Low-complexity cooperative active and passive beamforming multi-RIS-assisted communication networks

Abstract

Reconfigurable intelligent surface (RIS) is a groundbreaking technology that has a significant potential for sixth generation (6G) networks. Its unique capability to control wireless environments makes it an attractive option. However, the spatial diversity increased by assisting users with all deployed RISs, this investigation has two drawbacks the high complexity design, and the received signals by the far RISs are severely attenuated. Therefore, we propose a RIS selection strategy to select the proper RISs as a pre-stage before the joint beamforming between the base station (BS) and RISs to reduce the high complexity of joint beamforming optimization. Furthermore, the joint active and passive beamforming problem based on the selection is formulated. Hence, achieving spatial diversity by examining cooperation between passive beamforming of multi-hop RIS, leads to a challenging problem. To tackle this issue, we design an algorithm for the RIS selection scheme. Also, to relax the non-convexity of the proposed problem, we decouple the problem into solvable subproblems by utilizing the fractional programming (FP) and quadratic transform (QT) optimization methods. Simulation results have demonstrated through different user locations the effectiveness of the selection strategy in performance enhancement by 30% in the sum rate, besides an obvious reduction in the complexity cost than other techniques.

1 Introduction

Recent advancements in wireless communication technology, including 6G and 6G+, have garnered significant attention from researchers interested in the smart configuration of wireless networks in diverse environments. These advancements leverage the mmWave frequency spectrum, which is highly sensitive to environmental conditions. The objective is to build smart connections between devices and address environmental challenges using technologies such as artificial intelligence (AI), unmanned aerial vehicles (UAVs), cell-free massive multiple-input multiple-output (mMIMO), and metamaterial intelligent surfaces [1,2,3]. Among these technologies, reconfigurable intelligent surface plays a crucial role which includes a passive and cost-effective planar array of reflecting elements that can be adjusted to manipulate the phase shifts and amplitudes of electromagnetic waves. By collaboratively modifying the reflecting coefficients, RIS enhances the performance of non-line-of-sight links and overall system performance.

RIS is cost-effective and power-efficient compared to conventional transmitters or relays. Numerous studies have proposed models and analyzed performance metrics to improve wireless network coverage areas by deploying RIS and its reflective elements. In the beginning, the primary focus of the research was on exploring the potential improvements in wireless communications achieved by utilizing a single RIS. These studies investigated various aspects such as energy efficiency [4,5,6], weighted sum rate [7,8,9], network coverage [10,11,12]. The key outcomes of these studies underscore the substantial RIS deploying impact and leveraging its reflective elements to enhance network coverage.

Tons of papers have studied and designed RIS-assisted wireless networks because of the great impact of utilizing RIS capabilities in enhancing faded channels and blocking problems. The research begins by investigating joint active and passive beamforming (JAPBF) between BS and single RIS and then upgraded to utilizing cooperation of passive beamforming between multi-RISs in different topologies, i.e., parallel, cascaded, and hybrid ways [13,14,15,16,17,18,19].

1.1 Motivation and contribution

Unfortunately, assisting users with all deployed RISs is not an efficient way, because the signals reflected by the RISs are severely attenuated as shown in Fig. 1. Because of the mobility of users, in many cases, some RISs may be far from than user. By this motivation, it is critical to merge a new role as RIS selection (RIS-S) in multi-RIS assisting. RIS-S strategy is investigated through two schemes: a single selected RIS and a set of RISs to serve the user. RIS-S based on a single selected surface, authors focused on selecting the main and best-reflected path, [20,21,22,23,24]. Unlike RIS-S based on a set of selected surfaces, authors focused on choosing the best-reflected paths [25, 26]. Figures 1 and 2 show the conventional multi-hop RIS assisted without any selection and multi-hop RIS-assisted network after investigating the RIS-S strategy, respectively.

Fig. 1

The conventional multi-hop RIS

Fig. 2

The proposed RIS-S strategy in multi-hop RIS

The authors in [20] proposed two multi-RIS-aided schemes called exhaustive RIS-aided (ERA) and opportunistic RIS aided (ORA). It highlighted the advantages and drawbacks in the cases of exploiting all surfaces in assisting as ERA scheme or one scheduled RIS is exploited in assisting as ORA scheme. ORA scheme achieved low complexity of backhaul and fronthaul load. By investigating the parallel topology of multi-RIS assisted, [21] exploited jointly the RIS-S to active and passive beamforming (APBF) optimization by selecting the optimal combination between RISs and users. By assuming a RIS-S matrix includes all possible combinations set and by alternating manner RIS-S matrix is updated by the optimal combinations based on the APBF. Unlike in [21], cooperative optimization between the APBF and a RIS-S strategy is presented in [22], where the graph neural network (GNN) investigated to help in reaching the optimal RISs to construct the optimal topology of the wireless network. In a different way, authors in [23] optimized the outage probability and average symbol error probability by selecting the best RIS. The selection criteria of RIS-S were to select the RIS which has the best signal-to-noise ratio (SNR). Then, the authors extended this work to UAV assisted in case of multi-RISs distributed, by studying the optimal RIS placement to enhance the outage probability and the received signal. Hence, the selection strategy is based on the received signal power at UAV, not the destination, [24]. In [25], authors differ from the previous work by assuming a set of RISs is selected as the best surfaces for assisting users, instead of exploiting all surfaces. Motivated by minimizing the overall consumed power in the network, a dynamic controlling of involving surface in assisting i.e., RIS has binary status on–off, the energy efficiency of the network is maximized in [26].

Unlike [21] and [22], a RIS-S strategy proposed to be applied as a pre-stage before APBF optimization. This approach reduces the complexity of the optimization problem by selecting the proper RISs for each user. Moreover, multi-hop RIS investigated to enhance guiding the signal to the endpoint by mitigating the scattered path losses, because the signals reflected by the far RISs are severely attenuated.

Aforementioned, [21,22,23,24,25], authors utilized different optimization techniques to relax and decouple the problem into solved subproblems. In this study, the objective function involves a ratio of two functions. However, there are several other types of optimization problems that are commonly encountered to solve it such as convex optimization, stochastic optimization, and dynamic programming. Due to the non-convexity of the problem, fractional programming investigated to reformulate the problem into a convex and decouple the problem into subproblems that could be analyzed. Then quadratic transform applied to reshape the objective function in a way that is solved.

In summary, our contributions are:

We propose a RIS-S algorithm that specifies the best channels between the RISs and users based on a significant SNR. This selection role is a pre-stage before the APBF optimization, unlike previous works which selected the proper RISs.

• 1. In contrast to the previous work, all users are associated with a single RIS. We maximize the SINR, which by default leads to maximization of the sum rate by comprising the channel model and active beamforming (ABF) of BS, but it seems to be non-convex and multivariate. For the purpose of resolving this problem, by employing closed-form fractional programming methods, we are able to effectively decouple the original problem into three independent sub-problems. In a similar way [18], by introducing new variables, we are able to split the sum-of-logarithms-of-ratio problem into two terms, each of which may then be simplified independently using the quadratic transform.

• 2. Due to performing the RIS-S as a pre-stage before the APBF optimization, the complexity of the APBF optimization design is reduced.

The following sections will organize the remainder of the paper: In Sect. 2, both the model of the system and the RIS selection scheme are introduced and summarized in the proposed algorithm. In Sect. 3, we present the suggested problem formulation of the JAPBF technique based on RIS-S for multi-hop RIS-assisted multi-user wireless communication networks. The results of the simulation are presented in Sect. 4. In Sect. 4, we summarize our study.

2 Methods

2.1 System and channel model

In our proposed mmWave wireless communication network with multiple RISs, as shown in Fig. 2, establishing direct connections between the BS and users faces significant obstacles. Our approach is like a multi-arm network, where $N$ distributed RISs assist in transmitting data from the BS, equipped with $M$ antennas, to $K$ users, each with a single antenna. As mentioned earlier, we anchor the first RIS hop, denoted as RIS_l,1 for any l-th virtual route where $\left( {l = 1, 2, ..., L} \right),$ very close to the BS to ensure a line-of-sight (LoS) channel directly above the BS. Also, each virtual route represents $Q$ of RISs chain which equals $N/L$. The remaining RISs, denoted as RIS_l,n (i.e., RIS $n = 2, 3, ..., Q$), are distributed in a sequential manner to create multi-hop routes in a multi-arm configuration. Each RIS is composed of $N_{e}$ reflecting elements, with each element connected to a controller responsible for controlling the phase shifts.

In order to tackle the issue of multi-hop RIS routes and the presence of arbitrary reflection paths. We let ${\varvec{G}}_{l} \in {\text{C}}^{{N_{e} \times M}}$, ${\varvec{H}}_{l,n} \in {\text{C}}^{{N_{e} \times N_{e} }}$ and ${\varvec{g}}_{l,n,k} \in {\text{C}}^{{1 \times N_{e} }}$ indicate the channels from the BS to RIS_l,1, and RIS_l,n to RIS_l,n+1 and from RIS_L,Q to user k, respectively. The interpretation of the reflection coefficient matrix of RIS_l,n can be described as ${\Phi }_{l,n} \in {\text{C}}^{{N_{e} \times N_{e} }}$, for more specificity, it can be further represented as ${{\varvec{\Phi}}}_{l,n} { } = {\text{ diag}}({{\varvec{\upphi}}}_{{l,n{ }}} )$. ${{\varvec{\upphi}}}_{{l,n{ }}}$ = [$\phi_{{l,n,1{ }}}$, $\phi_{{l,n,2{ }}}$, …, $\phi_{{l,n,N_{e} { }}}$]^T is the reflection vector of the (l, n)-th RIS, where $\phi_{{l,n,n_{e} { }}}$ = ${\upbeta }_{{l,n,n_{e} { }}}$ $e^{{j\theta_{{l,n,n_{e} }} { }}}$, n_e = 1, 2, …, $N_{e}$. In addition, $\theta_{{l,n,n_{e} }} \in \left[ {0, 2\pi } \right]$ and $\beta_{{l,n,n_{e} }} \in \left[ {0, 1} \right]$ represents the alteration in phase and amplitude caused by the n_e-th reflecting element of RIS_l,n on the incoming signals. We assume that the elements of the RIS are fully reflective, which implies that $\beta = 1$.

$${\varvec{G}}_{l} = \sqrt {\beta_{{{\varvec{G}}_{l} }} } \left( {\sqrt {\frac{{F_{{{\varvec{G}}_{l} }} }}{{F_{{{\varvec{G}}_{l} }} + 1}}} {\varvec{G}}_{{l,{\text{LoS}}}} + \sqrt {\frac{1}{{F_{{{\varvec{G}}_{l} }} + 1}}} {\varvec{G}}_{{l,{\text{NLoS}}}} } \right).$$

(1)

where $\beta_{{{\varvec{G}}_{l} }}$ represents the free space path loss; $F_{{{\varvec{G}}_{l} }}$ is the Rician factor; the matrices ${\varvec{G}}_{{l,{\text{LoS}}}}$ and ${\varvec{G}}_{{l, {\text{NLoS}}}}$ are the line-of-sight (LoS) and non-LoS (NLoS) components of the channel ${\varvec{G}}_{l}$, respectively. By the same, we can express the channel $g_{l,Q,k}$ from RIS_L,Q to user $k$ as

$$\begin{aligned} {\varvec{g}}_{l,Q,k} & = \sqrt {\beta_{{{\varvec{g}}_{l,Q,k} }} } \left( {\sqrt {\frac{{F_{{{\varvec{g}}_{l,Q,k} }} }}{{F_{{{\varvec{g}}_{l,Q,k} }} + 1}}} g_{{l,Q,k{\text{LoS}}}} } \right. \\ & \quad \left. { + \sqrt {\frac{1}{{F_{{{\varvec{g}}_{l,Q,k} }} + 1}}} g_{{l,Q,k\;{\text{NLoS}}}} } \right) \\ \end{aligned}$$

(2)

where $\beta_{{g_{l,Q,k} }}$ represents the free space path loss; $F_{{{\varvec{g}}_{l,Q,k} }}$ is the Rician factor; the matrices ${\varvec{g}}_{{l,Q,k\;{\text{LoS}}}}$ and ${\varvec{g}}_{{l,Q,k\;{\text{NLoS}}}}$ are the LoS and NLoS components of the channel ${\varvec{g}}_{l,Q,k}$, respectively.

Similarly, the remaining channels can be established using the same procedure. Once they are properly installed, high-quality LoS channels can be formed between them. Consequently, it is possible to assume that the channel state information (CSI) for the link between the BS and the first RIS, as well as the inter-RIS links, is already known. In RIS-assisted systems, there are two methods to obtain CSI for the RIS-user channel, depending on whether the reflective elements of the RIS have sensing capabilities. As RISs are usually positioned in areas conducive to establishing the LoS connections with users, such as the facades of tall buildings, the CSI can be acquired using location-based channel estimation algorithms [27,28,29], even if the RIS is only purely passive.

2.2 RIS selection strategy algorithm

As aforementioned, RIS-S role adds ability to choose the optimal RISs for association with the UE, considering the channel conditions. Several studies have proposed channel estimation methods in the presence of RIS by analyzing the cascaded channel from the BS to RIS and from RIS to user equipment (UE), [29,30,31]. Efficient algorithms were proposed to estimate the RIS-related channels. For instance, in [30], a compressive sensing and deep learning approach was proposed by deploying a few active antenna elements on the RIS.

We assume that perfect CSI is available. Moreover, we assume that all users have the possibility to associate with all RISs simultaneously [18]. Due to the random scattering of signals, we assume the far surfaces can receive the signals through multi-hop RIS to mitigate the losses. Hence, the distributed RISs can simultaneously reflect the beam toward other surfaces and UEs by playing two roles routing and accessing. So, it could be divided into two groups:

• (1) Access RIS (ARIS): It is the RIS in the RIS chain. It reflects the beam to user equipment UE.

• (2) Routing RIS (RRIS): Its role is to focus the electromagnetic beams toward the next RIS hop. So, a chain of dynamically aligned RISs plays the dedicated role of routing. However, it minimizes interference efficiently.

As known to the reader, the interaction channel between two RISs is affected by some parameters such as the orientation angle of the surface, the incident angle of the beam, and the surface area. Also, the channel between the RIS and the user is affected by the arrival and departure of angles besides RIS and UE locations. Without loss of generality, all channels are assumed as Rician modeled, and we took into consideration the arrival and departure angles to include the orientation angles of the surface implicitly, while the routing ability between the surfaces is introduced and studied in [11, 19] to construct a chain of RISs due to guiding signals into poor coverage areas. On the basis, to simply represent the selection matrix between RISs and user $k$ is ${\varvec{s}}_{k} = \left[ {s_{k,1} , \ldots , s_{k,N} } \right] \in Z^{ 1 \times N}$ consisting of binary variables, and $s_{k,n}$ is defined as

$$s_{k,n} = \left\{ {\begin{array}{*{20}l} {1,} \hfill & {{\text{user}}\;k\;{\text{is}}\;{\text{assisted}}\;{\text{by}}\;{\text{RIS}}_{l,n} .} \hfill \\ {0,} \hfill & {{\text{otherwise}}{. }} \hfill \\ \end{array} } \right.$$

(3)

For clarity, the selection decision role depends on the SNR of the cascaded channel between the BS to RIS and the RIS to UE, respectively. By this way, we ensure that the selected RIS satisfies high received power and at the same time is closer to UE. However, the channel of the nearest surface for user $k$ may be a good path, this surface is far away from BS, and the received power will be attenuated.

Thus, specifying the $\gamma_{T}$ value is more critical to decide. We have two options to deal with this point. The threshold value could be a fixed or dynamic changed value. Option 1, $\gamma_{T}$ is a fixed value, it is not fair decision to select the more effective surfaces due to user mobility and channel condition variations. This means, if the fixed value is high and the surrounding channels are noisy, the selected surfaces are very low and may not be the best selection. So, practically the dynamic changed value is more convenient. On the basis, this study assumed that $\gamma_{T}$ is adaptively specified, by estimating the mean value between the highest and lowest values of possible $\gamma_{n,k}$ for user $k$. On the basis, the $\gamma_{T}$ and the number of selected surfaces are changed corresponding to each user $k$.

Initially, we assume that all surfaces have a possibility of reflection to user $k$, so the selection matrix ${\varvec{s}}_{k}$ could be one’s entries. SNR is estimated for each RIS to user $k$. Then, due to updating ${\varvec{s}}_{k}$, we define $\gamma_{T}$ as a SNR threshold. $s_{k,n}$ could be updated by “1” for all values higher than $\gamma_{T}$. In this way, we ensure that all the effective surfaces are selected, and the un-useful surfaces are suppressed.

The above steps are summarized in the RIS-S algorithm below.

Algorithm 1

RIS selection

2.3 JAPBF design for multi-hop RIS assisted networks

2.3.1 Problem formulation

In this section, we start by denoting the downlink signal from the BS to user $k$ as $x_{k}$ with $E\left[ {\left| {x_{k} } \right|^{2} } \right] = 1$, $k = 1, 2, \ldots K$. Subsequently, we can express the $K$ transmitted signals as

$${\varvec{x}} = \mathop \sum \limits_{k = 1}^{K} {\varvec{w}}_{k} x_{k}$$

(4)

where ${\varvec{w}}_{k} \in C^{M \times 1}$ indicates the active beamforming vector for user $k$. By modeling the channel, which incorporates all potential paths and routes originating from BS to user $k$ can be expressed as matrix ${\varvec{h}}_{k}$

$${\varvec{h}}_{k} = \left[ {{\varvec{A}},{\varvec{B}}} \right]{,}$$

(5)

where the vector ${\varvec{A}} \in C^{{\left( {1 \times Q} \right)}}$ represents all transmitted beams from BS to RIS_l,1 and the vector ${\varvec{B}} \in C^{{\left( {1 \times Q} \right)}}$ represents the paths and routes between RIS_l,n−1 to RIS_l,n and RIS_l,Q to user $k$

$${\varvec{A}} = \left[ {{\varvec{g}}_{l,1,k} {{\varvec{\Phi}}}_{l,1} {\mathbf{G}}_{l} , \ldots ,{\varvec{g}}_{L,1,k} {{\varvec{\Phi}}}_{L,1} {\mathbf{G}}_{L} } \right]$$

(6)

$$\begin{aligned} {\varvec{B}} & = \left[ {{\varvec{g}}_{l,2,k} } \right.\mathop \prod \limits_{i = n,n - 1, \ldots ,2} \left( {{{\varvec{\Phi}}}_{l,i} {\mathbf{H}}_{l,i - 1,i} } \right){{\varvec{\Phi}}}_{l,1} {\mathbf{G}}_{l} , \\ & \quad \ldots ,{\varvec{g}}_{L,2,k} \left. {\mathop \prod \limits_{i = n,n - 1, \ldots ,2} \left( {{{\varvec{\Phi}}}_{L,i} {\mathbf{H}}_{L,i - 1,i} } \right){{\varvec{\Phi}}}_{L,1} {\mathbf{G}}_{L} } \right]. \\ \end{aligned}$$

(7)

Then, the updated channel model, after selecting the effective paths and routes and excluding weak paths by investigating the selection matrix as

$${\varvec{h}}^{H}_{s,k} = {\varvec{s}}_{k} {\varvec{h}}^{H}_{k} {,}$$

(8)

based on, the k-th received signal corresponding to the updated channel form ${\varvec{h}}^{H}_{s,k}$ as

$$y_{k} = {\varvec{h}}^{H}_{s,k} {\varvec{x}}{ + }n_{k} {,}$$

(9)

where $n_{k} \sim C^{{N \left( {0, \sigma_{0}^{2} } \right)}}$ represents the additive white Gaussian noise (AWGN) at the k-th user. Additionally, the k-th user considers the interference caused by other users’ signals. Therefore, we can express the Signal-to-Interference-plus-Noise Ratio (SINR) at user $k$ as:

$$\gamma_{k} = \frac{{\left| {{\varvec{h}}_{s,k}^{H} {\varvec{w}}_{k} } \right|^{2} }}{{\mathop \sum \nolimits_{i = 1,i \ne k}^{K} \left| {{\varvec{h}}_{s,k}^{H} {\varvec{w}}_{i} } \right|^{2} + \sigma_{0}^{2} }} ,$$

(10)

with $\mathop \sum \limits_{k = 1}^{K} \left\| {{\varvec{w}}_{k} } \right\|^{2} \le { }P_{T}$ represents the power constraint, where $P_{T}$ is the maximum transmitted power at the BS. Maximizing the SINR relies on the BS leveraging spatial diversity of power to direct signals for user $k$, i.e., $\left\| {{\varvec{w}}_{k} } \right\|^{2} \le { }P_{T}$. Consequently, to maximize the SINR there is $\gamma_{k}^{\max } = \frac{{P_{T} \left\| {{\varvec{h}}_{s,k}^{H} } \right\|^{2} }}{{\sigma_{0}^{2} }}$ can be achieved, and as a result, we have $0 \le \gamma_{k} \le \gamma_{k}^{\max }$. Let ${\varvec{W}} = \left[ {{\mathbf{w}}_{1} , {\mathbf{w}}_{2} , \ldots .{\mathbf{w}}_{K} } \right] \in {\text{C}}^{M \times K}$ represent the transmit beamforming matrix. We aim to maximize the total user rate during downlink transmission by jointly designing the transmit beamforming matrix ${\varvec{W}}$ at the BS and the phase shift matrices at ${\varvec{\phi}}_{{l,n{ }}}$ each RIS_l,n, while ensuring compliance with the transmit power at the BS and the phase shift constraints of the RISs. In this way, the optimization problem can be represented as

$$\begin{array}{*{20}l} {\left( {{\text{P}}1} \right)\mathop {\max }\limits_{{{\varvec{W}},{\varvec{\phi}}_{1,1} , \ldots ,{\varvec{\phi}}_{L,Q} }} f_{1} \left( {{\varvec{W}},{\varvec{\phi}}_{1,1} , \ldots ,{\varvec{\phi}}_{L,Q} } \right) = \mathop \sum \limits_{k = 1}^{K} {\text{log}}(1 + \gamma_{k} ),} \hfill \\ {{\text{s.}}\,{\text{t}}{.}\;\left| {\phi_{{l,n,n_{e} }} } \right| = 1,} \hfill \\ {l = 1, \ldots ,L} \hfill \\ {n = 1, \ldots ,N} \hfill \\ {n_{e} = 1, \ldots ,N_{e} } \hfill \\ {\mathop \sum \limits_{k = 1}^{K} \parallel {\varvec{w}}_{k} \parallel^{2} \le P_{T} .} \hfill \\ \end{array}$$

(11)

(P1) is formulated to address the ABF at BS and the PBF for RIS_l,n, ${\varvec{W}}$, and ${\varvec{\phi}}_{1,1} , \ldots ,{\varvec{\phi}}_{L,Q}$, respectively. The objective is to minimize inter-user interference on the receiving end and maximize the sum rate for multiple users. However, due to the coupling of the objective variables in the joint APBF optimization problem, finding the optimal solution becomes challenging. In the following section, we will adopt a similar approach as in [18] to alleviate this challenge and obtain a sub-optimal solution for (P1).

2.3.2 Optimal active and passive beamforming

As previously mentioned, our proposed JAPBF aims to maximize the sum rate of multiple users while minimizing inter-user interference. However, we face challenges due to the interdependence between the variables ${\varvec{W}}$ and ${\varvec{\phi}}_{{l,n{ }}}$, as well as the presence of a sum-of-log-of-ratio term, $\mathop \sum \limits_{k = 1}^{K} {\text{log}}(1 + \gamma_{k} )$. To address these challenges efficiently, we adopt a similar approach as in [18], where we solve the optimized problem by introducing auxiliary variables. Additionally, we explore closed-form fractional programming techniques [32] to transform problem (P1) into a form with significantly reduced complexity. By employing the Lagrange dual transform, we can transform the problem (P1) while ensuring the equivalence of $\varrho_{k} = \gamma_{k}$ as

$$\begin{array}{*{20}l} {{\text{(P2)}}\mathop {\max }\limits_{{\varrho ,{\varvec{W}},{\varvec{\phi}}_{1,1} , \ldots ,{\varvec{\phi}}_{L,Q} }} f_{2} \left( {\varrho ,{\varvec{W}},{\varvec{\phi}}_{1,1} , \ldots ,{\varvec{\phi}}_{L,Q} } \right)} \hfill \\ {\quad = \mathop \sum \limits_{k = 1}^{K} \log \left( {1 + \varrho_{k} } \right) - \mathop \sum \limits_{k = 1}^{K} \varrho_{k} + \mathop \sum \limits_{k = 1}^{K} \frac{{\left( {1 + \varrho_{k} } \right)\gamma_{k} }}{{1 + \gamma_{k} }}} \hfill \\ {{\text{s.}}\,{\text{t}}{.}\;\left| {\phi_{{l,n,n_{e} }} } \right| = 1,l = 1,2, \ldots ,L,} \hfill \\ {n = 1,2, \ldots ,N,} \hfill \\ {n_{e} = 1,2, \ldots ,N_{e} ,} \hfill \\ {\mathop \sum \limits_{k = 1}^{K} \parallel {\varvec{w}}_{k} \parallel^{2} \le P_{T} ,} \hfill \\ {\varrho_{k} \ge 0,k = 1,2, \ldots ,K,} \hfill \\ \end{array}$$

(12)

The vector $\varvec{\varrho }$ = $\left[ {\varrho_{1} ,\varrho_{2} , \ldots .\varrho_{K} } \right]^{T}$ represents the auxiliary variable $\varrho_{k}$ for each user $k$. In problem (P2), by introducing the auxiliary variable ϱ and considering the QT technique, we can separate the summation-of-ratio log side $\mathop \sum \limits_{k = 1}^{K} {\text{log}}(1 + \gamma_{k} )$ into the summation of ratio $\mathop \sum \limits_{k = 1}^{K} \frac{{\left( {1 + \varrho_{k} } \right)\gamma_{k} }}{{\gamma_{k} }}$ and the summation of log $\mathop \sum \limits_{k = 1}^{K} \log \left( {1 + \varrho_{k} } \right)$ as mentioned in Eq. (12). This decomposition greatly facilitates the attainment of the optimal solution. Before analyzing the JAPBF for the two subproblems, we focus on the auxiliary variable ϱ in (P2), assuming the variables ${\varvec{W}}$ and ${\varvec{\phi}}_{1,1} , \ldots ,{\varvec{\phi}}_{L,Q}$ are fixed. Consequently, we can optimize the problem (P1) subject to $\varrho$. By expressing ${\mathbf{\varrho }}^{op} = \left[ { \varrho_{1}^{op} , \varrho_{2}^{op} , \ldots .., \varrho_{K}^{op} } \right]^{T}$ and using $\varrho_{k}^{op}$ instead of $\gamma_{k}$, we can reformulate Eq. (11) as

$$f_{2} \left( {\varrho^{op} ,{\varvec{W}},{\varvec{\phi}}_{1,1} , \ldots ,{\varvec{\phi}}_{L,Q} } \right) = f_{1} \left( {{\varvec{W}},{\varvec{\phi}}_{1,1} , \ldots ,{\varvec{\phi}}_{L,Q} } \right).$$

(13)

Based on this proposition, we proceed to find the optimal solution by introducing and reformulating the two separate subproblems, denoted as ${\varvec{W}}$ and ${\varvec{\phi}}_{1,1} , \ldots ,{\varvec{\phi}}_{L,Q}$.

(1) The analytical solution for the optimal active beamforming matrix ${\varvec{W}}$:

Assuming that ϱ is held constant, our attention is directed toward the third term in Eq. (12), which contains the variable to be optimized. Based on this premise, we can reformulate the corresponding optimization problem as follows

$$\begin{array}{*{20}l} {\mathop {\max }\limits_{{{\varvec{W}},\phi_{1,1} , \ldots ,\phi_{L,Q} }} = \mathop \sum \limits_{k = 1}^{K} \frac{{\tilde{\varrho }_{k} \left| {{\varvec{h}}_{s,k}^{H} \omega_{k} } \right|^{2} }}{{\mathop \sum \nolimits_{i = 1}^{K} \left| {{\mathbf{h}}_{s,k}^{H} \omega_{i} } \right|^{2} + \sigma_{0}^{2} }}} \hfill \\ {{\text{s.}}\,{\text{t}}{. }\;\left| {\phi_{{l,n,n_{e} }} } \right| = 1,} \hfill \\ {l = 1,2,..,L;} \hfill \\ {n = 1,2, \ldots ,N;} \hfill \\ {n_{e} = 1,2,..N_{e} ;} \hfill \\ {\mathop \sum \limits_{k = 1}^{K} {\varvec{w}}_{k}^{2} \le P_{T} ,} \hfill \\ \end{array}$$

(14)

where $\overline{\varrho }_{k} = \left( {1 + \varrho_{k} } \right)$ for k = 1, 2, …, K. To optimize ${\varvec{W}}$ at a given ${\mathbf{\varrho }}$ and ${\varvec{\phi}}_{1,1} , \ldots ,{\varvec{\phi}}_{L,Q}$,

$$\begin{array}{*{20}l} {{(}P{2}{\text{.2)}}} \hfill & {\mathop {\max }\limits_{{\varvec{W}}} \left( {\varvec{W}} \right) = \mathop \sum \limits_{k = 1}^{{\text{K}}} \frac{{\varrho_{k} \left| {{\varvec{h}}_{s,k}^{{\text{H}}} {\varvec{w}}_{k} } \right|^{2} }}{{\mathop \sum \nolimits_{i = 1}^{K} \left| {{\mathbf{h}}_{s,k}^{H} {\varvec{w}}_{i} } \right|^{2} + \sigma_{0}^{2} }}} \hfill \\ {} \hfill & {{\text{s}}\,{\text{t}}{.}\mathop \sum \limits_{k = 1}^{K} \parallel {\varvec{w}}_{k} \parallel^{2} \le P_{T} {.}} \hfill \\ \end{array}$$

(15)

As mentioned in (14), problem (P2.2) can be classified as a multiple-ratio fractional programming problem. To address this, we need to reformulate the problem as a biconvex optimization problem by introducing a new objective function that leverages the QT in [32], as

$$\begin{aligned} f_{2.2} \left( {{\varvec{W}},{\varvec{\chi}}} \right) & = \mathop {\max }\limits_{{{\varvec{W}},{\mathcal{X}}}} = \mathop \sum \limits_{k = 1}^{K} \sqrt {\overline{\varrho }_{k} } \left( {\chi_{k}^{*} {\mathbf{h}}_{s,k}^{H} {\varvec{w}}_{k} + {\varvec{w}}_{k}^{H} {\mathbf{h}}_{s,k} \chi_{k} } \right) \\ & \quad - \mathop \sum \limits_{k = 1}^{K} \left| {\chi_{k} } \right|^{2} \left( {\mathop \sum \limits_{i = 1}^{K} \left| {{\mathbf{h}}_{s,k}^{H} {\varvec{w}}_{i} } \right|^{2} + \sigma_{0}^{2} } \right), \\ & \quad {\text{s.}}\,{\text{t}}{.}\mathop \sum \limits_{k = 1}^{K} \parallel {\varvec{w}}_{k} \parallel^{2} \le P_{T} , \\ \end{aligned}$$

(16)

where ${\varvec{\chi}}$ is introduced as a new auxiliary variable, where ${\varvec{\chi}} = \left[ {{\upchi }_{1} ,{\upchi }_{2} ,{ } \ldots .,{\upchi }_{K} } \right]^{T}$ represents the auxiliary variable vector after transforming the optimization problem into a biconvex problem. Consequently, we can solve the biconvex problem by considering ${\varvec{W}}$ and ${\varvec{\chi}}$ as variables and addressing the following problem: finding the solution to (P2.2) while subject to the constraint of ${\varvec{W}}$. To deal with and analyze the problem in Eq. (15), we can investigate proposition 2 in [18], to represent the optimal beamforming vector ${\varvec{w}}_{k}$.

(2) The analytical solution for the optimal passive beamforming matrix ${\varvec{\phi}}_{1,1} , \ldots ,{\varvec{\phi}}_{L,Q}$:

The problem (P2) is transformed into a relaxed form, denoted as (P2.1), by introducing the auxiliary variable ${\mathbf{\varrho }}$. Unlike, consider the active beamforming matrix ${\varvec{W}}$ to be constant and concentrate on iteratively optimizing the reflection vector of each RIS. Our goal is to enhance the received signal at the user’s end by utilizing each RIS as both a relay and a scatter point, thereby increasing diversity. However, this approach poses challenges when it comes to optimizing the RIS phase shift matrix. Therefore, we treat the RIS phase shift matrix as an objective variable in the channel user $k$. Given that the reformulated channel is represented as

$${\varvec{h}}_{s,k}^{H} = \left( {\overline{\varvec{\phi }}_{{{\text{s l}},{\text{n}}}} {\varvec{\alpha}}_{{{\text{l}},{\text{n}}}} + {\varvec{\beta}}_{{{\text{l}},{\text{n}}}} } \right),$$

(17)

where $\overline{\varvec{\phi }}_{{{\text{s l}},{\text{n}}}}$ is the updated row vector after using the selected matrix ${\varvec{S}}_{k}$ that contains the reflecting vectors of ${\varvec{\phi}}_{l,n}^{H}$ by $Q{-} n + 1$ which indicates the times of appearance of reflecting vector in case of individually RIS_l,n optimization, i.e., $\overline{\varvec{\phi }}_{{{\text{ l}},{\text{n}}}} \in {\text{C}}^{{1 \times { }\left( {Q - n + 1} \right)N_{e} }}$. Furthermore, the matrix ${\varvec{\alpha}}_{{{\text{l}},{\text{n}}}}$ represents the portion of the channel that is associated with the optimized reflecting vector of the RIS_l,n, where ${\varvec{\alpha}}_{{{\text{l}},{\text{n}}}} \in {\text{C}}^{{{ }\left( {Q - n + 1} \right)N_{e} \times M}}$. On the other hand, the matrix ${\varvec{\beta}}_{{{\text{l}},{\text{n}}}}$ represents the residual portion of the channel that is not directly linked to the optimized reflecting variable, denoted as ${\varvec{\beta}}_{{{\text{l}},{\text{n}}}} \in {\text{C}}^{{{ }1 \times M}}$. Moving forward, we can focus on the part of the channel for user $k$, which includes the optimized reflecting vector of the RIS_l,n representing the route and path as follows:

$$\begin{aligned} {\varvec{\alpha}}_{l,n} & = \left[ {{\text{diag}}\left( {g_{l,n,k} } \right) \cdot {\varvec{R}}_{l,n} ,{\text{diag}}\left( {{\varvec{C}}_{l,n + 1} } \right) \cdot {\varvec{R}}_{l,n} ,} \right. \\ & \quad \ldots ,\left. {{\text{diag}}\left( {{\varvec{C}}_{l,Q} } \right) \cdot {\varvec{R}}_{l,n} } \right]^{T} , \\ \end{aligned}$$

(18)

where

$${\varvec{R}}_{l,n} = \left\{ {\begin{array}{*{20}l} {{\varvec{G}}_{l} ,} \hfill & {n = 1,} \hfill \\ {\mathop \prod \limits_{j = n,n - 1, \ldots ,2} \left( {{\varvec{H}}_{l,j - 1,j} {{\varvec{\Phi}}}_{l,j - 1} } \right){\varvec{G}}_{l} ,} \hfill & {n = 2, \ldots ,Q,} \hfill \\ \end{array} } \right.$$

(19)

Once we have reformulated ${\varvec{h}}_{s,k}^{H}$ for the passive beamforming scenario, the objective function in (13) can be represented as a function of $\phi_{l,n}$ as

$${\varvec{C}}_{l,n} = {\varvec{g}}_{l,n,k} \mathop \prod \limits_{i = n,n - 1, \ldots ,2} {{\varvec{\Phi}}}_{l,i} {\varvec{H}}_{l,i - 1,i}$$

(20)

$$\begin{aligned} {\varvec{\beta}}_{l,n} & = \mathop \sum \limits_{j = 1,j \ne l}^{L} \left( {{\varvec{g}}_{j,1,k} {{\varvec{\Phi}}}_{j,1} {\varvec{G}}_{j} } \right. \\ & \quad \left. {\left. { + \mathop \sum \limits_{m = 2}^{Q} {\varvec{g}}_{j,m,k} \mathop \prod \limits_{i = m,m - 1, \ldots ,2} \left( {{{\varvec{\Phi}}}_{j,i} {\varvec{H}}_{j,i - 1,i} } \right){{\varvec{\Phi}}}_{j,1} {\varvec{G}}_{j} } \right)} \right), \\ \end{aligned}$$

(21)

$$f_{3} \left( {\phi_{l,n} } \right) = \mathop \sum \limits_{k = 1}^{K} \frac{{\tilde{\varrho }_{k} \left| {\overline{\varvec{\phi }}_{{{\text{s l}},{\text{n}}}} {\mathbb{A}}_{l,n,k} + {\mathcal{B}}_{l,n,k} } \right|^{2} }}{{\mathop \sum \nolimits_{i = 1}^{K} \left| {\overline{\varvec{\phi }}_{{{\text{s l}},{\text{n}}}} {\mathbb{A}}_{l,n,i} + {\mathcal{B}}_{l,n,i} } \right|^{2} + \sigma_{0}^{2} }}.$$

(22)

$$\begin{aligned} f_{3.1} \left( {{\varvec{\phi}}_{l,n} ,{\varvec{\delta}}} \right) & = \mathop \sum \limits_{k = 1}^{K} \sqrt {\tilde{\varrho }_{k} } \left( {\delta_{k}^{*} \left( {\overline{\varvec{\phi }}_{{{\text{s l}},{\text{n}}}} {\mathbb{A}}_{l,n,k} + {\mathcal{B}}_{l,n,k} } \right)} \right. \\ & \quad + \left( {{\mathbb{A}}_{l,n,k}^{H} \overline{\varvec{\phi }}_{{{\text{s l}},{\text{n}}}}^{H} + {\mathcal{B}}_{l,n,k}^{*} } \right)\delta_{k} \\ & \quad - \mathop \sum \limits_{k = 1}^{K} \left| {\delta_{k} } \right|^{2} \left( {\mathop \sum \limits_{i = 1}^{K} \left| {\overline{\varvec{\phi }}_{{{\text{s l}},{\text{n}}}} {\mathbb{A}}_{l,n,k} + {\mathcal{B}}_{l,n,k} } \right|^{2} + \sigma_{0}^{2} } \right), \\ \end{aligned}$$

(23)

where $\delta_{1}$, $\delta_{2}$, …, $\delta_{K}$ are the newly introduced complex auxiliary variables and ${\varvec{\delta}} = \left[ {\delta_{1} , \delta_{2} , \ldots ,\delta_{K} } \right]^{T}$ is the auxiliary variable vector. Subsequently, as demonstrated in [32], optimizing f_3.1 $({\varvec{\phi}}_{l,n} , {\varvec{\delta}})$ with respect to ${\varvec{\phi}}_{l,n}$ is equivalent to solving the following problem with respect to and δ. By investigating proposition 3 in [18], we can analyze the problem in Eq. (22), to achieve the optimal expression $\phi_{{l,n,n_{e} }}$ of the reflecting vector ${\varvec{\phi}}_{l,n}$ is updated. In this way, we can ensure the convergence of the objective function, by updating the phase shift matrix of each RIS_l,n iteratively. From the complexity point of view, however, we compensate for the network performance by using more routes, the proposed scheme reduces the complexity of passive beamforming between deployed RISs. In this way, the convergence of the objective function is achieved by iteratively updating the phase shift of each RIS_l,n.

3 Results and discussion

To demonstrate the effectiveness of the proposed strategy and using $\gamma_{T}$ adaptively changed, we consider two different scenarios for user allocation with fixed RISs location, as shown in Figs. 3 and 4. This helps us understand how users impact the RIS-S scheme. The distribution of multiple RISs resembles the arms of an octopus, ensuring that the signal is injected toward the user’s side. In the mmWave frequency band, we assume that there are $N_{e}$ equals 49 reflected elements per surface. Using the Friis transmission law [33], we calculate the channels ${\varvec{G}}_{l}$ and ${\varvec{g}}_{l,k}$, with a Rician factor $F$ of 6, and the power of noise is set to − 85 dBm. Table 1 provides a summary of the simulation parameters.

Fig. 3

User allocation in 2 dimensions (case 1)

Fig. 4

User allocation in 2 dimensions (case 2)

Table 1 Simulation parameters

As shown in Fig. 3, case 1 illustrates a wide distance distribution of users. Based on the RIS selection, it is noted that the RIS 4 and RIS 6 suppressed. That is due to long distances between the surface to BS and UE, respectively. So, it means that these surfaces are not effective to involve in assisting users or enhance the communication network. On the contrary, the optimization complexity increases without noted utility. On the other hand, case 2 in Fig. 4 illustrates a short-distance distribution and the suppressed surfaces are RIS 5 and RIS 6 for the same reason.

Simulation results were simulated by MATLAB 2021b on Dell desktop 10^thg with RAM 16G and processor 3.75 GHz. We conducted an averaging across 300 independent realizations of wireless channels. For evaluating the proposed technique, we compared proposed strategy with one scheme employing RIS-S based on GNN [22] and two schemes: without RIS-S1 [10] and without RIS-S2 [18] which signify cooperation among all RISs with parallel and cascade topologies, respectively.

As shown in Figs. 5 and 6 the sum rate is measured in bit per channel use (bpcu) as a unit. Initially, the sum rate convergence of the RIS-S scheme appears superior to the benchmark schemes in two scenarios. However, while the proposed RIS-S scheme ultimately achieves better convergence compared to the GNN-based scheme, the GNN-based approach slightly outperforms at the sixth iteration. This variation arises from the fact that the GNN-based scheme conducts selection internally during the optimization process, unlike the proposed technique, which performs it as a preliminary stage. Ultimately, the RIS-S strategy yields the optimal sum rate value, even with changes in user locations. The selection matrix shifts are evident in Figs. 5 and 6. Additionally, two schemes without selection are executed in a simplified design from a path perspective. Consequently, each user is served by all selected RISs, while weak reflection paths may not be necessary for optimization. As noted in Figs.5 and 6 the convergence of all schemes stabilized after iteration 4 except GNN-based at iteration 6.

Fig. 5

Sum rate convergence of the proposed RIS-S scheme versus benchmark schemes (case 1)

Fig. 6

Sum rate convergence of the proposed RIS-S scheme versus benchmark schemes (case 2)

Figures 7 and 8 illustrate the influence of the number of elements concerning the selection strategy. It is widely acknowledged that augmenting the elements per surface improves the communication link. However, this improvement is constrained by factors such as surface location and orientation angle. Therefore, the most effective approach to enhance the communication link is to prioritize the selection process before expanding the surface area without notable benefits. As depicted in Figs. 7 and 8, the proposed RIS-S technique surpasses all other schemes. Moreover, it is noteworthy that the proposed technique achieves a higher sum rate compared to GNN-based methods, contrary to previous findings in Figs. 4 and 5. This discrepancy stems from the fact that GNN-based approaches select the optimal surface among deployed RISs, whereas the proposed technique selects a group or multiple surfaces from candidate RISs.

Fig. 7

Sum rate based on number of elements (N_e) for the proposed RIS-S scheme versus benchmark schemes (case 1)

Fig. 8

Sum rate based on number of elements (N_e) for the proposed RIS-S scheme versus benchmark schemes (case 2)

Figures 9 and 10 depict the time required for each scheme to converge and attain the optimal active weights and phase shifts of RISs in two scenarios. The RIS-S scheme exhibits the shortest convergence time, attributed to the optimization process being conducted between $S$ surfaces rather than $N$ RISs, as seen in benchmark schemes. Conversely, the GNN-based approach, despite performing the selection role, consumes more time to achieve optimal rates due to the involvement of selection within the optimization process. Furthermore, the two schemes without selection require extended time to reach optimal rates at each iteration, as JAPBF is established among all RISs. Ultimately, the significance of RIS-S as a preliminary stage before JAPBF optimization is performed, emphasizing its pivotal role.

Fig. 9

The consumed time needed for optimal rate (case 1)

Fig. 10

The consumed time needed for optimal rate (case 2)

We estimate the complexity of the proposed system as introduced in [18]. In Table 2, we note that two parameters that indicate the difference in comparison of complexity are the number of selected RISs ($S$) and the number of convergence iterations $\left( I \right).$ Parallel and cascaded topology utilization idea is based on using all deployed RISs ($N$) without checking if the surface is effective or not. Because the main object of parallel topology is achieving more spatial diversity by utilizing each surface to reflect directly to the user, also the cascaded topology constructs a chain of cascaded surfaces to guide the signal to the user. On the other hand, the RIS-S strategy facilitates choosing the more effective surfaces ($S$) for all deployed surfaces ($N$).

Table 2 Computational complexity comparison

At the most case, if all surfaces ($N$) are effective and this case is rarely, selection strategy works as the parallel topology. In addition, the convergence iteration number is affected by reaching the optimal maximum sum rate of the system, in order to compute and estimate the computational complexity of the JAPBF algorithm. Likewise the work proposed in [18] which represented the computational complexity as the same method for showing the complexity reduction. We can divide the estimation of complexity based on the active BF and the passive BF. To begin, the complexity values are $O \left( K \right)$ and $O \left( {M^{3} K} \right)$, respectively, as a result of updating the auxiliary’s variable ${\upchi }_{k}^{op}$ and calculating the complexity of $W$ for the BS. As a result, the complexity of active BF is $O \left( {M^{3} K + K} \right)$. Second, by calculating the auxiliary variable, the complexity equals $O \left( K \right)$ for all users $K$. Also, by iteratively updating the phase shift of each element of RIS_l,n, the complexity equals $O \left( {N_{e} } \right)$. In the same way, the complexity of passive BF is estimated to be $O \left( {N_{e} + K} \right)$. When $N$ RISs are used as multi-hop points, the complexity equals $O (\mathop \sum \limits_{n = 1}^{N} N_{e} + NK)$. Finally, we can estimate the overall computational complexity for JAPBF as $O \left( {\left( {(M^{3} + 2 + N)K + \mathop \sum \limits_{n = 1}^{N} N_{e} } \right)I_{c} } \right)$ where $I_{c}$ is the number of iterations needed to converge. However, our proposed model achieved the same computational complexity as [18], but the JAPBF algorithm converges and reaches the optimal solution faster.

4 Conclusion

We present a novel technique to joint active and passive beamforming (JAPBF) that differs from conventional methods. Instead of utilizing all deployed RISs, we introduce a new strategy for selecting RISs. This allows us to investigate the impact of spatial diversity gain by serving users with multiple RISs, while keeping the complexity low. By examining RIS-S before applying JAPBF, we can maximize the overall 8 achievable sum rate for users. The problem formulation is non-convex, so we employ fractional programming and quadratic transform techniques to relax and break down the problem into simpler subproblems. Through simulation results, we demonstrate that the JAPBF approach based on our proposed RIS-S strategy outperforms JAPBF approaches that rely on all RIS cooperation schemes. Additionally, our design significantly reduces computational complexity.

The following ideas are suggested for future research:

• RIS selection in case of more obstructed channel as Nakagami-m distribution.

• Studying the performance of the proposed RIS selection from the energy efficiency and spectral efficiency points to verify how the impact of utilized RIS reduction on the power consumption.