Maximising Achievable Rate Using Intelligent Reflecting Surface in 6G Wireless Communication Systems

Abstract

Intelligent reflecting surface (IRS) is a promising technique which aims to shift the paradigm of uncontrollable wireless environment to a controllable one by adding the function of reconfigurability using multiple passive reflecting elements. In this work, optimal beamforming design for maximising achievable rate with respect to variable location of the IRS is considered. In particular, a single-cell wireless system that employs an IRS to aid communication between the user and an access point (AP) equipped with multiple antennas is adopted. An optimisation problem is formulated which aims to maximise the achievable rate, subject to signal-to-interference-plus-noise ratio (SINR) constraint of each individual user as well as the total transmit power constraint at the AP. The problem is solved by jointly optimising the transmit beamforming using active aerial array at the AP and the reflection coefficients using passive phase shifting at the IRS. Since the original optimisation problem is strictly non-convex, the problem is solved optimally by solving a corresponding power minimisation problem. Rigorous simulations have been carried out and the results demonstrate that the IRS-enabled system outperforms benchmark systems and employs significantly fewer RF power amplifiers.

1. Introduction

Intelligent reflecting surface (IRS) recently sparked considerable interest as a potential method for improving the efficiency of wireless communication systems [1]. An IRS is a 2D planar meta-surface with the flexibility to embed numerous inexpensive and passive reflecting elements on a single meta-surface, giving it a direct line of sight (LoS) to the access points (AP) or base station (BS), making it easy to deploy on structures such as walls, ceilings, and tunnels. The Figure 1 shows a typical arrangement of IRS aided wireless communication system.

IRS can be used to boost radio systems’ capacity by lowering interference and raising the signal-to-noise ratio (SNR). IRS may efficiently cancel out interference and enhance the signal’s overall quality by adjusting the signal’s reflection angle and phase. Overall, the incorporation of IRS into the 6G standard has tremendous promise for enhancing the efficiency of radio networks, expanding their reach, and cutting costs for wireless providers [2]. The deployment and design of IRSs have been the subject of numerous studies aimed at improving wireless network performance in terms of capacity, signal quality, coverage and security [3].

In order to increase the SNR of an IRS-enabled wireless communication system, a joint optimisation framework for IRS placement and phase management is proposed in [4]. The SNR in turn will increase the achievable rate of the system. It demonstrates that the SNR of the system can be significantly improved by intelligently deciding the placement and phase shift of the IRS. In a similar approach in [5], the authors proposed an ideal design and deployment method for IRS in a wireless communication system which will ensure that the achievable rate of the system is maximised. In the proposed method, both the amount of phase shift and the IRS’s geometry are optimised using a genetic algorithm.

IRSs have emerged as a promising solution for minimising transmit power in wireless communication systems. Unlike conventional relay systems, IRS utilises passive signal reflection to direct energy efficiently toward receivers. In multi-user downlink scenarios, integrating IRS has been shown to significantly lower the transmit power required to meet specific signal-to-interference-plus-noise ration (SINR) targets, particularly as the number of reflecting elements increases [4]. For instance, Ref. [6] examined an IRS-assisted cognitive radio network and found that configurations with 8 to 32 IRS elements provided 20–32 dB performance improvements compared to systems without IRS. These findings highlight IRS’s potential to enhance energy efficiency without compromising system throughput.

IRS also contributes to improved bandwidth utilisation and spectral efficiency by reinforcing signal strength and coverage, thus supporting more effective frequency reuse. Research in mmWave communications reveals that enlarging the IRS array or increasing transmit power can dramatically enhance spectral efficiency—achieving up to a 3× gain at 5 MHz and up to 14× at 25 MHz bandwidths [7]. These benefits stem from IRS’s ability to mitigate signal degradation due to multipath and interference, resulting in higher data rates over existing spectrum resources. Therefore, IRS technology plays a critical role in optimising spectrum efficiency, which is vital for supporting the massive capacity demands of 6G networks [8].

From the viewpoint of network providers, quality of service (QoS) reflects the capability of the communication system to deliver consistent performance tailored to various user applications. Key QoS indicators include data throughput, latency, jitter, reliability, and outage probability, all of which are essential for fulfilling service-level agreements (SLAs) and ensuring user satisfaction. In wireless networks enhanced by IRS, delivering QoS becomes both more challenging and more promising. The IRS’s ability to reshape the wireless environment passively allows networks to adaptively steer signals, making it possible to satisfy individual QoS needs. For example, IRS can help sustain high SINR levels and reliable links even under non-line-of-sight (NLoS) conditions. Nevertheless, consistent QoS in IRS-assisted systems demands advanced solutions in beamforming, real-time reconfiguration, and optimisation that can address issues such as user mobility, fading, and resource allocation. Ultimately, service providers must strike a balance between maintaining QoS and optimising costs and energy consumption, positioning IRS as a transformative—yet technically demanding—tool for future 6G networks [9,10].

From the above discussion, it is evident that while some attempt has been made to improve achievable rate in an IRS-enabled system, a lot of other complexities remain unresolved and required further exploration and analytical solutions to these complex optmisation problems. In this work, the achievable rate at the user in an IRS-enabled wireless system is maximised, subject to each individual user’s SINR constraints as well as the total transmit power constraint at the AP. This non-convex problem sovled by optimally solving a power minimisation problem which achieved superior performance compared to benchmark systems.

The main contributions of this paper can be summarised as:

• To address a highly non-convex rate maximisation problem under QoS guarantees and total power constraints.
• To establish the solution equivalence between the rate maximisation and the corresponding power minimisation problems.
• To solve the power minimisation problem optimally.
• To demonstrate the effectiveness of the proposed solution approach through numerical simulations.
• To show the effect of IRS location on the performance of the system in simulation results.

The rest of this article is organised as follows: In Section 2, related works are discussed. Section 3 presents an IRS-enabled wireless communication system model. Then the rate maximisation problem is formulated based on the proposed model. For this, alternating optimisation algorithm is described in Section 4. Section 5 provides thorough numerical simulations and rigorous result analysis. Finally, in Section 6, the conclusions and future research ideas are presented.

2. Related Works

In the literature, several methods for increasing the achievable rate in a wireless network can be found. The authors in [11] considered a multi-user IRS-enabled wireless communication system and proposed a novel algorithm for channel estimation. The proposed algorithm demonstrates significant performance enhancement compared to a traditional wireless communication system without IRS and claims to consider practical deployment scenarios. However, for the proposed channel estimation algorithm, the paper does not provide a detailed discussion of the considered practical implementation issues such as hardware limitations, synchronisation requirements and processing overhead.

The study [12] explores how big multiple-input multiple-output (MIMO) systems may not be able to transmit wireless data as efficiently as large intelligent surfaces (LIS). The authors propose a new communication paradigm that makes advantage of a considerable quantity of passive reflecting properties in LIS in order to enhance the wireless channel for data transmission. Even though the idea of LIS has received a lot of attention lately, there is not much actual experience with deploying and employing LIS for wireless communication.

The major goal of the paper [13] is to give a thorough analysis of the application of IRS in massive MIMO communication systems. The study [14] offers a thorough overview of the most recent advances in secure communications supported by IRS. To address these issues and assess the viability and efficacy of the suggested solutions, additional study is needed.

Previous studies have explored enhancements in wireless communication performance through passive beamforming technologies, focusing on improvements in sum-rate, energy efficiency, and reliability. For instance, data rate enhancement was examined in [15], energy-aware design strategies in [16], phase-based reliability improvements without channel state information (CSI) in [17], and latency minimisation under imperfect CSI in [18]. However, joint optimisation of surface placement, phase control, and active beamforming in constrained indoor environments remains insufficiently addressed. To bridge this gap, this work adopts an alternating optimisation framework, where the active beamforming at the transmitter is optimally handled using maximum-ratio transmission (MRT) [19].

In addition to energy and spectrum efficiency, IRS significantly improves throughput and signal quality metrics like SNR and CNR. Alhamad et al. derived analytical throughput expressions and showed that when IRS functions as a transmitter, it outperforms traditional reflector models by about 1 dB [6]. Other studies suggest that the received SNR can increase with the square—or even fourth power—of the number of IRS elements, thereby enhancing communication reliability and performance [20]. Field trials further confirm these advantages, showing that IRS deployment can increase downlink throughput by 19 to 78 Mbps (up to 360 Mbps).

Moreover, IRS is particularly beneficial in NLoS environments, which are prevalent in high-frequency systems like mmWave and terahertz bands. When direct line-of-sight paths are obstructed, IRS can generate alternative paths by reflecting signals strategically, thus preserving link stability. Wang et al. demonstrated that IRS enhances received power and coverage in NLoS conditions, with performance gains scaling with the number of elements [21]. Additionally, IRS-enhanced deployments have shown improvements in reference signal received power (RSRP) by 3–10 dB and better signal quality in challenging environments such as urban shadows and indoor areas [22]. These attributes position IRS as a key enabler for robust and reliable wireless communication in future 6G networks.

The primary goal of the paper [23] is to present a thorough analysis of the application of IRS in the context of 6G wireless networks. The authors go into great detail about the different design difficulties that IRS in 6G networks faces, such as channel estimation, beamforming optimisation, and hardware limitations. It would have been beneficial to have heard more about standardisation initiatives and their possible effects on the practical application of the technology. The article [24] provides a complete account of current developments in IRS-assisted wireless power transfer. The study discusses the core concept of IRS. The paper [25] introduces IRS and some of their potential uses for them in wireless communication systems. There is no consideration of the potential benefits of the IRS in multi-antenna systems, such as massive MIMO. Future studies should go deeper into the potential benefits of IRS in multi-antenna systems.

A thorough summary of current developments in IRS technology is given in the work [26]; this paper’s limits and look for additional sources for more in-depth study or review of particular problems. For wireless communication systems that employ IRS, the study of the paper [27] proposes a hybrid active and passive beamforming technique (IRS). A comparison with already-existing alternatives is necessary to assess the inventiveness and efficacy of the proposed solution.

The authors in [28] proposed an innovative iterative estimation method for reconfigurable intelligent surface (RIS)-assisted near-field localisation. However, the complexity of the algorithm is quite high compared to other similar methods. The work in [29] proposes an interesting idea where IRS phase is optimised first and then a fair allocation of resources (both subcarrier and power per subcarrier) for an Orthogonal Frequency Division Multiplexing (OFDM) system. The performance analysis demonstrates that the algorithm maximises the date rate in a multiuser scenario. In order to achieve that, it develops a practical beamforming codebook to select the RIS configuration that maximises the SNR which in turn maximises the achievable rate. However, the proposed iterative method’s performance is inferior to well known semi-definite relaxation (SDR) method.

The work in [30] proposes two IRS elements grouping mechanism to reduce complexity of channel estimation with the aim of improving the achievable rate of the the uplink of a communication system. It also improves the energy efficiency of the system. In the first method, it combines the elements of sub-matrix as a single entity (such as 2 by 2 or 6 by 6). However, it neither provides insights on accuracy of such assumption, nor provides the robust argument for sub-matrix size. In the second method, using LoS between users transmission and reception, it reconstructs the channel matrix which reduces the estimation complexity and overhead. However, such assumption of LoS between users may not always be realistic.

3. System Model

Figure 1 demonstrates a single-cell network where downlink communications are employed with the assistance of IRS. It is utilised when an IRS is sent out to assist with communications across a certain frequency band between K single-antenna clients and a multi-antenna AP. The number K represents the collection of users. M and N represents the number of transmit antennas at the AP and the number of reflecting elements at the IRS, respectively [31]. If the received signal is reflected more than once by the IRS, then it is considered negligible due to huge amount of path loss and can be discarded without any significant loss in accuracy.

The channels between AP and user k is denoted by ${\mathbf{h}}_{r,k}^H\in {\mathbf{C}}^{1\times N}$. The channel between AP and IRS are denoted by $\mathbf{G}\in {\mathbf{C}}^{N\times M}$. The channel between IRS and the user k is denoted by ${\mathit{h}}_{d,k}^H\in {\mathbf{C}}^{1\times M}$ where k = $1,\dots ,K$. Let $\mathit{\theta }$ = $[{\theta }_1,\dots ,{\theta }_N]$ and is a diagonal matrix $\mathbf{\Theta }$ = $\mathrm{diag}({\beta }_1{e}^{j\theta 1},\dots ,{\beta }_N{e}^{j\theta N})$. Here, j denotes the imaginary unit value of the IRS’s reflection–coefficients matrix, where ${\theta }_n\in [0,2\pi )$ and ${\beta }_n\in [0,1]$ denote the phase shift angle and the amplitude of reflection coefficient of the $n^{th}$ element of the IRS, respectively. Hence, the AP-IRS link, the phase shift angle of IRS reflection, and the link between IRS and user are combined to form the AP-IRS-user composite channel [4].

In this paper, linear transmit precoding is used at the AP where each individual user is allocated a unique beamforming vector. Hence, x = ${\sum }_{k=1}^K{w}_k{s}_k$, can be used to represent the baseband broadcast signal that is complicated at AP where $w_k\in C^{M\times 1}$ represents the associated beamforming vector and $s_k$ are the transmitted data for the user k. It is presumed that $s_k$ represents a random variables with mean zero and unity variance, where k = 1, …, K. The combined received signal at user k through the AP-user as well as through the AP-IRS-user route can be written as [4]:

$$\begin{array}{c}\hfill y_k=({\mathit{h}}_{r,k}^H\mathbf{\Theta }\mathbf{G}+{\mathit{h}}_{d,k}^H)\sum _{j=1}^K{\mathit{w}}_j{s}_j+n_k,k=1,...,K\end{array}$$

(1)

where $n_k\sim \mathcal{CN}$ (0,${\sigma }_k^2$) is additive white Gaussian noise (AWGN) at the receiver of the user k. Hence, user k’s SINR can be expressed as:

$$\begin{array}{c}\hfill {\mathrm{SINR}}_k={\displaystyle \frac{|({\mathbf{h}}_{r,k}^H\mathbf{\Theta }\mathbf{G}+{\mathbf{h}}_{d,k}^H){\mathbf{w}}_k{|}^2}{{\sum }_{j\ne k}^K{(\mathbf{h}}_{r,k}^H\mathbf{\Theta }\mathbf{G}+{\mathbf{h}}_{d,k}^H{\left){\mathbf{w}}_j\right|}^2+{\sigma }_k^2}},\forall k.\end{array}$$

(2)

Consequently, the achievable rate of the user k can be defined as [4]:

$$\begin{array}{c}\hfill R_k={log}_2\left(1+{\mathrm{SINR}}_k\right).\end{array}$$

(3)

Next, the optimisation problem is formulated to maximise the rate of the user with worst achievable rate using IRS.

4. Problem Formulation

Let $\mathbf{W}$ = $[{\mathbf{w}}_1$, ⋯, ${\mathbf{w}}_K]\in C^{M\times K}$, ${\mathbf{H}}_r$ = $[{\mathbf{h}}_{r,1}$, ⋯, ${\mathbf{h}}_{r,K}]\in C^{N\times K}$, and ${\mathbf{H}}_d$ = $[{\mathbf{h}}_{d,1}$, ⋯, ${\mathbf{h}}_{d,K}]\in C^{M\times K}$. The rate maximisation problem is considered for the user with the worst wireless channel condition under total power constraint. The objective is to maximise the achievable rate by jointly optimising the transmit beamforming at the AP and the reflected beamforming direction at the IRS under individual SINR constraints for all users. Thus, the optimisation problem can be formulated as:

$$\begin{array}{cc}\hfill \left(\mathrm{P}0\right):\underset{\mathbf{W},\mathit{\theta }}{max}& \underset{\mathrm{k}}{min}{\mathrm{R}}_{\mathrm{k}}\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill \mathrm{s}.\mathrm{t}.& {\displaystyle \frac{|({\mathbf{h}}_{r,k}^H\mathbf{\Theta }\mathbf{G}+{\mathbf{h}}_{d,k}^{\mathbf{H}}){\mathbf{w}}_k{|}^2}{{\sum }_{j\ne k}^K{\left|({\mathbf{h}}_{r,k}^H\mathbf{\Theta }\mathbf{G}+{\mathbf{h}}_{d,k}^H){\mathbf{w}}_j\right|}^2+{\sigma }_k^2}}\ge {\gamma }_k,\forall k,\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill & \sum _{k=1}^K\left|\right|{\mathbf{w}}_k{\left|\right|}^2\le P_T,\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill & 0\le {\theta }_n\le 2\pi ,n=1,\dots ,N.\hfill \end{array}$$

(7)

Here, (5) is the QoS SINR constraint with ${\gamma }_k>0$ as the minimum required SINR of user k, and (6) is the total power constraint. The above problem is strictly non-convex due to the coupled variables in (4) and (5). This problem is solved by solving the corresponding power minimisation problem. The corresponding power minimisation solution can be formulated for the given optimal objective value $R_k^{*}$ of the rate maximisation problem (P0) as:

$$\begin{array}{cc}\hfill \left(\mathrm{P}1\right):\underset{\mathbf{W},\mathit{\theta }}{min}& \sum _{k=1}^K\left|\right|{\mathbf{w}}_k{\left|\right|}^2\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill \mathrm{s}.\mathrm{t}.& {\displaystyle \frac{|({\mathbf{h}}_{r,k}^H\mathbf{\Theta }\mathbf{G}+{\mathbf{h}}_{d,k}^{\mathbf{H}}){\mathbf{w}}_k{|}^2}{{\sum }_{j\ne k}^K{\left|({\mathbf{h}}_{r,k}^H\mathbf{\Theta }\mathbf{G}+{\mathbf{h}}_{d,k}^H){\mathbf{w}}_j\right|}^2+{\sigma }_k^2}}\ge {\gamma }_k^{*},\forall k,\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill & 0\le {\theta }_n\le 2\pi ,n=1,\dots ,N,\hfill \end{array}$$

(10)

where ${log}_2(1+{\gamma }_k^{*})=R_k^{*}$. Note that problem (P1) is a variation of problem (P0). Interestingly, it can be seen later that problem (P1) is easier to solve compared to problem (P0). The problem (P1) will be solved below and a solution will be established for correspondence between (P1) and (P0). Problem (P1) aims at minimising the total transmit power subject to the minimum required SINR ${\gamma }_k^{*}$ which is directly related to the achievable rate $R_k^{*}$.

Theorem 1.

Any optimal solution to the power minimisation problem (P1) is also optimal for problem (P0).

Proof. 

See Appendix A.    □

Let us now consider a single user system for simplicity, i.e., $K=1$. Simplifying the problem (P1), we get

$$\begin{array}{cc}\hfill \left(\mathrm{P}2\right):\underset{\mathbf{w},\mathit{\theta }}{min}& {\left|\right|\mathbf{w}\left|\right|}^2\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill \mathrm{s}.\mathrm{t}.& |({\mathbf{h}}_r^H\mathbf{\Theta }\mathbf{G}+{\mathbf{h}}_d^H){\mathbf{w}|}^2\ge \gamma {\sigma }^2,\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill & 0\le {\theta }_n\le 2\pi ,n=1,\dots ,N.\hfill \end{array}$$

(13)

Even after simplifying the problem as much as possible, the problem still remains a non-convex optimisation problem since the left-hand side of the above problem is not jointly concave with respect to $\omega$ and $\theta$. This challenge is overcome by the use of alternating optimisation and SDR.

4.1. Alternating Optimisation Technique

In this subsection, a different sub-optimal algorithm will be developed based on alternating optimisation which is demonstrated in Algorithm 1. Let us assume $\mathbf{w}$ = $\sqrt{\mathit{P}}\overline{\mathbf{w}}$, where $\mathit{P}$ denotes the transmit power at the AP and $\overline{\mathbf{w}}$ is the transmit beamforming vector. From (P2), the optimised transmit power can be written as: $\mathit{P}$* = ${\displaystyle \frac{\gamma {\sigma }^2}{\left|\right|({\mathbf{h}}_r^H\mathbf{\Theta }\mathbf{G}+{\mathbf{h}}_d^H)\overline{\mathbf{w}}{\left|\right|}^2}}$. And then the transmitted beamforming direction is optimised for given $\theta$. Here, the $n^{th}$ phase shift at the IRS can be expressed as:

$$\begin{array}{cc}\hfill {\theta }_n^{*}& ={\varphi }_0-arg\left({\mathbf{h}}_{n,r}^H\right)-arg\left({\mathbf{g}}_n^H\overline{\mathbf{w}}\right),\hfill \end{array}$$

(14)

where ${\mathbf{g}}_n^H$ denotes the $n^{th}$ row vector of $\mathbf{G}$, ${\mathbf{h}}_{n,r}^H$ denotes the $n^{th}$ element of ${\mathbf{h}}_r^H$ and the function $arg(·)$ is used to denote the phase (or argument) of a complex number.

It should be noted that ${\mathbf{g}}_n^H$$\overline{\mathbf{w}}$ is the combination of the transmitted beamforming direction and the AP-IRS channel. The fact that the derived phase, n, is independent of the amplitude of ${\mathbf{h}}_{n,r}$ is also significant to note. For fixed beamformimg direction $\overline{\mathbf{w}}$, (P2) can be expressed as:

$$\begin{array}{cc}\hfill \underset{\theta }{max}& |({\mathbf{h}}_r^H\mathbf{\Theta }\mathbf{G}+{\mathbf{h}}_d^H)\overline{\mathbf{w}}{|}^2\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill \mathrm{s}.\mathrm{t}.& 0\le {\theta }_n\le 2\pi ,n=1,\dots ,N.\hfill \end{array}$$

(16)

The following steps are used for the optimisation of transmitted power.

Algorithm 1: Alternating Optimisation Technique.

4.2. SDR

Now, SDR will be applied for solving problem (P2). It is known that MRT is the optimised transmit beamforming solution for problem (P2), i.e., $\mathbf{w}$* = $\sqrt{P}{\displaystyle \frac{{({\mathit{h}}_r^H\mathbf{\Theta }\mathbf{G}+{\mathit{h}}_d^H)}^H}{\left|\right|{\mathit{h}}_r^H\mathbf{\Theta }\mathbf{G}+{\mathit{h}}_d^H\left|\right|}}$, where P is the transmitted power of AP. Here, $\mathbf{w}$* is replaced by [4]:

$$\begin{array}{cc}\hfill \underset{P,\theta }{min}& P\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill \mathrm{s}.\mathrm{t}.& P\left|\right|{\mathbf{h}}_r^H\mathbf{\Theta }\mathbf{G}+{\mathbf{h}}_d^H{\left|\right|}^2\ge \gamma {\sigma }^2\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill & 0\le {\theta }_n\le 2\pi ,\forall n.\hfill \end{array}$$

(19)

Hence, reducing the transmitted power is equal to increasing the combined channel gain power and is expressed as:

$$\begin{array}{cc}\hfill \underset{\theta }{max}& \left|\right|{\mathbf{h}}_r^H\mathbf{\Theta }\mathbf{G}+{\mathbf{h}}_d^H{\left|\right|}^2\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill \mathrm{s}.\mathrm{t}.& 0\le {\theta }_n\le 2\pi ,\forall n.\hfill \end{array}$$

(21)

Let $\mathbf{v}$ = ${[{\mathbf{v}}_1,\dots ,{\mathbf{v}}_N]}^H$ where ${\mathbf{v}}_n=e^{j{\theta }_n},\forall n$. Then, let the change of variables ${\mathbf{h}}_r^H\mathbf{\Theta }\mathbf{G}\triangleq {\mathbf{v}}^H\mathrm{\Phi }$ where $\mathrm{\Phi }$ = $diag\left({\mathbf{h}}_r^H\right)\mathbf{G}\in C^{N\times M}$, we have $\left|\right|{\mathbf{h}}_r^H\mathbf{\Theta }\mathbf{G}+{\mathbf{h}}_d^H{\left|\right|}^2$ = $\left|\right|{\mathbf{v}}^H\mathrm{\Phi }+{\mathbf{h}}_d^H{\left|\right|}^2$. Thus, problem (4.21) is equal to [4]:

$$\begin{array}{cc}\hfill \underset{\mathrm{v}}{max}& {\mathbf{v}}^H\mathbf{\Phi }{\mathbf{\Phi }}^H\mathbf{v}+{\mathbf{v}}^H\mathbf{\Phi }{\mathbf{h}}_d+{\mathbf{h}}_d^H{\mathbf{\Phi }}^H\mathbf{v}+\left|\right|{\mathbf{h}}_d^H{\left|\right|}^2\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill \mathrm{s}.\mathrm{t}.& |{\mathbf{v}}_n{|}^2=1,\forall n.\hfill \end{array}$$

(23)

Here, this problem (22) and (23) is still a non-convex quadratically constrained quadratic program (QCQP), which can be rewritten as a homogeneous QCQP [4]. So, considering a new auxiliary variable t, the problem (22) and (23) is then equivalent to

$$R=\left[\begin{array}{cc}\mathbf{\Phi }{\mathbf{\Phi }}^H& \mathbf{\Phi }{\mathbf{h}}_d\\ {\mathbf{h}}_d^H{\mathbf{\Phi }}^H& 0\end{array}\right],\overline{\mathbf{v}}=\left[\begin{array}{c}\mathbf{v}\\ t\end{array}\right].$$

Note that ${\overline{\mathbf{v}}}^H\mathbf{R}\overline{\mathbf{v}}$ = tr($\mathbf{R}\overline{\mathbf{v}}{\overline{\mathbf{v}}}^H$). Let $\mathbf{V}=\overline{\mathbf{v}}{\overline{\mathbf{v}}}^H$, which has to be satisfied $\mathbf{V}$ > 0 and $rank\left(\mathbf{V}\right)=1$. So, it is given by:

$$\begin{array}{cc}\hfill \underset{\mathrm{V}}{\max }& tr\left(\mathbf{RV}\right)+\left|\right|{\mathit{h}}_d^H{\left|\right|}^2\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill \mathrm{s}.\mathrm{t}.& {\mathbf{V}}_{n,n}=1,n=1,\dots ,N+1,\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill & \mathbf{V}⪰\mathbf{0}.\hfill \end{array}$$

(26)

Problem (24)–(26) can be solved optimally by a well-known convex optimisation problem solving tool such as CVX [32], since it is a convex semidefinite program (SDP). As the relaxed issue (24) typically does not have a rank-one solution, i.e., rank($\mathbf{V}$) = 1, it follows that the problem (24)’s optimal objective value merely serves as the upper bound (22).

Note that the single-user system developed above can be extended to multi-user scenario following the approaches in [4].

5. Simulation and Results

The reflecting elements in the IRS have been assumed to be arranged as a rectangular array such that $N\triangleq N_x$   $N_z$, where $N_x$ and $N_z$ are the number of reflecting elements along the x-axis and the z-axis, respectively. In our simulations, $N_x$ is set to 5 and $N_z$ increases linearly with N [4]. The path loss is given by:

$$\begin{array}{c}\hfill L\left(d\right)=C_0{\left({\displaystyle \frac{d}{D_0}}\right)}^{-\alpha },\end{array}$$

(27)

where d is the distance of the individual link, $C_0$ denotes the path loss factor at the reference distance, $D_0=1$ m and $\alpha$ represents the path loss exponent. A Rician fading channel model is considered for small-scale fading. Thus, the channel $\mathbf{G}$ between AP and IRS is given by [4]:

$$\begin{array}{c}\hfill \mathbf{G}=\sqrt{{\displaystyle \frac{{\beta }_{AI}}{1+{\beta }_{AI}}}}{\mathbf{G}}^{LoS}+\sqrt{{\displaystyle \frac{1}{1+{\beta }_{AI}}}}{\mathbf{G}}^{NLoS}\end{array}$$

(28)

where ${\beta }_{AI}$ is the Rician channel factor, and ${\mathbf{G}}^{LoS}$ represents LoS component and ${\mathbf{G}}^{NLoS}$ represents the NLoS or Rayleigh fading channel components, respectively. In this case, this model is reducing to LoS channel component and we assume ${\beta }_{AI}\to \infty$ or ${\beta }_{AI}=0$. The channel $\mathbf{G}$ is multiplied by the square root of the path loss factor $L\left(d\right)$. The path loss component is given by ${\alpha }_{AI}$.

Let us assume that ${\alpha }_{Au}$ and ${\alpha }_{Iu}$ are the path loss components of the AP-user and IRS–user links as ${\alpha }_{Au}$ = $3.4$ and ${\beta }_{Au}=0$ for large distance channel and random LoS-scattering at the AP–user channel. It has also been assumed that all users have the same SINR target, i.e., ${\gamma }_k$ = $\gamma$, $\forall k$. For single-user system SNR target is set to be $\gamma$ = 10 dB and M = 4 where M is reflecting components.

According to Figure 2, the user on horizontal position which is parallel to the line connecting the AP and IRS with vertical distance of $d_{\nu }$ = 2 (m) between the two lines. d(m) is used as the horizontal line distance of AP and user. Hence, $d_1$ = $\sqrt{d^2+d_{\nu }^2}$ and $d_2$ = $\sqrt{{(d_0-d)}^2+d_{\nu }^2}$, where $d_1$ is the distance between AP and user and $d_2$ is IRS–user link distance. The Rician channel factors and the path loss exponents are set as ${\alpha }_{AI}$ = 2, ${\alpha }_{Iu}$ = $2.8$, ${\beta }_{Iu}$ = 0, and ${\beta }_{AI}$ = ∞, respectively [4].

Some other parameters are also defined for our system, i.e., $C_0$ = −30 dB, $d_0$ = 51 m and ${\sigma }_k^2$ = −80 dBm. The stopping criteria for alternating optimisation algorithm, e is set at ${10}^{-5}$.

The simulation has been carried out with Matlab R2023b. the optimisation is done using well known optimisation problem solving tool CVX. Further details of the tool can be found in [32]. For the simulation, a Macbook air with an M3 chip and 24 GB RAM was used.

In this section, the performance of several schemes have been compared, namely: (1) SDR: The solution is obtained by applying Gaussian randomisation techniques and SDR [4]; (2) Alternating optimisation: This method is described above in the Section 4.1; (3) AP-user MRT: It is set as $\overline{\mathbf{w}}$ = ${\displaystyle \frac{{\mathbf{h}}_d}{\left|\right|{\mathbf{h}}_d\left|\right|}}$ for achieving MRT based on the AP–user direct channel [33]; (4) AP-IRS MRT: It is solved by setting $\overline{\mathbf{w}}$ = ${\displaystyle \frac{\mathbf{g}}{\left|\right|\mathbf{g}\left|\right|}}$ for achieving MRT based on the AP-IRS rank-one channel with ${\mathbf{g}}^H$ denoting any row in $\mathbf{G}$ [33]; (5) Random phase shift: Here, $\mathit{\theta }$ is randomly set in the range [0, 2$\pi$] and then MRT is performed at the AP [4]; (6) Without IRS: It is solved as $\mathbf{w}$ = $\sqrt{\gamma {\sigma }^2}{\displaystyle \frac{{\mathbf{h}}_d}{\left|\right|{\mathbf{h}}_d{\left|\right|}^2}}$ [4].

In the above, the SDR technique plays a vital role in solving the non-convex optimisation problem that arises when jointly designing the IRS phase shifts and AP beamforming. Specifically, by reformulating the problem as QCQP and introducing an auxiliary matrix variable, the difficult rank-one constraint is relaxed into a convex form. This makes the problem solvable using convex optimisation tools such as CVX which is mentioned before.

Alongside this, two intuitive yet practical beamforming schemes are investigated: AP-user MRT and AP-IRS MRT. The AP-user MRT aligns the transmit beamforming vector with the direct AP-to-user channel. This approach is particularly effective when the user is close to the AP and the direct link dominates. However, in environments where the direct path is blocked such as indoor scenarios with thick walls its performance degrades significantly. In contrast, the AP-IRS MRT steers the AP’s beamforming vector toward the IRS instead of the user, assuming that the reflected path dominates. This method proves useful when the user is near the IRS and can benefit from a strong reflected signal. Nevertheless, neither of these two MRT strategies jointly optimise the IRS phase shifts and AP beamforming, resulting in suboptimal performance when the channel conditions are more balanced or complex.

In addition to these schemes, a random phase shift strategy is also considered as a low-complexity benchmark. In this case, the phase shifts applied by the IRS elements are randomly selected from a uniform distribution over $[0,2\pi )$, and the AP performs MRT based on the combined AP-user and AP-IRS-user channel. Although this method requires no channel state information or optimisation, it can still provide a modest performance gain due to signal reflections. However, the improvement scales only linearly with the number of IRS elements N, whereas optimised designs based on SDR or alternating optimisation achieve a quadratic ($N^2$) gain. This clearly highlights the advantage of jointly optimising the active and passive beamforming components.

Finally, a conventional Without IRS is considered, where the IRS is disabled and the system relies solely on the direct AP-to-user transmission. The AP applies MRT using only the direct channel. This setup serves as a useful baseline performance reference point to demonstrate the limitations in terms of coverage and power efficiency, especially when the direct link is either weak or blocked. Simulation results consistently show that the required transmit power in this case is significantly higher, particularly for users located far from the AP or in NLoS environments.

(1) Transmit Power at AP vs. AP-User Distance: In Figure 3, we have evaluated the horizontal distance between the AP and user against the transmit power required for every scheme. Since signal attenuation is higher for the system without the IRS, it can be seen that the user from the AP requires a higher transmit power at the AP. The transmit power requirement can be reduced by installing an IRS. However, the work also demonstrated that in an IRS-enabled wireless network, transmit power efficiency is not always guaranteed at every AP–user distance. When the user is placed further away from the AP and nearer to the IRS, it enables to get a stronger reflected signal from the IRS. So, the user who is closer to either the AP $(d=22$ m) or IRS ($d=46$ m), for example, uses less transmitted power than the user who is much far from both distance ($d=39$ m). This scenario also indicates that the signal coverage may be successfully enhanced by using just a passive IRS rather than installing a second AP or active relay. For instance, with the same transmit power of 12 dBm, the network’s coverage is roughly 33 m without the IRS, but by implementing the suggested combined beamforming designs with an IRS, this value is increased to be above 50 m.

Note that although the proposed alternating optimisation technique outperforms the existing schemes, it is more computationally demanding than the other schemes. For example, the ‘SDR’ method does not solve the problem iteratively and stops optimisation after the first round. On the other hand, the ‘alternating optimisation’ technique continues optimising the variables iteratively until a reasonable accuracy is obtained. Let us assume that the computational complexity of problem (P1) is in the order of $\mathcal{O}\left(Q_1\left(n\right)\right)$ and that of problem (P2) is $\mathcal{O}\left(Q_2\left(n\right)\right)$, where n is the number of optimisation variables. Then the complexity order of the ‘SDR’ method is $\mathcal{O}\left(Q_1\left(n\right)\right)+\mathcal{O}\left(Q_2\left(n\right)\right)$, whereas the complexity of the ‘alternating optimisation’ technique is in the order of $t\left(\mathcal{O}\left(Q_1\left(n\right)\right)+\mathcal{O}\left(Q_2\left(n\right)\right)\right)$, where t is the number of iterations. The ‘random phase shift’ method does not optimise the IRS phase-shift variable $\mathbf{\Theta }$ and hence requires significantly lower computational tasks. The MRT and the ‘without IRS’ schemes are much simpler schemes and have even lower computational complexities. However, with the emergence of modern computational power combined with machine learning, computational resources are no longer a major bottleneck for many applications. For scenarios where system efficiency is a priority, the proposed ‘alternating optimisation’ is the preferred method at affordable costs.

(2) Transmit Power at AP versus AP-IRS Distance: In Figure 3, the asymptotic performances of IRS passive beamforming with respect to the movement of IRS’s position horizontally has been analysed. In Figure 3 the distance between AP and user (e.g., d = 70) is constant. Then the IRS’s position is moved form (e.g., $d_0$ = 30 to 70). It can be seen that the transmit power of AP is maximum at the starting point which have gradually decreased through the starting point to the position near the user. The more the IRS is moved nearer to the user’s position, the lower the required transmit power. The alternating optimisation algorithm demonstrates a constant power consumption to a certain point and then it starts gradually decreasing at low power transmission at AP with respect to the distance between AP and IRS.

In Figure 4, the distance between AP and user (e.g., $d=50$) is constant; then, the IRS’s position has moved form (e.g., $d_0$ = 30 to 70). It can be seen that the transmit power of AP is maximum at the starting point and at the ending point. When the IRS’s position moved to the user’s position, then the power consumption became gradually low. It can also be seen that the mirror reflection in the next movement. The transmit power became 14 dBm when the distance between IRS and user was the lowest. The more the IRS approaches the user’s position, the lower the transmit power. The alternating optimisation technique has shown a constant power consumption initially to a certain point and then it started gradually decreasing near the user’s position and then it started increasing like a mirror reflection when it started moving far from the user’s position. Finally, it again shows a constant transmission-power at AP with respect to the distance of AP-IRS.

In Figure 5, the distance between AP and user (e.g., $d=30$) is constant. Then the IRS’s position changed form (e.g., $d_0$ = 30 to 70). It can be seen that the transmit power of AP is at the minimum at the starting point. When the IRS’s position moved far from the user’s position, then the power consumption became gradually higher. Here, the transmit power became nearly 29 dBm when the distance between IRS and user was the highest. The farther the IRS is from the user’s position, the higher the transmit power. Here, alternating optimisation technique has shown initially a little increment in power consumption to a certain point and then it has shown a constant power transmission at AP with respect to the distance of AP-IRS.

6. Conclusions

In this paper, an IRS-enabled wireless system is designed for a 6G wireless communication system which exploits the passive reflective properties on IRS element to maximise the achievable rate. A highly non-convex rate maximisation problem under QoS guarantee and total power constraints is considered. The solution equivalent between the rate maximisation and corresponding power minimisation is demonstrated by solving the power minimisation problem optimally. A thorough analytical solution is described before the effectiveness of the proposed method is demonstrated via rigorous simulation work. Through detailed analysis of the results, it has been demonstrated that that the proposed method outperforms not only the benchmark methods such as random phase shift but also other well-known techniques such as SDR and MRT. The user movement is analysed at various AP-IRS distances, and through this, the optimal positioning of the IRS is also explored for most effective performance.

The work can be further extended by incorporating challenging scenarios where extracting actual channel state information which is very difficult to obtain in real time. For example, optimal design for the vehicular network scenarios with fast-moving vehicles equipped with IRS can be considered. Also, the theoretical performance bound for the achievable rate in the considered scenario is still unknown. Analysing the theoretical bounds could provide useful insights for optimal design.