technologieswere designed mainly to adapt to or compensate the random and time-varying wireless channelsonly, but have very limited control over them, thus leaving an ultimate barrier uncleared inachieving ultra-reliable and ultra-high-capacity wireless systems in the future. Recently, intel-ligent reflecting surface (IRS) has emerged as an appealing solution to tackle this issue. Bydynamically tuning its large number of reflecting elements (or so-called passive beamforming),IRS is able to “reconfigure” wireless channels and refine their realizations and/or distributions[2]–[4], rather than adapting to them only in the traditional approach. In addition, IRS elementsdo not require transmit or receive radio frequency (RF) chains as they simply reflect the incidentsignal as a passive array, thus drastically reducing the hardware cost and energy consumptionas compared to traditional active transceivers and relays. Thus, by efficiently integrating IRSsinto future wireless networks, a quantum-leap improvement in capacity and energy efficiency isanticipated over today’s wireless systems.Due to the great potential of IRS, its performance has been recently studied in the literatureunder different wireless system setups, such as IRS-aided multi-antenna/multiple-input multiple-output (MIMO) system [5], [6], orthogonal frequency division multiplexing (OFDM) system[7], [8], non-orthogonal multiple access (NOMA) system [9], [10], multi-cell network [11],[12], simultaneous wireless information and power transfer [13], [14], mobile edge computing[15], [16], physical-layer security [17], [18], unmanned aerial vehicle (UAV) communication[19], [20], and so on. However, all of these works consider one or multiple distributed IRSs,which assist in the wireless communication between the base station (BS) and users with onlyone single signal reflection by each IRS. This simplified approach, however, generally results insuboptimal performance. This is because by properly deploying IRSs, strong line-of-sight (LoS)channels can be achieved for inter-IRS links, which can provide more pronounced cooperativepassive beamforming (CPB) gains over the conventional single-IRS assisted system. Inspired bythis, the authors in [21] first proposed a double-IRS system, where a single-antenna BS serves asingle-antenna user through a double-reflection link with two cooperative IRSs deployed near the … zy x … … BSUser 1 User 2 User k … User K Used active beam Used passive beam … Unused beams
Fig. 1. A multi-IRS aided massive MIMO system with the MBMH routing via joint BS/IRS active/passive beamforming.
BS and user, respectively. It was shown in [21] that this system provides a CPB gain that increases quartically with the total number of IRS reflecting elements, thus is significantly higher thanthe quadratic growth of the passive beamforming gain in the conventional single-IRS link. Theauthors in [22]–[24] further extended [21] to address the more practical Rician fading channeland multi-antenna/multi-user setups. Specifically, the authors in [22] proposed two differentchannel estimation schemes for the double-IRS aided single-user system under arbitrary and LoS-dominant inter-IRS channels, respectively. In [23], the authors studied the channel estimationproblem in a more challenging double-IRS aided multi-user MIMO system with coexisting single-and double-reflection links. Furthermore, the passive beamforming optimization for the two IRSsunder this system was studied in [24]. Despite of the above recent works, the general multi-IRS aided multi-user communication system with multi-hop (i.e., more than two hops) signalreflections has not been investigated in the literature yet. Under this general setup with moreavailable IRSs in the network, different end-to-end LoS paths can be achieved between the BSand multiple remote users at the same time via multi-hop signal reflections by different groups ofIRSs selected. This thus gives rise to a new cooperative multi-beam multi-hop (MBMH) routing design problem, where the selected IRSs and their beam-routing paths for different users arejointly optimized with the active/passive beamforming at the BS/IRSs to maximize the receivedsignal power at all users.In this paper, we study this new MBMH routing problem for the downlink communication in a massive MIMO system, where a BS equipped with a large number of active antennas transmitsindependent messages to a set of remote single-antenna users simultaneously over the samefrequency band, aided by multiple distributed IRSs as shown in Fig. 1. In [25], by consideringonly a single user in this system, we have derived the optimal single-beam multi-hop routingsolution. However, different from [25], a new challenge arises in our considered MBMH routingdesign in this paper, which is to avoid the inter-user/path interference due to undesired scatteringby the IRSs that serve for different users/paths, especially when there exist LoS channels betweenthem. This thus leads to a new type of path separation constraints among different users, wherethe IRSs selected for different users/paths should avoid having LoS channels with each other. Thisstringent constraint thus makes the MBMH routing problem in this paper more challenging tosolve, as compared to its single-beam special case in [25] without the inter-user/path interferenceconsidered. Moreover, unlike [1] and [25] where the continuous active/passive beamforming isassumed for the ease of exposition, in this paper we consider the more practical design basedon beamforming codebook, which consists of only a finite number active/passive beamformingdirections at the BS/IRSs, as shown in Fig. 1.To solve the proposed MBMH routing problem in a multi-IRS aided massive MIMO network,we first derive the optimal BS/IRS active/passive beamforming solution in their respectivecodebooks for given beam-routing paths of the users, by exploiting the high angular resolutionof the massive MIMO BS and the inter-IRS LoS channels, respectively. Next, we show that theresultant MBMH routing problem is NP-complete by recasting it into an equivalent neighbor-disjoint path optimization problem in graph theory. To deal with this challenging problem, arecursive algorithm is proposed to partially enumerate the feasible MBMH routing solutions. Bytuning its parameter, the proposed algorithm can strike a flexible balance between performanceand complexity. It is also shown that in the special case of continuous passive beamforming atthe IRSs, the MBMH routing problem can be solved in a more efficient manner by the proposedalgorithm. Numerical results show that our proposed algorithm can find the near-optimal MBMHrouting solution with low computational complexity and outperforms other benchmark schemes.It is also revealed that the optimal MBMH routing solution varies considerably with the numberof reflecting elements as well as the size of passive beamforming codebook at each IRS.The rest of this paper is organized as follows. Section II presents the system model. SectionIII presents the optimal BS/IRS active/passive beamforming design and the problem formulationfor our considered MBMH routing optimization. Section IV presents the proposed solution to this problem based on graph theory. Section V presents the simulation results to show theperformance of the proposed scheme as compared to other benchmark schemes. Finally, SectionVII concludes this paper and discusses future work.The following notations are used in this paper. Bold symbols in capital letter and small letterdenote matrices and vectors, respectively. The conjugate, transpose and conjugate transpose ofa vector or matrix are denoted as ( · ) ∗ , ( · ) T and ( · ) H , respectively. R n ( C n ) denotes the set ofreal (complex) vectors of length n . For a complex number s , s ∗ and | s | denote its conjugate andamplitude, respectively. For a vector a ∈ C n , diag( a ) denotes an n × n diagonal matrix whoseentries are given by the elements of a ; while for a square matrix A ∈ C n × n , diag( A ) denotesan n × vector that contains the n diagonal elements of A . (cid:107) a (cid:107) denotes the Euclidean normof the vector a . (cid:98)·(cid:99) denotes the greatest integer less than or equal to its argument. | A | denotesthe cardinality of a set A . j denotes the imaginary unit, i.e., j = − . For two sets A and B , A ∪ B denotes the union of A and B . ∅ denotes an empty set. (cid:12) and ⊗ denote the Hadamardproduct and Kronecker product, respectively. O ( · ) denotes the order of complexity.II. S YSTEM M ODEL
As shown in Fig. 1, we consider a massive MIMO downlink system, where J distributed IRSsare deployed to assist in the communications from a multi-antenna BS to K remote single-antenna users. Assume that the BS is equipped with N B (cid:29) K active antennas, while eachIRS is equipped with M passive reflecting elements. Without loss of generality and for easeof practical implementation, we assume that the BS serves the K users by selecting K beamsfrom a predefined codebook, denoted as W B , which consists of N B orthogonal and unit-powerbeams, where N B can be arbitrarily large in massive MIMO. For the purpose of exposition, weconsider the challenging scenario where the BS-user direct links are severely blocked for allthe K users considered in this paper. As such, the BS can only communicate with each userthrough a multi-reflection signal path that is formed by a set of IRSs associated with the user.To mitigate the potential inter-user interference during the multi-hop signal reflection, the signalpaths for all K users should be sufficiently separated and thus each IRS is associated with atmost one user. For convenience, we denote the sets of users and IRSs as K (cid:44) { , , · · · , K } and J (cid:44) { , , · · · , J } , respectively. To maximize the reflected signal power by each selectedIRS and ease the hardware implementation, we set the reflection amplitude of all its elementsto the maximum value of one. As such, the reflection coefficient matrix of each IRS j, j ∈ J is given by Φ j = diag { e jθ j, , e jθ j, , · · · , e jθ j,M } ∈ C M × M , and its passive beamforming vector isdenoted as θ j = diag( Φ j ) ∈ C M × . The passive beamforming vector of each IRS is assumedto be selected from a codebook W I , i.e., θ j ∈ W I , ∀ j ∈ J , and W I consists of D = 2 b beampatterns, where b denotes the number of controlling bits for W I . For convenience, we refer tothe BS and user k, k ∈ K as nodes 0 and J + k in the system, respectively. Accordingly, wedefine H ,j ∈ C M × N B , j ∈ J as the channel from the BS to IRS j , g Hj,J + k ∈ C × M , j ∈ J asthat from IRS j to user k , and S i,j ∈ C M × M , i, j ∈ J , i (cid:54) = j as that from IRS i to IRS j . Forease of exposition, we assume that the passive reflecting elements of each IRS in J are arrangedin a uniform rectangular array (URA) parallel to the x - z plane, while the BS applies a uniformlinear array (ULA) parallel to the y -axis, as shown in Fig. 1. The antenna and element spacingat the BS and each IRS is assumed to be d A and d I , respectively. The numbers of elements ineach IRS’s horizontal and vertical directions are assumed to be M and M , respectively, with M M = M .Let d i,j , i (cid:54) = j denote the distance between nodes i and j , for which some reference trans-mitting/reflecting elements of the BS/IRSs are selected without loss of generality. To ensurethe far-field propagation between any two nodes, we assume that d i,j ≥ d , ∀ i (cid:54) = j , where d denotes the minimum distance to satisfy this condition. Then, by carefully deploying the J IRSs, LoS dominant propagation may be achieved between some pair of nodes i and j if d i,j ispractically small (but larger than d ). To simplify the active and passive beamforming designs aswell as enhance the strength of the multi-reflection signal paths, we only exploit the LoS linksin the system for the multi-hop signal reflection. Then, to describe the LoS condition betweenany two nodes i (BS/IRS) and j (IRS/user) in the considered system, we define a binary LoScondition indicator l i,j ∈ { , } . In particular, l i,j = 1 indicates that the link between nodes i and j consists of an LoS link; otherwise, l i,j = 0 . In addition, we set l i,i = 0 , ∀ i and thus, l i,j = l j,i , ∀ i, j . In this paper, to focus on the new MBMH routing design, we assume that theLoS condition indicators l i,j ’s are known and constant after deploying the IRSs, while how toacquire such knowledge in practice is an interesting problem to be addressed in our future work.Based on the LoS condition between any two nodes in the considered system, a multi-hop LoSlink can be established between the BS and each user k, k ∈ K by properly selecting a subset ofassociated IRSs. For example, if l ,i = l i,j = l j,J + k = 1 , i, j ∈ J , we can select IRSs i and j asthe associated IRSs of user k , which successively reflect its intended signal from the BS towardits receiver. The IRSs that are not associated with any user in K are turned off to minimize the scattered interference in the system.Next, we characterize the LoS channel between any two nodes in the system (if any), whichis modeled as the product of array responses at their two sides. For convenience, we define thefollowing steering vector function, e ( φ, N ) = [1 , e − jπφ , · · · , e − jπ ( N − φ ] T ∈ C N × , (1)where N denotes the number of elements in a ULA, and φ denotes the phase difference betweenthe observations at two adjacent elements. Obviously, e ( φ, N ) is a periodic functions of φ andhas a period of 2. Hence, we restrict φ ∈ [0 , in the sequel of this paper. If φ ≥ or φ < ,we set φ as φ − (cid:98) φ (cid:99) . Then, the array response at the BS is expressed as a B ( ϑ ) = e (cid:16) d A λ sin ϑ, N B (cid:17) , (2)where ϑ denotes the angle-of-departure (AoD) relative to the BS antenna boresight, and λ denotesthe carrier wavelength. For the URA at each IRS, its array response is expressed as the Kroneckerproduct of two steering vector functions in the horizontal and vertical directions, respectively,i.e., a I ( ϑ a , ϑ e ) = e (cid:16) d I λ sin ϑ e cos ϑ a , M (cid:17) ⊗ e (cid:16) d I λ cos ϑ e , M (cid:17) , (3)where ϑ e and ϑ a denote its elevation angle-of-arrival (AoA)/AoD and azimuth AoA/AoD, respec-tively. Then, we define ϑ ,j as the AoD from the BS to IRS j , ϕ aj,i / ϕ ej,i as the azimuth/elevationAoA at IRS j from node i (BS or IRS), and ϑ ai,j / ϑ ei,j as the azimuth/elevation AoD from IRS i tonode j (IRS or user). The above AoAs and AoDs can be estimated by exploiting the geometricrelationship of the BS, IRSs and users in the system [21] or by integrating sensors to the IRSs[3].Based on the above, we define ˜ h j, = a B ( ϑ ,j ) and ˜ h j, = a I ( ϕ aj, , ϕ ej, ) for the LoS channelfrom the BS to IRS j, j ∈ J , ˜ s i,j, = a I ( ϑ ai,j , ϑ ei,j ) and ˜ s i,j, = a I ( ϕ aj,i , ϕ ej,i ) for that fromIRS i to IRS j, i, j ∈ J , as well as ˜ g j,J + k = a I ( ϑ aj,J + k , ϑ ej,J + k ) for that from IRS j to user k, j ∈ J , k ∈ K . Then, if l ,j = 1 , the BS-IRS j channel is expressed as H ,j = √ βd ,j e − j πd ,jλ ˜ h j, ˜ h Hj, , j ∈ J , (4)where β ( < denotes the LoS path gain at the reference distance of 1 meter (m). Similarly, if l i,j = 1 , i, j ∈ J , the IRS i -IRS j channel is given by S i,j = √ βd i,j e − j πdi,jλ ˜ s i,j, ˜ s Hi,j, , i, j ∈ J , i (cid:54) = j. (5) Finally, if l j,J + k = 1 , the IRS j -user k channel is expressed as g Hj,J + k = √ βd j,J + k e − j πdj,J + kλ ˜ g Hj,J + k , j ∈ J , k ∈ K . (6)Based on (4)-(6), we can characterize the multi-hop LoS channel between the BS and each user k, k ∈ K , with the given reflection path and BS/IRS active/passive beamforming. Specifically,let Ω ( k ) = { a ( k )1 , a ( k )2 , · · · , a ( k ) N k } , k ∈ K denote the reflection path from the BS to user k , where N k ( ≥ and a ( k ) n ∈ J denote the number of associated IRSs for user k and the index of the n -th associated IRS, with n ∈ N k (cid:44) { , , · · · , N k } , respectively. For convenience, we define a ( k )0 = 0 and a ( k ) N k +1 = J + k, k ∈ K , corresponding to the BS and user k , respectively. Then, toensure that each IRS in N k only reflects user k ’s information signal at most once, the followingconstraints should be met: a ( k ) n ∈ J , a ( k ) n (cid:54) = a ( k ) n (cid:48) , ∀ n, n (cid:48) ∈ N k , n (cid:54) = n (cid:48) , k ∈ K . (7)Moreover, each constituent link of Ω ( k ) , along with the BS-IRS a ( k )1 link and the IRS a ( k ) N k -user k link, should consist of an LoS link, i.e., l a ( k ) n ,a ( k ) n +1 = 1 , ∀ n ∈ N k ∪ { } , k ∈ K . (8)Furthermore, to avoid the scattered inter-user interference, we consider that there is no directLoS link between any two nodes belonging to different reflection paths (except the commonnode or the BS). Thus, we have l a ( k ) n ,a ( k (cid:48) ) n (cid:48) = 0 , a ( k ) n (cid:54) = a ( k (cid:48) ) n (cid:48) , ∀ n, n (cid:48) (cid:54) = 0 , k, k (cid:48) ∈ K , k (cid:54) = k (cid:48) . (9)Thus, each Ω ( k ) is a feasible path if and only if the constraints in (7)-(9) are satisfied. Given K feasible paths Ω ( k ) , k ∈ K , we define w k ∈ C N × , k ∈ K as the BS active beamformingdesign for user k , with w k ∈ W B . Then, the BS-user k effective channel is expressed as h ,J + k (Ω ( k ) ) = g Ha ( k ) Nk ,J + k Φ a ( k ) Nk (cid:16) (cid:89) n ∈N k ,n (cid:54) = N k S a ( k ) n ,a ( k ) n +1 Φ a ( k ) n (cid:17) H ,a ( k )1 w k , k ∈ K , (10)which depends on both the CPB design for the N k selected IRSs and the active beamformingdesign w k for the BS. By substituting (4)-(6) into (10) and rearranging the terms in it, we obtain h ,J + k (Ω ( k ) ) = e − j(cid:36) k κ (Ω ( k ) ) (cid:16) N k (cid:89) n =1 A ( k ) n (cid:17) ˜ h Ha ( k )1 , w k , k ∈ K , (11) The methods and results in this paper are extendible to the more general path separation constraints, e.g., without q -hop LoSlink between any two reflection paths, with q ≥ . where A ( k ) n = ˜ s Ha ( k )1 ,a ( k )2 , Φ a ( k )1 ˜ h a ( k )1 , if n = 1˜ s Ha ( k ) n ,a ( k ) n +1 , Φ a ( k ) n ˜ s a ( k ) n − ,a ( k ) n , if ≤ n ≤ N k − g Ha ( k ) Nk ,J + k Φ a ( k ) Nk ˜ s a ( k ) Nk − ,a ( k ) Nk , if n = N k , (12) (cid:36) k = πλ (cid:80) N k n =0 d a ( k ) n ,a ( k ) n +1 is proportional to the end-to-end transmission distance, and κ (Ω ( k ) ) = ( √ β ) N k +1 N k (cid:81) n =0 d a ( k ) n ,a ( k ) n +1 (13)denotes the cascaded LoS path gain between the BS and user k under the path Ω ( k ) , which turnsout to be the product of the LoS path gains of all constituent links in Ω ( k ) .Thus, the BS-user k equivalent channel power, | h ,J + k (Ω ( k ) ) | , is expressed as | h ,J + k (Ω ( k ) ) | = β N k +1 N k (cid:81) n =1 | A ( k ) n | · | ˜ h Ha ( k )1 , w k | N k (cid:81) n =0 d a ( k ) n ,a ( k ) n +1 , k ∈ K . (14)Based on (14), we can obtain the optimal active/passive beamforming design at the BS/IRSswith a given MBMH routing solution, whereby the MBMH routing problem can be formulated,as detailed in the next section.III. O PTIMAL B EAMFORMING D ESIGN AND P ROBLEM F ORMULATION
In this section, we first derive the optimal active and passive beamforming design to maximizeeach | h ,J + k (Ω ( k ) ) | , k ∈ K in (14) under a given reflection path Ω ( k ) . With the optimal beam-forming design, we then formulate the MBMH routing problem to further optimize Ω ( k ) , k ∈ K . A. Optimal Active and Passive Beamforming Design
First, it is observed from (14) that for any given reflection path for user k , to maximize | h ,J + k (Ω ( k ) ) | , the magnitude of each A ( k ) n and ˜ h Ha ( k )1 , w k should be maximized, subject to thecodebook constraints at each IRS and the BS, respectively. First, given the codebook W I at eachIRS, consider that an IRS j reflects the signal from its last node i to the next node r . We denote by θ I ( i, j, r ) its corresponding optimal passive beamforming vector, which can be obtained byenumerating all beam patterns in W I , i.e., θ I ( i, j, r ) = arg max θ ∈W I | ˜ s Hj,r, diag( θ ) ˜ h j, | if i = 0arg max θ ∈W I | ˜ g Hj,J + k diag( θ )˜ s i,j, | if r = J + k arg max θ ∈W I | ˜ s Hj,r, diag( θ )˜ s i,j, | otherwise . ∀ i, j, r (15)In particular, if the continuous passive beamforming with b → ∞ is applied by each IRS, asall array responses have unit-modulus entries, (15) can be simplified as θ I ( i, j, r ) = ˜ s j,r, (cid:12) ˜ h ∗ j, if i = 0˜ g j,J + k (cid:12) ˜ s ∗ i,j, if r = J + k ˜ s j,r, (cid:12) ˜ s ∗ i,j, otherwise . ∀ i, j, r (16)Accordingly, in the reflection path of user k , the passive beamforming of each IRS a ( k ) n , n ∈ N k should be set as θ a ( k ) n = diag( Φ a ( k ) n ) = θ I ( a ( k ) n − , a ( k ) n , a ( k ) n +1 ) . (17)Note that in the special case of continuous IRS beamforming with b → ∞ , by substituting (16)and (17) into (12), we have A ( k ) n = M, ∀ n ∈ N k , k ∈ K .To gain more useful insights into the optimal passive beamforming solutions at each IRS in(15) and (16), we consider that both nodes i and r are IRSs, which corresponds to the third casein (15). Then, according to (3), it can be shown that [26] ˜ s Hj,r, diag( θ j )˜ s i,j, = a HI ( ϑ aj,r , ϑ ej,r )diag( θ j ) a I ( ϕ aj,i , ϕ ej,i )= ( a HI ( ϑ aj,r , ϑ ej,r ) (cid:12) a TI ( ϕ aj,i , ϕ ej,i )) θ j = (cid:16) e H (cid:16) d I λ φ (1) i,j,r , M (cid:17) ⊗ e H (cid:16) d I λ φ (2) i,j,r , M (cid:17)(cid:17) θ j , (18)where φ (1) i,j,r (cid:44) sin ϑ ej,r cos ϑ aj,r − sin ϕ ej,i cos ϕ aj,i and φ (2) i,j,r (cid:44) cos ϑ ej,r − cos ϕ ej,i . Similarly, it can beverified that (18) also holds if node i is the BS (the first case in (15)) or node r is a user (thesecond case in (15)). It follows from (18) that if b → ∞ , the optimal passive beamforming in(16) can be rewritten as θ I ( i, j, r ) = e (cid:16) d I λ φ (1) i,j,r , M (cid:17) ⊗ e (cid:16) d I λ φ (2) i,j,r , M (cid:17) , which perfectly alignsthe horizontal and vertical directions at the same time. Motivated by this observation, we set θ j = θ (1) I ( i, j, r ) ⊗ θ (2) I ( i, j, r ) in (18), where θ (1) I ( i, j, r ) and θ (2) I ( i, j, r ) denote the horizontal and vertical passive beamforming vectors for IRS j ,respectively. Then, we can obtain ˜ s Hj,r, diag( θ j )˜ s i,j, = (cid:16) e H (cid:16) d I λ φ (1) i,j,r , M (cid:17) ⊗ e H (cid:16) d I λ φ (2) i,j,r , M (cid:17)(cid:17) · (cid:16) θ (1) I ( i, j, r ) ⊗ θ (2) I ( i, j, r ) (cid:17) = (cid:16) e H (cid:16) d I λ φ (1) i,j,r , M (cid:17) θ (1) I ( i, j, r ) (cid:17) · (cid:16) e H (cid:16) d I λ φ (2) i,j,r , M (cid:17) θ (2) I ( i, j, r ) (cid:17) . (19)It is noted from (19) that the passive beamforming of IRS j can be decoupled into horizontaland vertical IRS passive beamforming. Accordingly, we define W (1) I and W (2) I as the codebooksfor the horizontal and vertical IRS passive beamforming, respectively. The numbers of controllingbits for W (1) I and W (2) I are denoted as b and b , respectively, which satisfy b + b = b . Hence,the IRS codebook W I can be decomposed as W I = { θ | θ = θ ⊗ θ , θ ∈ W (1) I , θ ∈ W (2) I } , (20)while the optimal IRS passive beamforming in (15) can be computed as θ I ( i, j, r ) = θ (1) I ( i, j, r ) ⊗ θ (2) I ( i, j, r ) , where θ (1) I ( i, j, r ) = arg max θ ∈W (1) I (cid:12)(cid:12)(cid:12)(cid:12) e H (cid:16) d I λ φ (1) i,j,r , M (cid:17) θ (cid:12)(cid:12)(cid:12)(cid:12) , (21) θ (2) I ( i, j, r ) = arg max θ ∈W (2) I (cid:12)(cid:12)(cid:12)(cid:12) e H (cid:16) d I λ φ (2) i,j,r , M (cid:17) θ (cid:12)(cid:12)(cid:12)(cid:12) . (22)Note that compared to the joint three-dimensional (3D) beam search in (15), the complexity ofbeam search can be greatly reduced from O (2 b ) to O (2 b + 2 b ) by separately solving (21) and(22).Accordingly, in the reflection path of user k, k ∈ K , the passive beamforming of each IRS a ( k ) n , n ∈ N k in (17) can be rewritten as θ I ( a ( k ) n − , a ( k ) n , a ( k ) n +1 ) = θ (1) I ( a ( k ) n − , a ( k ) n , a ( k ) n +1 ) ⊗ θ (2) I ( a ( k ) n − , a ( k ) n , a ( k ) n +1 ) , (23)which can be simplified as θ I ( a ( k ) n − , a ( k ) n , a ( k ) n +1 ) = e (cid:16) d I λ φ (1) a ( k ) n − ,a ( k ) n ,a ( k ) n +1 , M (cid:17) ⊗ e (cid:16) d I λ φ (2) a ( k ) n − ,a ( k ) n ,a ( k ) n +1 , M (cid:17) , (24)in the case of continuous passive beamforming at each IRS with b and b → ∞ .Next, we focus on the optimal active beamforming design for the BS, which should maximizethe amplitude of ˜ h Ha ( k )1 , w k , k ∈ K in (14). To this end, we define w B ( j ) = arg max w ∈W B | ˜ h Hj, w | , j ∈ J , (25) as the optimal active beamforming solution for the BS to transmit the beam to IRS j , which isobtained by enumerating all beam patterns in the codebook at the BS, W B . In particular, if thecontinuous active beamforming is applied at the BS, (25) becomes equivalent to the maximum-ratio transmission (MRT) based on ˜ h j, , i.e., w B ( j ) = ˜ h j, / (cid:107) ˜ h j, (cid:107) , k ∈ K . (26)With this definition, the BS active beamforming in the reflection path of user k should be set as w k = w B ( a ( k )1 ) e j(cid:36) k , k ∈ K , (27)where the effective phases (cid:36) k , k ∈ K of the K reflection paths are compensated at the BS.It is worth noting that if N B is sufficiently large, the MRT-based beamforming in (26) ensuresthat the power of the information signal for each user k, k ∈ K overwhelms that of the inter-userinterference in the BS-IRS a ( k )1 link, i.e., the first link in Ω ( k ) . This is because with a large N B , theBS antenna array has a practically high angular resolution. If all first-hop IRSs in the reflectionpaths for the K users, i.e., IRS a ( k )1 , k ∈ K , are sufficiently separated in the angular domain, thefollowing asymptotically favorable propagation for massive MIMO [27] can be achieved: N B | ˜ h Ha ( k )1 , w k | = 1 , k ∈ K , N B | ˜ h Ha ( k )1 , w k (cid:48) | ≈ , k, k (cid:48) ∈ K , k (cid:54) = k (cid:48) . (28)The asymptotically favorable propagation in (28) may also be achieved with practical finite-size codebooks, e.g., the discrete Fourier transform (DFT)-based codebook (see Section V fordetails). This is because when the codebook size N B is sufficiently large, the codebook will havea high resolution, such that the selected beam patterns are close to the MRT-based beamformingin (26). Hence, the inter-user interference can be approximately nulled in the first link of eachreflection path Ω ( k ) by properly selecting W B . Furthermore, since the path separation constraintsin (9) ensure that the scattered inter-user interference in the subsequent links of Ω ( k ) is wellmitigated, user k is approximately free of inter-user interference, while achieving the maximumend-to-end channel power with the BS via IRSs’ passive beamforming in (23) and the BS’sactive beamforming in (27).Under the above optimal beamforming designs, we define ˜ A ( k ) n as the maximum value of A ( k ) n in (12) by following (23). It is worth noting that ˜ A ( k ) n depends on the AoAs/AoDs between nodes a ( k ) n − and a ( k ) n , as well as those between nodes a ( k ) n and a ( k ) n +1 . Besides, it also depends on the numbers of controlling bits for the IRS codebooks, i.e., b and b . In particular, with increasing b or b , the resolution of IRS codebook can be improved, thus resulting in a larger ˜ A ( k ) n . In thespecial case of continuous IRS beamforming with b and b → ∞ , we have ˜ A ( k ) n = M , whichis regardless of AoAs/AoDs. It follows that the effect of AoAs and AoDs diminishes when theresolution of IRS codebooks is sufficiently high. By substituting (23) and (28) into (14), themaximum BS-user k equivalent channel power is given by | h ,J + k (Ω ( k ) ) | = β N k +1 N B N k (cid:81) n =1 | ˜ A ( k ) n | N k (cid:81) n =0 d a ( k ) n ,a ( k ) n +1 , k ∈ K . (29)It is observed from (29) that besides the conventional active BS beamforming gain of N B ,a new multiplicative CPB gain of (cid:81) N k n =1 | ˜ A ( k ) n | is also achieved for each BS-user k equivalentchannel. As previously discussed, if b or b is small, this multiplicative CPB gain will dependheavily on the AoAs and AoDs between any two consecutive nodes in Ω ( k ) . However, if both b and b are sufficiently large, this CPB gain can be greatly enhanced and approaches its maximumvalue, M N k . In general, there exists a fundamental trade-off between maximizing the CPB gainversus the end-to-end path gain, i.e., κ (Ω ( k ) ) in (13) (or minimizing the end-to-end path loss κ − (Ω ( k ) ) ), as the former monotonically increases with N k , while the latter generally decreaseswith N k . Besides this trade-off, there exists another trade-off in balancing all | h ,J + k (Ω ( k ) ) | ’s fordifferent users in K . Specifically, due to the practically finite number of IRSs and LoS paths inthe system as well as the path separation constraints in (9), maximizing the channel power forone user generally reduces the number of feasible paths for the other users. Particularly, if thenumber of users is large, some users may be denied access due to the lack of feasible paths.As such, the optimal MBMH routing design should reconcile the above trade-offs and take intoaccount the resolution of practical IRS codebooks, so as to achieve the optimum performanceof all K users in a fair manner. B. Problem Formulation
In this paper, we aim to maximize the minimum signal-to-noise-plus-interference ratio (SINR)achievable by the K users, by optimizing the reflection paths Ω ( k ) , k ∈ K , subject to thefeasibility constraints in (7)-(9). Due to the well mitigated inter-user interference at each user’s receiver, this is equivalent to maximizing the minimum BS-user effective channel power, i.e., min k ∈K | h ,J + k (Ω ( k ) ) | . The optimization problem is thus formulated as(P1) max { Ω ( k ) } k ∈K min k ∈K | h ,J + k (Ω ( k ) ) | s.t. (7)-(9) . (30)However, (P1) is a combinatorial optimization problem due to its integer and coupled variables.In addition, ˜ A ( k ) n , k ∈ K in h ,J + k (Ω ( k ) ) are functions of AoAs and AoDs in the network if b or b is small, while they become a constant M in the case of continuous IRS beamforming,as considered in [1] and [25]. Thus, it is challenging to obtain the optimal solution to (P1) viastandard optimization methods in general, especially in the case with a small b or b . To tacklethis challenging problem, we reformulate it as an equivalent graph-optimization problem whichis then solved, as detailed in the next section.IV. P ROPOSED S OLUTION TO (P1)In this section, we first reformulate (P1) as an equivalent problem in graph theory under thegeneral case with finite b and b , and thereby show that it is NP-complete. Then, a parametrizedrecursive algorithm is proposed to efficiently solve this problem sub-optimally in general. Finally,we show that (P1) can be more efficiently solved by the proposed algorithm in the special caseof continuous IRS beamforming with b and b → ∞ . A. Problem Reformulation via Graph Theory
Obviously, in (P1), it is equivalent to minimizing the maximum | h ,J + k (Ω ( k ) ) | − among all k ∈ K . Based on (14), we have | h ,J + k (Ω ( k ) ) | − = d ,a ( k )1 βN B · N k (cid:89) n =1 d a ( k ) n ,a ( k ) n +1 β | ˜ A ( k ) n | , k ∈ K . (31)Then, by taking the logarithm of (31), (P1) becomes equivalent to min { Ω ( k ) } k ∈K max k ∈K F (Ω ( k ) ) , s.t. (7)-(9) , (32)where F (Ω ( k ) ) = ln d ,a ( k )1 βN B + N k (cid:88) n =1 ln d a ( k ) n ,a ( k ) n +1 β | ˜ A ( k ) n | . (33)Next, we recast problem (32) as an equivalent problem in graph theory subject to the con-straints (7)-(9). Following the similar procedures in [1] and [25], we construct a directed andunweighted graph G = ( V , E ) . The vertex set V consists of all nodes in the system, i.e., V = { , , , · · · , J + K } . Furthermore, we consider that each of the K beams can only berouted outwards from one IRS i to a farther IRS j from the BS with d j, > d i, , i, j ∈ J , so asto reach its intended user as quickly as possible. Hence, the edge set E is defined as E = { (0 , j ) | l ,j = 1 , j ∈ J } ∪ { ( i, j ) | l i,j = 1 , d j, > d i, , i, j ∈ J }∪ { ( j, J + k ) | l j,J + k = 1 , j ∈ J , k ∈ K} , (34)i.e., there exists an edge from vertex i to vertex j if and only if an LoS path exists between themand d j, > d i, , except that vertex j corresponds to a user, i.e., j = J + k, k ∈ K . Thus, we have | E | = (cid:80) J + Ki =0 (cid:80) J + Kj =0 l i,j . Note that (34) ensures that there is no circle in G , i.e., G is a directacyclic graph (DAG). Given the constructed graph G , any reflection path from the BS to user k corresponds to a path from node 0 to node J + k in G . However, different from the beam routingproblems in [1] and [25] with continuous IRS beamforming with b and b → ∞ , it is difficultto assign a weight to each edge in G to recast problem (32) as an equivalent graph-optimizationproblem. This is because each ˜ A ( k ) n in (33) is associated with three vertices, i.e., vertices a ( k ) n − , a ( k ) n and a ( k ) n +1 , but each edge in G is only associated with two vertices. However, in [1] and[25], we have ˜ A ( k ) n = M , which greatly simplifies the weight assignment in G .To resolve the above issues, a new DAG of higher dimension, denoted as G = ( V, E ) , shouldbe constructed from G . Specifically, besides vertex and vertices J + k, k ∈ K , we create avertex in G for each edge in G ; while for every two edges in G that share a common vertex,we create an edge between their corresponding vertices in G . The resulting graph G is knownas the line graph of G in graph theory. Mathematically, for G , its vertex set V is given by V = { v i,j | ( i, j ) ∈ E } ∪ { , J + 1 , J + 2 , · · · , J + K } . (35)Obviously, we have | V | = | E | + K + 1 . The edge set E is given by E = { (0 , v ,j ) | j ∈ J } ∪ { ( v j,J + k , J + k ) | j ∈ J , k ∈ K} ∪ { ( v i,j , v j,r ) | i, j, r ∈ V } . (36)It follows from (35) and (36) that the edge (0 , v ,j ) ( ( v j,J + k , J + k ) ) indicates that there existsan LoS path from the BS (IRS j ) to IRS j (user k ). Moreover, the edge ( v i,j , v j,r ) indicates thatthere exist two pairwise LoS paths from node i and node r via IRS j . In this new graph G , someedges in E involve three vertices, thus making the weight assignment possible. To determinethe edge weights in G , we first rewrite F (Ω ( k ) ) , k ∈ K in (33) as F (Ω ( k ) ) = ln d ,a ( k )1 √ βN B + ln d a ( k ) Nk ,J + k √ β + N k (cid:88) n =1 ln d a ( k ) n − ,a ( k ) n d a ( k ) n ,a ( k ) n +1 β | ˜ A ( k ) n | , k ∈ K , (37) by rearranging the terms in it. Accordingly, the weight of each edge in E is set as follows: W (0 , v ,j ) = ln d ,j √ βN B , W ( v j,J + k , J + k ) = ln d j,J + k √ β ,W ( v i,j , v j,r ) = ln d i,j d j,r β | ˜ s Hj,r, diag( θ I (0 ,j,r )) ˜ h j, | if i = 0ln d i,j d j,r β | ˜ g Hj,J + k diag( θ I ( i,j,J + k )) ˜ s i,j, | if r = J + k ln d i,j d j,r β | ˜ s Hj,r, diag( θ I ( i,j,r )) ˜ s i,j, | otherwise . (38)Note that the above weights may be negative, e.g., when M , b and b are practically large, suchthat the argument of the logarithm in (38) is smaller than one.With the constructed line graph G , we can establish a one-to-one correspondence between eachpath from vertex to vertex J + k, k ∈ K in G and that in G . For example, if a path in G isgiven by → v , → v , → v ,J +1 → J + 1 , then it corresponds to the path → → → J + 1 in G and thus a reflection path from the BS to user 1 via IRSs 1 and 3. In particular, the sumof edge weights of any path from vertex to vertex J + k, k ∈ K in G is equal to F (Ω ( k ) ) , if itscorresponding path in G is Ω ( k ) . Since G is a DAG, it is easy to verify that G is also a DAG.Thus, for any path in G , its corresponding path in G can automatically satisfy the constraintsin (7)-(8). To handle the more challenging constraint (9), we present the following definitions. Definition 1:
Neighbor-disjoint paths refer to the paths in a graph which do not have anycommon or neighboring vertices except their starting points.According to Definition 1, the constraints in (9) can be satisfied if the K paths from vertex to vertices J + k, k ∈ K in G are neighbor-disjoint. As such, problem (32) is equivalent tothe following graph-optimization problem, denoted as (P2).(P2) Find K paths from vertex to vertices J + k, k ∈ K in G , respectively, such thatthe length of the longest path (i.e., the path with the maximum sum of edge weights) isminimized and their corresponding paths in G are neighbor-disjoint. Note that neighbor-disjoint routing design has been previously studied in various multi-hopwireless networks, such as ad-hoc networks and wireless sensor networks, for the purpose ofload balancing or interference mitigation [28], [29]. However, most of these works only focuson discovering a set of neighbor-disjoint paths through different medium access control (MAC)layer protocols, but not from an optimal routing design perspective. A common routing designis by utilizing the shortest path algorithm to sequentially update the paths for the K users [28]. Specifically, after deriving the shortest path for a user in G , the nodes in its corresponding pathin G (except node 0) and their neighbors are removed. Then, a new line graph G is constructedto determine the shortest path for the next user, so as to satisfy (9). However, as will be shownin Section V, this sequential update design generally yields suboptimal paths and even fails toreturn feasible paths. This is because the set of feasible paths for the current user criticallydepends on the optimized paths for the previous users. In fact, it has been proved in [29] thatfinding K neighbor-disjoint paths in G is NP-complete even in the case of K = 2 . As such,(P2) remains a challenging problem, which will be solved next. B. Proposed Solution to (P2)
The basic idea of the proposed solution to (P2) is by first finding Q ( ≥ candidate shortestpaths from node 0 to each node J + k, k ∈ K in G (thus G ). Given these candidate shortest paths,we further construct a new path graph , based on which a recursive algorithm is performed topartially enumerate the feasible paths and select the best one as the solution to (P2), as specifiedbelow.
1) Step 1: Find the candidate shortest paths.
First, for the nodes 0 and J + k, k ∈ K in G ,we invoke the Yen’s algorithm [30] to find Q candidate shortest paths between them. If the totalnumber of paths between the two nodes is less than Q , we assume that there exist additionalvirtual paths between them with infinite sum of edge weights. For convenience, we denote by p ( q ) k and c ( q ) k , k ∈ K , q ≤ Q the q -th candidate shortest path between vertices 0 and J + k andits sum of edge weights, respectively. Let P = { p ( q ) k , k ∈ K , q ≤ Q } be the set of all candidateshortest paths. The time complexity for this step is O ( KQ | V | ( | E | + | V | log | V | )) [30].
2) Step 2: Construct the path graph.
Next, we construct a new undirected graph G p = ( V p , E p ) ,where each vertex in V p corresponds to one candidate shortest path obtained in Step 1 (thustermed as path graph), i.e., V p = { v ( p ( q ) k ) | k ∈ K , q ≤ Q } . Hence, we have | V p | = KQ . Bythis means, we can establish a one-to-one mapping between any path in P and one vertex in G p . Moreover, since there also exists a one-to-one mapping between any path in G (thus in P )and a path in G , each vertex in G p also corresponds to a unique path in G . In particular, thevertex v ( p ( q ) k ) in G p corresponds to a path from vertex 0 to vertex J + k in G . For example, asshown in Fig. 2, if the path → v , → v , → v ,J +1 → J + 1 is the second candidate shortestpath from node 0 to node J + 1 in G and also included in P (e.g., Q = 2 ), then it correspondsto the vertex v ( p (2)1 ) in G p , which thus corresponds to the path → → → J + 1 in G . J +1 J +2 J +3 J +4 (1) G v ( ) p p v ( ) p v ( ) p v ( ) p v ( ) p p p p V p ,1 V p ,2 V p ,3 V p ,4 G p with Q =2 v v v J +1 v p v J +1 v v v v J +2 v J +3 v J +4
01 3 J +124 J +2 5 J +3 6 J +4 G Fig. 2. An example of the constructed graphs with J = 6 and K = 4 , where the corresponding paths and vertices are markedby the same color. Based on this fact, for any two vertices in G p , we add an edge between them if and only iftheir corresponding paths in G are neighbor-disjoint. As | V p | = KQ , we need to execute thisprocedure KQ ( KQ − / times; while in each execution, we need to check the connectivitybetween any two vertices in the two corresponding paths in G , respectively, so as to determinewhether they are neighbor-disjoint or not. Since the number of vertices in any path in G shouldnot exceed | V | = J + K +1 , the worst-case complexity of this step is given by O ( K Q ( J + K ) ) .Finally, we assign each vertex v ( p ( q ) k ) in G p with a weight, which is equal to the sum of edgeweights of its corresponding path in G , i.e., c ( q ) k , obtained in Step 1.To relate the path graph G p to (P2), we first introduce the following definitions. Definition 2: A K -partite graph refers to a graph whose vertices can be partitioned into K disjoint sets, such that there is no edge between any two vertices within the same set. Definition 3:
A clique is a subset of vertices of an undirected graph, such that every twodistinct vertices in the clique are adjacent.Based on Definitions 2 and 3, we can verify the following facts, which specify the relationshipamong G p , G and G . Fact 1: G p is a K -partite graph, with the k -th disjoint set given by V p,k = { v ( p ( q ) k ) | q ≤ Q } , k ∈ K . Fact 2:
For K neighbor-disjoint paths from vertex 0 to vertices J + k, k ∈ K in G , if theircorresponding paths in G are included in P , they correspond to a clique of size K in G p .Fact 1 can be proved by noting that for any two vertices in each V p,k , k ∈ K , their corre-sponding paths in G should not be neighbor-disjoint, since they share the same end vertex J + k . For Fact 2, it can be easily verified based on Definition 3 and the definition of G p . For example, in Fig. 2, G p is a 4-partite graph and consists of two cliques of size 4, i.e., ( v ( p (1)1 ) , v ( p (1)2 ) , v ( p (1)3 ) , v ( p (1)4 )) and ( v ( p (2)1 ) , v ( p (1)2 ) , v ( p (1)3 ) , v ( p (1)4 )) , each corresponding to 4neighbor-disjoint paths in G . The two vertices v ( p (1)1 ) and v ( p (2)1 ) in V p, are not connected astheir corresponding paths in G share the same end vertex J + 1 .According to Facts 1 and 2, we aim to solve the following clique search problem, denoted as(P3).(P3) Find a clique of size K in a K -partite graph G p , whose maximum vertex weight isminimized. The optimal clique for (P3) corresponds to the best solution to (P2) among the paths in P .Thus, if Q is set to be sufficiently large, such that the optimal paths from node 0 to each node J + k, k ∈ K are included in P , the proposed algorithm ensures to find an optimal solution to (P2)(and hence (P1)), if (P3) is optimally solved. Accordingly, by tuning the value of its parameter Q , the proposed algorithm can flexibly balance between its performance and complexity.
3) Step 3: Clique enumeration.
To find the optimal solution to (P3), we can enumerate allcliques of size K in G p and then compare their respective maximum vertex weights. However,finding all cliques of size K in a graph is also an NP-complete problem in general when K > [30]. As such, we propose a recursive algorithm to achieve this purpose by leveraging the K -partite property of G p , thereby optimally solving (P3).Specifically, we will show that each clique of size K in G p can be recursively constructedbased on the cliques of smaller sizes. Note that its K vertices must be selected from the K disjoint sets V p,k , k ∈ K , respectively. Without loss of optimality, we assume that its k -th vertexis selected from V p,k . Accordingly, let Ω r , r ≤ K denote the set of all cliques of size r in G p ,with the s -th vertex of each clique in Ω r selected from V p,s , s = 1 , , · · · , r . Obviously, we have Ω = V p, . Moreover, for each clique (of size r ) in Ω r , r ≤ K − , if there exists a vertex in V p,r +1 which is adjacent to all vertices in this clique, then a new clique (of size r + 1 ) in Ω r +1 can be constructed by appending the vertex to this clique. As such, based on the initial conditionfor Ω and the recursion for Ω r , r ≤ K − , all cliques of size K in G p can be enumerated in theset Ω K , which requires the worst-case complexity of O ( Q K ) . To further reduce complexity, itis noted that when a clique of size K − is constructed, among all feasible vertices in V p,K , weonly need to append the vertex with the lowest weight to it. This is because the cliques obtainedby appending other feasible vertices cannot yield a lower maximum vertex weight. Thus, the worst-case complexity of the above recursive algorithm can be reduced to O ( Q K − ) . In fact,since the number of feasible vertices may significantly decrease when increasingly larger cliquesare constructed (owing to the more stringent adjacency constraint), the actual complexity of theproposed recursive enumeration is much lower than O ( Q K − ) , as will be shown in Section V.Denote by C i the i -th clique (of size K ) in Ω K after the enumeration. For each C i ∈ Ω K , wecan obtain the maximum vertex weight among all of its K vertices, denoted as c i = max v ( p ( q ) k ) ∈C i c ( q ) k . Thus, the best clique in Ω K can be obtained as C i (cid:63) , with i (cid:63) (cid:44) arg min i c i . The main procedures ofthe proposed clique enumeration method for solving (P3) are summarized in Algorithm 1, wherea function “R EC E NUM ” is defined and recursively called to achieve the recursive enumeration.
4) Step 4: Map and output.
Finally, a generally suboptimal MBMH routing solution with afinite value of Q can be obtained by mapping the K vertices in C i (cid:63) to K neighbor-disjoint pathsin G . The process of solving (P2) is summarized in Algorithm 2. The worst-case complexity ofAlgorithm 2 is given by the sum of the complexity of the first three steps, i.e., O ( KQ | V || E | + KQ | V | log | V | + K Q ( J + K ) + Q K − ) .It is worth noting that if Ω K = ∅ with a given Q after performing Algorithm 1, this indicatesthat (P3) is infeasible. To obtain a feasible clique of size K , the value of Q can be increasedto enlarge the solution set of (P3). However, if (P3) is still infeasible even with the maximumallowable Q , then it can be claimed that (P2) (thus (P1)) is infeasible. As such, some userswould be denied access to the considered system. In this case, the proposed algorithms can helpdetermine the optimal user selection and the reflection paths for the selected users. Specifically,let K (cid:48) be the maximum number of users that can be granted access to the considered system,which is given by the maximum value of k such that Ω k (cid:54) = ∅ . Then, the selected users andtheir reflection paths can be obtained by mapping the best clique (of size K (cid:48) ) in Ω K (cid:48) to K (cid:48) neighbor-disjoint paths in G . C. Special Case with Continuous IRS Beamforming
If the continuous beamforming with b and b → ∞ is applied at each IRS, (P1) can be moreefficiently solved based on G , without the need of constructing its line graph G . Specifically,since we have ˜ A ( k ) n = M in this case, (31) becomes | h ,J + k (Ω ( k ) ) | − = M N B N k (cid:89) n =0 d a ( k ) n ,a ( k ) n +1 M β , k ∈ K . (39) Algorithm 1
Proposed Clique Enumeration Method for Solving (P3) Initiate r = 1 and a clique C = ∅ . Execute R EC E NUM ( r, C ) and obtain Ω K . Compare the maximum vertex weights for all obtained cliques in Ω K , i.e., c i ’s, and determinethe best clique C i (cid:63) . function R EC E NUM ( r, C ) if r = K then Among all vertices in V p,K which are adjacent to every vertex in C , append the vertex with the lowest weight to C and obtain a new clique of size K , C (cid:48) . Add C (cid:48) to the set Ω K . else Initialize s = 1 . while s ≤ Q do if C = ∅ or the s -th vertex in V p,r is adjacent to every vertex in C then Append this vertex to C and obtain a clique of size r , C (cid:48) . Add C (cid:48) to the set Ω r and execute R EC E NUM ( r + 1 , C (cid:48) ) . end if Update s = s + 1 . end while end if end function By taking the logarithm of (39) and discarding irrelevant constant terms therein, (P1) becomesequivalent to min { Ω ( k ) } k ∈K max k ∈K N k (cid:88) n =0 ln d a ( k ) n ,a ( k ) n +1 M √ β , s.t. (7)-(9) . (40)Similarly as in Section IV-A, we construct the DAG G = ( V , E ) . However, according to(40), we can directly assign a weight to each edge ( i, j ) in E , denoted as W ( i, j ) = ln d i,j M √ β . Asa result, (P2) reduces to finding K neighbor-disjoint paths from vertex to vertices J + k, k ∈ K in G , respectively, such that the length of the longest path is minimized. To solve this simplifiedproblem, Algorithm 2 can be similarly applied. The only difference is that the input graph is G instead of G . Algorithm 2
Proposed Algorithm for Solving (P2) Input the line graph G and the number of candidate shortest paths for each user, Q . Find Q candidate shortest paths from vertex 0 to each vertex J + k, k ∈ K by invoking theYen’s algorithm and determine the path set P = { p ( q ) k , k ∈ K , q ≤ Q } . Construct the path graph G p = ( V p , E p ) with the following steps based on P . a) Make a vertex for each path in P , i.e., V p = { v ( p ( q ) k ) | k ∈ K , q ≤ Q } . b) Add an edge between any two vertices in G p if their corresponding paths in G are neighbor-disjoint to determine E p . c) Assign each vertex v ( p ( q ) k ) in G p with the weight c ( q ) k . Obtain C i (cid:63) by performing Algorithm 1. Map the K vertices in C i (cid:63) to K neighbor-disjoint paths in G and output them. z ( m ) y (m) x (m)
453 5550454040 35 BSIRS User (a) 3D plot
40 45 50 55 60 x (m) y ( m )
123 456 7 891011 12 1314 1516 1718 1920 212223 24User 4User 1User 2 User 3BS (b) Graph representationFig. 3. Simulation setup.
V. N
UMERICAL R ESULTS
In this section, we provide numerical results to evaluate our proposed MBMH routing design.We focus on an indoor multi-IRS aided system (e.g., in a smart factory), as shown in Fig. 3(a).The heights of the BS and all users are assumed to be 3 m and 1.5 m, respectively, while theheights of all IRSs are randomly generated between 1.5 m and 3 m. The system is assumed tooperate at a carrier frequency of 5 GHz. Thus, the carrier wavelength is λ = 0 . m and theLoS path gain at the reference distance 1 m is β = ( λ/ π ) = − . dB. Based on the LoS probability specified in [31], we consider that there is an LoS link between two nodes i and j ,i.e., l i,j = 1 , i, j ∈ V , if its occurrence probability is greater than 0.99, or d i,j ≤ d A = λ/ and d I = λ/ , respectively.Moreover, we set the minimum distance for far-field propagation as d = G , is shown in Fig. 3(b).The numbers of elements in each IRS’s horizontal and vertical dimensions are set to be identicalas M (cid:44) √ M = M = M . The BS is equipped with N B = 32 antennas. We use the N B -pointDFT-based codebook as the BS’s codebook W B , which equally divides the spatial domain [0 , into N B sectors. Specifically, let w B,i ∈ C N B × denote the i -th beam pattern in W B . We have w B,i = 1 √ N B e (cid:16) i − N B , N B (cid:17) , i = 1 , , · · · , N B . (41)It is verified via simulation that with the deployment of IRSs in Fig. 3(a) and the codebook in(41), the asymptotically favorable propagation in (28) can be achieved for all links between theBS (node 0) and the possible first-hop IRSs (the neighbors of node 0 in G ). The numbers ofcontrolling bits for each IRS’s codebooks in the horizontal and vertical dimensions, i.e., W (1) I and W (2) I , are assumed to be identical as b (cid:44) b/ b = b . Thus, the number of beampatterns in W (1) I and W (2) I is identical to D (cid:44) b = √ D . Similarly as the BS, the D -pointDFT codebook is used for W (1) I and W (2) I . Let θ (1) I,i and θ (2) I,i denote the i -th beam patterns in W (1) I and W (2) I , respectively. Then, we have θ (1) I,i = θ (2) I,i = e (cid:16) i − D , M (cid:17) , i = 1 , , · · · , D . (42)In the proposed recursive algorithm, the number of candidate shortest paths for each node J + k or user k, k ∈ K is set to Q = 20 .First, Fig. 4 shows the optimized reflection paths for all users under different numbers of IRSreflecting elements and controlling bits for the IRS codebook in each dimension, i.e., M and b . In Fig. 4(a), by utilizing the Bellman-Ford algorithm [30] for the shortest path problem on G , we plot the optimal reflection path for each user without the path separation constraints in(9) under M = 20 (i.e., M = 400 ) and b = 7 (i.e., b = 14 ) bits. It is observed that thereexist LoS links between the paths for users 1, 2 and 3, as highlighted in dashed lines, whichmay result in severe scattered inter-user interference. Thus, the proposed algorithm is neededto obtain a feasible MBMH routing solution to (P1) that meets (9). In Figs. 4(b)- 4(d), we plot
40 45 50 55 60 x (m) y ( m )
13 456 7 891011 12 131416 1718 1920 212223 24 152 7User 1User 2 User 3BS User 4 (a) M = 20 , b = 7 , without (9)
40 45 50 55 60 x (m) y ( m ) (b) M = 20 , b = 7 , with (9)
40 45 50 55 60 x (m) y ( m )
18 7 1316 239
23 456 81011 1214 15171920 2122 24User 1User 2 User 3BS User 4 (c) M = 20 , b = 4 , with (9)
40 45 50 55 60 x (m) y ( m )
23 456 7 891011 12 1314 1516 1718 1920 212223 2423 592216 311 24 126 7 810 12 1314 15171820 21User 1User 2 User 3BS User 4 (d) M = 27 , b = 7 , with (9)Fig. 4. Optimized reflection paths under different setups. the optimized MBMH routing solutions by the proposed algorithm subject to (9). By comparingFig. 4(b) with Fig. 4(a), it is observed that the paths for users 2 and 3 are changed due to thepath separation constraints in (9). As a result, their effective channel powers with the BS aresacrificed in order to yield K neighbor-disjoint paths between the BS and all users. On the otherhand, by comparing Fig. 4(b) with Fig. 4(c), it is observed that for a given M , increasing theresolution of the IRS codebook may lead to different optimized paths. This is expected sincewith a larger b , each IRS has a higher degree of freedom in controlling the direction of thereflected signal, which may result in different reflection paths. Next, by comparing Fig. 4(b) withFig. 4(d), it is observed that when b = 7 , the optimized paths for some users, e.g., users 1 and
4, may go through more IRSs under M = 27 than those under M = 20 . This is due to thedifferent dominating effects of the end-to-end path loss and the CPB gain in maximizing theusers’ effective channel powers with the BS as b becomes large. In particular, as M = 20 ,minimizing the end-to-end path loss is dominant over maximizing the CPB gain. However, as M increases to , maximizing the CPB gain becomes more dominant. Since the CPB gainmonotonically increases with the hop count of the reflection path when b is large, the optimizedreflection paths generally go through more IRSs. Number of Candidate Shortest Paths per User, Q -50-49-48-47-46 E ff e c t i v e C hanne l P o w e r ( d B ) M =30M =29M =28 Fig. 5. Max-min channel power versus the number of candidate shortest paths, Q . Fig. 5 shows the maximized minimum (max-min) BS-user channel power among all usersby the proposed recursive algorithm versus the number of candidate shortest paths per user Q ,under b = 7 and M = 28 , and . It is observed that the max-min BS-user channel poweris monotonically non-decreasing with Q , since increasing Q enlarges the size of the solutionset of (P3). It can be verified that the performance of the proposed algorithm cannot be furtherimproved by increasing Q when Q ≥ , and under M = 28 , and , respectively.This implies that the optimal solution to (P2) (thus (P1)) is likely to be found by the proposedalgorithm. It is also observed that a larger Q is needed to find a feasible or a closer-to-optimalsolution to (P2) as M increases. The reason is that for any given Q , the Q candidate shortest pathsfor each user generally go through more IRSs with increasing M due to the more significanteffect of CPB gain. As a result, the paths for different users in K are more likely to be close to each other and thus may violate (9). Thus, Q generally increases with M to yield a feasibleor better solution to (P3). Nonetheless, it is worth mentioning that even with a large Q (e.g., Q ≥ ), the running time of the proposed clique enumeration method in Algorithm 1 is onlyaround 0.06 seconds, which is very low for practical implementation.
20 22 24 26 28 30 32
Number of IRS Reflecting Elements in each Dimension, M -85-80-75-70-65-60 E ff e c t i v e C hanne l P o w e r ( d B ) Proposed AlgorithmMaximum Path Loss MinimizationMinimum CPB Gain MaximizationSequential Update (a) b = 4
20 22 24 26 28 30 32
Number of IRS Reflecting Elements in each Dimension, M -70-65-60-55-50-45-40 E ff e c t i v e C hanne l P o w e r ( d B ) Proposed AlgorithmMaximum Path Loss MinimizationMinimum CPB Gain MaximizationSequential Update (b) b = 7 Fig. 6. Max-min channel power versus number of IRS reflecting elements in each dimension, M . Next, Fig. 6 shows the max-min BS-user channel power among all users by different schemesversus the number of IRS reflecting elements in each dimension, M , under b = 4 and b = 7 .For performance comparison, we consider the following three benchmark schemes. The first benchmark is the sequential update scheme, as mentioned at the end of Section IV-A. As itsperformance critically depends on the order of the update for the users, we enumerate all possible K ! orders and show its best performance. The second benchmark minimizes the maximum pathloss among all BS-user LoS links, while the third benchmark maximizes the minimum CPB gainamong all BS-user LoS links. Their corresponding reflection paths can be obtained by assumingunit CPB gain and unit end-to-end path loss, i.e., | ˜ A ( k ) n | = 1 , ∀ n, k and κ (Ω ( k ) ) = 1 , ∀ k , in ourproposed algorithm, respectively.First, it is observed from Fig. 6(a) that when b = 4 or the resolution of the IRS codebook islow, the sequential update scheme can yield the same performance as the proposed algorithm.However, the second and third benchmarks are observed to achieve a much worse performanceas compared to the proposed algorithm, which even degrades as M increases. This is becausethe former fails to take into account the effect of AoAs and AoDs in the network when b Number of controlling bits for IRS codebook in each Dimension, b -160-140-120-100-80-60-40 E ff e c t i v e C hanne l P o w e r ( d B ) Continuous IRS BeamformingProposed AlgorithmMaximum Path Loss MinimizationMinimum CPB Gain MaximizationSequential Update
Fig. 7. Max-min channel power versus the number of controlling bits for IRS codebook in each dimension, b . is small, as discussed at the end of Section III-A; while the latter overestimates the effect ofpassive beamforming gain and overlooks that of end-to-end path loss. On the other hand, as b increases to , it is observed from Fig. 6(b) that the sequential update scheme yields a worseperformance than the proposed algorithm when M = 26 and even fails to output feasible pathswhen M > . Different from Fig. 6(a), the second and third benchmarks are observed to achievethe same performance as the proposed algorithm when M ≤ and M ≥ , respectively. Thisis because the CPB gain is greatly improved with increasing b and the effect of AoAs and AoDsdiminishes. As such, the CPB gain and end-to-end path loss can dominate the BS-user effectivechannel power when M is large and small, respectively. However, when M is moderate, e.g., M = 27 and , these two schemes are observed to yield worse performance than the proposedalgorithm, which strikes a better trade-off between maximizing the CPB gain and minimizingthe end-to-end path loss.Finally, in Fig. 7, we plot the max-min BS-user channel power by the proposed algorithm andthe above three benchmark schemes versus the number of controlling bits for IRS codebook ineach dimension, b , under M = 28 . In addition, we also show the performance by the continuousIRS beamforming with b → ∞ . It is observed that the continuous IRS beamforming yieldsthe largest max-min BS-user channel power among all schemes considered. This is because themaximum passive beamforming gain can be achieved at each selected IRS for any given reflection paths, i.e., ˜ A ( k ) n = M, ∀ n, k . Nonetheless, as b increases, it is observed that the performanceof the proposed algorithm improves and ultimately achieves a performance very close to thecontinuous IRS beamforming as b ≥ . In contrast, the sequential update scheme is observed toonly output feasible reflection paths as b = 4 , and . Although the second benchmark yieldsthe same performance as our proposed algorithm when b = 6 , its performance becomes worsethan ours when b decreases or increases. The reason is that it fails to consider the effect ofAoAs and AoDs in the network as b is small or reconcile the trade-off with the CPB gain as b is large. The third benchmark is observed to achieve a worse performance than our proposedalgorithm as b ≥ , since it overlooks the end-to-end path loss. However, it yields the sameperformance as the proposed algorithm when b is extremely small, e.g., b = 1 and . Thisindicates that the AoAs and AoDs in the network or the placement of IRSs may dominate theeffective BS-user channel powers in this regime. All the above observations are consonant withour analysis provided at the end of Section III-A.VI. C ONCLUSIONS
This papers studies a new MBMH routing problem for a multi-IRS aided massive MIMOsystem, where cascaded LoS links are established between the multi-antenna BS and multipleusers by exploiting the cooperative signal reflections of selected IRSs. We present the optimalactive and passive beamforming solutions at the BS and each selected IRS, respectively. However,under the stringent path separation constraints for avoiding the inter-user interference, the MBMHrouting problem is NP-complete and challenging to solve. To derive a high-quality subopti-mal solution without incurring prohibitive complexity, we propose a parameterized recursivealgorithm for this problem by leveraging graph theory. It is shown that both the number ofIRS reflecting elements and size of IRS beamforming codebook can greatly impact the optimalMBMH routing solution as well as the achievable max-min BS-user channel power. In particular,the optimal MBMH routing design should take into account the AoAs and AoDs in the system ifthe size/resolution of IRS beamforming codebook is not large. Besides, there exists a fundamentaltrade-off between minimizing the end-to-end path loss and maximizing the CPB gain, which havedifferent dominating effects under different numbers of IRS reflecting elements.This paper can be extended in several promising directions for future work. First, it is inter-esting to study the MBMH routing problem under the general multi-path channel model. In thiscase, the MBMH routing problem becomes more challenging to be solved as the beamforming design cannot be simplified by assuming the LoS inter-IRS channels. Moreover, how to efficientlyfind the optimal active/passive beamforming solution in this case without assuming any priorchannel knowledge is also challenging. Second, the considered MBMH routing problem maybecome infeasible as the number of users is large or some users are close to each other inlocation. More sophisticated MBMH routing solution with relaxed path separation constraints isthus needed to yield more feasible reflection paths while mitigating the inter-user interferenceeffectively. R EFERENCES [1] W. Mei and R. Zhang, “Cooperative multi-beam routing for multi-IRS aided massive MIMO,” 2020. [Online]. Available:https://arxiv.org/pdf/2011.02354.pdf[2] Q. Wu and R. Zhang, “Towards smart and reconfigurable environment: Intelligent reflecting surface aided wireless network,”
IEEE Commun. Mag. , vol. 58, no. 1, pp. 106–112, Jan. 2020.[3] Q. Wu, S. Zhang, B. Zheng, C. You, and R. Zhang, “Intelligent reflecting surface aided wireless communications: Atutorial,” 2020. [Online]. Available: https://arxiv.org/pdf/2007.02759.pdf[4] E. Basar et al. , “Wireless communications through reconfigurable intelligent surfaces,”
IEEE Access , vol. 7, pp. 116 753–116 773, Sep. 2019.[5] Q. Wu and R. Zhang, “Intelligent reflecting surface enhanced wireless network via joint active and passive beamforming,”
IEEE Trans. Wireless Commun. , vol. 18, no. 11, pp. 5394–5409, Nov. 2019.[6] S. Zhang and R. Zhang, “Capacity characterization for intelligent reflecting surface aided MIMO communication,”
IEEEJ. Sel. Areas Commun. , vol. 38, no. 8, pp. 1823–1838, Aug. 2020.[7] Y. Yang, B. Zheng, S. Zhang, and R. Zhang, “Intelligent reflecting surface meets OFDM: Protocol design and ratemaximization,”
IEEE Trans. Commun. , vol. 68, no. 7, pp. 4522–4535, Jul. 2020.[8] Y. Yang, S. Zhang, and R. Zhang, “IRS-enhanced OFDMA: Joint resource allocation and passive beamformingoptimization,”
IEEE Wireless Commun. Lett. , vol. 9, no. 6, pp. 760–764, Jun. 2020.[9] B. Zheng, Q. Wu, and R. Zhang, “Intelligent reflecting surface-assisted multiple access with user pairing: NOMA orOMA?”
IEEE Commun. Lett. , vol. 24, no. 4, pp. 753–757, Apr. 2020.[10] T. Hou, Y. Liu, Z. Song, X. Sun, Y. Chen, and L. Hanzo, “Reconfigurable intelligent surface aided NOMA networks,”
IEEE J. Sel. Areas Commun. , vol. 38, no. 11, pp. 2575–2588, Nov. 2020.[11] C. Pan, H. Ren, K. Wang, W. Xu, M. Elkashlan, A. Nallanathan, and L. Hanzo, “Multicell MIMO communications relyingon intelligent reflecting surfaces,”
IEEE Trans. Wireless Commun. , vol. 19, no. 8, pp. 5218–5233, Jun. 2020.[12] W. Mei and R. Zhang, “Performance analysis and user association optimization for wireless network aided by multipleintelligent reflecting surfaces,” 2020. [Online]. Available: https://arxiv.org/pdf/2009.02551.pdf[13] Q. Wu and R. Zhang, “Joint active and passive beamforming optimization for intelligent reflecting surface assisted SWIPTunder QoS constraints,”
IEEE J. Sel. Areas Commun. , vol. 38, no. 8, pp. 1735–1748, Aug. 2020.[14] C. Pan, H. Ren, K. Wang, M. Elkashlan, A. Nallanathan, J. Wang, and L. Hanzo, “Intelligent reflecting surface aidedMIMO broadcasting for simultaneous wireless information and power transfer,”
IEEE J. Sel. Areas Commun. , vol. 38,no. 8, pp. 1719–1734, Aug. 2020. [15] T. Jiang and Y. Shi, “Over-the-air computation via intelligent reflecting surfaces,” in Proc. IEEE Global Commun. Conf. ,Waikoloa, HI, USA, Dec. 2019.[16] T. Bai, C. Pan, Y. Deng, M. Elkashlan, A. Nallanathan, and L. Hanzo, “Latency minimization for intelligent reflectingsurface aided mobile edge computing,”
IEEE J. Sel. Areas Commun. , vol. 38, no. 11, pp. 2666–2682, Nov. 2020.[17] M. Cui, G. Zhang, and R. Zhang, “Secure wireless communication via intelligent reflecting surface,”
IEEE WirelessCommun. Lett. , vol. 8, no. 5, pp. 1410–1414, Oct. 2019.[18] X. Yu, D. Xu, Y. Sun, D. W. K. Ng, and R. Schober, “Robust and secure wireless communications via intelligent reflectingsurfaces,”
IEEE J. Sel. Areas Commun. , vol. 38, no. 11, pp. 2637–2652, Nov. 2020.[19] H. Lu, Y. Zeng, S. Jin, and R. Zhang, “Enabling panoramic full-angle reflection via aerial intelligent reflecting surface,”in
Proc. IEEE Int. Conf. Commun. Workshop , Dublin, Ireland, Jun. 2020.[20] S. Fang, G. Chen, and Y. Li, “Joint optimization for secure intelligent reflecting surface assisted UAV networks,”
IEEEWireless Commun. Lett. , 2020, early access.[21] Y. Han, S. Zhang, L. Duan, and R. Zhang, “Cooperative double-IRS aided communication: Beamforming design and powerscaling,”
IEEE Wireless Commun. Lett. , vol. 9, no. 8, pp. 1206–1210, Aug. 2020.[22] C. You, B. Zheng, and R. Zhang, “Wireless communication via double IRS: Channel estimation and passive beamformingdesigns,”
IEEE Wireless Commun. Lett. , 2020, early access. [Online]. Available: https://arxiv.org/pdf/2008.11439.pdf[23] B. Zheng, C. You, and R. Zhang, “Efficient channel estimation for double-IRS aided multi-user MIMO system,” 2020.[Online]. Available: https://arxiv.org/pdf/2011.00738.pdf[24] B. Zheng, C. You, and R. Zhang, “Double-IRS assisted multi-user MIMO: Cooperative passive beamforming design,”2020. [Online]. Available: https://arxiv.org/pdf/2008.13701.pdf[25] W. Mei and R. Zhang, “Cooperative beam routing for multi-IRS aided communication,”
IEEE Wireless Commun. Lett. ,2020, early access. [Online]. Available: https://arxiv.org/pdf/2010.13589.pdf[26] C. You, B. Zheng, and R. Zhang, “Fast beam training for IRS-assisted multiuser communications,”
IEEE Wireless Commun.Lett. , vol. 9, no. 11, pp. 1845–1849, Nov. 2020.[27] H. Q. Ngo, E. G. Larsson, and T. L. Marzetta, “Aspects of favorable propagation in massive MIMO,” in
Proc. IEEE Eur.Signal Process. Conf , Lisbon, Portugal, Sep. 2014, pp. 76–80.[28] J.-Y. Teo, Y. Ha, and C.-K. Tham, “Interference-minimized multipath routing with congestion control in wireless sensornetwork for high-rate streaming,”
IEEE Trans. Mobile Comput. , vol. 7, no. 9, pp. 1124–1137, Sep. 2008.[29] S. Waharte and R. Boutaba, “On the probability of finding non-interfering paths in wireless multihop networks,” in