Drone-Mounted Intelligent Reflecting Surface-Assisted Multiple-Input Multiple-Output Communications for 5G-and-Beyond Internet of Things Networks: Joint Beamforming, Phase Shift Design, and Deployment Optimization

Abstract

In 5G-and-beyond (B5G) Internet of Things (IoT) networks, the integration of intelligent reflecting surfaces (IRSs) with millimeter-wave (mmWave) multiple-input multiple-output (MIMO) techniques can significantly improve signal quality and increase network capacity. However, a single fixed IRS lacks the dynamic adjustment capability to flexibly adapt to complex environmental changes and diverse user demands, while mmWave MIMO is constrained by limited coverage. Motivated by these challenges, we investigate the application of drone-mounted IRS-assisted MIMO communications in B5G IoT networks, where multiple IRS-equipped drones are deployed to provide real-time communication support. To fully exploit the advantages of the proposed MIMO-enabled air-to-ground integrated information transmission framework, we formulate a joint optimization problem involving beamforming, phase shift design, and drone deployment, with the objective of maximizing the sum of achievable weighted data rates (AWDRs). Given the NP-hard nature of the problem, we develop an iterative optimization algorithm to solve it, where the optimization variables are tackled in turn. By employing the quadratic transformation technique and the Lagrangian multiplier method, we derive closed-form solutions for the optimal beamforming and phase shift design strategies. Additionally, we optimize drone deployment by using a distributed discrete-time convex optimization approach. Finally, the simulation results show that the proposed scheme can improve the sum of AWDRs in comparison with the state-of-the-art schemes.

1. Introduction

1.1. Background

With the advancement of millimeter-wave (mmWave) and massive multiple-input multiple-output (MIMO) techniques, 5G-and-beyond (B5G) Internet of Things (IoT) networks exhibit significant improvements in network capacity, bandwidth, and connection density [1]. In B5G IoT networks, the mmWave technique enables transmission rates with higher frequency, while the MIMO technique increases signal transmission efficiency and reliability through the use of multiple antenna systems [2,3,4]. Nevertheless, in regions where the deployment of base stations presents significant challenges, B5G IoT networks face difficulty in achieving effective and reliable coverage due to the limited propagation range of mmWave signals and their susceptibility to obstacles [5]. Facing the above challenges, intelligent reflecting surfaces (IRSs), as an emerging wireless communication technology, have been introduced in B5G IoT networks. By optimizing the signal propagation path, IRSs allow for low-cost, energy-efficient dense deployment in areas with weak signal coverage, significantly improving spectral efficiency and communication quality [6]. Moreover, IRSs can reflect signals from remote base stations, and their precise signal reflection control helps to reduce signal attenuation and interference. Therefore, the combined use of mmWave MIMO and IRSs can further enhance the transmission performance of B5G IoT networks and effectively improve the quality of service (QoS) for cellular users (CUEs).

While the integration of an IRS with MIMO has demonstrated considerable potential in enhancing the transmission performance of B5G IoT networks, the static deployment of an IRS significantly constrains its adaptability and flexibility in remote regions characterized by limited infrastructure [7,8,9]. To overcome the limitation of fixed IRSs, an IRS can be installed on drones, leveraging their mobility for flexible signal coverage and dynamic adjustment [10]. Using a drone-mounted IRS can actively regulate space electromagnetic waves to establish stable communication links in various geographic locations and communication scenarios, thereby bypassing obstacles such as buildings and trees [11]. In this situation, it can effectively alleviate the communication interruption problem caused by obstacles between the base station and CUEs so as to provide real-time communication support for B5G IoT networks [12,13,14].

1.2. Related Works and Motivations

The mmWave MIMO technique can provide high-capacity and high-speed communication in B5G IoT networks. By integrating it with IRSs, signal coverage can be effectively enhanced, and QoS can be improved. Specifically, early studies primarily focused on channel modeling and basic transmission characteristics. For example, the authors in [15,16] investigated the channel estimation problem of IRS-assisted mmWave MIMO systems by exploiting the sparsity of mmWave channels in the angular domain, laying the foundation for the optimization and application of mmWave MIMO systems. To satisfy the low-power requirements, the authors in [17,18] proposed a joint active and passive beamforming design scheme for a single-user mmWave downlink communication system, optimizing both the IRS reflection coefficients and the base station’s beamforming to achieve optimal signal quality. Inspired by the mentioned works, the authors in [19,20,21,22] considered multiple IRS-assisted multi-user mmWave MIMO systems, studying the impact of the optimal placement and topology of the IRS on the channel capacity. Additionally, given the challenges in the practical applications of IRS-and-MIMO techniques, the issue of robust transmission with statistical channel state information assistance was explored in [23]. Furthermore, some efforts have been devoted to non-orthogonal multiple access mechanisms [24], hardware-based hybrid precoding [25], and phase shift optimization schemes [26,27].

However, due to the weak penetration performance of mmWave, the IRS-assisted MIMO technique faces challenges (e.g., a high path loss and signal degradation) in practical applications. As an essential component of B5G IoT networks, drones facilitate line-of-sight (LoS) transmissions by providing additional aerial coverage, thus addressing the limited propagation range of IRS-assisted MIMO systems. Currently, drones can serve as relay nodes or carry IRSs to assist communication in B5G IoT networks [28,29,30,31]. As a relay node, a drone must perform tasks such as signal demodulation, processing, and modulation, which consumes its limited resources and energy [32]. Moreover, relay drones typically require a higher transmission power to ensure the quality of signal transmission. In contrast, drones equipped with IRSs can offer significant advantages in terms of energy efficiency [33,34,35]. This approach works by adjusting the phase of the IRS reflection units to direct the signal towards the optimal propagation path, thereby reducing signal attenuation and interference, as well as effectively enhancing signal quality and coverage. In [36,37], the authors designed a joint trajectory planning and beamforming scheme for drone-mounted IRS-assisted MIMO communications, aiming to maximize both the spectral efficiency and energy efficiency of B5G IoT networks. Building upon this, the authors in [38] introduced an anchoring mechanism and improved network performance by optimizing the anchoring position and data collection strategy. Additionally, the authors in [39,40,41,42,43] investigated the effects of hover mode on system performance in drone-mounted IRS-assisted MIMO communications, based on which multi-dimensional resources were optimized.

The aforementioned studies promote the application of drone-mounted IRS-assisted MIMO communications into B5G IoT networks. Nevertheless, some problems remain to be investigated. In our considered scenarios, the main motivations are summarized as follows:

• First, the studies in [15,16,17,18,19] explored the combined application of IRS and MIMO techniques to enhance transmission performance. However, these studies overlooked the potential performance improvements achieved through air-to-ground integrated collaboration. Air-to-ground integrated networks can significantly enhance the coverage and robustness of B5G IoT networks by leveraging aerial and terrestrial resources, which is particularly important for mmWave MIMO communications. As described above, due to the limited penetration capability of mmWave signals, they are highly susceptible to significant degradation in environments with obstacles. In such scenarios, it is essential to consider deploying drone-mounted IRSs to extend the signal propagation range.
• Second, although the authors in [36,37,38] considered installing IRSs on drones to enhance signal coverage, they only examined the single-input single-output (SISO) scenario. Compared to SISO systems, MIMO systems can increase data transmission rates and system capacity by transmitting and receiving signals through multiple antennas within the same frequency band. Additionally, MIMO provides improved interference resistance and signal quality, thereby enhancing the reliability and performance of B5G IoT networks. Thus, by intelligently controlling the wireless transmission environment, IRS-assisted MIMO systems offer new degrees of freedom for optimizing B5G IoT networks. Moreover, most existing works implicitly assume that the IRS is deployed at an optimal location, without considering the optimization of IRS deployment [44]. However, the placement of the IRS affects the signal reflection angle and link quality. Considering this practical problem, the deployment location of the IRS on drones should be optimized.
• Finally, most existing studies have primarily focused on scenarios involving a single IRS, and there is an urgent need to further investigate resource optimization in environments with multiple IRS units. Specifically, the deployment of a single IRS in B5G IoT networks does not fully enhance the QoS for CUEs. The collaboration of multiple IRS units provides greater flexibility and control by exploiting multiple reflection paths to improve transmission performance. However, in air-to-ground integrated communication scenarios supported by multiple IRS units, optimizing multi-dimensional resources (e.g., beamforming and phase shift design) under resource constraints to boost network performance remains a complex and critical challenge that demands further in-depth research.

1.3. Contributions

As mentioned above, the current research on B5G IoT networks rarely addresses the optimization of IRS deployment and the number of IRS units. Additionally, the integration of an IRS with the mmWave MIMO technique and air-to-ground integrated collaborative communication remains underexplored. To bridge these gaps, we consider a multi-drone cooperative communication scenario equipped with an IRS, and based on this, we propose an air-to-ground integrated information transmission framework enabled by MIMO. By optimizing multi-dimensional resources, we aim to maximize the sum of the achievable weighted data rates (AWDRs) of B5G IoT networks. The contributions of this paper are summarized as follows:

• First, we present a novel approach to enhancing coverage performance in B5G IoT networks through the integration of multiple drones equipped with IRSs. Specifically, we introduce a MIMO-enabled air-to-ground transmission framework that leverages multiple IRS units, mounted on hovering drones, to optimize the reflection of mmWave signals from remote base stations. Unlike existing approaches, our method jointly optimizes beamforming, phase shift design, and deployment strategies to maximize the sum of AWDRs.
• Second, since the optimization problem of maximizing the sum of AWDRs is difficult to solve directly, we decouple the optimization variables and iteratively optimize the beamforming, phase shift design, and multi-drone deployment strategies. Specifically, we derive closed-form solutions for the optimal beamforming and phase shift design strategies. Furthermore, to address deployment optimization, we tackle the problem of finding the optimal solution in discrete space by employing a distributed discrete-time convex optimization approach, which can obtain the optimal deployment strategy for the drones.
• Finally, we conduct extensive simulations to evaluate the performance of the proposed AWDR sum maximization scheme. The simulation results demonstrate that the proposed AWDR sum maximization scheme outperforms the state-of-the-art schemes [7,10,11] in terms of the sum of AWDRs. Specifically, by increasing the number of IRS units and introducing the air-to-ground integrated information transmission framework, the sum of AWDRs is improved by 122.3% and 93.5%, respectively. Moreover, we investigate the impact of key parameters on the sum of AWDRs, including the number of CUEs, the number of IRS elements, and the maximum transmission power.

1.4. Organization

The rest of this paper is arranged as follows: Section 2 introduces the system model and describes in detail the formulated optimization problem. In Section 3, we propose the AWDR sum maximization scheme. Then, Section 4 presents the simulation results. Finally, Section 5 concludes this paper.

2. System Model and Problem Formulation

2.1. IRS-Assisted mmWave MIMO Communication Model

Figure 1 illustrates the considered IRS-assisted mmWave MIMO communication model, consisting of one remote base station, U CUEs, and B IRS units. The mmWave MIMO communication offers abundant spectrum resources and high data transmission rates. However, mmWave MIMO signals suffer from significant propagation losses, especially when the LoS path is obstructed, leading to substantial degradation in signal coverage and communication quality. To address this issue, this paper proposes the use of an IRS to enhance the downlink transmission of mmWave MIMO signals. Specifically, an IRS is installed on a hovering drone, and a controller intelligently adjusts the phase and amplitude of the reflected signal, thereby improving the communication quality and coverage between the drone and ground CUEs.

It is assumed that communication between the remote base station and U single-antenna CUEs is obstructed by obstacles, with no direct path available. In this scenario, the remote base station must rely on the drone-mounted IRS to reflect the signal to the CUEs, enabling data transmission in an air-to-ground scenario. To extend the signal transmission range, directional signal transmission and reception are employed. The remote base station is assumed to be equipped with a uniform linear array (ULA) consisting of A antenna elements. Considering that some signals need to bypass obstacles, B IRS units are deployed in B5G IoT networks, with each unit comprising $C_x$ horizontal elements and $C_y$ vertical elements, where $C=C_x\times C_y$, defined as $\mathcal{C}=\left\{1,\dots ,C\right\}$. The set of CUEs is defined as $\mathcal{U}=\left\{1,\dots ,U\right\}$, the set of antenna elements is defined as $\mathcal{A}=\left\{1,\dots ,A\right\}$, and the set of IRS units is defined as $\mathcal{B}=\left\{1,\dots ,B\right\}$.

In B5G IoT networks, we define three key reference point coordinates: the coordinates $r_{\mathrm{BS}}$ of the remote base station, the coordinates $r_{\mathrm{IRS}}^b$ of the b-th $\left(\forall b\in \mathcal{B}\right)$ drone-mounted IRS, and the coordinates $r_{\mathrm{MU}}^u$ of the u-th $\left(\forall u\in \mathcal{U}\right)$ CUE. In this paper, $r_{\mathrm{BS}}$ is a fixed value, $r_{\mathrm{MU}}^u=\left(x_{\mathrm{MU}}^u,y_{\mathrm{MU}}^u,z_{\mathrm{MU}}^u\right)$, and $r_{\mathrm{IRS}}^b=\left(x_{\mathrm{IRS}}^b,y_{\mathrm{IRS}}^b,z_{\mathrm{IRS}}^b\right)$. We define the distance between the remote base station and the b-th drone-mounted IRS as $d\left(r_{\mathrm{BS}},r_{\mathrm{IRS}}^b\right)$. Additionally, we define the distance between the b-th drone-mounted IRS and the u-th CUE as $d\left(r_{\mathrm{MU}}^u,r_{\mathrm{IRS}}^b\right)$, which is given by

$$\begin{array}{c}d\left(r_{\mathrm{MU}}^u,r_{\mathrm{IRS}}^b\right)=\sqrt{{\left(x_{\mathrm{MU}}^u-x_{\mathrm{IRS}}^b\right)}^2+{\left(y_{\mathrm{MU}}^u-y_{\mathrm{IRS}}^b\right)}^2+{\left(z_{\mathrm{MU}}^u-z_{\mathrm{IRS}}^b\right)}^2}.\hfill \end{array}$$

(1)

2.2. Air-to-Ground Channel Model

As discussed in [45], the channels between the remote base station and the b-th drone-mounted IRS, as well as between the b-th drone-mounted IRS and the u-th CUE, can be expressed as

$$\begin{array}{ccc}{\mathbf{W}}_b\hfill & =& \sqrt{d{\left(r_{\mathrm{BS}},r_{\mathrm{IRS}}^b\right)}^{-\delta }·h_0}\hfill \\ \hfill & \times & \left(\sqrt{{\displaystyle \frac{R_1}{R_1+1}}}{W}_b^{\mathrm{LoS}}+\sqrt{{\displaystyle \frac{1}{R_1+1}}}{W}_b^{\mathrm{NLoS}}\right),\hfill \end{array}$$

(2)

and

$$\begin{array}{ccc}{\mathbf{h}}_{b,u}\hfill & =& \sqrt{d{\left(r_{\mathrm{MU}}^u,r_{\mathrm{IRS}}^b\right)}^{-\omega }·h_0}\hfill \\ \hfill & \times & \left(\sqrt{{\displaystyle \frac{R_2}{R_2+1}}}{h}_{b,u}^{\mathrm{LoS}}+\sqrt{{\displaystyle \frac{1}{R_2+1}}}{h}_{b,u}^{\mathrm{NLoS}}\right),\hfill \end{array}$$

(3)

where ${\mathbf{W}}_b$ is the channel between the remote base station and the b-th drone-mounted IRS, ${\mathbf{W}}_b\in {\mathbb{C}}^{A\times C}$; ${\mathbf{h}}_{b,u}$ is the channel between the b-th drone-mounted IRS and the u-th CUE, ${\mathbf{h}}_{b,u}\in {\mathbb{C}}^{C\times 1}$; $\delta$ is the path loss index from the remote base station to the b-th drone-mounted IRS; $\omega$ is the path loss index from the b-th drone-mounted IRS to the u-th CUE; $h_0$ is the path loss index per unit distance; $R_1$ and $R_2$ are the Rician factors of two separate links [22]; $W_b^{\mathrm{LoS}}$ and $W_b^{\mathrm{NLoS}}$ are the LoS and non-LoS (NLoS) components of the channel between the remote base station and the b-th drone-mounted IRS, respectively; and $h_{b,u}^{\mathrm{LoS}}$ and $h_{b,u}^{\mathrm{NLoS}}$ are the LoS and NLoS components of the channel between the b-th drone-mounted IRS and the u-th CUE, respectively.

The signal is transmitted from the remote base station, reflected by the drone-mounted IRS units, and eventually reaches the U CUEs. The desired signal of the u-th CUE is given by

$$\begin{array}{c}{S}_u=\sum _{b=1}^B{\mathbf{h}}_{b,u}^{\mathrm{H}}{\mathbf{\Phi }}_b^{\mathrm{H}}{\mathbf{W}}_b{\mathbf{p}}_u{m}_u,\hfill \end{array}$$

(4)

where $m_u$ is the signal transmitted from the remote base station to the u-th CUE; ${\mathbf{p}}_u$ is the beamforming matrix used by the remote base station to transmit signal $m_u$ to the u-th CUE, ${\mathbf{p}}_u\in {\mathbb{C}}^{A\times 1}$; and ${\mathbf{\Phi }}_b$ is the phase shift matrix of the b-th drone-mounted IRS, ${\mathbf{\Phi }}_b=\sqrt{\eta }\mathrm{diag}\left({\left[{\theta }_{b,1,}\dots ,{\theta }_{b,C}\right]}^{\mathrm{T}}\right)$, where $\eta$ is the reflection coefficient. In ${\mathbf{\Phi }}_b$, we can obtain ${\theta }_{b,c}=e^{{\phi }_{b,c}i}$, where i is the imaginary unit, with $i=\sqrt{-1}$, and ${\phi }_{b,c}$ is the phase shift. Since this paper assumes that the incident signal energy in the scenario is not absorbed, we can conclude that $\eta =1$. In addition, the beamforming strategy is defined as $\mathcal{P}=\left[{\mathbf{p}}_1,\dots ,{\mathbf{p}}_U\right]$, and we have $\mathrm{tr}\left(\mathcal{P}{\mathcal{P}}^{\mathrm{H}}\right)\le P_{max}$, where $P_{max}$ is the maximum transmission power. Furthermore, considering that signals from other CUEs may cause co-channel interference to the target CUE (i.e., the u-th CUE), it is necessary to model the interference $I_u$ experienced by the u-th CUE. The co-channel interference $I_u$ can be expressed as

$$\begin{array}{c}{I}_u=\sum _{b=1}^B{\mathbf{h}}_{b,u}^{\mathrm{H}}{\mathbf{\Phi }}_b^{\mathrm{H}}{\mathbf{W}}_b\sum _{j=1,j\ne u}^U{\mathbf{p}}_j{m}_j,\hfill \end{array}$$

(5)

where j is the j-th $\left(j\in \mathcal{U}\right)$ CUE that also occupies this channel simultaneously.

Moreover, the total signal $Z_u$ received by the u-th CUE is the sum of the signals generated by each reflecting element of the drone-mounted IRS after the remote base station’s transmitted signal is reflected by the drone-mounted IRS. We have

$$\begin{array}{ccc}{Z}_u\hfill & =& S_u+I_u+n_u\hfill \\ \hfill & =& \sum _{b=1}^B{\mathbf{h}}_{b,u}^{\mathrm{H}}{\mathbf{\Phi }}_b^{\mathrm{H}}{\mathbf{W}}_b{\mathbf{p}}_u{m}_u\hfill \\ \hfill & +& \sum _{b=1}^B{\mathbf{h}}_{b,u}^{\mathrm{H}}{\mathbf{\Phi }}_b^{\mathrm{H}}{\mathbf{W}}_b\sum _{j=1,j\ne u}^U{\mathbf{p}}_j{m}_j+n_u,\hfill \end{array}$$

(6)

where $n_u$ is the noise received by the u-th CUE, which follows the circularly symmetric complex Gaussian (CSCG) distribution, i.e., $n_u\sim \mathcal{C}\mathcal{N}\left(0,{\sigma }_n^2\right)$, where ${\sigma }_n^2$ is the noise power.

2.3. Problem Formulation

As mentioned above, the signal-to-interference-plus-noise ratio (SINR) $SIN{R}_u$ of the u-th CUE is given by

$$\begin{array}{c}SIN{R}_u={\displaystyle \frac{S_u{S}_u^{\mathrm{H}}}{{\sum }_{j=1,j\ne u}^U{I}_u{I}_u^{\mathrm{H}}+{\sigma }_u^2}},\hfill \end{array}$$

(7)

where $S_u{S}_u^{\mathrm{H}}={\left|S_u\right|}^2$ is the power of the desired signal for the u-th CUE, and $I_u{I}_u^{\mathrm{H}}={\left|I_u\right|}^2$ is the interference power. Then, substituting (4) and (5) into (7), $SIN{R}_u$ can be further rewritten as

$$\begin{array}{c}SIN{R}_u={\displaystyle \frac{{\left|{\sum }_{b=1}^B{\mathbf{h}}_{b,u}^{\mathrm{H}}{\mathbf{\Phi }}_b^{\mathrm{H}}{\mathbf{W}}_b{\mathbf{p}}_u\right|}^2}{{\sum }_{j=1,j\ne u}^U{\left|{\sum }_{b=1}^B{\mathbf{h}}_{b,u}^{\mathrm{H}}{\mathbf{\Phi }}_b^{\mathrm{H}}{\mathbf{W}}_b{\mathbf{p}}_j\right|}^2+{\sigma }_u^2}}.\hfill \end{array}$$

(8)

To reflect the varying priorities and importance of different CUEs, we introduce a weight factor $w_u$ to represent the priority of the u-th CUE. By optimizing the beamforming $\mathcal{P}$, phase shift ${\mathbf{\Phi }}_b$, and deployment ${\mathbf{Ø}}_b$, the problem for maximizing the sum of AWDRs can be mathematically formulated as

$$\begin{array}{cc}\underset{\mathcal{P},{\mathbf{\Phi }}_b,{\mathbf{Ø}}_b}{max}\hfill & f_1\left(\mathcal{P},{\mathbf{\Phi }}_b,{\mathbf{Ø}}_b\right)=\sum _{u=1}^U{w}_u{log}_2\left(1+SIN{R}_u\right)\hfill \end{array}$$

(9a)

$$\begin{array}{cc}\mathrm{s}.\mathrm{t}.\hfill & \mathrm{tr}\left(\mathcal{P}{\mathcal{P}}^{\mathrm{H}}\right)\le P_{max},\hfill \end{array}$$

(9b)

$$\begin{array}{cc}\hfill & {\theta }_{b,c}\in F_{\mathrm{cont}},\forall b\in \mathcal{B},\forall c\in \mathcal{C},\hfill \end{array}$$

(9c)

$$\begin{array}{cc}\hfill & \left(x_{\mathrm{IRS}}^{min},y_{\mathrm{IRS}}^{min}\right)\le \left(x_{\mathrm{IRS}}^b,y_{\mathrm{IRS}}^b\right)\le \left(x_{\mathrm{IRS}}^{max},y_{\mathrm{IRS}}^{max}\right),\hfill \end{array}$$

(9d)

where $F_{\mathrm{cont}}=\left\{{\theta }_{b,c}=e^{{\phi }_{b,c}i}\left|{\phi }_{b,c}\in \left[0,2\pi \right)\right.\right\}$, and $F_{\mathrm{cont}}$ is the continuous phase. Since the phase variation of the drone-mounted IRS is constrained by hardware design and control accuracy, it can only take discrete values. In addition, $F_{\mathrm{disc}}$ is defined as the discrete phase of the drone-mounted IRS, $F_{\mathrm{disc}}=\left\{{\theta }_{b,c}=e^{i{\phi }_{b,c}}\left|{\phi }_{b,c}\in {\left\{{\displaystyle \frac{2\pi w}{2^q}}\right\}}_{w=0}^{2^q-1}\right.\right\}$, where q is the phase resolution. Furthermore, $\left(x_{\mathrm{IRS}}^{min},y_{\mathrm{IRS}}^{min}\right)$ and $\left(x_{\mathrm{IRS}}^{max},y_{\mathrm{IRS}}^{max}\right)$ are the feasible ranges of the horizontal and longitude locations of the drone-mounted IRS units, respectively. The deployment strategy is defined as ${\mathbf{Ø}}_b=\left(x_{\mathrm{IRS}}^b,y_{\mathrm{IRS}}^b\right)$.

The formulated AWDR sum maximization problem is non-convex and has the characteristics of non-deterministic polynomial (NP)-hard. Therefore, to solve this problem, we redefine the objective function as

$$\begin{array}{c}\underset{\mathcal{P},{\mathbf{\Phi }}_b,{\mathbf{Ø}}_b,\mathbf{\alpha }}{max}{f}_2\left(\mathcal{P},{\mathbf{\Phi }}_b,{\mathbf{Ø}}_b,\mathbf{\alpha }\right)={\displaystyle \frac{1}{ln2}}\sum _{u=1}^U{w}_{u}ln\left(1+{\alpha }_u\right)-w_u{\alpha }_u+{\displaystyle \frac{w_{u}SIN{R}_u\left(1+{\alpha }_u\right)}{1+SIN{R}_u}}.\hfill \end{array}$$

(10)

where $\mathbf{\alpha }={\left[{\alpha }_1,\dots ,{\alpha }_U\right]}^{\mathrm{T}}$ is the auxiliary vector introduced by the Lagrangian dual transform [10]. To make these two objective functions equivalent, it is sufficient to ensure that their optimal objective values are equal. Given $SIN{R}_u$, let $f_2$ be a concave differentiable function with respect to ${\alpha }_u$. From this, the optimal solution ${\alpha }_u^{\ast }$ can be derived as

$$\begin{array}{c}{\displaystyle \frac{\partial f_2}{\partial {\alpha }_u}}=0\Rightarrow {\alpha }_u^{\ast }=SIN{R}_u.\hfill \end{array}$$

(11)

Substituting ${\alpha }_u^{\ast }$ into (10), we simplify by eliminating common terms and applying the logarithmic base change, yielding (12):

$$\begin{array}{ccc}\hfill & & \underset{\mathcal{P},{\mathbf{\Phi }}_b,{\mathbf{Ø}}_b,\mathbf{\alpha }}{max}{f}_2\left(\mathcal{P},{\mathbf{\Phi }}_b,{\mathbf{Ø}}_b,\mathbf{\alpha }\right)\hfill \\ \hfill & =& {\displaystyle \frac{1}{ln2}}\sum _{u=1}^U{w}_{u}ln\left(1+SIN{R}_u\right)-w_{u}SIN{R}_u+{\displaystyle \frac{w_{u}SIN{R}_u\left(1+SIN{R}_u\right)}{1+SIN{R}_u}}\hfill \\ \hfill & =& {\displaystyle \frac{1}{ln2}}\sum _{u=1}^U{w}_{\mathrm{u}}ln\left(1+SIN{R}_u\right)=\sum _{u=1}^U{w}_u{log}_2\left(1+SIN{R}_u\right).\hfill \end{array}$$

(12)

According to (12), we can verify that the two objective functions are equivalent and expressed by (13):

$$\begin{array}{ccc}\hfill \underset{\mathcal{P},{\mathbf{\Phi }}_b,{\mathbf{Ø}}_b}{max}{f}_1\left(\mathcal{P},{\mathbf{\Phi }}_b,{\mathbf{Ø}}_b\right)& =& {\displaystyle \sum _{u=1}^U}{w}_u{log}_2\left(1+SIN{R}_u\right)\Rightarrow \hfill \\ \underset{\mathcal{P},{\mathbf{\Phi }}_b,{\mathbf{Ø}}_b,\alpha }{max}{f}_2\left(\mathcal{P},{\mathbf{\Phi }}_b,{\mathbf{Ø}}_b,\alpha \right)\hfill & =& {\displaystyle \frac{1}{ln2}}{\displaystyle \sum _{u=1}^U}{w}_{u}ln\left(1+{\alpha }_u\right)-w_u{\alpha }_u+{\displaystyle \frac{w_{u}SIN{R}_u\left(1+{\alpha }_u\right)}{1+SIN{R}_u}}.\hfill \end{array}$$

(13)

Based on the analysis above, solving (9a) is equivalent to solving (10). To handle the logarithmic terms in (9a) more effectively, we give ${\alpha }_u$ in advance. Then, the problem for maximizing the sum of AWDRs can be reformulated as

$$\begin{array}{cc}\underset{\mathcal{P},{\mathbf{\Phi }}_b,{\mathbf{Ø}}_b}{max}\hfill & f_3\left(\mathcal{P},{\mathbf{\Phi }}_b,{\mathbf{Ø}}_b\right)=\sum _{u=1}^U{\displaystyle \frac{{\omega }_u\left(1+{\alpha }_u\right)SIN{R}_u}{1+SIN{R}_u}}\hfill \end{array}$$

(14a)

$$\begin{array}{cc}\mathrm{s}.\mathrm{t}.\hfill & \mathrm{tr}\left(\mathcal{P}{\mathcal{P}}^{\mathrm{H}}\right)\le P_{max},\hfill \end{array}$$

(14b)

$$\begin{array}{cc}\hfill & {\theta }_{b,c}\in F_{\mathrm{cont}},\forall b\in \mathcal{B},\forall c\in \mathcal{C},\hfill \end{array}$$

(14c)

$$\begin{array}{cc}\hfill & \left(x_{\mathrm{IRS}}^{min},y_{\mathrm{IRS}}^{min}\right)\le \left(x_{\mathrm{IRS}}^b,y_{\mathrm{IRS}}^b\right)\le \left(x_{\mathrm{IRS}}^{max},y_{\mathrm{IRS}}^{max}\right).\hfill \end{array}$$

(14d)

3. AWDR Sum Maximization Scheme

In B5G IoT networks, we propose an AWDR sum maximization scheme for the downlink of IRS-assisted mmWave MIMO communications. This scheme involves three key optimization variables: the beamforming of the remote base station, the phase shift of the drone-mounted IRS, and the deployment of multiple drones. Since the formulated AWDR sum maximization problem is difficult to solve directly, this section decouples the three optimization variables and develops an effective iterative optimization algorithm to solve this non-convex optimization problem, as shown in Figure 2.

3.1. Joint Beamforming and Phase Shift Design

In Section 3.1, we focus on solving the joint beamforming and phase shift design problem, which is embedded in the developed iterative optimization algorithm. When the deployment ${\mathbf{Ø}}_b$ is given, this joint optimization problem is also NP-hard, which makes it impossible to obtain the globally optimal solution in polynomial time. Due to this reason, we further decouple this joint optimization problem and iteratively solve $\mathcal{P}$ and ${\mathbf{\Phi }}_b$.

3.1.1. Beamforming Optimization

In this stage, it is assumed that the phase shift ${\mathbf{\Phi }}_b$ and deployment ${\mathbf{Ø}}_b$ have been determined. Then, for simplified notation, we define

$$\begin{array}{c}{\tilde{\mathbf{h}}}_u^H=\sum _{b=1}^B{\mathbf{h}}_{b,u}^{\mathrm{H}}{\mathbf{\Phi }}_b^{\mathrm{H}}{\mathbf{W}}_b.\hfill \end{array}$$

(15)

Substituting (15) into (8), we have

$$\begin{array}{cc}\underset{\mathcal{P}}{max}\hfill & f_4\left(\mathcal{P}\right)=\sum _{u=1}^U{\displaystyle \frac{{\overline{\alpha }}_u{\left|{\tilde{\mathbf{h}}}_u^H{\mathbf{p}}_u\right|}^2}{{\sum }_{j=1}^U{\left|{\tilde{\mathbf{h}}}_u^{\mathrm{H}}{\mathbf{p}}_j\right|}^2+{\sigma }_u^2}}\hfill \end{array}$$

(16a)

$$\begin{array}{cc}\mathrm{s}.\mathrm{t}.\hfill & \mathrm{tr}\left(\mathcal{P}{\mathcal{P}}^{\mathrm{H}}\right)\le P_{max},\hfill \end{array}$$

(16b)

where ${\overline{\alpha }}_u=w_u\left(1+{\alpha }_u\right)$. Problem $f_4\left(\mathcal{P}\right)$ is a multi-ratio fractional programming problem [46], where the objective is to maximize $SIN{R}_u$, subject to the constraint (9b). To solve $f_4\left(\mathcal{P}\right)$, we employ the quadratic transformation (QT) technique and introduce the auxiliary variable $\mathit{\beta }$, leading to the following result:

$$\begin{array}{ccc}{f}_5\left(\mathcal{P},\mathit{\beta }\right)\hfill & =& \sum _{u=1}^{U}2\sqrt{{\overline{\alpha }}_u}\Re \left\{{\beta }_u^{\ast }{\tilde{\mathbf{h}}}_u^{\mathrm{H}}{\mathbf{p}}_u\right\}\hfill \\ \hfill & -& {\left|{\beta }_u\right|}^2\left(\sum _{j=1}^U{\left|{\tilde{\mathbf{h}}}_u^{\mathrm{H}}{\mathbf{p}}_j\right|}^2+{\sigma }_u^2\right),\hfill \end{array}$$

(17)

where $\mathit{\beta }={\left[{\beta }_1,\dots ,{\beta }_U\right]}^{\mathrm{T}}$, and $\Re \left\{·\right\}$ represents the real part. Let $f_5$ be a concave differentiable function with respect to ${\beta }_u$. From this, the optimal solution ${\beta }_u^{\ast }$ can be derived as

$$\begin{array}{c}{\displaystyle \frac{\partial f_5}{\partial {\beta }_u}}=0\Rightarrow {\beta }_u^{\ast }={\displaystyle \frac{\sqrt{{\overline{\alpha }}_u}{\tilde{\mathbf{h}}}_u^{\mathrm{H}}{\mathbf{p}}_u}{{\sum }_{j=1}^U{\left|{\tilde{\mathbf{h}}}_u^{\mathrm{H}}{\mathbf{p}}_j\right|}^2+{\sigma }_u^2}},\hfill \end{array}$$

(18)

where the optimal solution ${\mathbf{p}}_u^{\ast }$ for the given ${\beta }_u$ can be obtained by using the Lagrangian multiplier method, which can be expressed as

$$\begin{array}{c}{\mathbf{p}}_u^{\ast }=\sqrt{{\overline{\alpha }}_u}{\beta }_u\left(\mu {\mathbf{I}}_A+{\sum }_{j=1}^U{\left|{\beta }_j\right|}^2{\tilde{\mathit{h}}}_j{\tilde{\mathit{h}}}_j^{\mathrm{H}}\right){\tilde{\mathit{h}}}_u,\hfill \end{array}$$

(19)

where $\mu >0$ is the Lagrangian multiplier for (9b), and ${\mathbf{I}}_A$ is the unit matrix, ${\mathbf{I}}_A=\mathrm{diag}\left(\underset{A}{\underbrace{1,1,\dots ,1}}\right)$. Then, we rewrite (19) as

$$\begin{array}{c}{\mathcal{P}}^{\ast }={\left(\mu {\mathbf{I}}_A+\mathbf{X}\right)}^{-1}\mathbf{Y}\mathrm{Z},\hfill \end{array}$$

(20)

where $\mathbf{X}={\sum }_{j=1}^U{\left|{\beta }_j\right|}^2{\tilde{\mathit{h}}}_j{\tilde{\mathit{h}}}_j^{\mathrm{H}}$, $\mathbf{Y}=\left[{\tilde{\mathit{h}}}_1,{\tilde{\mathit{h}}}_2,\dots ,{\tilde{\mathit{h}}}_U\right]$, and $\mathbf{Z}=\mathrm{diag}\left(\left[\sqrt{{\overline{\alpha }}_1}{\beta }_1,\dots ,\sqrt{{\overline{\alpha }}_U}{\beta }_U\right]\right)$. The first-order derivative of $\mathrm{tr}\left(\mathcal{P}{\mathcal{P}}^{\mathrm{H}}\right)$ with respect to $\mu$ is given by

$$\begin{array}{ccc}{\displaystyle \frac{\partial \mathrm{tr}\left(\mathcal{P}{\mathcal{P}}^{\mathrm{H}}\right)}{\partial \mu }}\hfill & =& \mathrm{tr}\left({\left({\displaystyle \frac{\partial \mathrm{tr}\left(\mathcal{P}{\mathcal{P}}^{\mathrm{H}}\right)}{\partial \mathcal{P}}}\right)}^{\mathrm{H}}{\displaystyle \frac{\partial \mathcal{P}}{\partial \mu }}\right)\hfill \\ \hfill & =& -2\mathrm{tr}\left({\mathcal{P}}^{\mathrm{H}}{\left(\mu {\mathbf{I}}_A+\mathbf{X}\right)}^{-1}{\left(\mu {\mathbf{I}}_N+\mathbf{X}\right)}^{-1}\mathbf{Z}\right)\hfill \\ \hfill & =& -2\mathrm{tr}\left({\mathcal{P}}^{\mathrm{H}}{\left(\mu {\mathbf{I}}_A+\mathbf{X}\right)}^{-1}\mathcal{P}\right).\hfill \end{array}$$

(21)

In (21), since the matrix $\mu {\mathbf{I}}_A+\mathbf{X}$ is positive definite, its inverse matrix ${\left(\mu {\mathbf{I}}_A+\mathbf{X}\right)}^{-1}$ is also positive definite. In this situation, we know that ${\left(\mu {\mathbf{I}}_A+\mathbf{X}\right)}^{-1}$ is non-negative, i.e., $-2\mathrm{tr}\left({\mathcal{P}}^{\mathrm{H}}{\left(\mu {\mathbf{I}}_A+\mathbf{X}\right)}^{-1}\mathcal{P}\right)\le 0$. It is observed that, as $\mu$ increases, the value of $\mathrm{tr}\left(\mathcal{P}{\mathcal{P}}^{\mathrm{H}}\right)$ decreases, showing that (20) monotonically decreases as $\mu$ increases. Therefore, under the given power constraint, the optimal value of ${\mu }^{\ast }$ is given by

$$\begin{array}{c}{\mu }^{\ast }=\left\{\mu >0:\mathrm{tr}\left(\mathcal{P}{\mathcal{P}}^{\mathrm{H}}\right)=P_{max}\right\}.\hfill \end{array}$$

(22)

3.1.2. Phase Shift Design Optimization

We perform phase adjustment on (15) and the diagonalization of the channel vector, which allows us to rewrite it as

$$\begin{array}{c}{\tilde{\mathbf{h}}}_u^{\mathrm{H}}{\mathbf{p}}_j=\sqrt{\eta }\sum _{b=1}^B{\mathit{\theta }}_b^{\mathrm{H}}\mathrm{diag}\left({\mathbf{h}}_{b,u}^{\mathrm{H}}\right){\mathbf{W}}_b{\mathbf{p}}_j,\hfill \end{array}$$

(23)

where ${\mathit{\theta }}_b={\left[{\theta }_{b,1},\dots ,{\theta }_{b,C}\right]}^{\mathrm{T}}$. We define ${\mathbf{K}}_{b,u,j}=\sqrt{\eta }\mathrm{diag}\left({\mathbf{h}}_{b,u}^{\mathrm{H}}\right){\mathbf{W}}_b{\mathbf{p}}_j$, and (16a) and (16b) can be rewritten as

$$\begin{array}{cc}\underset{{\mathit{\theta }}_b}{max}\hfill & f_6\left({\mathit{\theta }}_b\right)=\sum _{u=1}^U{\displaystyle \frac{{\overline{\alpha }}_u{\left|{\sum }_{b=1}^B{\mathit{\theta }}_b^{\mathrm{H}}{\mathbf{K}}_{b,u,u}\right|}^2}{{\left|{\sum }_{b=1}^B{\mathit{\theta }}_b^{\mathrm{H}}{\mathbf{K}}_{b,u,j}\right|}^2+{\sigma }_u^2}}\hfill \end{array}$$

(24a)

$$\begin{array}{cc}\mathrm{s}.\mathrm{t}.\hfill & {\left|{\theta }_{b,c}\right|}^2=1,\forall b\in \mathcal{B},\forall c\in \mathcal{C}.\hfill \end{array}$$

(24b)

For ease of calculation, we define

$$\begin{array}{c}\mathbf{\Theta }={\left[{\mathit{\theta }}_1,{\mathit{\theta }}_2,{\mathit{\theta }}_3,\dots ,{\mathit{\theta }}_B\right]}^{\mathrm{T}},\hfill \end{array}$$

(25)

and

$$\begin{array}{c}{\mathbf{K}}_{u,j}=\left[{\mathbf{k}}_{1,u,j},{\mathbf{k}}_{2,u,j},\dots {\mathbf{k}}_{B,u,j}\right].\hfill \end{array}$$

(26)

Then, $\mathrm{tr}\left({\mathbf{\Theta }}^{\mathrm{H}}{\mathbf{K}}_{u,j}\right)$ can be calculated as

$$\begin{array}{c}\mathrm{tr}\left({\mathbf{\Theta }}^{\mathrm{H}}{\mathbf{K}}_{u,j}\right)=\sum _{b=1}^B{\mathit{\theta }}_b^{\mathrm{H}}{\mathbf{k}}_{b,u,j}.\hfill \end{array}$$

(27)

On this basis, (24a) can be rewritten as

$$\begin{array}{ccc}\underset{\tilde{\mathit{\theta }}}{max}{f}_7\left(\tilde{\mathit{\theta }}\right)\hfill & =& \sum _{u=1}^U{\displaystyle \frac{{\overline{\alpha }}_u{\left|\mathrm{tr}\left({\mathbf{\Theta }}^{\mathrm{H}}{\mathbf{K}}_{u,u}\right)\right|}^2}{{\sum }_{j=1}^U{\left|\mathrm{tr}\left({\mathbf{\Theta }}^{\mathrm{H}}{\mathbf{K}}_{u,u}\right)\right|}^2+{\sigma }_u^2}}\hfill \\ \hfill & =& \sum _{u=1}^U{\displaystyle \frac{{\overline{\alpha }}_u{\left|{\tilde{\mathit{\theta }}}^{\mathrm{H}}{\tilde{\mathit{k}}}_{u,u}\right|}^2}{{\sum }_{j=1}^U{\left|{\tilde{\mathit{\theta }}}^{\mathrm{H}}{\tilde{\mathit{k}}}_{u,j}\right|}^2+{\sigma }_u^2}},\hfill \end{array}$$

(28)

where $\tilde{\mathit{\theta }}=\mathrm{vec}\left(\mathbf{\Theta }\right)$ and ${\tilde{\mathit{k}}}_{u,j}=\mathrm{vec}\left({\mathbf{K}}_{u,j}\right)$.

By introducing the auxiliary vector æ, we can obtain the corresponding QT equation of (28) as

$$\begin{array}{ccc}\underset{\tilde{\mathit{\theta }}, \mathbf{\ae }}{max}{f}_8\left(\tilde{\mathit{\theta }},\mathbf{\ae }\right)\hfill & =& \sum _{u=1}^{U}2\sqrt{{\overline{\alpha }}_u}\Re \left\{{\widehat{\rho }}_u{\tilde{\mathit{\theta }}}^{\mathrm{H}}{\tilde{\mathit{k}}}_{u,u}\right\}\hfill \\ \hfill & -& {\left|{\rho }_u\right|}^2\left(\sum _{j=1}^U{\left|{\tilde{\mathit{\theta }}}^{\mathrm{H}}{\tilde{\mathit{k}}}_{u,j}\right|}^2+{\sigma }_u^2\right),\hfill \end{array}$$

(29)

where æ $={\left[{\rho }_1,\dots {\rho }_U\right]}^{\mathrm{T}}$. By using the Lagrangian multiplier gradient method, we can obtain

$$\begin{array}{c}{\displaystyle \frac{\partial f_8}{\partial {\rho }_u}}=0\Rightarrow {\rho }_u^{\ast }={\displaystyle \frac{\sqrt{{\overline{\alpha }}_u}{\tilde{\mathit{\theta }}}^{\mathrm{H}}{\tilde{\mathit{k}}}_{u,u}}{{\sum }_{j=1}^U{\left|{\tilde{\mathit{\theta }}}^{\mathrm{H}}{\tilde{\mathit{k}}}_{u,j}\right|}^2+{\sigma }_u^2}}.\hfill \end{array}$$

(30)

We can obtain (31) by optimizing $\tilde{\mathit{\theta }}$.

$$\begin{array}{c}\underset{\tilde{\mathit{\theta }}}{max}{f}_9\left(\tilde{\mathit{\theta }}\right)=-{\tilde{\mathit{\theta }}}^{\mathrm{H}}\mathbf{M}\tilde{\mathit{\theta }}+2\Re \left\{{\tilde{\mathit{\theta }}}^H\mathbf{g}\right\}-\sum _{u=1}^U{\left|{\rho }_u\right|}^2{\sigma }_u^2,\hfill \end{array}$$

(31)

where $\mathbf{M}={\sum }_{u=1}^U\left({\left|{\rho }_u\right|}^2{\sum }_{j=1}^U{\tilde{\mathit{k}}}_{u,j}{\tilde{\mathit{k}}}_{u,j}^{\mathrm{H}}\right)$ and $\mathbf{g}={\sum }_{u=1}^U\sqrt{{\overline{\alpha }}_u}{\rho }_u^{\ast }{\tilde{\mathit{k}}}_{u,u}$. By relaxing the constraint in (9c) and removing the constant term in $f_9\left(\tilde{\mathit{\theta }}\right)$, the objective function is transformed into

$$\begin{array}{cc}\underset{\tilde{\mathit{\theta }}}{max}\hfill & f_{10}\left(\tilde{\mathit{\theta }}\right)=-{\tilde{\mathit{\theta }}}^{\mathrm{H}}\mathbf{M}\tilde{\mathit{\theta }}+2\Re \left\{{\tilde{\mathit{\theta }}}^{\mathrm{H}}\mathbf{g}\right\}\hfill \end{array}$$

(32a)

$$\begin{array}{cc}\mathrm{s}.\mathrm{t}.\hfill & {\left|{\tilde{\theta }}_u\right|}^2\le 1,\forall u=1,2,\dots ,C_{\mathrm{tot}},\hfill \end{array}$$

(32b)

where $C_{\mathrm{tot}}=BC$. We construct the Lagrangian function of $f_{10}\left(\tilde{\mathit{\theta }}\right)$ as

$$\begin{array}{ccc}\mathcal{L}\left(\tilde{\mathit{\theta }},\mathit{\zeta }\right)\hfill & =& -{\tilde{\mathit{\theta }}}^{\mathrm{H}}\mathbf{M}\tilde{\mathit{\theta }}+2\Re \left\{{\tilde{\mathit{\theta }}}^{\mathrm{H}}\mathbf{g}\right\}\hfill \\ \hfill & -& \sum _{u=1}^{C_{\mathrm{tot}}}{\zeta }_u\left({\tilde{\mathit{\theta }}}^{\mathrm{H}}{\mathbf{e}}_u{\mathbf{e}}_u^{\mathrm{H}}\tilde{\mathit{\theta }}-1\right).\hfill \end{array}$$

(33)

Thus, it can be concluded that

$$\begin{array}{c}{f}_{\mathcal{L}}\left(\tilde{\mathit{\theta }},\mathit{\zeta }\right)=f_{10}\left(\tilde{\mathit{\theta }}\right)-\sum _{u=1}^{C_{\mathrm{tot}}}{\zeta }_u\left({\tilde{\mathit{\theta }}}^{\mathrm{H}}{\mathbf{e}}_u{\mathbf{e}}_u^{\mathrm{H}}\tilde{\mathit{\theta }}-1\right),\hfill \end{array}$$

(34)

where $\mathit{\zeta }={\left[{\zeta }_1,{\zeta }_2,\dots ,{\zeta }_{C_{\mathrm{tot}}}\right]}^{\mathrm{T}}$ and ${\mathbf{e}}_u\in {\mathbb{R}}^{C_{\mathrm{tot}}\times 1}$. To deal with the constraint ${\left|{\tilde{\theta }}_u\right|}^2\le 1$, we introduce the Lagrangian multiplier ${\zeta }_u\ge 0$ to construct the dual function as follows:

$$\begin{array}{cc}\underset{\mathit{\zeta }}{min}\hfill & f_{\mathcal{D}}\left(\mathit{\zeta }\right)=\underset{\tilde{\mathit{\theta }}}{sup}{f}_{\mathcal{L}}\left(\tilde{\mathit{\theta }},\mathit{\zeta }\right)\hfill \end{array}$$

(35a)

$$\begin{array}{cc}\mathrm{s}.\mathrm{t}.\hfill & {\zeta }_u\ge 0, \forall u=1,2,\dots ,C_{\mathrm{tot}}.\hfill \end{array}$$

(35b)

Accordingly, we can obtain the optimal ${\tilde{\mathit{\theta }}}^{\ast }$, which is given by

$$\begin{array}{c}{\displaystyle \frac{\partial f_{\mathcal{L}}}{\partial \tilde{\mathit{\theta }}}}=0\Rightarrow {\tilde{\mathit{\theta }}}^{\ast }={\left(\mathbf{M}+\sum _{u=1}^{C_{\mathrm{tot}}}{\zeta }_u{\mathbf{e}}_u{\mathbf{e}}_u^{\mathrm{H}}\right)}^{-1},\hfill \end{array}$$

(36)

and

$$\begin{array}{c}\mathbf{g}=\mathbf{D}\left(\mathit{\zeta }\right)\mathbf{g},\hfill \end{array}$$

(37)

where $\mathbf{D}\left(\mathit{\zeta }\right)={\left(\mathbf{M}+\mathrm{diag}\left(\mathit{\zeta }\right)\right)}^{-1}$.

Substituting (36) and (37) into (35a), we have

$$\begin{array}{c}{f}_{\mathcal{D}}\left(\mathit{\zeta }\right)={\mathbf{g}}^{\mathrm{H}}\mathbf{D}\left(\mathit{\zeta }\right)\mathbf{g}+\mathrm{tr}\left(\mathrm{diag}\left(\mathit{\zeta }\right)\right),\hfill \end{array}$$

(38)

where $f_{\mathcal{D}}\left(\mathit{\zeta }\right)$ can be regarded as a semi-definite programming (SDP) problem. According to the above analysis, the following optimization problems can be solved by using CVX toolboxes:

$$\begin{array}{cc}\underset{\phi ,\mathit{\zeta }}{max}\hfill & \phi -\mathrm{tr}\left(\mathrm{diag}\left(\mathit{\zeta }\right)\right)\hfill \end{array}$$

(39a)

$$\begin{array}{cc}\mathrm{s}.\mathrm{t}.\hfill & \left[\begin{array}{c}\mathbf{M}+\mathrm{diag}\left(\mathit{\zeta }\right),\mathbf{g}\\ {\mathbf{g}}^{\mathrm{H}},-\phi \end{array}\right]⪰0,\hfill \end{array}$$

(39b)

where $\phi$ is a scalar, and (39b) indicates that the matrix is semi-definite.

To simplify the derivative calculation, we apply the chain rule, which yields the following result:

$$\begin{array}{ccc}{\displaystyle \frac{\partial f_{\mathcal{D}}}{\partial {\zeta }_u}}\hfill & =& 1-\mathrm{tr}\left[\mathbf{D}\left(\mathit{\zeta }\right)\mathbf{g}{\mathbf{g}}^{\mathrm{H}}\mathbf{D}\left(\mathit{\zeta }\right){\displaystyle \frac{\partial \left(\mathbf{M}+\mathrm{diag}\left(\mathit{\zeta }\right)\right)}{\partial {\zeta }_u}}\right]\hfill \\ \hfill & =& 1-{\left[\mathbf{D}\left(\mathit{\zeta }\right)\mathbf{g}{\mathbf{g}}^{\mathrm{H}}\mathbf{D}\left(\mathit{\zeta }\right)\right]}_{u,u}.\hfill \end{array}$$

(40)

The form of (40) can be transformed as follows:

$$\begin{array}{c}\left[\mathbf{D}\left(\mathit{\zeta }\right)\mathbf{g}\right]·{\left[\mathbf{D}\left(\mathit{\zeta }\right)\mathbf{g}\right]}^{\mathrm{H}}=\tilde{\mathit{\theta }}·{\tilde{\mathit{\theta }}}^{\mathrm{H}}=1.\hfill \end{array}$$

(41)

Then, the optimal ${\mathit{\zeta }}^{\ast }$ is expressed as

$$\begin{array}{c}{\mathit{\zeta }}^{\ast }=\left\{{\zeta }_u\ge 0:\left[\mathbf{D}\left(\mathit{\zeta }\right)\mathbf{g}\right]·{\left[\mathbf{D}\left(\mathit{\zeta }\right)\mathbf{g}\right]}^{\mathrm{H}}=1\right\}.\hfill \end{array}$$

(42)

Equation (42) demonstrates that the solution obtained using the constraint relaxation method ensures the satisfaction of the unit modulus constraint ${\tilde{\mathit{\theta }}}^{\mathrm{H}}{\mathbf{e}}_u{\mathbf{e}}_u^{\mathrm{H}}\tilde{\mathit{\theta }}=1$.

3.2. Deployment Optimization

Similarly, in this stage, we assume that $\mathcal{P}$ and ${\mathbf{\Phi }}_b$ have been given in advance. Then, a distributed discrete-time convex optimization approach is employed to optimize the drone deployment [47]. Specifically, this paper solves the optimal solution coordinates by using an error term analysis, where the objective function is the sum of U convex functions, representing U CUEs. Our goal is to find the optimal coordinates $\left({\widehat{x}}_{\mathrm{IRS}}^b,{\widehat{y}}_{\mathrm{IRS}}^b\right)$ of the b-th drone-mounted IRS, i.e., minimize the distance $d\left(r_{\mathrm{MU}}^u,r_{\mathrm{IRS}}^b\right)$ between the b-th drone-mounted IRS and the u-th CUE. In this situation, the sum of AWDRs can be maximized. Mathematically, we can formulate the above problem as

$$\begin{array}{c}\underset{x,y\in {\mathbb{R}}^n}{min}f\left(x,y\right)={\sum }_{u=1}^U\sqrt{{\left(x_{\mathrm{MU}}^u-x_{\mathrm{IRS}}^b\right)}^2+{\left(y_{\mathrm{MU}}^u-y_{\mathrm{IRS}}^b\right)}^2+{\left(z_{\mathrm{MU}}^u-z_{\mathrm{IRS}}^b\right)}^2}.\hfill \end{array}$$

(43)

In (43), to simplify the calculation, the distance in space is calculated using the Euclidean metric, but this does not account for the potential presence of obstacles. As discussed in [48], parameters such as building height can be introduced to further align with the real-world scenarios of B5G IoT networks. It is noted that $z_{\mathrm{IRS}}^b$ and $z_{\mathrm{MU}}^u$ are fixed values. Subsequently, according to the push-sum framework [49], and by incorporating the gradient descent method to handle the associated closed convex set constraints, we initialize the coordinates of the b-th drone-mounted IRS. On this basis, we calculate the gradient value of the objective function $f\left(x,y\right)$ as

$$\begin{array}{c}\nabla f\left(x,y\right)=\left[{\displaystyle \frac{\partial f}{\partial x}},{\displaystyle \frac{\partial f}{\partial y}}\right],\hfill \end{array}$$

(44)

where

$$\begin{array}{c}{\displaystyle \frac{\partial f}{\partial x}}=\sum _{u=1}^U{\displaystyle \frac{x_{\mathrm{MU}}^u-x_{\mathrm{IRS}}^b}{\sqrt{{\left(x_{\mathrm{MU}}^u-x_{\mathrm{IRS}}^b\right)}^2+{\left(y_{\mathrm{MU}}^u-y_{\mathrm{IRS}}^b\right)}^2+{\left(z_{\mathrm{MU}}^u-z_{\mathrm{IRS}}^b\right)}^2}}},\hfill \end{array}$$

(45)

and

$$\begin{array}{c}{\displaystyle \frac{\partial f}{\partial y}}=\sum _{u=1}^U{\displaystyle \frac{y_{\mathrm{MU}}^u-y_{\mathrm{IRS}}^b}{\sqrt{{\left(x_{\mathrm{MU}}^u-x_{\mathrm{IRS}}^b\right)}^2+{\left(y_{\mathrm{MU}}^u-y_{\mathrm{IRS}}^b\right)}^2+{\left(z_{\mathrm{MU}}^u-z_{\mathrm{IRS}}^b\right)}^2}}}.\hfill \end{array}$$

(46)

In each iteration, the b-th drone-mounted IRS updates its coordinates according to (47) and (48).

$$\begin{array}{c}{x}_{\mathrm{IRS}}^b\left(k+1\right)=pro{j}_{\left(x_{\mathrm{IRS}}^{min},x_{\mathrm{IRS}}^{max}\right)\times \left(y_{\mathrm{IRS}}^{min},y_{\mathrm{IRS}}^{max}\right)}\left(x_{\mathrm{IRS}}^b\left(k\right)-\gamma \left(\nabla f\left(x_{\mathrm{IRS}}^b\left(k\right),y_{\mathrm{IRS}}^b\left(k\right)\right)+\lambda \left(k\right)\nabla v_u\left(x_{\mathrm{IRS}}^b\left(k\right),y_{\mathrm{IRS}}^b\left(k\right)\right)\right)\right),\hfill \end{array}$$

(47)

and

$$\begin{array}{c}{y}_{\mathrm{IRS}}^b\left(k+1\right)=pro{j}_{\left(x_{\mathrm{IRS}}^{min},x_{\mathrm{IRS}}^{max}\right)\times \left(y_{\mathrm{IRS}}^{min},y_{\mathrm{IRS}}^{max}\right)}\left(y_{\mathrm{IRS}}^b\left(k\right)-\gamma \left(\nabla f\left(x_{\mathrm{IRS}}^b\left(k\right),y_{\mathrm{IRS}}^b\left(k\right)\right)+\lambda \left(k\right)\nabla v_u\left(x_{\mathrm{IRS}}^b\left(k\right),y_{\mathrm{IRS}}^b\left(k\right)\right)\right)\right).\hfill \end{array}$$

(48)

In (47) and (48), k is the k-th iteration, $pro{j}_{\left(x_{\mathrm{IRS}}^{min},x_{\mathrm{IRS}}^{max}\right)\times \left(y_{\mathrm{IRS}}^{min},y_{\mathrm{IRS}}^{max}\right)}$ indicates operation on a $\left(x_{\mathrm{IRS}}^{min},x_{\mathrm{IRS}}^{max}\right)\times \left(y_{\mathrm{IRS}}^{min},y_{\mathrm{IRS}}^{max}\right)$ projection, $\gamma$ is the step size, and $\lambda \left(k\right)$ is the Lagrangian multiplier, which can be updated as

$$\begin{array}{c}\lambda \left(k+1\right)=\lambda \left(k\right)+\tau max\left\{0,v_u\left(x_{\mathrm{IRS}}^b\left(k+1\right),y_{\mathrm{IRS}}^b\left(k+1\right)\right)\right\},\hfill \end{array}$$

(49)

where $\tau$ is the step size of the Lagrangian multiplier, and $v_u\left(x_{\mathrm{IRS}}^b\left(k+1\right),y_{\mathrm{IRS}}^b\left(k+1\right)\right)$ can be calculated as

$$\begin{array}{c}{v}_u\left(x_{\mathrm{IRS}}^b\left(k+1\right),y_{\mathrm{IRS}}^b\left(k+1\right)\right)=\left\{\left(x_{\mathrm{IRS}}^{min}-x\right),\left(x-x_{\mathrm{IRS}}^{max}\right),\left(y_{\mathrm{IRS}}^{min}-y\right),\left(y-y_{\mathrm{IRS}}^{max}\right)\right\}.\hfill \end{array}$$

(50)

In (50), we take the maximum value of these four values (i.e., $\left(x_{\mathrm{IRS}}^{min}-x\right)$, $\left(x-x_{\mathrm{IRS}}^{max}\right)$, $\left(y_{\mathrm{IRS}}^{min}-y\right)$, and $\left(y-y_{\mathrm{IRS}}^{max}\right)$). Therefore, $\tau$ increases only if $v_u\left(x_{\mathrm{IRS}}^b\left(k+1\right),y_{\mathrm{IRS}}^b\left(k+1\right)\right)\ge 0$. By using (50), we can guarantee that the deployment of the drone-mounted IRS units is within the feasible range.

Additionally, we perform an error analysis, where the squared norm of the error term is defined as

$$\begin{array}{c}{∥e\left(k+1\right)∥}^2={∥x_{\mathrm{IRS}}^b\left(k+1\right)-x_{\mathrm{IRS}}^b\left(k\right),y_{\mathrm{IRS}}^b\left(k+1\right)-y_{\mathrm{IRS}}^b\left(k\right)∥}^2.\hfill \end{array}$$

(51)

To ensure the convergence of the deployment optimization algorithm, we define the iteration threshold as $\epsilon$. When ${∥e\left(k+1\right)∥}^2\le \epsilon$, the deployment optimization algorithm is considered to have converged, and the iteration can be stopped. The detailed procedures are summarized in Algorithm 1. In Algorithm 1, we have $\tau =0.01$. The specific reasons are as follows: According to the theory of constrained optimization, the Lagrangian multiplier $\tau$ should gradually increase during the iteration process to enforce the satisfaction of the constraint conditions [50]. However, an excessively large increment may result in oscillations within the algorithm, whereas an overly small increment could potentially decelerate convergence. If the change in $\tau$ is significantly larger than the gradient magnitude (e.g., 0.1), it can accelerate convergence but may also increase the constraint violation, causing oscillations that require additional iterations to correct, thus reducing overall efficiency. However, if the increment is too small (e.g., 0.001), it may fail to respond to constraint changes in complex scenarios, such as high-density users or dynamic obstacles. Based on these considerations, an increment of 0.01 is chosen, which matches the gradient magnitude and ensures a reasonable update step size, balancing both the rapid response in the initial stages and long-term stability.

Algorithm 1 Deployment Optimization Algorithm

1: Initialization   2:      Set $k=0$, $x_{\mathrm{IRS}}^b\left(0\right)$, $y_{\mathrm{IRS}}^b\left(0\right)$, $\gamma$, $\tau$, and $\epsilon$.   3: repeat   4:      According to (45) and (46), we calculate ${\displaystyle \frac{\partial f}{\partial x}}$ and ${\displaystyle \frac{\partial f}{\partial y}}$, respectively.   5:      if $v_u\left(x_{\mathrm{IRS}}^b\left(k+1\right),y_{\mathrm{IRS}}^b\left(k+1\right)\right)\ge 0$ then   6:       $\tau \left(k+1\right)=\tau \left(k\right)+0.01.$   7:      else    8:       $\tau \left(k+1\right)=\tau \left(k\right).$   9:      end if 10:      Equation (49) is used to update $\lambda \left(k+1\right)$. 11:      Equations (47) and (48) are used to update $x_{\mathrm{IRS}}^b\left(k+1\right)$ and $y_{\mathrm{IRS}}^b\left(k+1\right)$. 12:      $k=k+1.$ 13: until Convergence 14: Output the optimal deployment optimization strategy.

4. Performance Evaluation

4.1. Simulation Parameters

In this section, the performance of the proposed AWDR sum maximization scheme is evaluated through simulation experiments. The proposed scheme is compared with three state-of-the-art schemes: (A) Scheme 1 [7]; (B) Scheme 2 [10]; and (C) Scheme 3 [11]. Specifically, Scheme 1 proposes a UAV downlink communication network empowered by an IRS to enhance signal quality for multiple users by dynamically adjusting the cascade link. This scheme employs block coordinate descent for resource allocation and reflecting design. Scheme 2 presents a distributed IRS-assisted multi-user mmWave system, where multiple IRSs are utilized to improve mmWave signal coverage when direct base station–user links are unavailable. A joint active and passive beamforming problem is formulated for weighted sum-rate maximization, and an alternating iterative algorithm is proposed to solve the non-convex problem. Scheme 3 investigates the impact of key factors (e.g., IRS reflecting elements’ power radiation patterns, mutual coupling among IRS elements, and IRS coverage area) on drone-enabled systems and designs an IRS-aided data transmission scheme for B5G IoT networks. Additionally, the scheme that optimizes beamforming and deployment is used as the benchmark. In the benchmark scheme, the phase shifts are set randomly, which emphasizes the necessity of optimizing the phase shifts. For the simulations, the key parameters are outlined as follows: $A=32$, $B=2$, $C=\left[20,120\right]$, $U=\left[10,100\right]$, $P_{max}=\left[1,10\right]$ Watt, $\tau =0.01$, $\gamma =0.1$, $\left(x_{\mathrm{IRS}}^{min},y_{\mathrm{IRS}}^{min}\right)=\left(0,0\right)\mathrm{km}$, $\left(x_{\mathrm{IRS}}^{max},y_{\mathrm{IRS}}^{max}\right)=\left(2,2\right)\mathrm{km}$, $z_{\mathrm{IRS}}^b=0.2\mathrm{km}$, and $z_{\mathrm{MU}}^u=0\mathrm{km}$. The CUEs are randomly distributed in the $4\mathrm{km}\times 6\mathrm{km}$ range, and the noise power is $-85\mathrm{dBm}$. The simulation result is the average value after 1000 runs.

4.2. Performance Comparison

Figure 3 illustrates the curves of the sum of AWDRs as a function of the number of CUEs for different schemes. The results show that the sum of AWDRs increases significantly with the number of CUEs for all schemes. Overall, the sum of AWDRs shows an approximately linear growth trend with respect to the number of CUEs. However, local fluctuations occur due to the influence of the algorithm’s initial values and environmental complexity. The observed results are the average values after 1000 runs. Specifically, compared to Scheme 1, increasing the number of drone-mounted IRS units improves the sum of AWDRs by 122.3%. Furthermore, compared to Scheme 2, under the same number of CUEs, the introduction of the air-to-ground integrated information transmission framework improves the sum of AWDRs by 93.5%. Finally, for Scheme 3, optimizing the deployment of multiple drones significantly enhances the sum of AWDRs. Therefore, the proposed AWDR sum maximization scheme demonstrates superior transmission performance in both sparse and dense user scenarios.

Figure 4 presents the curves of the sum of AWDRs as a function of the number of IRS elements for different schemes. In Figure 4, it is evident that, as the number of IRS elements increases, the sum of AWDRs increases for all schemes. This trend is similar to that in Figure 3. This result indicates that both increasing the number of IRS units and the number of elements in each drone-mounted IRS can significantly enhance the transmission performance of air-to-ground integrated networks. However, it is noted that an increase in the number of IRS elements also leads to higher algorithmic complexity. When the number of IRS elements is large, further increasing the number of elements does not lead to a significant improvement in the sum of AWDRs. Under this condition, a trade-off exists between the number of IRS elements and algorithmic complexity, thereby affecting the optimization of the sum of AWDRs. Therefore, optimizing the sum of AWDRs by balancing the number of IRS elements and algorithmic complexity will be a key issue in future research.

Figure 5 shows the variation in the sum of AWDRs as a function of the maximum transmission power. It can be observed that the sum of AWDRs increases with the increase in the maximum transmission power for all compared schemes. However, the rate of increase in the sum of AWDRs gradually slows down as the maximum transmission power increases. Specifically, when the maximum transmission power exceeds 7 Watt, further increases in the transmission power do not significantly improve the transmission performance of air-to-ground integrated networks. This is because the remote base station has already optimized the beamforming design during data transmission. Moreover, regardless of the increase in the transmission power, the proposed AWDR sum maximization scheme consistently achieves a higher achievable weighted sum rate than Scheme 1, Scheme 2, and Scheme 3. This result is consistent with the simulation results shown in Figure 3 and Figure 4, thereby further confirming the effectiveness of the proposed AWDR sum maximization scheme.

Figure 6 plots the running time versus the number of simulation runs. It can be observed that the running time of our proposed scheme is the longest, while the benchmark scheme exhibits the shortest running time. However, as indicated by the results in Figure 3, Figure 4 and Figure 5, although the benchmark scheme has the lowest computational complexity, its performance is the poorest. We improve the transmission performance by slightly increasing the complexity. Furthermore, in this paper, we use the running time as an indicator to evaluate the computational complexity, without providing a theoretical analysis of the computational complexity of the proposed scheme. In future work, we will conduct a more in-depth analysis.

4.3. Impact of Key Parameters

Figure 7 plots the impact of the number of IRS elements on the sum of AWDRs under different numbers of CUEs. Since the x-axis represents the number of CUEs and the y-axis represents the sum of AWDRs, the sum of AWDRs increases linearly with the number of CUEs. However, as shown in the figure legend, the sum of AWDRs increases non-linearly with the number of IRS elements. As the number of IRS elements increases, the rate of improvement in the sum of AWDRs gradually slows down. Similar to the analysis in Figure 4, increasing the number of IRS elements also leads to an increase in the algorithm complexity. Furthermore, when the number of CUEs is relatively small, the effect of increasing the number of IRS elements on network performance improvement is not significant. In this situation, a smaller number of IRS elements can still ensure basic transmission quality. Considering this, in practical deployment, it is essential to not only consider the total number of IRS elements but also to account for the number of elements in each drone-mounted IRS. Therefore, determining how to achieve higher transmission performance while maintaining low algorithmic complexity has become a key research challenge. Additionally, Figure 8 demonstrates the impact of the number of IRS elements on the sum of AWDRs under different maximum transmission powers. The results are consistent with those in Figure 5. This confirms that optimizing the beamforming and phase shift design can effectively improve the sum of AWDRs.

5. Conclusions

In this paper, we present a MIMO-enabled air–ground integrated information transmission framework for B5G IoT networks. First, multiple drone-mounted IRS units are employed to enhance the QoS for CUEs. Subsequently, by optimizing beamforming, phase shift design, and drone deployment, we formulate a problem aimed at maximizing the sum of AWDRs. To solve this problem efficiently, we devise an iterative optimization algorithm. At each iteration, this algorithm jointly optimizes beamforming and phase shift design, yielding closed-form solutions, and it determines the optimal deployment strategy by using a distributed discrete-time convex optimization approach. The simulation results demonstrate that the proposed scheme outperforms state-of-the-art schemes in terms of the sum of AWDRs.

In future research, we will further explore how to optimize the real-time application of the proposed solutions, particularly in dynamic user location scenarios. To ensure that the B5G IoT network remains stable and efficient despite changes in user location, we plan to introduce more advanced dynamic adjustment mechanisms. Additionally, we aim to conduct simulations to test and evaluate the transmission performance in mobile environments, thus providing more reliable and timely services to mobile users. Furthermore, since the studied problem belongs to the class of NP-hard problems, the solution cannot be obtained in polynomial time. In our future work, we will derive the closed-form expression of the studied problem for small-scale scenarios instead of the numerical solution obtained through iterative optimization.