To satisfy the demands of a thousand-fold increase network capacity, several advanced technologies were proposed in the past decade, including massive multiple-input multiple-output (MIMO), millimeter wave (mmWave) communications, and ultra-dense networks [1, 2, 3, 4, 5]. However, the energy consumption and hardware cost of the above technologies have been drastically increased due to the substantial power-hungry radio-frequency (RF) chains regained in MIMO/mmWave systems and a large number of pico/macro base stations (BSs) deployed in ultra-dense networks ,. To tackle the above issue, intelligent reflecting surface (IRS) has been recently proposed as a promising and energy-efficient solution to improve the wireless system performance cost-effectively [8, 9, 10, 11].
IRS is a programmable planar surface consisting of a large number of square metallic patch units, each of which can be digitally controlled independently to introduce different reflection amplitudes, phases, polarizations, and frequency responses on the incident signals , . For example, for a 1-bit control command, there are two phase responses, namely and , which can be realized by fabricating two different patch widths in a single-layered dielectric board . The main benefits of bringing IRS in the future wireless networks are discussed as follows. First, each metallic patch unit is able to dynamically adjust its reflecting coefficients with the help of a smart controller such that the desired signals and interfering signals can be added constructively and destructively at the desired receivers, respectively. For instance, the results in 
showed that for a single-user IRS-aided systems, the received signal-to-noise ratio (SNR) increases quadratically with the number of reflecting elements,, at the IRS, i.e., which is also known as the squared power gain. As for multiuser systems, the multiuser interference can be significantly suppressed by jointly optimizing the BS transmit beamforming and the IRS phase shift matrix. Second, due to the small structure size of a metallic patch unit, a typical IRS is capable of attaching hundreds of such metallic patch units in practice, thereby providing a significant beamforming gain for improving system performance. Third, since each unit in the IRS is a simple passive component composed of PIN-diodes without the need of active RF chains, the power consumption of the PIN-diode is much lower than that of an active antenna with RF chain. In fact, experiments conducted recently in  has shown that for a large IRS consisting of reflecting elements, the total power consumption is only .
Due to above appealing benefits, there have been considerable work on the development of IRS in wireless communication systems. The existing research works about IRS include channel estimation, joint passive beamforming (i.e., IRS phase shift matrix optimization), and BS transmit beamforming optimization. To fully reap the benefits of the IRS in wireless networks, acquiring accurate channel state information (CSI) is indispensable[16, 17, 18]. Once the BSs have obtained the CSI, the applications of IRS to different systems have been studied to enhance their performance with different performance design objectives ,[19, 20, 21, 22]. Different from the conventional precoding adopted at the BS only, the joint optimization of the BS transmit beamforming and the IRS phase shift matrix in IRS-aided systems is necessary to fully unleash the potential of IRS . For example, an IRS-aided single-cell wireless system was studied in , where the authors aimed at minimizing the transmit power at the BS by jointly optimizing the BS transmit beamforming and the IRS passive phase matrix under the assumption that the phase shifts at the IRS can be continuously adjusted. It was then extended to the practical case , where each of the reflecting elements can take only finite discrete phase shift values and the results unveiled that the squared power gain can still be achieved in this case. Besides information transmission, the applications of IRS is also appealing for substantially improving the performance of wireless power transfer systems as shown in ,,. Besides, a combination of symbol-level precoding and IRS techniques for a multiuser system was studied in , and a significant performance gain was obtained by the enhanced capability in mitigating. Furthermore, it was shown in  that artificial noise can be leveraged to improve the secrecy rate in the IRS-assisted secrecy communication, especially in presence of multiple eavesdroppers.
In the past decades, CoMP techniques have attracted great attention due to its ability of suppressing the intercell interference caused by the widely deployed pico-and macro-cells . As specified by the Third Generation Partnership Project (3GPP), there are mainly two CoMP transmission techniques: coordinated scheduling/coordinated beamforming coordinated multipoint (CS/CB-CoMP) transmission technique and joint processing coordinated multipoint (JP-CoMP) transmission technique . For the CS/CB-CoMP transmission technique, the user data is only available at one serving BS while the user scheduling and beamforming optimization are made with coordination among the BSs. In contrast, for the JP-CoMP transmission technique, the user data is available at all BSs in the multicell network, and the BSs are capable of transmitting the same data streams to one user simultaneously , . Note that the concept of JP-CoMP is similar to that of Cell-Free Massive MIMO with the same objective to achieve coherent processing across geographically distributed BSs so as to improve the system throughput ,. For Cell-Free Massive MIMO systems, the structure is relatively simple, where many single-antenna access points (APs) simultaneously serve a much smaller number of single-antenna users. However, for JP-CoMP systems, the transmitters can be equipped with multiple antennas that simultaneously support substantial multi-antenna users systems to improve the spectral efficiency. Furthermore, rather than deploying substantial APs in Cell-Free Massive MIMO systems, only one BS needs to be deployed in one cell in JP-CoMP systems, which is considerably cost-effective and energy-efficient. The question is whether the combination of JP-CoMP technique and IRS can provide symbiotic benefits. However, this research is still in its infancy, which motivates this work.
In this paper, we study an IRS-aided JP-CoMP downlink transmission in a multiple-user MIMO system, where multiple multi-antenna BSs serve multiple multi-antenna cell-edge users with the help of an IRS. Specifically, since cell-edge users suffer severe propagation loss due to the long distances between them and the BSs, we deploy an IRS in the cell-edge region to help the BSs to serve multiple cell-edge users. Note that an IRS can be attached to a building to provide a high probability in establishing line-of-sight (LoS) propagation for the BS-IRS link and IRS-user link, as shown in Fig.1. By exploiting JP-CoMP, joint transmission can be performed among all BSs to serve the desired cell-edge users. It is observed from Fig. 1 that each cell-edge user receives the superposed signals, one is from the BSs-user link and the other is from the BSs-IRS-user link. By carefully adapting the IRS phase shifts, multiuser interference in the system can be further suppressed. In addition, we compare the system performance between the considered JP-CoMP system and small-cell systems with multicell cooperation (i.e, CS/CB-CoMP systems). It is expected that by fully exploiting the user data, the intercell interference caused by the multiple BSs could be further suppressed by JP-CoMP, thereby achieving better performance than CS/CB-CoMP. However, it is still unknown, how much performance gain of JP-CoMP system can be achieved compared to that of CS/CB-CoMP systems with the help of IRS. In this paper, we study two different systems, namely the single user system and the multiuser system, and propose two different low-complexity suboptimal resource allocation algorithms, respectively. The simulation results demonstrate the superiority of our proposed IRS-aided JP-CoMP design, and show that our proposed IRS-aided JP-CoMP design can achieve significantly higher performance gain compared to the existing IRS-aided CS/CB-CoMP design. To the best of our knowledge, the JP-CoMP downlink transmission system assisted by the IRS has not been studied in the literature yet. The main contributions of this paper are summarized as follows:
We study a multicell network consisting of multiple users, multiple BSs, and one IRS. The BSs are connected by a central processor for a joint data processing, and the IRS is deployed at the cell-edge region for enhancing data transmission to the users. Taking into account the fairness among the users, the goal of this paper is to maximize the minimum achievable rate of the cell-edge users by jointly optimizing the transmit beamforming matrix at the BSs and the phase shift matrix at the IRS.
The formulated joint design problem is shown to be a non-convex optimization problem, which is difficult to solve optimally in general. As a result, we first transform the max-min achievable rate problem into an equivalent form based on the mean-square error (MSE) method. Then, we consider two scenarios: the single-user system and the multiuser system. For the single-user system, the BS transmit beamforming is optimally solved by the dual subgradient method when the IRS phase shift matrix is fixed, and the IRS phase shift matrix design problem is addressed by the Majorization-Minimization (MM) method when the BS transmit beamforming is fixed. Based on these two solutions, an efficient suboptimal iterative resource allocation algorithm based on alternating optimization is proposed. For the multiuser system, since the above algorithm for the single-user systems can not be applied, we transform the transmit beamforming into a second-order cone programming (SOCP) for a fixed IRS phase shift matrix, which can be efficiently solved by the interior point method. In addition, for the fixed transmit beamforming matrix, the IRS phase shift matrix is optimized based on the semidefinite relaxation (SDR) technique. Then, an efficient iterative algorithm is also proposed to alternately to optimize transmit beamforming matrix and IRS phase shift matrix.
Extensive simulations are conducted which demonstrate that with the assistance of an IRS, a significant throughput gain can be achieved compared to that without an IRS. In addition, our results also show that the proposed IRS-aided JP-CoMP design is superior to the IRS-aided CS/CB-CoMP design in terms of max-min rate.
The rest of this paper is organized as follows: Section II introduces the system model and problem formulation. In Sections III and IV, we study the IRS-aided single user and multiuser systems, respectively. Numerical results are provided in Section V, and the paper is concluded in Section VI.
Boldface lower-case and upper-case letter denote column vector and matrix, respectively. Transpose, conjugate, and transpose-conjugate operations are denoted by, , and , respectively. stands for the set of complex matrices. and , respectively, denote the identity matrix and zero matrix. For a square matrix , , , , and respectively, stand for its trace, determinant, inverse, and rank, while indicates that matrix is positive semi-definite. represents the th diagonal element of the matrix . denotes the gradient of the function with respect to . denotes the real part of a complex number. is a Hadamard product operator. is the expectation operator. . denotes a vector with each element being the phase of the corresponding element in . denotes the diagonalization operation. represents the vectorization operation. Big denotes the computational complexity notation. and stand for the Frobenius norm and the Eucliden norm, respectively. For a complex value , denotes the imaginary unit. In addition, denotes a circularly symmetric complex Gaussian vector with mean and covariance matrix .
Ii System Model And Problem Formulation
Ii-a System Model
Consider an IRS-aided JP-CoMP downlink transmission network, which consists of BSs, cell-edge users, and one IRS as shown in Fig. 1. We assume that each BS is equipped with transmit antennas, each cell-edge user is equipped with receiver antennas, and the IRS has reflecting elements. Denote the sets of BSs, users, and reflecting elements as , , and , respectively. We assume that the size of the considered overall area is small so that the delay between two paths are very small and can be neglected [30, 31]. Let , , and , respectively, denote the complex equivalent baseband channel matrix between the -th user and BS , between BS and the IRS, and between the IRS and the -th user, , .
Mathematically, the transmitted signals by BS , , is given by111In a JP-CoMP systems, the BSs are connected to a central processing for data and information exchange among BSs so that each user can be served by all the BSs simultaneously.
where represents desired data streams for user satisfying , stands for the transmit beamforming matrix for user by BS . Different from the traditional CSI estimation methods, which the channel estimation is performed on the receiver side with substantial processing units, however, each reflecting element at the IRS is passive without powerful processing units. As a result, the reflecting coefficients at the IRS and transmit pilots at the BS are jointly designed for acquiring CSI in single-cell systems [16, 17, 18]. In particular, the CSI for IRS-aided multicell systems can be directly obtained via activating one BS while turning off the other BSs in a take turn manner. As such, we assume that the CSI for all the channel links are perfectly known by the central processor. As shown in Fig. 1, each user receives not only the desired signals from the BSs, but also the reflected signals by the IRS. Note that different from CS/CB-CoMP multicell systems, where each user data is only available at one serving BS, each user data in JP-CoMP multicell systems is available at all BSs. The received signal at user is thus given by
where represents the phase shift matrix adopted at the IRS, where and , respectively, denote the amplitude reflection coefficient and phase shift of the -th reflecting element, is the received noise with denoting the noise power at each antenna. For the sake of low implementation complexity, in this paper, each element of the IRS is designed to maximize the signal reflection, i.e., , .
For notational simplicity, we define , , and . Then, we can rewrite (2) as
As such, the achievable data rate (nat/s/Hz) of user is given by
Ii-B Problem Formulation
In this paper, to guarantee the user fairness, we aim at maximizing the minimum achievable rate of the users by jointly optimizing the downlink transmit beamforming and the IRS phase shift matrix, subject to transmit power constraints at the BSs222To characterize the fundamental performance limits of JP-CoMP IRS-aided systems, we assume that the capacity backhaul links from the BSs to the central processor is sufficient for data information exchange among BSs ,.. Accordingly, the problem can be formulated as
where denotes the maximum BS transmit power. Although constraint (6) is convex and (7) is linear with respect to , it is challenging to solve problem due to the coupled transmit beamforming matrix and the phase shift matrix in (5). In general, there is no efficient method to solve problem optimally. To facilitate the solution development, we first transform problem into an equivalent form denoted by based on the mean-square error (MSE) method . Specifically, the achievable rate in (4) can be viewed as a data rate for a hypothetical communication system where user estimates the desired signal with an estimator , the estimated signal is given by
As such, the MSE matrix is given by
By introducing additional variables and , , we then have the following theorem:
Theorem 1: Problem is equivalent333Here, “equivalent” means both problems share the same optimal solution. to , which is shown as below:
Proof: Please refer to Appendix A.
Although introduces additional variables and , the new problem structure facilitates the design of a computationally efficient suboptimal algorithm. In the following, we first consider a single cell-edge user system, where the transmit beamforming matrix and phase shift matrix are obtained based on the dual subgradient method and majorization-minimization method, respectively. Then, we consider the joint IRS phase shift and transmit beamforming optimization problem in the multiuser system which is then handled by applying the SOCP and SDR techniques, respectively.
Iii Single Cell-edge User System
In this section, we consider a single cell-edge user system, namely . For notational simplicity, we drop user index in this section. Then, the problem for the single-user system can be simplified as
Although simplified, is still difficult to handle due to the coupled optimization variables in the objective function of . However, we observe that both and are concave with respect to the objective function of . In addition, variable does not exist in the constraint set and the variable only appears in constraint (14).
By applying the standard convex optimization technique, setting the first-order derivative of the objective function of with respective to and to zero, the optimal solutions of and can be respectively obtained as
To address the coupled transmit beamforming matrix and phase shift matrix, we first decouple into two sub-problems, namely transmit beamforming optimization with the fixed phase shift matrix and phase shift matrix optimization with the fixed transmit beamforming matrix, and then an iterative method is proposed based on the alternating optimization .
Iii-a Transmit Beamforming Matrix Optimization with Fixed Phase Shift Matrix
We first consider the first sub-problem of , denoted as , for optimizing the BS transmit beamforming matrix by assuming that the IRS phase shift matrix is fixed. By dropping the irrelevant constant term , the transmit beamforming matrix optimization problem can be simplified as
Problem is a standard convex optimization problem which can be solved by the convex tools such as CVX . Instead of relying on the generic solver with high computational complexity, we propose an efficient approach based on the Lagrangian dual subgradient method. Note that it can be readily checked that problem satisfies the Slater’s condition, thus, strong duality holds and its optimal solution can be obtained via solving its dual problem . In the following, we solve by solving its dual problem. Specifically, by introducing dual variable , corresponding to constraint (13), we have the Lagrangian function of given by
Accordingly, the dual function of is given by
Setting the first-order derivative of with respect to to zero yields
By collecting and stacking above equations, the optimal transmit beamforming matrix can be obtained as
where is given by
and is given by
Next, we address the corresponding dual problem, which is given by
It can be seen that the dual problem has no additional constraints. In addition, with any fixed dual variable , the optimal transmit beamforming matrix can be directly solved in a closed-form as in (21). As such, we propose an efficient method, namely subgradient method, to solve the dual problem . The update rule of parameters is given by
where superscript denotes the iteration index and represents the positive step size for updating . The detailed descriptions of the dual subgradient method are summarized in Algorithm 1.
Iii-B Phase Shift Matrix Optimization with Fixed Transmit Beamforming
Next, we consider the second sub-problem of , denoted as , for optimizing the phase shift matrix, , by assuming that the transmit beamforming matrix, , is fixed. By dropping the constant term , the phase shift matrix optimization problem can be simplified as
Problem is a non-convex optimization problem due to the non-convex objective function. To address this issue, by expanding and , we have
where , , , and .
Similarly, we have
where and . As such, we can equivalently transform the objective function of as (by dropping constants and )
Define , where , and . Additionally, we have the following identities 
Then, we can rewrite in (29) as
As a result, problem is equivalent to
Problem is non-convex due to the unit-modulus constraints in (32). Here, we handle based on the MM method, which guarantees at least a locally optimal solution with a low computational complexity ,. The key idea of using the MM algorithm lies in constructing a sequence of convex surrogate functions. Specifically, at the -th iteration, we need to construct an upper bound function of , denoted as , that satisfies the following three properties ,: (a) ; (b) ; (c) , where (a) denotes that is an upper-bounded function of , (b) represents that and have the same solutions at point , and (c) indicates and have the same gradient at point .
Note that in (31), we can see that and are semidefinite matrices, and it can be readily checked that is also a semidefinite matrix. In the sequence, we have the following lemma:
Lemma 1: Based on , at the -th iteration, the surrogate function for a quadratic function can be expressed as
is the maximum eigenvalue of. Therefore, at any -th iteration, we solve the following problem
Since , at the -th iteration, we can rewrite as
where . Obviously, the optimal solution to minimize problem is given by
The details of the proposed MM method are summarized in Algorithm 2.
Iii-C Overall Algorithm and Complexity Analysis
Based on the solutions to two sub-problems, an efficient iterative algorithm is proposed, which is summarized in Algorithm 3. The complexity analysis of Algorithm 3 is given as below. In step 3, the complexity of computing is . In step 4, the complexity of computing is . In step 5, the complexity of computing is , where is number of iterations required for updating ,. In step 6, the complexity of computing the maximum eigenvalue, i.e., , of is , and the complexity of computing is , then the total complexity of Algorithm 2 is , where is the total number of iterations required by Algorithm 2 to converge. Therefore, the total complexity of Algorithm 3 is , where represents the total number of iterations required by Algorithm 3 to converge.
Iv Multiple Cell-Edge Users System
In this section, we consider the multiuser scenario shown in Fig. 1. To handle problem , it can be seen in Appendix A, the optimal and can be directly obtained from (59) and (60), respectively. Similar to the single-user system, we decompose into two sub-problems, namely transmit beamforming matrix optimization with the fixed phase shift matrix and performing phase shift matrix optimization with the fixed transmit beamforming matrix. Note that the proposed MM method and the dual subgradient method in the single-user system cannot be applied to the multiuser system due to constraint (10) in . However, in the following, we resort to SOCP technique to solve the transmit beamforming matrix optimization sub-problem, and the SDR technique to address the phase shift matrix optimization sub-problem.
Iv-a SOCP for Transmit Beamforming Matrix Optimization
By fixing the phase shifts at the IRS, the transmit beamforming optimization problem is
It is not difficult to observe that is a convex optimization problem and can be transformed into an semidefinite program (SDP) problem. According to , the SOCP has a much lower worst-case computational complexity than that of the SDP method by applying the interior-point method to solve problem . We have the following theorem:
Theorem 2: Problem is equivalent to the following SOCP problem:
Proof: Please refer to Appendix B.
Therefore, is a standard SOCP problem, which can be optimally solved by the interior point method .
Iv-B SDR Technique for Phase Shift Matrix Optimization
Next, by fixing the transmit beamforming matrix, the phase shift matrix optimization problem, denoted by , can be formulated as
Problem is non-convex due to the non-convex constraint (10). To tackle this non-convex problem, the SDR technique is applied. By using and , we have
As a result, we can rewrite as
where . Similar to (31), define , where , and