I Introduction
Massive multipleinput multipleoutput (MIMO) is a promising technology to meet the ever growing demands for higher throughput and better qualityofservice of the fifthgeneration (5G) and beyond wireless networks [20, 13, 16]. However, the high hardware complexity and cost required by massive MIMO systems are still the main roadblock that hinders its implementation in practice, especially in the high frequency band such as millimeter wave (mmWave) bands [2, 33]. Moreover, due to unfavorable propagation conditions, the link between the base station (BS) and users might be highly vulnerable to blockages, thus making the communication unstable and inefficient [23, 1]. Therefore, it is necessary to develop new spectrum and energy efficient technologies for massive MIMO systems.
In order to achieve high beamforming gains with lowcost systems, intelligent reflecting surface (IRS), also known as large intelligent surface (LIS), has been proposed as a promising technology in recent years [15, 9, 14, 30]. IRS is a planar array made of newly developed metamaterial, consisting of a large number of costeffective and energyefficient passive reflecting elements [30, 10]. Each reflecting element is controlled by a smart micro controller so that the incident signal can be reflected with reconfigurable amplitudes and phase shifts. By adaptively tuning the phase shifts of reflecting elements, IRSs are able to enhance the received signal power or suppress the cochannel interference for desired users, thus improving the coverage and performance of wireless systems [27].
IRSaided wireless communications have attracted much attention recently [26, 25, 19, 28, 27, 7, 29, 24, 8, 11, 31, 4, 32, 21, 6, 17]. In [26, 25], it was shown that for a singleuser scenario, the IRSassisted system can obtain a received signal power gain in the order of , compared with the conventional massive MIMO system that achieves a received signal power gain in the order of . Here denotes the number of antennas at the transmitter and denotes the number of reflecting elements at the IRS. Such an improvement is due to the fact that IRS works as the receiver and transmitter simultaneously. In [19], authors studied the problem of maximizing the minimum signaltointerferenceplusnoise ratio (SINR) in a multiuser scenario. It was empirically shown that for the multiuser case, IRSassisted systems can offer massive MIMO like gains with a much fewer number of active antennas. Most prior works on joint active and passive beamforming, e.g. [28, 27, 26, 19, 25, 7], either focus on the scenario where only a single IRS is employed or the scenario where multiple IRSs are deployed to serve a single user. The scenario where multiple IRSs are used to serve multiple users has been rarely investigated. The work [29] studied the joint beamforming problem in such a scenario, and their aim is to minimize the total transmit power under an individual SINR constraint for each user.
In this paper, we consider the scenario where multiple IRSs equipped with a large number of passive reflecting elements are deployed to assist the BS to simultaneously serve a small number of singleantenna users. Our objective is to maximize the minimum SINR at users by jointly optimizing the transmit precoding vector at the BS and phase shift parameters at IRSs. An important theoretical finding made in this paper is that when the phase shift parameters of an IRS are optimally tuned to serve a certain user, due to the asymptotic orthogonality among channel vectors associated with different users, this IRS will become interferencefree to other users. Such a property is referred to as automatic interference cancelation (AIC), and is proved to hold valid asymptotically when for both the lineofsight (LOS)dominated IRSuser channel and the Rayleigh IRSuser channel. This property is important as it can help avoid complicated interIRS interference management. By utilizing this property, the maxmin problem can be converted into an IRSuser association problem whose objective is to determine which IRSs are assigned for each user. Our theoretical analysis reveals that the proposed solution attains an SINR in the order of , i.e., it scales quadratically with the number of reflecting elements. Also, this result suggests that even with a moderate number of active antennas at the BS, increasing the number of passive reflecting elements can help achieve massive MIMO like gains, thus significantly reducing the energy consumption at the BS.
The rest of the paper is organized as follows. In Section II, the system model and the joint active and passive beamforming problem are discussed. The proposed joint active and passive beamforming method is provided in Section III. The automatic interference cancelation property is discussed and proved in Section IV. The theoretical analysis of our proposed solution is presented in Section V. Simulation results are provided in Section VI, followed by concluding remarks in Section VII.
Ii System Model and Basic Assumptions
Consider an IRSassisted massive MIMO system, where multiple IRSs are deployed to assist the BS equipped with antennas to simultaneously serve () singleantenna users in the same timefrequency resource. Suppose IRSs are employed, and the number of reflecting units at each IRS is denoted by (). Let denote the channel from the BS to the th IRS, and denote the channel from the th IRS to the th user. In this paper, we neglect the direct link from the BS to each user in order to simplify our analysis and better understand the impact of IRSs on the system performance. Also, in this paper, we assume

The channel between the BS and each IRS is lineofsight (LOS) dominated and has a rankone structure.
This assumption is very likely to be met in practice because the IRS is usually installed on the facade of a high rise building in the vicinity of the BS [19]. As a result, the channel matrix between the BS and the IRS is dominated by the LOS component. Particularly, for mmWave frequency bands, measurement campaigns have shown that the power of LOS path is much higher than the sum of power of NLOS paths [18]. Thus the channel can be wellapproximated as a rankone matrix. Specifically, the BSIRS channel can be expressed as
(1) 
where is the complex gain, , , and are the associated azimuth angle of arrival (AoA), elevation AoA, and azimuth angle of departure (AoD) respectively, and () is the array response vector associated with the IRS (BS). Due to the rankone structure of the BSIRS channel, we need to assume

The number of IRSs is no less than the number of users, i.e. .
Otherwise it is impossible for the BS to serve users simultaneously. On the other hand, as a costeffective and energyefficient technology, it is envisioned that, in future wireless networks, IRSs are densely deployed to provide favorable propagation environments. In addition, we assume

The transmit array response vectors are near orthogonal to each other, i.e. for .
Such an assumption is reasonable because to enhance the signal coverage, IRSs are usually deployed surrounding the BS and as a result, the AoD parameters are expected to be sufficiently separated from each other. Particularly, when a large number of antennas are employed at the BS, the nearorthogonality property holds valid even for slightly separated AoD parameters .
With the aid of a smart micro controller, each element of the IRS can independently reflect the incident signal with a reconfigurable phase shift [28]. Therefore the composite channel between the BS and the th user is given by
(2) 
where
(3) 
is the phaseshift matrix associated with the th IRS, in which denotes the phase shift associated with the th passive element of the th IRS. The signal received at the th user can then be expressed as
(4) 
where with is the transmit beamforming vector for the th user, is the transmitted symbol for the
th user which is assumed to be independent and identically distributed (i.i.d.) random variables with unit variance,
is the transmit power allocated for the th user and we have the constraint , in which is the total transmit power, and denotes the additive complex Gaussian noise with zero mean and variance . Therefore, the signal to interference plus noise ratio (SINR) of the th user is given by(5) 
Assuming the knowledge of the global channel state information, our objective is to devise the transmit precoding vectors , the transmit power allocation , and the phase shift matrices to maximize the minimum SINR among SINRs associated with all users. Such a problem can be formulated as a maxmin problem:
s.t.  
(6) 
In the following, we will show that IRS equipped with a large number of passive elements enjoys an appealing automatic interference cancelation property. By resorting to such a property, the above maxmin problem can be converted into an IRSuser association problem, which can be analytically solved.
Iii Proposed Method
Our proposed method can be divided into two steps. In the first step, given passive beamforming parameters , we optimize active beamforming parameters, i.e. the transmit precoding vectors and the transmit power . The optimal active beamforming parameters are then substituted into (6) to obtain a passive beamforming problem. In the second step, we proposed an effective and analytical solution to this passive beamforming problem.
Iiia Active Beamforming Optimization
Given , we now discuss how to solve the active beamforming problem. Note that when the IRS parameters are fixed, this problem is simplified as a maxmin SINR problem for conventional massive MIMO systems, and its solution has already been proposed in [3, 22]. To gain insight into the problem, let us focus on the transmit power allocation problem first by assuming the precoding vectors are fixed. The transmit power allocation problem can be cast as
s.t.  (7) 
By introducing an auxiliary variable , the problem (7) can be rewritten as
s.t.  
(8) 
Using the KarushKuhnTucker(KKT) conditions and noting that all inequalities of (8) become equalities at the optimal point, it can be shown [3] that the optimal solution and satisfy the following conditions
(9)  
(10)  
(11)  
(12)  
(13)  
(14)  
(15) 
where and are optimal Lagrange dual variables. From (9), (10), and (11), we can arrive at
(16) 
The above KKT condition (11) indicates that the transmit precoding vector can be found via
s.t.  (17) 
whose optimal solution is given by
(18) 
Substituting (18) into (11), we have
(19) 
Based on (16) and (19), can be obtained via a fixedpoint iteration method. Specifically, we first randomly generate as the initial point. At the th iteration, we calculate
(20) 
and then update
(21) 
After is obtained, can then be calculated from (16) and (19):
(22) 
Now, define , , and
Equation (9) can be expressed as . Therefore the optimal transmit power can be obtained as
(23) 
IiiB Passive Beamforming Optimization
From (22), the joint beamforming problem (6) can be simplified as a passive beamforming problem:
s.t.  (24) 
Note that in the above optimization, both the composite channel vectors and are dependent on . Our objective is to devise the phase shift parameters to maximize the SINR . Such a problem, however, is challenging due to the nonconvexity of the objective function. Moreover, cannot be expressed as an explicit function of , which further complicates the problem. In the following, we will convert the above problem into a more amiable form which helps us gain insight into the problem.
Substituting (1) into (2), the composite channel can be expressed as
(25) 
where
(26)  
(27)  
(28) 
From (25), we see that the composite channel from the BS to the th user, , is a linear combination of the transmit array response vectors , with each transmit array response vector weighted by . The weight is referred to as the “passive beamforming gain” from the th IRS to the th user. Substituting (25) into the objective function of (24), we arrive at
(29) 
where , and .
Since the steering vectors are mutually orthogonal with each other (see A3), there exists a matrix such that
is a unitary matrix, i.e.
. Using this fact and the Woodbury matrix identity, we have(30) 
where
Therefore the passive beamforming problem (24) becomes
s.t.  (31) 
Denote
(32) 
as the passive beamforming matrix with its th entry given by . Clearly, to maximize the SINR , on one hand, we wish to make the passive beamforming gains as large as possible; on the other hand, we wish to be orthogonal to each other such that the crossinterference term is minimized. A key insight here is that when the phase shift parameters of the th IRS are optimally tuned to serve the th user, i.e. is maximized, due to the asymptotic orthogonality among channel vectors associated with different users, this IRS will be like nonexistent and interferencefree to other users, i.e. for . In this case, each row of the passive beamforming matrix contains only a single nonzero element, and as a result, its columns are mutually orthogonal. Such a property, termed as automatic interference cancelation (AIC), will be rigorously proved in Section IV. This property is important as it can help avoid complicated interIRS interference management.
Based on the AIC property, a simple yet effective solution is to divide IRSs into disjoint groups, with parameters of each group of IRSs optimally tuned to serve one particular user. This problem is referred to as a user association problem. This solution is effective because, on one hand, it can achieve a reasonably large for each ; on the other hand, the crossinterference term disappears due to the orthogonality among . To formulate this user association problem, define as the maximum achievable passive beamforming gain. From (26), it can be easily verified that is given by
(33) 
which is attained when parameters associated with the th IRS are set as
(34) 
where represents the argument of the complex number , and denote the th entry of and , respectively. Let . The IRSuser association problem can be formulated as
s.t.  
(35) 
where entries of are chosen either to be or 0, denotes the th row of , and the constraint is due to the AIC property, i.e. when the phase shift parameters of the th IRS are optimally tuned to serve a certain user, this IRS will become interferencefree to other users. Altogether, is constructed by keeping only one element for each row of and setting the rest elements of equal to zero. Since we can choose any element from each row of , the number of feasible solutions is up to , from which we need to choose the one that maximizes the objective function in (35).
The simplest method to solve (35) is to exhaustively search all feasible solutions. When and are small, say , the computational complexity of this exhaustive search scheme is still practical. For the scenario where a large number of IRSs are deployed, we develop a greedy algorithm to search for the best IRSuser association. Specifically, we first choose the largest passive beamforming gain from , say is the largest (in terms of magnitude) in . Based on this result, the th IRS is allocated to the th user. Next, we nullify the th IRS and the th user by setting entries on the th row and the th column of equal to zero. Thus we obtain an updated passive beamforming gain matrix, denoted as . Then we choose the largest element (in terms of magnitude) in , say , and let the th IRS assigned to the th user. Again, the passive beamforming gain matrix is updated by setting entries of the th row and the th column of equal to zero. This procedure is repeated until each user is served by an IRS. If the number of IRSs is larger than the number of users, then we need to assign extra IRSs to users. We first set the th rows of equal to zero and obtain a new matrix . Then, we choose the largest entry (in terms of magnitude) in , say , and let the th IRS assigned to the th user. Next, the passive beamforming gain matrix is updated by setting the th row of equal to zero. This procedure is repeated until all IRSs are assigned. Although this greedy algorithm is not guaranteed to yield the optimal solution, it has a very low computational complexity.
Iv Automatic Interference Cancelation
In the previous section, we propose an effective solution that converts the passive beamforming problem into a user association problem, based on the property that when the th IRS is optimally tuned to serve the th user, this IRS will become nonexistent (i.e. interferencefree) to other users. Such a property is referred to as the AIC property. In this section, we consider two typical IRSuser channel models, namely, an LOSdominated channel model and a Rayleigh fading channel model. We will show that the AIC property holds asymptotically for both scenarios when .
Iva LOSDominated IRSUser Channel
Suppose the channel between the th IRS and the th user () is dominated by the LOS component, which is usually the case for mmWave communications [12]. In this case, can be written as
(36) 
where is the complex gain between the th IRS and the th user, () is the associated azimuth (elevation) AoD, and is the array response vector associated with the IRS. Assume a uniform planar array (UPA) is used at each IRS, the steering vector can be expressed as
where () denotes the number of elements along the horizontal (vertical) axis, , and is the coordinate of the reflecting element. From (26), we know that is given by
(37) 
where denotes the phase shift parameter of the th IRS’s reflecting element at the coordinate , and
(38) 
It is clear that is maximized when
(39) 
When the parameters of the th IRS are optimally tuned to serve the th user, we now show that this IRS becomes interferencefree to other users in an asymptotic sense, i.e. as for any . We have
(40) 
where
(41) 
Suppose the locations of different users are well separated such that and , . Then we have
(42) 
Hence the AIC property holds in an asymptotic sense as .
IvB Rayleigh Channel
For the scenario where there is no dominant propagation along an LOS between the IRS and the user, the channel can be modeled as an independent and identically distributed (i.i.d.) Rayleigh fading channel [27], i.e.
(43) 
where is a factor that depends on the distance between the th IRS and the th user. According to (34), the passive beamforming gain is maximized when
(44) 
where and denote the th entry of and , respectively. The can then be calculated as
(45) 
where . Since are i.i.d. with
, according to Khinchin’s law of large numbers, we have
(46) 
for any , i.e.
(47) 
when . Therefore, the AIC property holds asymptotically for the Rayleigh channel.
V Performance Analysis
In this section, we provide a theoretical analysis of our proposed method. Specifically, given the global channel state information and , we analyze the maxmin SINR attained by our proposed solution. According to (31), the maxmin SINR is given by
(48) 
In the previous section, we have shown that the AIC property holds asymptotically for both LOSdominated and Rayleigh fading channels. In other words, when the parameters of the th IRS are optimally tuned to serve the th user, this IRS is interferencefree to other users. As a result, vectors are orthogonal to each other, and we have
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