Joint Active/Passive Beamforming Optimization via RWMMSE and Gradient Projection for Downlink IRS-Aided MU-MISO Systems

Metamaterials are significantly advancing the way we wirelessly transmit electromagnetic signals between communicating devices. Specifically, intelligent reflecting surfaces (IRS), a metamaterial-derived technology with multiple passive reconfigurable reflecting elements, facilitate a wide array of novel opportunities. The proper reconfiguration of its elements, which is the primary task towards achieving these prospects, has been previously explored by many researchers, where most proposals are characterized by high computational complexity. This study builds upon previous research on downlink IRS-assisted multi-user multiple-input single-output (MISO) systems by introducing innovative methods to accelerate their operations. Specifically, we introduce a weighted sum rate problem and utilize the reduced weighted minimum mean square error (RWMMSE) algorithm to determine active precoders, alongside the employment of gradient projection (GP) method to enhance passive beamforming techniques. These two techniques undergo a novel restructuring process, where the RWMMSE framework, originally designed for MIMO systems, is transformed to work with a MISO system. The GP framework is enhanced with a Barzilai-Borwein method for the step size selection that ensures a fast convergence of the proposed algorithm, resulting in lower complexity. Ultimately, the numerical findings confirm our assertion, demonstrating the efficacy of our suggested algorithm in achieving performance similar to benchmark methods while significantly reducing complexity.

SECTION I. Introduction

Wireless communication encounters numerous challenges, including fading, scattering, and shadowing [1], [2], which complicate the transmission and reception of signals. In addition, the inherent nature of free propagation exposes transmitted information to eavesdropping, resulting in security concerns [2], [3], [4], and enables the interference of signals with each other. This results in further problems even within the same network operator. Various potential solutions have been proposed to address these issues, such as ultradense networks [5], [6], [7], [8], multiple-input multiple-output (MIMO) [9], [10], [11], [12], [13], [14], and cell-free massive MIMO [15], [16], [17], [18], [19], [20], [21], [22]. Although these solutions offer significant relief to the aforementioned challenges, they are primarily reactive, as they adjust their operations based on current channel conditions.

In recent years, both academia and the industry have explored the feasibility of manipulating channel conditions to suit specific requirements. These investigations have resulted in the emergence of artificial metamaterials [23], [24], [25], [26], [27], which are distinguished by their capability for real-time reconfiguration. Various technologies derived from metamaterials, including dynamic metasurface antennas [28], [29], intelligent transmitting surfaces [30], and intelligent reflecting surfaces (IRSs) [25], [31], [32], [33], have exploited this property to gain extra control over various wireless channel conditions, which consequently open up to several use cases such as coverage extensions, increasing the rank of the channel, which is crucial for spatial multiplexing, providing an alternative path in scenarios where the direct path is blocked, etc.

Moreover, these technologies can be creatively combined with numerous others to generate diverse sets [28], [30], [34] tailored to specific objectives. Among the various feasible combinations, the most prevalent one involves the integration of IRS with multi-antenna systems owing to their wide range of applications, including enhancing spectral efficiency, enabling interference mitigation, ensuring secure communication [35], and enhancing spatial multiplexing. This study primarily focuses on the IRS-supported downlink multi-user (MU) multiple-input single-output (MISO) system.

Briefly, the IRS is a metamaterial-based technology that involves a planar array consisting of many nearly passive reflecting elements. The small size of these elements allows them to be packed in a multitude within a small aperture. The IRS is also equipped with a controller, which is usually connected via a dedicated link to the base station (BS), to fetch instructions on how to coordinate the operation of each element. Subsequently, in most use cases, the IRS is deployed at the site of the BS and in the vicinity of the receiver, specifically by being plated on the surfaces of user-surrounding structures, such as furniture, ceilings, and building facades. [34]. When applied to aid MU MIMO communication, the system necessitates the simultaneous optimization of both the active (BS precoders) and passive (IRS phase shifts) beamformers, which is a highly non-convex challenging task, to realize its full capabilities. Previous studies have introduced noteworthy techniques for addressing this issue [36], [37], [38]; however, most such methods entail considerable computational complexity throughout their execution.

In a prior study [36], innovative methods were introduced to enhance the weighted sum rate (WSR) of an IRS-supported MU downlink MISO system. This involved the iterative optimization of the BS transmit precoders using the weighted minimum mean square error (WMMSE) algorithm [39] while adjusting the phase shifts at the IRS using either block coordinate descent (BCD) or manifold optimization (MO). Despite their impressive performance in maximizing WSR, these techniques are plagued by a significant increase in complexity, particularly with increase in the number of BS transmit antennas, thereby rendering them impractical. Speaking of MO, it is worth mentioning that this scheme has been adopted by several other works to solve various IRS phase-sift-related problems [40], [41], [42]. Specifically, the work [40], which remarkably reduced the computational complexity of alternating optimization (AO) in [43] for finding the optimal IRS phase shifts in the IRS-aided point-to-point downlink MIMO system, proposes two low-complexity sub-optimal schemes based on AO and MO. Similarly, the WMMSE technique has also been used to design the active beamformers of several other IRS-aided systems equipped with discrete-phased [44] and/or active [45] IRS, and operating under imperfect channel state information (CSI) [46].

Because of its high efficiency in tackling problems with deeply coupled optimization parameters, the AO technique has also been used in [38], [47] to optimize the phases of the single-user (SU) IRS-aided downlink MISO system, aiming at minimizing the total transmit power of the AP while ensuring that the target user enjoys its desired signal-to-noise ratio (SNR). The idea behind this is to control the reflections at the IRS in such a way that the reflected signal adds constructively with that from the direct link upon reaching the user. Moreover, these works respectively proposed semi-definite relaxation (SDR) and branch-and-bound (BnB) techniques, which, despite providing quite good performance, similarly suffer from high computation complexity.

In an attempt to lower complexity, a previous work [48] utilized cross-entropy (CE) and gradient projection (GP) methods to enhance the phase shifts optimization of IRS, aiming to maximize the spectral efficiency (SE) of SU downlink MISO systems. GP functions similar to a gradient descent (GD) approach, which necessitates the determination of search direction and step size at each stage to converge towards a global minimum. In contrast, CE exhibits similarities to a genetic ant colony algorithm. Although GP exhibited a simpler internal structure compared to CE, both techniques remain overly complex for real-world system implementation.

A. Contributions

The high complexity of these schemes is undoubtedly one of the major factors that cause hesitancy in deploying IRS-aided MIMO systems in practice. In light of the drawbacks of these existing schemes, efforts to find highly performant algorithms with low complexity remain imperative in order to realize affordable and sustainable wireless communication in the future. Considering this objective, this study builds upon previous works by employing a combination of the GP and WMMSE techniques to iteratively find the passive beamformer at the IRS and active beamformer at the BS in the downlink MU-MISO system supported by the IRS. Specifically, the active beamformers are optimized using a simplified and low-complexity reduced WMMSE (RWMMSE) algorithm [49], which is a modified variant of the original WMMSE approach. As it is shown in the later sections, RWMMSE tremendously manages to cut down its computational complexity by achieving two essential breakthroughs. First, it decreases the dimensions of the WMMSE’s invertible matrix from the number of BS transmit antennas, which is substantially large for the envisioned massive MIMO systems, to the number of user receive antennas, which is arguably small. Secondly, it manages to circumvent the lagrangian power constraint, which is normally computed by a repetitive line search procedure, by reformulating its unconstrained objective function such that all the constraints are eliminated or smoothly embedded within the objective function itself.

Conversely, this study optimizes the passive beamformer using an improved GP algorithm, which adeptly determines the search direction and step size without requiring iterative procedures. We show in the subsequent sections how these parameters are updated in each iteration by using information about their current state and that of their immediate previous state. To summarize everything, here is the bulleted list of our contributions.

• First, this study is one of the early, if not the first, attempt to adopt the RWMMSE technique in the IRS-aided system. Although the original scheme was also used for a MU system, we extend this effort not only to a more challenging IRS-aided system that has more sets of channels between the transmitter and the receiver but also transform it for an equivalent MISO system.

• Because the effectiveness of the GP algorithm depends almost entirely on the choice of the step size and search direction, we design an efficient strategy for selecting a step size that ensures fast convergence, which consequently leads to low complexity.

• Finally, we provide extensive numerical results that demonstrate the faster convergence as well as the effectiveness of the proposed scheme in achieving approximately the same performance as that of existing schemes but with substantially reduced complexity.

This paper is organized as follows. Section II discusses the system model considered in this work and proceeds further to problem formulation. The details of the proposed solution and analysis of its computation complexity are presented in Section III. Section IV provides in-depth numerical results, which compare the performance of the proposed algorithm with existing schemes. Finally, we provide concluding remarks in Section V.

Notations: This paper adopts the following notations. Scalars are denoted by lowercase letters. Bold lowercase (x) and uppercase (X) letters denote vectors and matrices, respectively. $\mathbb {C}$ denotes the complex domain. The complex conjugate, transpose, and conjugate transpose are denoted by the superscripts $*$ , T, and H, respectively. For a given matrix X, we use $\mathop {\mathrm {Tr}}\nolimits \left ({{ \mathbf {X} }}\right) $ and $\mathbf {X}^{-1}$ to denote the trace and inverse, respectively. The argument and magnitude of x are denoted by $ \arg { \left ({{ x }}\right) } $ and $ \left |{{ x }}\right |$ , respectively. The Euclidean norm, absolute value, and i-th element of vector x are given by $\| \mathbf {x} \|$ , $\left |{{ \mathbf {x} }}\right |$ , and $ [\mathbf {x}]_{i}$ , respectively. An $ N \times N $ diagonal matrix containing $ {a}_{1},\cdots,{a}_{N} $ as its diagonal elements is denoted by $ {\text {diag}}({a}_{1},\cdots,{a}_{N}) $ . $\frac {\partial g\left ({{x}}\right)}{\partial x}$ denotes the derivative of the $ g\left ({{x}}\right) $ with respect to (w.r.t) x.

SECTION II. System Model and Problem Formulation

In this work, we consider the communication system depicted in Fig. 1, where an IRS equipped with N reflecting elements assists the transmission of data signals from an M-antenna BS. The BS uses a maximum power P to transmit K streams of data to K single-antenna users. Although IRS can be used for various purposes, e.g., interference cancellation, wireless power transfer, reflecting modulation, etc., its primary purpose in our case is to intelligently reflect the incident signals such that they add constructively upon reaching the user. The users receive the unit-power transmitted symbols via flat-fading channels, which are perfectly known to the BS. Specifically, the channel of user k to the BS and IRS is denoted as $ \mathbf {h}_{\text {d},k}\in \mathbb {C}^{M \times 1} $ and $ \mathbf {h}_{\text {r},k}\in \mathbb {C}^{N \times 1}$ , respectively, whereas that between IRS and the BS is denoted as $ \mathbf {G}\in \mathbb {C}^{N \times M} $ . The diagonal matrix $ \boldsymbol {\Theta } = \text {diag} \left ({{ \theta _{1}, \theta _{2}, \cdots, \theta _{N} }}\right) \in \mathbb {C}^{N \times N} $ collects the unit-amplitude reflection coefficients of all the IRS elements, where $\theta _{n}=e^{j \vartheta _{n}}$ and $\vartheta _{n}$ denotes the phase shift of the n-th element. Before transmission, each unit-power symbol $s_{k}, \forall {k= 1,\cdots,K}$ , where $s_{k}$ denotes the k-th user’s symbol, undergoes a precoding process with its corresponding precoder $\mathbf {f}_{k} \in \mathbb {C}^{M \times 1}$ . The precoded symbols are then added to create a transmitted signal:\begin{equation*} \mathbf {d} = \sum _{k=1} ^{K} \mathbf {f}_{k}s_{k}. \tag {1}\end{equation*} Then, the signal that user k receives is given by\begin{equation*} y_{k} = \left ({{ \mathbf {h}_{\text {r},k}^{H} \boldsymbol {\Theta } \mathbf {G} + \mathbf {h}_{\text {d},k}^{H} }}\right) \mathbf {d} + z_{k}, \tag {2}\end{equation*} where $ z_{k} $ is a white Gaussian noise with zero mean and variance $ \sigma _{k}^{2} $ . We then let $\boldsymbol {\theta } = \left [{{ \theta _{1}, \theta _{2}, \cdots, \theta _{N} }}\right]^{H} \in \mathbb {C}^{N \times 1}$ and $ \mathbf {E}_{k} = \text {diag} \left ({{ \mathbf {h}_{\text {r},k}^{H} }}\right) \mathbf {G} \in \mathbb {C}^{N \times M}$ , which help to recast the composite channel between the BS and user k as $\mathbf {h}_{k} = \mathbf {h}_{\text {d},k} + \mathbf {E}_{k}^{H} \boldsymbol {\theta }$ , thereby simplifying the analysis to be conducted thereafter. To begin with, the signal-to-interference-plus-noise ratio (SINR) of user k is compactly expressed as\begin{equation*} SINR_{k}= \frac {\vert \mathbf {h}^{H}_{k} \mathbf {f}_{k} \vert ^{2}}{\sum _{j \neq k}^{K} \vert \mathbf {h}^{H}_{k} \mathbf {f}_{j} \vert ^{2} + \sigma _{k}^{2}}. \tag {3}\end{equation*} One of the main metrics when dealing with MU-MIMO communication systems is the SINR. This is crucial because it provides a measure of how strongly the user’s desired signal is compared with both interference and receiver noise. However, analyzing such systems is sometimes complicated because of interference. Nevertheless, the WSR, which in the case of this study is expressed as\begin{equation*} R \left ({{ \mathbf {F}, \boldsymbol {\theta } }}\right) = \sum _{k = 1} ^{K} \omega _{k} \log _{2} \left ({{ 1 + SINR_{k} }}\right), \tag {4}\end{equation*} with $\omega _{k}$ representing the priority of user k and $\mathbf {F} = \left [{{ \mathbf {f}_{1}, \mathbf {f}_{2}, \cdots, \mathbf {f}_{K} }}\right] \in \mathbb {C}^{M \times K}$ , can be used to address the communication systems of these characteristics. Therefore, in this work, we aim to maximize the system’s WSR through the optimization of F and $\boldsymbol {\theta }$ . This problem can be formulated as follows:\begin{align*} \mathcal {P}(A): \quad & \max _{ \mathbf {F}, \boldsymbol {\theta }} \quad R \left ({{ \mathbf {F}, \boldsymbol {\theta } }}\right) \tag {5a}\\ & ~\mathrm {s.t.} \quad \vert {\theta _{n}}\vert = 1, \forall {n = 1, \cdots, N}, \tag {5b}\\ & \hphantom {~\mathrm {s.t.} \quad } \sum _{k = 1} ^{K} \| \mathbf {f} _{k} \|^{2} \leq P. \tag {5c}\end{align*}

FIGURE 1.

The IRS-assisted MU MISO system.

Note that the constraint (5b) makes the problem $ \mathcal {P}(A)$ non-convex, subsequently making the design of its solution challenging. To the best of our knowledge, there is no global optimal solution for non-convex problems. Therefore, the following section presents a proficient AO-based algorithm designed to produce a locally optimal solution to $\mathcal {P}(A)$ .

SECTION III. Proposed Solution to Problem $ \mathcal {P}(A)$

The challenge in addressing $\mathcal {P}(A)$ arises not only from its non-convex nature, but also from the strong interdependence between $\boldsymbol { \theta }$ and F, rendering the problem particularly intricate to solve. As mentioned earlier, the two-block version of the BCD, the AO, is highly effective in tackling such issues. Essentially, AO fixes one parameter while iteratively optimizing the other until the objective function converges. Thus, this study employs the same AO technique to devise a solution for the single-variable subproblem while keeping the others constant.

A. Solution for Active Beamformer

The subproblem of optimization of F for fixed $ \boldsymbol {\theta } $ is given by:\begin{align*} \mathcal {P}(B): \quad & \max _{ \mathbf {F}} \quad R \left ({{ \mathbf {F}, \boldsymbol {\theta } }}\right) \tag {6a}\\ & \mathrm {s.t.} \quad \sum _{k = 1} ^{K} \| \mathbf {f} _{k} \|^{2} \leq P. \tag {6b}\end{align*} $ \mathcal {P}(B) $ can be solved using a WMMSE technique [39]. This technique has high computational complexity, accumulated through the iterative computation of the Lagrangian variable for power constraints and repetitive inversion of matrices with high dimensions. Moreover, the complexity is in the cubic order of the number of transmit antennas, which is expected to grow even higher in massive MIMO systems.

This work uses the RWMMSE technique to address this complexity issue on the optimization of F. Although an extensive discussion of the RWMMSE was provided in [49], for convenience, we briefly outline specific fundamental ideas while addressing this particular subproblem. The basic premise of the RWMMSE is built upon two observations of nontrivial stationary points—does not result in a zero WSR. The initial observation suggests that any nontrivial stationary point $ \mathbf {f}^{\star }_{i}$ , must satisfy $\mathbf {f}^{\star }_{i} = \mathbf {H} ^{H} \mathbf {x}_{i}$ for certain unique $\mathbf {x}_{i} \in \mathbb {C}^{K \times 1}$ . Using this property, $ {\mathcal {P}}({B}) $ is recast to\begin{align*} & \max _{ \mathbf {X} } \quad \sum _{k = 1} ^{K} \omega _{k} \log _{2} \left ({{ 1 + \frac {\vert \mathbf {h}_{k} ^{H} \mathbf {H}^{H} \mathbf {x}_{k} \vert ^{2}}{\sum _{j \neq k}^{K} \vert \mathbf {h}_{k}^{H} \mathbf {H}^{H} \mathbf {x}_{j} \vert ^{2} + \sigma _{k}^{2} } }}\right) \tag {7a}\\ & ~\mathrm {s.t.} \quad \sum _{k = 1} ^{K} \| \mathbf {H}^{H} \mathbf {x} _{k} \|^{2} \leq P, \tag {7b}\end{align*} where $ \mathbf {H} = \left [{{ \mathbf {h}_{1}, \mathbf {h}_{2}, \cdots, \mathbf {h}_{K} }}\right]^{H} \in \mathbb {C}^{K \times M} $ and $\mathbf {X} = \left [{{ \mathbf {x}_{1}, \mathbf {x}_{2}, \cdots, \mathbf {x}_{K} }}\right] $ . This serves as the first major advantageous difference of RWMMSE from WMMSE in the sense that (7) significantly reduces the dimensions of the parameter $\mathbf {X} \in \mathbb {C}^{K \times K}$ compared to that of F in ${\mathcal {P}}({A}) $ . Nevertheless, applying WMMSE to solve (7) remains computationally expensive. This is because constraint (7b) requires the adoption of a line search procedure for the computation of the Lagrangian parameter to ensure the satisfaction of this constraint. To address this challenge, we introduce the second observation, which asserts that any nontrivial stationary point of $ \mathcal {P}(B) $ must fulfill the power constraint (6b) exactly. Consequently, leveraging this insight, we derive the following unconstrained problem.\begin{equation*} \max _{ \mathbf {X} } \quad \sum _{k = 1} ^{K} \omega _{k} \log _{2} \left ({{ 1 + \frac { \left |{{ \bar {\mathbf {h}}_{k} \mathbf {x}_{k} }}\right |^{2} }{ a_{k} } }}\right), \tag {8}\end{equation*} which necessitates appropriate adjustments in the relationship between $ \mathbf {f} ^{\star }_{k} $ and $ \mathbf {x}_{k}$ , expressed as $ \mathbf {f}^{\star }_{k} = \sqrt { \beta } \mathbf {H}^{H} \mathbf {x}_{k}$ . Note that, in (8), $ a_{k} = \sum _{j \neq k}^{K} \left |{{ \bar {\mathbf {h}}_{k} \mathbf {x}_{j} }}\right |^{2} + \frac {\sigma _{k}^{2}}{P} \sum _{i=1}^{K} \mathop {\mathrm {Tr}}\nolimits \left ({{ \bar {\mathbf {H}} \mathbf {x}_{i} \mathbf {x}_{i}^{H} }}\right)$ is used for simplification of the notations, $ \beta = \frac {P}{ \sum _{k=1}^{K} \mathop {\mathrm {Tr}}\nolimits \left ({{ \bar {\mathbf {H}} \mathbf {x}_{k} \mathbf {x}_{k}^{H} }}\right) }$ is the scaling factor, $\bar {\mathbf {H}} = \mathbf {H} \mathbf {H}^{H} \in \mathbb {C}^{K \times K}$ , and $\bar {\mathbf {h}}_{k} = \mathbf {h}_{k} ^{H} \mathbf {H}^{H} \in \mathbb {C}^{1 \times K} $ . Notice how the power constraint (7b) is included in the objective function (8), representing the second significant advantage of RWMMSE over WMMSE. This approach circumvents the necessity for computationally expensive line search procedures. Using Lemma 4.1 of [50], problem (8) can be reformulated into its equivalent unconstrained weighted sum-mean square error (MSE) problem $ \mathcal {P}(C) $ as:\begin{equation*} \mathcal {P}(C): \quad \min _{ u_{k}, \chi _{k}, \mathbf {x}_{k}, \forall k} \quad r\left ({{u_{k}, \chi _{k}, \mathbf {x}_{k} }}\right), \tag {9}\end{equation*} where $r\left ({{u_{k}, \chi _{k}, \mathbf {x}_{k} }}\right) = \sum _{k=1}^{K} \omega _{k} \bigl (\chi _{k} \left |{{ 1 - u_{k}^{*} \bar {\mathbf {h}}_{k} \mathbf {x}_{k} }}\right |^{2} + \chi _{k}\left |{{ u_{k} }}\right |^{2} a_{k} - \log _{2} \chi _{k} \bigr)$ ; $ \left \{{{ u_{k} }}\right \}_{k=1}^{K} $ and real-valued $ \left \{{{ \chi _{k} }}\right \}_{k=1}^{K} $ are the introduced auxiliary variables, which aid the problem transformation. The solutions for these variables derived in Appendices A and B, using the BCD method and first-order optimality condition, are given in closed form as:\begin{align*} u_{k} & = \left ({{ \left |{{ \bar {\mathbf {h}}_{k} \mathbf {x}_{k} }}\right |^{2} + a_{k} }}\right)^{-1} \bar {\mathbf {h}}_{k} \mathbf {x}_{k}, \tag {10}\\ \chi _{k} & = \left ({{ 1 - u_{k}^{*} \bar {\mathbf {h}}_{k} \mathbf {x}_{k} }}\right)^{-1}, \tag {11}\end{align*} respectively. Similarly, the solution of $\mathbf {x}_{k}$ , whose derivation is given in Appendix C, is obtained by fixing the variables $\left \{{{ u_{k} }}\right \}_{k=1}^{K}$ and $\left \{{{ \chi _{k} }}\right \}_{k=1}^{K}$ and solving for $ \mathcal {P}(C)$ , which leads to\begin{align*} \mathbf {x}_{k}& = \omega _{k} \chi _{k} u_{k} \Bigg (\sum _{i=1}^{K} \frac { \sigma _{i}^{2} }{P} \omega _{i} \chi _{i} \left |{{ u_{i} }}\right |^{2} \bar {\mathbf {H}} \\ & \quad \qquad \qquad + \sum _{j=1}^{K} \omega _{j} \chi _{j} \left |{{ u_{j} }}\right |^{2} \bar {\mathbf {h}}_{j}^{H} \bar {\mathbf {h}}_{j} \Biggl)^{-1} \bar {\mathbf {h}}_{k}^{H}. \tag {12}\end{align*} To further reduce the iterative processes, the solution for $\mathbf {x}_{k}, \forall k$ can be collectively computed for all users by the following procedure. By letting $ \gamma = \sum _{i=1}^{K} \frac { \sigma _{i}^{2} }{P} \omega _{i} \chi _{i} \left |{{ u_{i} }}\right |^{2}$ , $ \hat { \mathbf {U} } = \text {diag} \left ({{ u_{1}, u_{2}, \cdots, u_{K} }}\right)$ , and $ \hat { \boldsymbol {\chi } } = \text {diag} \left ({{ \omega _{1} \chi _{1}, \omega _{2} \chi _{2}, \cdots, \omega _{K} \chi _{K} }}\right)$ , (12) can be rewritten in terms of X as follows:\begin{align*} \mathbf {X} & = \left ({{ \gamma \bar {\mathbf {H}} + \bar {\mathbf {H}} \hat {\mathbf {U}} \hat {\boldsymbol {\chi }} \hat {\mathbf {U}}^{H} \bar {\mathbf {H}} }}\right)^{-1} \bar {\mathbf {H}} \hat {\mathbf {U}} \hat {\boldsymbol {\chi }} \tag {13}\\ & = \hat {\mathbf {U}} \left ({{ \gamma \hat {\boldsymbol {\chi }}^{-1} + \hat {\mathbf {U}}^{H} \bar {\mathbf {H}} \hat {\mathbf {U}} }}\right)^{-1}. \tag {14}\end{align*} Notably, it has been reported that the solution (14) not only gets rid of the expensive iterative process but also reduces the dimension of the invertible matrix in MU-MIMO systems, especially when the number of streams is less than the total number of receive antennas [49]. To finalize, Algorithm 1 summarizes the procedure for optimizing F using RWMMSE. On top of the users’ priorities, $\omega _{k}$ ’s, the algorithm also accepts the composite matrix H and the properly initialized X that satisfies the constraint (7b). Subsequently, the auxiliary variables $ u_{k} $ and $ \chi _{k}$ , together with X, are iteratively updated using their corresponding update rules in steps 1, 1, and 1 until the objective function converges. When convergence is achieved, we obtain the BS precoders using $ \mathbf {f}_{k} = \sqrt { \beta } \mathbf {H}^{H} \mathbf {x}_{k} $ .

Algorithm 1 Optimization of F by RWMMSE

Input:

$\omega _{k}$ , $ \forall k $ , H, and X.

Output:

X and $ \mathbf {f}_{k}, \forall {k}$ .

1:

Initialization: Generate $ \bar {\mathbf {H}} = \mathbf {H}\mathbf {H}^{H} $ .

2:

Set $ \chi _{k} = 1 $ .

3:

repeat

4:

$ \chi _{k}^{'} = \chi _{k} $

5:

Find $ u_{k} $ , $\forall k$ using (10).

6:

Find $ \chi _{k} $ , $\forall k$ using (11).

7:

Find X using (14).

8:

until $ \left |{{\sum _{k=1}^{K} {\omega _{k}} \log {\chi _{k}} - \sum _{k=1}^{K} {\omega _{k}} \log {\chi ^{'}_{k}} }}\right | \leq \epsilon _{f} $

9:

Find $ \mathbf {f}_{k} = \sqrt {\beta } \mathbf {H}^{H} \mathbf {x}_{k}, \forall {k}$ .

B. Solution for Passive Beamformer

After obtaining the active beamformer F, it is fixed to allow the optimization of the passive beamformer $ \boldsymbol {\theta } $ . Fixing F transforms the two-variable problem $\mathcal {P}(A)$ into\begin{align*} \mathcal {P}(D): \quad & \max _{ \boldsymbol {\theta }} \quad f \left ({{ \boldsymbol {\theta } }}\right) \tag {15}\\ & ~\mathrm {s.t.} \quad \vert {\theta _{n}}\vert = 1, \forall {n = 1,\cdots,N}, \tag {16}\end{align*} where\begin{align*} & f(\boldsymbol {\theta }) \\ & = \sum \limits _{k=1}^{K} \omega _{k}{\log _{2} \left ({{ 1 + { \frac { \left |{{ \boldsymbol {\theta }^{H} \mathbf {e}_{k,k} + r_{k,k} }}\right | ^{2}} { \sum _{j \neq k}^{K} \left |{{ \boldsymbol {\theta }^{H} \mathbf {e}_{k,j} + r_{k,j} }}\right | ^{2} + \sigma _{k}^{2} } } }}\right)} \tag {17}\end{align*} serves as a single-variable objective function with $ \mathbf {e}_{k,j} = \mathbf {E}_{k} \mathbf {f}_{j}\in \mathbb {C}^{N \times 1} $ and a scalar $ r_{k,j} = \mathbf {h}_{\text {d},k}^{H} \mathbf {f}_{j}\in \mathbb {C}^{1 \times 1} $ . Notably, $f(\boldsymbol {\theta })$ is both continuous and differentiable; therefore, various GD-based methods can be used to optimize $\mathcal {P}(D)$ . For example, the work [36] used Riemannian conjugate gradient (RCG), whereas [48] deployed GP on the SU version of $\mathcal {P}(D)$ in an attempt to reduce RCG’s complexity. Because of the implicit simple structure of GP, this study extends it to the MU systems and enhances its step size selection at each iteration using a Barzilai-Borwein (BB) step size selection procedure [51].

Specifically, given a point $ \boldsymbol {\theta }^{i} $ of the i-th iteration, the GP needs the search direction $ \boldsymbol { \rho }^{i} $ to descend by a step size $ \alpha ^{i} \gt 0 $ to obtain the next point $ \boldsymbol {\theta }^{i+1} $ . This is numerically expressed as follows:\begin{equation*} \boldsymbol {\theta }^{i+1} = \boldsymbol {\theta }^{i} + \alpha ^{i} \boldsymbol { \rho }^{i},\end{equation*} where the search direction is given by $ \boldsymbol { \rho } ^{i} = -\frac {\partial }{\partial \boldsymbol {\theta }^{i}} f(\boldsymbol {\theta }^{i})$ 1$= -2\sum _{k=1}^{K} \omega _{k} g_{k}(\boldsymbol {\theta }^{i})$ with\begin{align*} g_{k}(\boldsymbol {\theta }) & = \frac {\sum _{j=1}^{K} \mathbf {e}_{k,j} \mathbf {e}_{k,j}^{H} \boldsymbol {\theta } + \sum _{j=1}^{K} \mathbf {e}_{k,j} r_{k,j}^{*} } { \sum _{j=1}^{K} \left |{{ \boldsymbol {\theta }^{H} \mathbf {e}_{k,j}^{*} + r_{k,j}^{*} }}\right |^{2} + \sigma _{k}^{2} } \\ & \quad - \frac {\sum _{j\neq k}^{K} \mathbf {e}_{k,j} \mathbf {e}_{k,j}^{H} \boldsymbol {\theta } + \sum _{j \neq k}^{K} \mathbf {e}_{k,j} r_{k,j}^{*} } { \sum _{j \neq k}^{K} \left |{{ \boldsymbol {\theta }^{H} \mathbf {e}_{k,j} + r_{k,j} }}\right |^{2} + \sigma _{k}^{2} } \tag {18}\end{align*} being the Euclidean gradient. Unlike $\boldsymbol { \rho } ^{i}$ , the choice for $ \alpha _{i} $ is not that straightforward. For example, [48] proposed the use of the reciprocal of $4\lambda _{max}$ , where $\lambda _{max}$ is the maximum eigenvalue of $ \mathbf {E}_{k}\mathbf {E}_{k}^{H}$ , for the step size of an SU MISO system. This procedure is however 2-fold challenging when adopted in our study. First, we observe that obtaining $\lambda _{max}$ from the large-dimensional matrix $ \mathbf {E}_{k}\mathbf {E}_{k}^{H} \in \mathbb {C}^{N \times N} $ results in high complexity. Second, an extension to the MU scenario must recompute $\lambda _{max}$ for each user, which leads to even higher complexity. This issue is addressed by adopting a BB-inspired step size selection technique [51]. Essentially, the BB technique chooses its step size by solving the problem $ \displaystyle \min _{\alpha ^{i}} \| \boldsymbol {\nu }^{i} - \alpha \boldsymbol {\psi }^{i} \|^{2}$ , where $ \boldsymbol {\psi }^{i} = \frac {\partial }{\partial \boldsymbol {\theta ^{i}}} f(\boldsymbol {\theta }^{i}) - \frac {\partial }{\partial \boldsymbol {\theta }^{i-1}} f(\boldsymbol {\theta }^{i-1}) $ and $ \boldsymbol {\nu }^{i} = \boldsymbol {\theta }^{i} - \boldsymbol {\theta }^{i-1}$ , which yield the step size\begin{align*} \alpha _{i}= \begin{cases} \displaystyle \frac {1}{{c}\| \frac {\partial }{\partial \theta ^{i}} f(\boldsymbol {\theta }^{i}) \|}, & \text {if}~ i= 0 \\ \displaystyle \frac {{\boldsymbol {\nu }^{i}}^{H}{\boldsymbol {\psi }^{i}}}{\|{\boldsymbol {\psi }^{i}}\|}, & \text {otherwise}, \end{cases} \tag {19}\end{align*} with ${c} \in (0,1)$ denoting a scaling factor at the first iteration. This choice of step size is motivated by the ability to obtain a two-point approximation of the secant equation of the quasi-Newton method, which is two-fold advantageous. First, it is computationally inexpensive, and second, it is insensitive to ill-conditioning [52]. Algorithm 2 summarizes the procedures to solve $ \mathcal {P}(D) $ using the proposed GP method.

Algorithm 2 Optimization of Optimization of $\mathcal {P} \left ({{ D }}\right)$ by GP

Input:

$ \omega _{k} $ , $ \mathbf {E}_{k} ~ \forall {k} $ , F, and $ \boldsymbol {\theta }$ .

Output:

$\boldsymbol {\theta }$ .

1:

Set $ i=0 $ and $ \boldsymbol {\theta }^{i} =\boldsymbol {\theta } $ .

2:

repeat

3:

Find the derivative $ \frac {\partial }{\partial \boldsymbol {\theta }^{i}} f(\boldsymbol {\theta }^{i})$ .

4:

Update $ \alpha ^{i} $ by using (19).

5:

$ \boldsymbol {\varpi } \leftarrow \boldsymbol {\theta }^{i} - {\alpha }^{i} \frac {\partial }{\partial \boldsymbol {\theta }^{i}} f(\boldsymbol {\theta }^{i})$

6:

$ \boldsymbol {\theta }^{i+1} \leftarrow e^{j\arg (\boldsymbol {\varpi })} $

7:

$ i \leftarrow i + 1 $

8:

until $\left |{{ f(\boldsymbol {\theta }^{i})-f(\boldsymbol {\theta }^{i-1}) }}\right | \leq \epsilon _{\theta }$

Putting everything together, we present in Algorithm 3 the summary of the proposed AO scheme, which we henceforth refer to as RWMMSE-GP.

Algorithm 3 Proposed RWMMSE-GP Method

Input:

$\omega _{k}$ , $ \mathbf {h}_{\text {r},k} $ , $ \mathbf {h}_{\text {d},k} $ , $\forall {k}$ , G.

Output:

$\boldsymbol {\theta }$ , F.

1:

Initialization: Generate X and $ \boldsymbol {\theta }$ randomly such that $ \sum _{k=1}^{K} \mathop {\mathrm {Tr}}\nolimits (\bar { \mathbf {H} } \mathbf {x}_{k} \mathbf {x}_{k}^{H}) \leq {P}$ .

2:

Set $ \ell = 0 $ .

3:

Calculate WSR, $ R^{\ell } $ .

4:

repeat

5:

$ \ell \leftarrow \ell + 1 $

6:

Update passive beamformer, $ \boldsymbol {\theta } $ , using Algorithm 2.

7:

Update active beamformer, F, using Algorithm 1.

8:

Compute WSR, $ C^{\ell } $ .

9:

until $ \left |{{ R^{\ell } - R^{\ell -1} }}\right | \leq \epsilon _{R}$

C. Complexity Analysis:

In this subsection, we evaluate the complexity of the proposed RWMMSE-GP method by examining its floating point operations (FLOPS) [53], [54]. We begin by presenting the complexity of the first half of the proposed scheme, i.e., RWMMSE, which starts with the generation of the matrix $ \bar {\mathbf {H}}$ , whose complexity is given as $ \mathcal {O}(MK^{2}) $ . The next dominant complexity of RWMMSE, which is given by $ \mathcal {O}(I_{R}K^{3})$ , where $ I_{R} $ denotes the number of iterations needed for RWMMSE convergence, is accumulated from the matrix inversion operation during the update of X. The second half of the proposed algorithm uses GP to optimize the IRS phase shifts, whose complexity is dominated by the computation of the search direction $ \frac {\partial }{\partial \boldsymbol {\theta }^{i}} f(\boldsymbol {\theta }^{i}) $ . This process requires a complexity of $ \mathcal {O}(I_{G}K^{2}N^{2})$ , where $ I_{G} $ denotes the number of iterations needed for GP convergence. Therefore, the total complexity of the proposed RWMMSE-GP scheme is $ \mathcal {O} \left ({{I_{O} MK^{2} + I_{R}K^{3} + I_{G}K^{2}N^{2} }}\right)$ , with $ I_{O} $ denoting the number of iterations required for the outer AO loop to converge.

For comparison, we adopt BCD and AO based on WMMSE and MO [36]. Conveniently, we abbreviate this AO as “WMMSE-MO,” whose complexity is given as $ \mathcal {O} (\left ({{ I_{\lambda } + I_{W} }}\right) KM^{3} + I_{MO} K^{2}N^{2}) $ with $ I_{\lambda }$ , $ I_{W}$ , and $ I_{MO} $ denoting the accumulated number of iterations for searching the Lagrangian power constraint $ \lambda $ , convergence of WMMSE, and convergence of MO, respectively. Similarly, the total computational complexity of the BCD algorithm is given by $ \mathcal {O}(I_{A}(2KMN+ KM^{2}) + I_{O}K^{2}N^{2})$ , where $I_{A}$ denotes the number of iterations required for the Armijo search.

SECTION IV. Numerical Evaluation

This section presents the simulation results of our proposed algorithm. We selected WMMSE-MO and BCD [36] as the benchmark schemes. Furthermore, a system that randomly selects the phases of each IRS element (Random phase) and an unassisted system (No IRS) are adopted to serve as the lower-bound benchmark system.

A. Parameter Setup

Most of the system parameters used in this numerical evaluation are similar to those employed in [36]. This work considers a single-cell system where a BS equipped with $ M = 16 $ antennas positioned at $ (0, 0) $ serves $ K = 4 $ single-antenna users, which are randomly located within a 10 m-radius circle centered at $ (200~\text {m}, 30~\text {m}) $ . This communication, whose setup is shown in Fig. 2, is assisted by an IRS fixed at $ (200~\text {m},0) $ and equipped with $ N = 200 $ reflecting elements. All the associated channels are provided by the mmWave clustered channel model with a limited number of propagation paths [55]. Since IRS is normally deployed in the sight of both BS and users, it is reasonable to assume that its associated channels, i.e., G and $ \mathbf {h}_{\text {r},k} $ follow Rician small-scale fading with a Rician factor $ \kappa = 10$ , whereas the direct link channels $ \mathbf {h}_{ \text {d},k}$ , which are mostly in deep fade, are assumed to follow Rayleigh small-scale fading. Furthermore, the large-scale fading, whose square root in linear scale is multiplied by the generated channels, is expressed as\begin{align*} PL \left ({{ \partial }}\right) [\text {dBm}] = \begin{cases} \displaystyle 32.6 + 36.7 \log \left ({{ \partial }}\right), & \text {NLoS case} \\ \displaystyle 35.6 + 22 \log \left ({{ \partial }}\right), & \text {LoS case}, \end{cases} \tag {20}\end{align*} where $ \partial $ is the link distance in meters. Moreover, the priority of user k is computed using\begin{equation*} \omega _{k} = \frac {1}{ \eta _{k} \sum _ {j=1} ^{K} 1/ \eta _{j} }, \tag {21}\end{equation*} where $ \eta _{j} = 10^{0.1~PL \left ({{ \partial _{d,j} }}\right)} $ and $ \partial _{d,k} $ is the distance between the BS and user k. Finally, the system bandwidth, user’s noise power, and BS’ transmit power are set to 180 kHz, $ \sigma _{k}^{2} = -170~\text {dBm/Hz}$ , and 20 dBm, respectively, whereas the proposed algorithm’s control parameters are given as $ \epsilon _{f} = \epsilon _{R} = \epsilon _{\theta } = 10^{-5} $ and $ c = \frac {1}{6} $ .

FIGURE 2.

Simulation setup.

B. WSR Comparison

We start by analyzing the convergence behavior of the RWMMSE-GP, WMMSE-MO, and BCD algorithms for $N = 200$ , $M = 16$ , $K = 4$ , and $P= 20~\text {dBm} $ . In Fig. 3, we observe that the proposed algorithm converges much faster with slightly higher performance than the WMMSE-MO and BCD algorithms. Quantitatively, the proposed algorithm converges within 100 iterations, whereas WMMSE-MO and BCD algorithms converge after 1000 and 1800 iterations, respectively. This fast convergence obviously comes from the joint effort of both the RWMMSE and GP techniques in reducing the number of iterative processes as well as the choice of more relevant values for algorithm parameters, such as step sizes at each iteration. It is worth mentioning that the convergence of BCD and WMMSE-MO was reported in their original work to be within 100 iterations. This was achieved with a low-resource network configured with $ N = 100$ , $ P = 0~\text {dBm}$ , and $ M = 4 $ . Unfortunately, when the resources of the network increase, as depicted in this work, these schemes start to suffer and converge very slowly. Moreover, because the convergence speed is inversely proportional to the complexity, the proposed algorithm is expected to require lower complexity than the existing schemes. This is demonstrated in the next subsection.

FIGURE 3.

Convergence behavior for $N = 200$ , $M = 16$ , $K = 4$ , and $ P = 20~\text {dBm} $ .

Next, in Fig. 4, we investigate the achievable WSRs of the various schemes with respect to the number of IRS elements N. In this case, we set $ K=4$ , $ M = 16$ , and $ P = 20~\text {dBm} $ . First, as evident, the expected trend of performance improvement of the proposed scheme, BCD, and WMMSE-MO for the increase in the number of IRS elements is observed. Additionally, despite requiring a lesser number of iterations for convergence, the proposed scheme is observed to attain almost the same performance as the benchmark schemes for the entire range of N. This clearly illustrates the efficiency of the proposed method in achieving remarkable performance while reducing complexity.

FIGURE 4.

WSR vs. N for $ P = 20~\text {dBm}$ , $ M = 16$ , and $K=4$ .

We then evaluate the effect of the number of transmit antennas, M, on the performance of the system. Fig. 5 compares the performance of the proposed, benchmark, and lower bound schemes for $M=\{4, 8, 16, 32, 64\}$ . It is observed that as the number of transmit antennas increases, the performance of all schemes improves as well. Note that the proposed scheme still demonstrates its effectiveness of achieving almost the same performance as that of the benchmark schemes for the entire range of M. Additionally, although the beamforming and diversity gains arising from the increased number of transmit antennas make the WSRs of the lower-bound schemes increase, their achievable performance is way lower compared to that of the proposed and benchmark schemes, each of which is assisted by a properly configured IRS.

FIGURE 5.

WSR vs. M for $ P = 20~\text {dBm}$ , $ N = 200$ , and $ K=4$ .

Fig. 6 presents the achievable WSR of various algorithms for different transmit power $ P=\{0 - 30~\text {dBm}\} $ when $ K=4$ , $ N = 200$ , and $ M = 16 $ . As shown in previous results, the performance of the benchmark schemes is observed to be approximately the same as that of the proposed schemes. Similarly, because of the assistance obtained from the IRS, the performance of the proposed and benchmark schemes is higher than that of the lower-bound schemes. Lastly, we generally observe from Fig. 6 that the performance of all schemes improves with the BS transmit power.

FIGURE 6.

WSR vs. P for $ K=4$ , $ N = 200$ , and $ M = 16 $ .

Fig. 7 compares the performance of different algorithms by shifting the IRS location from ($170~\text {m}, 0~\text {m}$ ) to ($205~\text {m}, 0~\text {m}$ ) while keeping the rest of the parameters at their default values. In Fig. 7, we observe that the achievable WSR decreases when the IRS moves far away from the users, whereas the highest WSR is achieved when the IRS gets closest to the users, in this case, at ($195~\text {m}, 0~\text {m}$ ). Furthermore, the proposed algorithm is observed to attain slightly higher performance than the benchmark algorithms over the entire range of IRS positions.

FIGURE 7.

Effect of IRS location for $N = 200$ , $M = 16$ , $K = 4$ , and $P= 20~\text {dBm} $ .

C. Complexity Comparison

This subsection provides a complexity comparison between the proposed algorithm and benchmark schemes in terms of both FLOPS and running time. We begin by presenting in Fig. 8 the FLOPS of the proposed algorithm, WMMSE-MO, and BCD schemes for different numbers of IRS elements N. We set $ K=4$ , $ M = 16$ , and $ P = 20~\text {dBm} $ . In Fig. 8, we observe that the proposed algorithm requires significantly lower complexity than that required by WMMSE-MO and BCD schemes. For example, for $N=50$ , the complexity reduction ratios of the proposed algorithm compared with WMMSE-MO and BCD schemes are approximately 95.9% and 86.4%, respectively. Nearly the same trend is observed for all considered sizes of IRS, which validates the fact that the fast convergence of the proposed algorithm results in a lower complexity. Specifically, the avoidance of the iterative process in RWMMSE and the Barzilai-Borwein choice of step size in GP contributes much to the quick convergence of the proposed scheme.

FIGURE 8.

FLOPS vs. N for $ P = 20~\text {dBm}$ , $M = 16$ , and $ K=4$ .

Finally, the runtime complexities of the benchmark and the proposed algorithms for generating the results in Fig. 4 are evaluated and presented in Fig. 9. This complexity, which is accrued by MATLAB R2023a deployed on a 32 GB RAM Windows 10 computer equipped with a 12th Gen Intel(R) Core(TM) i7-12700 processor clocking at 2.10 GHz, represents the average total time needed by each algorithm to converge. We assume a system with different numbers of IRS elements for a fixed $ K=4$ , $ M = 16$ , and $ P = 20~ \text {dBm} $ . As anticipated, the runtime of all methods increases with the size of the IRS. This is attributed to the heightened computational workload in configuring each IRS element’s phase. Furthermore, it is evident from Fig. 9 that the proposed method achieves the shortest runtime across all IRS sizes. For example, when $N=50$ , the running times for the proposed scheme are 80.9% and 96.3% less than that of the BCD and WMMSE-MO schemes, respectively. Moreover, when $N=300$ , the running time for the proposed scheme is 84.2% less than both BCD and WMMSE-MO schemes.

FIGURE 9.

Runtime vs. N for $P=20~\text {dBm}$ , $M = 16$ , and $K=4$ .

SECTION V. Conclusion

In this study, we investigated a downlink IRS-aided MU-MISO system. We formulated a WSR problem for this system, comprising two highly coupled optimizable parameters, i.e., active and passive beamformers. We proposed an AO technique to decouple the parameters and optimize them independently in an alternate manner. Unlike many existing techniques, which use WMMSE to optimize the active beamformer, we adopted a low-complexity RWMMSE technique for this purpose. The RWMMSE scheme attains this efficiency by the dimension reduction of the invertible matrices, circumvention of the Lagrangian power constraint, and the avoidance of many iterative processes. We further proposed GP for the optimization of the passive beamformer instead of using high-complexity algorithms like MO. Similarly, the proposed GP is enhanced with a robust BB method for the choice of step size, which speeds up its convergence.

The simulation results demonstrated that the proposed scheme achieved nearly the same performance as benchmark schemes with a significantly reduced running time and computational complexity. For example, it has been shown that even when the number of IRS elements sextuple from 50 to 300, the complexity reduction ratio of the proposed scheme over WMMSE-MO and BCD slightly decreased from 95.9% and 86.4% to 75.4% and 81.5%, respectively. Interestingly, this complexity reduction comes at zero cost in terms of achievable WSR.

It is noted that the remarkable performance achieved by the proposed algorithm is obtained with the assumption of perfect CSI. Thus, further investigation into the potential applications of the proposed algorithm in the context of imperfect CSI represents an intriguing avenue for future research.

Appendix A Solution of $\mathbf {u}_{k}$ for Fixed $\chi _{k}$ and $x_{k}$

The solution for $u_{k}$ with fixed $\chi _{k}$ and $x_{k}$ is found by solving $\frac {\partial }{\partial u_{k}} r\left ({{u_{k}, \chi _{k}, \mathbf {x}_{k} }}\right) = 0 $ for $u_{k}$ , i.e.,\begin{align*} & \frac {\partial }{\partial u_{k}} \sum _{k=1}^{K} \omega _{k} \chi _{k} \left |{{ 1 - u_{k}^{*} \bar {\mathbf {h}}_{k} \mathbf {x}_{k} }}\right |^{2} + \frac {\partial }{\partial u_{k}} \sum _{k=1}^{K} \omega _{k} \chi _{k} \left |{{ u_{k} }}\right |^{2} a_{k} \\ & \quad \qquad \qquad \qquad \qquad - \frac {\partial }{\partial u_{k}} \sum _{k=1}^{K} \omega _{k} \log _{2} \chi _{k} = 0. \tag {22}\end{align*} Note that $\frac {\partial }{\partial u_{k}} \sum _{k=1}^{K} \omega _{k} \chi _{k} \left |{{ 1 \!-\! u_{k}^{*} \bar {\mathbf {h}}_{k} \mathbf {x}_{k} }}\right |^{2} \!=\! 2\omega _{k} \chi _{k} u_{k} \left |{{ \bar {\mathbf {h}}_{k} \mathbf {x}_{k} }}\right |^{2} - 2\omega _{k} \chi _{k} \bar {\mathbf {h}}_{k} \mathbf {x}_{k}$ , $\frac {\partial }{\partial u_{k}} \sum _{k=1}^{K} \omega _{k} \chi _{k} \left |{{ u_{k} }}\right |^{2} a_{k} = 2\omega _{k} \chi _{k} u_{k} a_{k}$ , and $\frac {\partial }{\partial u_{k}} \sum _{k=1}^{K} \omega _{k} \log _{2} \chi _{k} = 0 $ . Combining these derivatives together according to (22) and solving for $u_{k}$ leads to the solution in (10).

Appendix B Solution of $\chi _{k}$ for Fixed $u_{k}$ and $\mathbf {x}_{k}$

The solution of $\chi _{k}$ is obtained in a similar manner to that of $u_{k}$ . Observe that the derivative of each term of $r\left ({{u_{k}, \chi _{k}, \mathbf {x}_{k} }}\right)$ w.r.t $\chi _{k}$ is given by\begin{align*} \frac {\partial }{\partial \chi _{k}} \sum _{k=1}^{K} \omega _{k} \chi _{k} \left |{{ 1 - u_{k}^{*} \bar {\mathbf {h}}_{k} \mathbf {x}_{k} }}\right |^{2} & = \omega _{k} \left |{{ 1 - u_{k}^{*} \bar {\mathbf {h}}_{k} \mathbf {x}_{k} }}\right |^{2}. \tag {23}\\ \frac {\partial }{\partial \chi _{k}} \sum _{k=1}^{K} \omega _{k} \chi _{k} \left |{{ u_{k} }}\right |^{2} a_{k}& = \omega _{k} \left |{{ u_{k} }}\right |^{2} a_{k}. \tag {24}\\ \frac {\partial }{\partial \chi _{k}} \sum _{k=1}^{K} -\omega _{k} \log \chi _{k} & = -\frac {\omega _{k}}{\chi _{k}}. \tag {25}\end{align*} Note that we use the natural logarithm in (25) due to its analytical simplicity. Next, sum (23)–(25) to obtain\begin{equation*} \frac {1}{\chi _{k}} = \left |{{ 1 - u_{k}^{*} \bar {\mathbf {h}}_{k} \mathbf {x}_{k} }}\right |^{2} + \left |{{ u_{k} }}\right |^{2} a_{k}. \tag {26}\end{equation*} Finally, by substituting $u_{k}$ with its solution in (10) and applying standard mathematical procedures, (26) simplifies to (11).

Appendix C Solution of $\mathbf {x}_{k}$ for Fixed $\chi _{k}$ and $u_{k}$

Similarly, the initial step to solving for $\mathbf {x}_{k}$ is to compute the derivative $\frac {\partial }{\partial \mathbf {x}_{k}} r\left ({{u_{k}, \chi _{k}, \mathbf {x}_{k} }}\right) = 0$ , i.e.,\begin{align*}& \frac {\partial }{\partial \mathbf {x}_{k}}\sum _{k=1}^{K} \omega _{k} \chi _{k} \left |{{ 1 - u_{k}^{*} \bar {\mathbf {h}}_{k} \mathbf {x}_{k} }}\right |^{2} + \frac {\partial }{\partial \mathbf {x}_{k}}\sum _{k=1}^{K} \omega _{k} \chi _{k} \left |{{ u_{k} }}\right |^{2} a_{k} \\ & \quad \qquad \qquad \qquad \qquad - \frac {\partial }{\partial \mathbf {x}_{k}}\sum _{k=1}^{K} \omega _{k} \log _{2} \chi _{k} = 0. \tag {27}\end{align*} Consequently, the first, second, and third terms of (27) are respectively given by\begin{align*} & \frac {\partial }{\partial \mathbf {x}_{k}}\sum _{k=1}^{K} \omega _{k} \chi _{k} \left |{{ 1 - u_{k}^{*} \bar {\mathbf {h}}_{k} \mathbf {x}_{k} }}\right |^{2} \\ & = 2 \omega _{k} \chi _{k} \left |{{ u_{k} }}\right |^{2} \bar {\mathbf {h}}_{k}^{H} \bar {\mathbf {h}}_{k} \mathbf {x}_{k} - 2 \omega _{k} \chi _{k} u_{k} \bar {\mathbf {h}}_{k}^{H}, \tag {28}\\ & \frac {\partial }{\partial \mathbf {x}_{k}}\sum _{k=1}^{K} \omega _{k} \chi _{k} \left |{{ u_{k} }}\right |^{2} \left ({{ \sum _{j \neq k}^{K} \left |{{ \bar {\mathbf {h}}_{k} \mathbf {x}_{j} }}\right |^{2} + \frac {\sigma _{k}^{2}}{P} \sum _{i=1}^{K} \mathop {\mathrm {Tr}}\nolimits \left ({{ \bar {\mathbf {H}} \mathbf {x}_{i} \mathbf {x}_{i}^{H} }}\right) }}\right) \\ & = 2 \sum _{j \neq k}^{K} \omega _{j} \chi _{j} \left |{{ u_{j} }}\right |^{2} \bar {\mathbf {h}}_{j}^{H} \bar {\mathbf {h}}_{j} \mathbf {x}_{k} + 2 \sum _{i = 1}^{K} \frac {\sigma _{i}^{2}}{P} \omega _{i} \chi _{i} \left |{{ u_{i} }}\right |^{2} \bar {\mathbf {H}} \mathbf {x}_{k}, \tag {29}\end{align*} and\begin{equation*} \frac {\partial }{\partial \mathbf {x}_{k}}\sum _{k=1}^{K} \omega _{k} \log _{2} \chi _{k} = 0. \tag {30}\end{equation*} Note that for the sake of convenience, we replaced $a_{k}$ in (29) with its definition. By substituting (28), (29), and (30) back into (27) and solving for $\mathbf {x}_{k}$ , we obtain the solution in (12).