Robust Energy-Efficient Transmission for Cell-Free Massive MIMO Systems with Imperfect CSI

Abstract

In this paper, we investigate a long-term power minimization problem of cell-free massive multiple-input multiple-output (MIMO) systems. To address this issue and to ensure the system queue stability, we formulate a dynamic optimization problem aiming to minimize the average total power cost in a time-varying system under imperfect channel conditions. The problem is then converted into a real-time weighted sum rate maximization problem for each time slot using the Lyapunov optimization technique. We employ approximation techniques to design robust sparse beamforming, which enables energy savings of the network and mitigates channel uncertainty. By applying direct fractional programming (DFP) and alternating optimization, we can obtain a locally optimal solution. Our DFP-based algorithm minimizes the average total power consumption of the network while satisfying the quality of service requirements for each user. Simulation results demonstrate the rapid convergence of the proposed algorithm and illustrate the tradeoff between average network power consumption and queue latency.

1. Introduction

Cell splitting and frequency reuse technologies have significantly improved the spectrum efficiency of cellular networks, which provides robust support for the rapid development of mobile communication systems [1,2]. However, the continuous reduction in cellular cell area leads to increasing inter-cell interference and handover complexity, which hinders the enhancement of mobile communication system performance. To address these challenges, cell-free networks have emerged as a viable solution for redefining the future architecture of cellular networks [3,4,5].

The cell-free distributed massive multiple-input multiple-output (MIMO) architecture revolutionized the conventional cell concept by introducing the notion of “user-centric” [6]. In this network architecture, numerous distributed access points (APs) are deployed, which leads to reducing distances between users and APs and facilitating spatial macro-diversity gain. Further, uniform coverage is achieved across the entire area. Such favorable propagation characteristics further reduce interference among users, thereby considerably enhancing the quality of service (QoS) of the served users [7].

Since we have difficulties dealing with the co-channel interference in a cell-free distributed massive MIMO system, the performance of the massive MIMO systems largely depends on advanced beamforming designs at the APs, which greatly rely on the full channel state information (CSI) at the APs. In cell-free massive MIMO systems, particularly high-mobility scenarios, the increasing number of antennas employed by the APs and channel aging pose challenges in accurately estimating the CSI, inevitably leading to channel estimation errors. However, the existing studies on beamforming designs of distributed massive MIMO systems have primarily focused on theoretical aspects with perfect CSI [8,9,10]. As a result, performance degradation is inevitable in the presence of imperfect CSI. Reference [11] pointed out that the information-theoretic analysis based on the ideal CSI of cell-free distributed massive MIMO systems was not accurate. The traditional linear minimum mean square error algorithms (LMMSE) [12] can obtain an accurate estimation performance, while the complexity of computing for APs was too high. Reference [13] compared the performance of the beamforming designs with and without the robustness under the imperfect CSI, which showed that the robust beamforming design outperforms the one without the robustness. Against this background, the investigation of robust beamforming design is of significant necessity to cope with the channel uncertainty.

The dramatic increase in the number of mobile terminals inevitably brings about a large number of AP deployments, where the high-precision hardware RF circuits will consume huge amounts of energy, and the energy consumption problem will always exist and become increasingly severe. Energy-efficient transmission strategies based on static models were investigated in [14,15,16,17,18], where power and throughput were optimized while related control policies including queue stability and queue latency of the system were not taken into consideration. As a matter of fact, queue stability and queue latency are crucial indicators for measuring system performance. Reference [19] shows that an essential tradeoff exists between the power consumption and the queue latency. Therefore, a stochastic problem considering the balance between the average power consumption and the average queue latency in a time-varying system with different performance constraints should be put enough emphasis on. To tackle this problem, the well-known Lyapunov theory proves to be highly beneficial. The authors in [20] extensively investigated the Lyapunov optimization theory, providing its applications in communication systems. Several studies have explored this stochastic optimization algorithm [21,22,23,24]. Although the application of the Lyapunov optimization theory in communication and queuing systems is not novel, few studies have considered its application to distributed massive MIMO systems. The spatial multiplexing leads to the coupled traffic queues of different users. Using the Lyapunov technique, the difficulty in solving the weighted sum rate maximization problem is increasing. An effective virtual water-filling algorithm was proposed in [25] to solve the weighted sum rate maximization problem. Although the scenario of this work was single-cell co-located massive MIMO systems, which is easy to model, it still inspired us with the application of the Lyapunov technique to solve the random queue data in massive MIMO systems. The problem of low-latency and ultra-reliable communication in 5G (5th Generation Mobile Communication Technology) millimeter wave massive MIMO systems was investigated in [26]. They established a dynamic scheduling scheme with latency constraints but overlooked the total power consumption problem. Reference [27] developed a robust MIMO downlink scheduling using the flow control algorithm, assuming an infinite backlog at the base stations (BS). The authors in [28] proposed a beamforming design to minimize the average power consumption in a time-varying C-RAN system, where they set the estimation CSI to be perfect. All in all, these studies all simplified the channel model and the system performance constraints, neglecting the complicated non-linear, non-convex relationship between power consumption, channel uncertainty, QoS constraints, queue stability, resource allocation, and total throughput in a practical system.

Motivated by the aforementioned factors, we propose a robust beamforming design for a time-varying cell-free distributed massive MIMO system that accounts for the channel uncertainty. Our orientation is to minimize the average network power consumption while considering random traffic arrivals, which ensures a high QoS for each user. The major contributions of this paper can be concluded as follows:

• We investigate the problem of minimizing the long-term average total power consumption in a time-varying high-mobility cell-free massive MIMO system with channel uncertainty. Therein, we consider random traffic arrivals, achievable rate limits, and channel uncertainty.
• We transform the investigated optimization problem into a weighted sum rate maximization problem for each slot via the prestigious Lyapunov optimization technique. We employ a successive convex approximation (SCA) method on the objective function to generate sparse beamforming. Additionally, we modify the objective function via bounding and approximation transformation to combat channel uncertainty. Furthermore, we employ the convex–concave procedure (CCP) to linearize the non-convex QoS constraints.
• We apply a direct fractional programming (DFP) technique to transform the weighted sum rate maximization problem into a tractable form and demonstrate its convergence. Compared with the traditional WMMSE algorithm, our proposed algorithm converges in fewer iterations. Moreover, we show the tradeoff between average power consumption and the queue latency.

We organize the rest of the paper as follows. In Section 2, we describe the system model and formulate the problem. Section 3 transforms the problem above. We propose an algorithm to solve the problem in Section 4. Section 5 shows and analyzes the simulation results. Section 6 draws the conclusion of this paper.

Notation: In this paper, matrices and column vectors are denoted by the upper and lower case boldface letters, respectively. The $P\times P$ identity matrix is denoted by ${\mathrm{I}}_{\mathrm{P}}$. ${\mathbb{C}}^{\mathit{M}}$ and ${\mathbb{C}}^{\mathit{M}\times \mathit{N}}$ represent the complex M-dimensional vectors and complex $\mathit{M}\times \mathit{N}$ matrices, respectively. The conjugation, transpose, and conjugate transpose are denoted by ${\left(·\right)}^{\ast }$, ${\left(·\right)}^T$, and ${\left(·\right)}^H$, respectively. $\Re \left(a\right)$ is the real part of a. Ensemble expectation is denoted by $\mathbb{E}(·)$. $\mathbf{c}\sim \mathcal{C}\mathcal{N}(x,\mathbf{Y})$ represents a random vector $\mathbf{c}$, which is a vector obeying complex Gaussian distribution with an average vector x and covariance matrix $\mathbf{Y}$.

2. System Model and Problem Formulation

We consider a cell-free massive MIMO system, shown as Figure 1, where the APs are responsible for data transmission only and the central processor unit (CPU) is responsible for data processing. L APs equipped with M transmitting antennas are used to serve K single-antenna users. The bandwidth of the system is denoted by $\mathit{D}$.

2.1. Channel Model

The system is assumed to operate in time division duplexing (TDD) mode, which allows the reciprocity between the uplink and downlink channels. The channel reciprocity leverages the CSI obtained by the uplink training to assist the downlink transmission [29,30]. A time-slot structure is adopted, as shown in Figure 1, where each time slot consists of $N_b$ blocks. The first block is dedicated to uplink training, while the remaining blocks are utilized for downlink transmission aided by the CSI estimated in the first block. The channel vector from all APs to the user k in block n at slot t ${\mathbf{h}}_{k,n}\left(t\right)\in {\mathbb{C}}^{ML\times 1}$ can be expressed as follows:

$${\mathbf{h}}_{\mathit{k},\mathit{n}}^{\mathit{H}}\left(\mathit{t}\right)={\alpha }_{\mathit{k},\mathit{n}}{\mathbf{h}}_{1,\mathit{n}}^{\mathit{H}}\left(\mathit{t}\right){\mathbf{G}}^{\mathit{H}}+\sqrt{1-{\alpha }_{\mathit{k},\mathit{n}}^2}\left({\mathbf{m}}_{\mathit{k}}^{\mathit{T}}\odot {\mathbf{c}}_{\mathit{k},\mathit{n}}^{\mathit{T}}\right){\mathbf{G}}^{\mathit{H}},$$

(1)

where ${\alpha }_{\mathit{k},\mathit{n}}\in (0,1)$, ${\mathbf{m}}_{\mathit{k}}\in {\mathbb{R}}^{\mathit{ML}\times 1}$ and $\mathbf{G}\in {\mathbb{C}}^{ML\times ML}$ are, respectively, the temporal correlation coefficient between ${\mathbf{h}}_{k,n}\left(t\right)$ and ${\mathbf{h}}_{\mathrm{k},\mathit{1}}\left(t\right)$ related to the user moving velocity, the deterministic vector with nonnegative elements, and the deterministic unitary matrix, which can all be estimated via thet channel sounding process [13]. ${\mathbf{c}}_{\mathit{k},\mathit{n}}\in {\mathbb{C}}^{ML\times 1}$ is the random vector representing the channel uncertainty, which can be estimated as ${\mathbf{c}}_{\mathit{k},\mathit{n}}\sim \mathcal{C}\mathcal{N}(0,{\mathbf{I}}_{\mathit{ML}})$. When $\alpha$ approximates to 0, the model can describe the high-speed scenarios. When $\alpha$ is close to 1, the model can model quasi-static scenarios. The subscript n is omitted for simplicity in the following.

2.2. Downlink Transmission Model

Let ${\mathbf{w}}_{\mathit{k}}\left(\mathit{t}\right)=\left[{\mathbf{w}}_{\mathit{k}}^1\left(\mathit{t}\right),{\mathbf{w}}_{\mathit{k}}^2\left(\mathit{t}\right),\cdots ,{\mathbf{w}}_{\mathit{k}}^{\mathit{L}}\left(\mathit{t}\right)\right]\in {\mathbb{C}}^{\mathit{ML}\times 1}$ be the transmit beamforming vector over all the APs for the users at slot t, where ${\mathbf{w}}_{\mathit{k}}^{\mathit{l}}\in {\mathbb{C}}^{\mathit{M}\times 1}$ is the beamforming vector transmitted from AP l to the user k at slot t. If the AP l does not serve the user k, we have ${\mathbf{w}}_{\mathit{k}}^{\mathit{l}}\left(t\right)=0$.

Then, the received signal ${\mathit{e}}_{\mathit{k}}\left(t\right)$ of the user k on block n at slot t can be expressed as

$$e_{\mathit{k}}\left(t\right)={\mathbf{h}}_{\mathit{k}}^{\mathit{H}}\left(t\right){\mathbf{w}}_{\mathit{k}}\left(t\right){\mathit{s}}_{\mathit{k}}\left(t\right)+\sum _{\mathit{j}\ne \mathit{k}}^{\mathit{K}}{\mathbf{h}}_{\mathit{k}}^{\mathit{H}}\left(t\right){\mathbf{w}}_{\mathit{j}}\left(t\right){\mathit{s}}_{\mathit{j}}\left(t\right)+n_{\mathit{k}}\left(t\right),$$

(2)

where ${\mathit{s}}_{\mathit{k}}\sim \mathcal{C}\mathcal{N}(0,1)$, ${\mathit{n}}_{\mathit{k}}\sim \mathcal{C}\mathcal{N}(0,{\sigma }^2)$ are the intended signal and the receiver noise for the user k, respectively. The SINR for the user k can be expressed as

$${\mathrm{SINR}}_k\left(t\right)=\frac{{\left|{\mathbf{h}}_k^H\left(\mathit{t}\right){\mathbf{w}}_k\left(t\right)\right|}^2}{{\displaystyle \sum _{j\ne k}^K}{\left|{\mathbf{h}}_k^H\left(\mathit{t}\right){\mathbf{w}}_j\left(t\right)\right|}^2+{\sigma }^2}.$$

(3)

The achievable rate for the user k is given by

$${\mathit{R}}_k\left(t\right)={log}_2\left(1+{\mathrm{SINR}}_{\mathit{k}}\left(t\right)\right).$$

(4)

2.3. AP Selection

For energy saving and carbon reduction, we adopt a power-saving model design to generate sparse beamforming vectors [16]. The power consumption model of each AP contains the transmitting mode and the sleeping mode. The power consumption of AP l can be expressed as

$$\begin{array}{c}\hfill P^l=\left\{\begin{array}{c}{\lambda }^l{P}^{tx}+P_{\Delta }^l\mathrm{if}\mathrm{AP}\mathit{l}\mathrm{is}\mathrm{in}\mathrm{the}\mathrm{active}\mathrm{mode},\hfill \\ P_s^l\mathrm{if}\mathrm{AP}\mathit{l}\mathrm{is}\mathrm{in}\mathrm{the}\mathrm{sleep}\mathrm{mode},\hfill \end{array}\right.\end{array}$$

(5)

where $\lambda$ is the power transfer efficiency coefficient; $P^{tx}={\sum }_{k=1}^K{∥{\mathbf{w}}_k^l\left(t\right)∥}_2^2$ is the transmit power of AP l; and $P_{\Delta }^l$ and $P_s^l$ are the minimum power to keep AP active and the low power consumption of AP in sleep mode, respectively. Thus, the total power consumption can be calculated using

$$\begin{array}{c}\hfill P_{\mathit{total}}=\sum _{l=1}^L\left(\left(\sum _{k=1}^K{\lambda }_k^l{∥{\mathbf{w}}_{\mathit{k}}^{\mathit{l}}\left(\mathit{t}\right)∥}_2^2+P_s^l\right)+\parallel \sum _{k=1}^K{∥{\mathbf{w}}_{\mathit{k}}^l\left(\mathit{t}\right)∥}_2^2{\parallel }_0{P}_{\Delta }^l\right).\end{array}$$

(6)

2.4. Queue Stability and Problem Formulation

We assume that $\mathbf{X}(\mathit{t}) = \left[{\mathrm{X}}_1\left(t\right),{\mathrm{X}}_2\left(t\right),\cdots ,{\mathrm{X}}_K\left(t\right)\right]$ estimated as $\mathbb{E}\left({\mathrm{X}}_k\left(t\right)\right)={\eta }_k$ is the random vector of stochastic traffic data arrivals (bits) at the end of slot t and the queue state information for the users is $\mathbf{Q}(\mathit{t}) = \left[{\mathrm{Q}}_1\left(t\right),{\mathrm{Q}}_2\left(t\right),\cdots ,{\mathrm{Q}}_K\left(t\right)\right]$. We can update the queue state information according to

$$Q_k(t+1)=\max \left\{\left(Q_k\left(t\right)-{\phi }_k\left(t\right)\right),0\right\}+{\mathrm{X}}_k\left(t\right),$$

(7)

where ${\phi }_k\left(t\right)=\mathit{D}{\mathit{R}}_k\left(t\right)$ represents the traffic departure. Before modelling the impacts of AP selection and beamforming policy on the average queue latency and the average total network energy cost, we firstly introduce the definition of queue stability as follows [20]:

Definition 1. 

(Queue stability): A stochastic queue $\mathbf{Q}\left(t\right)$ is strongly stable if

$$\underset{\mathit{T}\to \infty }{lim\sup }\frac{1}{T}\sum _{t=0}^{T-1}\mathbb{E}\left(\mathbf{Q}\left(\mathit{t}\right)\right)<\infty .$$

(8)

The stability of the network of queues can be ensured if all individual queues of the network are stable.

By using AP selection and robust beamforming, we target to keep the network of queues stable and minimize the average network energy cost while meeting the QoS requirements for each user. Thus, we can formulate the problem as

$$\begin{array}{cc}\hfill & \underset{{\mathbf{w}}_k^l\left(t\right)}{min}{\overline{\mathit{P}}}_{\mathit{total}}=\underset{\mathit{T}\to \infty }{lim}\frac{1}{T}\sum _{t=0}^{T-1}\mathbb{E}\left(P_{\mathit{total}}\right)<\infty ,\hfill \end{array}$$

(9a)

$$\begin{array}{cc}\hfill & \mathrm{s}.\mathrm{t}.R_k\left(t\right)\ge {\gamma }_k,\forall \mathit{k}\in \mathcal{K},\hfill \end{array}$$

(9b)

$$\begin{array}{cc}\hfill & \underset{\mathit{T}\to \infty }{lim\sup }\frac{1}{T}\sum _{t=0}^{T-1}\mathbb{E}\left(\mathbf{Q}\left(\mathit{t}\right)\right)<\infty ,\hfill \end{array}$$

(9c)

$$\begin{array}{cc}\hfill & \sum _{k=1}^K{∥{\mathbf{w}}_{\mathit{k}}^l\left(\mathit{t}\right)∥}_2^2\le P_{\mathit{max}},\forall l\in L,\hfill \end{array}$$

(9d)

where the conditional expectation of the network power consumption is related to the random sparse beamforming vectors. Equation (9b) is the constraint on the instantaneous achievable rate to guarantee the QoS requirements of each user, where $\gamma$ represents the minimal achievable rate for each user. Equation (9c) is the network stability constraint. Equation (9d) is the power consumption constraint for each AP, where $P_{\mathit{max}}$ is the max power consumption for each AP. The stochastic traffic arrivals and the time-varying channel conditions are unpredictable in practice. It is impossible to obtain the optimal solution due to the stochasticity of the traffic arrivals and the channel conditions. To deal with this issue, we will employ the Lyapunov optimization technique [28,31], which can convert this optimization problem into a real-time one at each slot.

3. Problem Reformulation via the Lyapunov Optimization Technique

In this section, we will resort to the Lyapunov optimization technique to transform the long-term optimization problem presented in (9) into a real-time one. The Lyapunov function is defined as $L\left(\mathbf{Q}\left(t\right)\right)=\frac{1}{2}{\displaystyle \sum _{k=1}^K}{Q}_k{\left(t\right)}^2$, which indicates the queue congestion state information in the APs. The system is kept stable by continually pushing the Lyapunov function to a lower congestion state. Hence, we, respectively, define the one-step conditional Lyapunov drift and the Lyapunov drift-plus-penalty function as [20]

$$\Delta \left(\mathbf{Q}\left(t\right)\right)=\mathbb{E}\left[L\left(\mathbf{Q}\right(t+1\left)\right)-L\left(\mathbf{Q}\right(t\left)\right)\left|\mathbf{Q}\right(t)\right],$$

(10)

and

$$\Delta \left(\mathbf{Q}\left(t\right)\right)+\mathit{V}\mathbb{E}\left[P_{\mathit{total}}\right|\mathbf{Q}\left(t\right)],$$

(11)

where V serves as a positive control parameter. We can switch the parameter V to regulate the tradeoff between the average energy cost and the average queue latency. A larger V indicates that the optimization will place more emphasis on minimizing the average power consumption. On the other side, a small V means that the queue stability in optimization will receive more weight. Let $p^{\ast }$ represent the theoretical optimal value of (9), and then, the connection between the queue stability and the Lyapunov drift-plus-penalty function can be demonstrated in Theorem 1 [20].

Theorem 1. 

(Lyapunov Optimization Theory): We assume that constants Γ, Φ, and V are positive for all slot t and the values of $\mathbf{Q}\left(t\right)$ are random. Then, the Lyapunov drift-plus-penalty function meets

$$\Delta \left(\mathbf{Q}\left(t\right)\right)+\mathit{V}\mathbb{E}\left[P_{\mathit{total}}\right|\mathbf{Q}\left(t\right)]\le \Gamma +V{p}^{\ast }-\mathsf{\Phi }\sum _k{\mathit{Q}}_k\left(t\right),$$

(12)

and then, all queues ${\mathit{Q}}_k\left(t\right)$ are strongly stable. The average penalty of power consumption meets

$$\underset{\mathit{T}\to \infty }{lim\sup }\frac{1}{T}\sum _{t=0}^{T-1}\mathbb{E}\left(P_{\mathit{total}}\right)\le \frac{\Gamma }{V}+p^{\ast }.$$

(13)

Proof. 

See Theorem 4.2 in [20].   □

The results of the Lyapunov optimization theory force us to minimize (11) to maintain the stability of the network of queues to the maximum. Instead of simply minimizing (11), our policy aims to minimize the upper bound of it, which is introduced as the Lemma 1.

Lemma 1. 

(Upper bound of lyapunov drift-plus-penalty function): The upper bound of the drift-plus-penalty function for any time slot t can be written as follows under any control policy.

$$\Delta \left(\mathbf{Q}\left(t\right)\right)+\mathit{V}\mathbb{E}\left[P_{\mathit{total}}\right|\mathbf{Q}\left(t\right)]\le \Gamma +\mathit{V}\mathbb{E}\left[P_{\mathit{total}}\right|\mathbf{Q}\left(t\right)]+\sum _{k=1}^K{Q}_k\left(t\right)\mathbb{E}[{\mathrm{X}}_k\left(t\right)-{\phi }_k\left(t\right)|\mathbf{Q}\left(t\right)],$$

(14)

where Γ is a positive constant and satisfies $\Gamma \ge \frac{1}{2}{\displaystyle \sum _{k=1}^K}\mathbb{E}[{\mathrm{X}}_k^2\left(t\right)-{\phi }_k^2\left(t\right)|\mathbf{Q}\left(t\right)]$ for any slot t.

Proof. 

See Appendix A.   □

According to the method of opportunistically minimizing an expectation [20], minimizing $\mathbb{E}\left[f\right(t\left)\right|\mathbf{Q}\left(t\right)]$ is equivalent to minimizing $f\left(t\right)$ with the observation of $\mathbf{Q}\left(t\right)$[28]. Moreover, ${\displaystyle \sum _{k=1}^K}{Q}_k\left(t\right){\mathrm{X}}_k\left(t\right)$ and $\Gamma$ in (14) are not influenced by the policy at slot t. Consequently, problem (9) can be simplified as follows:

$$\begin{array}{cc}\hfill & \underset{{\mathbf{w}}_k^l\left(t\right)}{\max }\sum _{k=1}^K{Q}_k\left(t\right){\phi }_k\left(t\right)-V{P}_{\mathit{total}},\hfill \end{array}$$

(15a)

$$\begin{array}{cc}\hfill & \mathrm{s}.\mathrm{t}.R_k\left(t\right)\ge {\gamma }_k,\forall \mathit{k}\in \mathcal{K},\hfill \end{array}$$

(15b)

$$\begin{array}{cc}\hfill & \sum _{k=1}^K{∥{\mathbf{w}}_{\mathit{k}}^l\left(\mathit{t}\right)∥}_2^2\le P_{\mathit{max}},\forall l\in L.\hfill \end{array}$$

(15c)

However, the objective function in (15) is NP-hard. Therefore, obtaining the globally optimal solution to (15) using effective methods within polynomial time is exceedingly challenging. Instead of pursuing the global optimal solution, we will strive to develop an algorithm that yields sub-optimal solutions for problem (15).

4. Robust Beamforming Design Based on Fractional Programming

The objective function and constraints of this optimization problem are non-convex, which brings some challenges. In this section, we will solve the weighted sum rate maximization problem by using fractional programming. We will drop the slot index t to simplify the expression in the formulation below.

4.1. Group Sparse Beamforming Formulation

Because of the existence of ${\ell }_0$-norm, the total power consumption is non-convex. In compressed sensing theory, the ${\ell }_0$-norm can be approximated as a convex weighted ${\ell }_1$-norm problem. In view of this, the power consumption can be approximated as follows:

$$\sum _{l=1}^L\left(\sum _{k=1}^K{\lambda }_k^l{∥{\mathbf{w}}_k^l∥}_2^2+{\rho }^l\sum _{k=1}^K{∥{\mathbf{w}}_k^l∥}_2^2{P}_{\Delta }^l\right),$$

(16)

where ${\rho }^l$ is the weight associated with AP l and the user k, and it can be updated as follows:

$${\rho }^l=\frac{1}{{\displaystyle \sum _{k=1}^K}{∥{\mathbf{w}}_{\mathit{k}}^l∥}_2^{2u}+{\theta }^u},$$

(17)

where u is a positive exponent and $\theta$ is adaptively chosen according to

$$\theta =\max \left\{\left({\min }_{k,l}{∥{\mathbf{w}}_k^l∥}_2^2\right),\tau \right\},$$

(18)

where $\tau$ is a very small positive number that prevents the denominator from being 0, and thus, the total power consumption in the objective function is transformed into a convex item [32,33]. We can find that when the AP l consumed more power, the weight indicator ${\rho }^l$ will turn small. Therefore, when the AP l consumed little power in the last iteration, the weight indicator ${\rho }^l$ will increase, which leads to the degradation of the power consumption of the AP l in the next iteration. After some iterations, the power consumption of the AP l will be very small and the AP l will be turned into the sleep mode, and that is how we obtain the sparsity of the beamforming.

4.2. Robust Beamforming Design and Fractional Programming

Firstly, we can approximate the sum rate as [30]

$$\begin{array}{cc}\hfill & {\displaystyle \sum _{k=1}^K}{R}_k\le \sum _{k=1}^K{\log }_2\left(1+\mathbb{E}\left(\frac{{\mathbf{w}}_{\mathit{k}}^H{\mathbf{h}}_k{\mathbf{h}}_k^H{\mathbf{w}}_k}{{\displaystyle \sum _{j\ne k}^K}{\mathbf{w}}_j^H{\mathbf{h}}_k{\mathbf{h}}_k^H{\mathbf{w}}_j+{\sigma }^2}\right)\right)\hfill \\ \hfill & \approx \sum _{k=1}^K{\log }_2\left(1+\frac{{\mathbf{w}}_{\mathit{k}}^H\mathbb{E}\left({\mathbf{h}}_k{\mathbf{h}}_k^H\right){\mathbf{w}}_k}{{\displaystyle \sum _{j\ne k}^K}{\mathbf{w}}_j^H\mathbb{E}\left({\mathbf{h}}_k{\mathbf{h}}_k^H\right){\mathbf{w}}_j+{\sigma }^2}\right).\hfill \end{array}$$

(19)

According to [13], we have

$$\mathbb{E}\left({\mathbf{h}}_k{\mathbf{h}}_k^H\right)={\alpha }_k^2{\mathbf{h}}_{1,n}{\mathbf{h}}_{1,\mathit{n}}^{\mathit{H}}+\left(1-{\alpha }_k^2\right)\mathbf{G}\mathrm{diag}({\mathbf{d}}_k){\mathbf{G}}^{\mathit{H}}={\mathbf{P}}_k^H{\mathbf{P}}_k,$$

(20)

where ${\mathbf{d}}_k={\mathbf{m}}_k\odot {\mathbf{m}}_k$, and $\mathbf{P}$ can be expressed as

$${\mathbf{P}}_k=\left[\begin{array}{c}{\alpha }_k^2{\mathbf{h}}_{1,\mathit{n}}^{\mathit{H}}\\ \sqrt{1-{\alpha }_k^2}\mathrm{diag}({\mathbf{d}}_k){\mathbf{G}}^{\mathit{H}}\end{array}\right].$$

(21)

Thus, the optimal problem can be approximated as follows:

$$\begin{array}{cc}\hfill & \underset{{\mathbf{w}}_k^l}{\max }{Q}_k{\displaystyle \sum _{k=1}^K}{\log }_2\left(1+\frac{{\mathbf{w}}_{\mathit{k}}^H{\mathbf{P}}_k^H{\mathbf{P}}_k{\mathbf{w}}_k}{{\displaystyle \sum _{j\ne k}^K}{\mathbf{w}}_j^H{\mathbf{P}}_k^H{\mathbf{P}}_k{\mathbf{w}}_j+{\sigma }^2}\right)-\frac{\mathit{V}}{\mathit{D}}{P}_{\mathit{total}},\hfill \end{array}$$

(22a)

$$\begin{array}{cc}\hfill & \mathrm{s}.\mathrm{t}.\frac{{\mathbf{w}}_{\mathit{k}}^H{\mathbf{P}}_k^H{\mathbf{P}}_k{\mathbf{w}}_k}{{\displaystyle \sum _{j\ne k}^K}{\mathbf{w}}_j^H{\mathbf{P}}_k^H{\mathbf{P}}_k{\mathbf{w}}_j+{\sigma }^2}\ge 2^{{\gamma }_k-1},\forall \mathit{k}\in \mathcal{K},\hfill \end{array}$$

(22b)

$$\begin{array}{cc}\hfill & \sum _{k=1}^K{∥{\mathbf{w}}_{\mathit{k}}^l∥}_2^2\le P_{\mathit{max}},\forall l\in L.\hfill \end{array}$$

(22c)

By using the CCP and first-order Taylor expansion, we can transform the QoS constraints into a convex form as follows:

$$\begin{array}{cc}\hfill & \underset{{\mathbf{w}}_k^l}{\max }{Q}_k{\displaystyle \sum _{k=1}^K}{\log }_2\left(1+\frac{{\mathbf{w}}_{\mathit{k}}^H{\mathbf{P}}_k^H{\mathbf{P}}_k{\mathbf{w}}_k}{{\displaystyle \sum _{j\ne k}^K}{\mathbf{w}}_j^H{\mathbf{P}}_k^H{\mathbf{P}}_k{\mathbf{w}}_j+{\sigma }^2}\right)-\frac{\mathit{V}}{\mathit{D}}{P}_{\mathrm{total}},\hfill \\ \hfill & \mathrm{s}.\mathrm{t}.2^{{\gamma }_k-1}\left(\sum _{j\ne k}^K{\mathbf{w}}_j^H{\mathbf{P}}_k^H{\mathbf{P}}_k{\mathbf{w}}_j+{\sigma }^2\right)\hfill \end{array}$$

(23a)

$$\begin{array}{cc}\hfill & -\left(2\Re \left\{{\mathbf{w}}_{\mathit{k}}^{H,\left(n\right)}{\mathbf{P}}_k^H{\mathbf{P}}_k{\mathbf{w}}_k\right\}-{\left|{\mathbf{P}}_k{\mathbf{w}}_k^{\left(n\right)}\right|}^2\right)\le 0,\forall \mathit{k}\in \mathcal{K},\hfill \end{array}$$

(23b)

$$\begin{array}{cc}\hfill & \sum _{k=1}^K{∥{\mathbf{w}}_{\mathit{k}}^l∥}_2^2\le P_{\mathit{max}},\forall l\in L.\hfill \end{array}$$

(23c)

where ${\mathbf{w}}_k^{\left(n\right)}$ denotes the optimal solution obtained from the previous time of iteration. This is an NP-hard problem, and it is difficult to find a globally optimal solution. However, by resorting to direct fractional programming, we can obtain a stationary point via the alternative optimization principle.

Lemma 2. 

Quadratic transform: We focus on the following optimization problem:

$$\underset{x}{\max }\sum _{k=1}^K\frac{{\left|A_k\left(x\right)\right|}^2}{B_k\left(x\right)}.$$

(24)

By introducing the auxiliary variables $\mathbf{Y}=({\mathrm{y}}_1,{\mathrm{y}}_2\cdots {\mathrm{y}}_K)$, the problem can be reformulated as

$$\underset{x,y}{\max }\sum _{k=1}^K\left(2\Re \left(y_k^{\ast }{A}_k\left(x\right)\right)\right)-y_k^{\ast }{B}_k\left(x\right)y_k.$$

(25)

It is evident that (25) is equivalent to (24) [34].

After the quadratic transformation, the problem can be reformulated as

$$\begin{array}{cc}\hfill & \underset{{\mathbf{w}}_k^l,{\mathit{y}}_k}{\max }f(\mathbf{w},\mathbf{Y}),\hfill \\ \hfill & \mathrm{s}.\mathrm{t}.2^{{\gamma }_k-1}\left(\sum _{j\ne k}^K{\mathbf{w}}_j^H{\mathbf{P}}_k^H{\mathbf{P}}_k{\mathbf{w}}_j+{\sigma }^2\right)\hfill \end{array}$$

(26a)

$$\begin{array}{cc}\hfill & -\left(2\Re \left\{{\mathbf{w}}_{\mathit{k}}^{H,\left(n\right)}{\mathbf{P}}_k^H{\mathbf{P}}_k{\mathbf{w}}_k\right\}-{\left|{\mathbf{P}}_k{\mathbf{w}}_k^{\left(n\right)}\right|}^2\right)\le 0,\forall \mathit{k}\in \mathcal{K},\hfill \end{array}$$

(26b)

$$\begin{array}{cc}\hfill & \sum _{k=1}^K{∥{\mathbf{w}}_{\mathit{k}}^l∥}_2^2\le P_{\mathit{max}},\forall l\in L,\hfill \end{array}$$

(26c)

where

$$\begin{array}{cc}\hfill f\left(\mathbf{w},\mathbf{Y}\right)=& \sum _{k=1}^K{Q}_{k}log\left(1+\Re \left\{y_k^{\ast }{\mathbf{P}}_k{\mathbf{w}}_k\right\}-y_k^{\ast }\left({\sigma }^2+\sum _{j\ne k}^K{\mathbf{w}}_j^H{\mathbf{P}}_k^H{\mathbf{P}}_k{\mathbf{w}}_k\right)y_k\right)\hfill \\ \hfill & -\frac{V}{D}{P}_{\mathrm{total}},\hfill \end{array}$$

(27)

and where ${\mathit{y}}_k^{\ast }$ represents the conjugation of y.

We follow Algorithm 4 in [34] to maximize f over $\mathbf{w}$ and $\mathit{y}$ iteratively. The optimal solution of $\mathit{y}$ for a fixed $\mathbf{w}$ is as follows:

$$y_k=\frac{{\mathbf{P}}_k{\mathbf{w}}_k}{{\sigma }^2+{\displaystyle \sum _{j\ne k}^K}{\mathbf{w}}_j^H{\mathbf{P}}_k{\mathbf{P}}_k{\mathbf{w}}_j}.$$

(28)

Because the outer logarithmic function is non-decreasing and concave, and the internal function is concave, the compound functions of them are proven to be convex. Thus, the optimization problem (26) is a convex problem of $\mathbf{w}$. For any fixed $\mathit{y}$, the optimal $\mathbf{w}$ can be easily obtained. Our proposed DFP algorithm is shown in the Algorithm 1 below.

Algorithm 1: DFP-based robust sparse beamforming algorithm.

5. Numerical Simulation Results and Discussion

In this section, we evaluate the performance of the proposed algorithm based on simulation results. Each AP utilizes M antennas to serve K mobile users. We randomly generate the users in a circle with a radius of 1500 m. The temporal correlation coefficient ${\alpha }_{\mathit{k},\mathit{n}}$, the deterministic vector ${\mathbf{m}}_{\mathit{k}}$, the deterministic unitary matrix $\mathbf{G}$, and ${\mathbf{h}}_{1,\mathit{n}}$ are assumed to be known at the APs for all users k. Some other parameters are shown in

Table 1. Simulation parameters.

Parameter	Value
Number of APs L	7
Distance of APs T	800 m
Number of antennas employed in each AP	4
Number of user	5
Number of antenna at user $N_{\mathrm{r}}$	1
Convergence tolerance	${10}^{-3}$
Gaussian noise variance ${\sigma }_z^2$	$-160\mathrm{dBm}/\mathrm{Hz}$
SINR threshold	$10\mathrm{dB}$
the number of blocks within one slot $N_b$	10
system bandwidth D	${10}^6\mathrm{Hz}$
reweighted coefficient u	4/3
reweighted coefficient $\tau$	${10}^{-10}$
Power constraints of each AP $P_{max}$	0.8 W

.

Figure 2 demonstrates the convergence of our proposed sparse beamforming algorithm when $K=5$. The target SINR is set to 10 dB. After about eight iterations, AP1, AP2, and AP6 are in the active mode, while the other four APs are turned into the sleep mode, indicating the sparsity property of the transmit beamforming vectors.

Figure 3 illustrates the sum-rate convergence results of the proposed DFP algorithm and the well-known WMMSE algorithm for two different scenarios. It is shown that the DFP algorithm we proposed outperforms the traditional WMMSE algorithm. The DFP algorithm achieves a sum rate of 8.4 bits/Hz within only three iterations while the WMMSE algorithm needs six iterations. Moreover, our proposed DFP algorithm achieves a higher sum rate than the WMMSE algorithm. The reason is that the solution of the DFP obtained using the CVX toolbox is more precise than the closed-form solution of the WMMSE.

Figure 4 and Figure 5 present an analysis of the average queue latency and the average network energy consumption with respect to the control parameter V for different mean arrival rates $\eta$, respectively. Higher mean traffic arrival rates result in longer average latency and increased network power consumption, which can be attributed to the requirement for more power to timely transmit a larger number of traffic arrivals. The average network power consumption exhibits a decreasing trend along with V for a given mean arrival rate. With a significant rise in V, the rate of power loss begins to diminish. On the contrary, a larger value of V leads to an approximately linear increase in average queue length, which degrades the delay performance.

This occurs because a higher value of V prioritizes network power consumption over delay performance. Thus, the parameter V measures the tradeoff between power consumption and delay performance. Furthermore, Figure 5 demonstrates the performance of the proposed beamforming algorithm with AP selection consistently outperforming that without AP selection. When the parameter V is small, the system gives a higher priority to minimizing the average queue delay to keep the stability of the system. More APs have to be turned on to transmit the traffic data, so the difference between the beamforming design with AP selection and that without AP selection is small. While with the parameter V increasing, the system gives a higher priority to minimizing the average power consumption so more APs are turned into the sleep mode. Therefore, we can find that the more the parameter V is, the more the beamforming design with the AP selection outperforms that without the AP selection.

Figure 6 compares the power-delay tradeoffs for different mean arrival rates $\eta$, where by changing the control parameter V, we obtained different tradeoff points. The average network power consumption exhibits a decreasing and convex relationship with the average delay. With a small delay, a slight increase in the delay requirement can yield significant power savings, while for an excessively large delay, the additional delay results in negligible power savings. Additionally, the AP selection strategy provides a more favorable power-delay tradeoff. In the regime of small delays, the proposed algorithm offers versatile and effective means for balancing the power-delay tradeoff, as even a slight relaxation in the delay requirement can lead to considerable energy savings. The system necessitates the activation of more APs with a stricter delay requirement (e.g., less and equal to 0.75 slots), which makes the advantages of AP selection insufficiently prominent. Consequently, a suitably chosen control parameter V enables the desired power-delay tradeoff.

6. Conclusions

This paper addressed the tradeoff between total power consumption and queue stability by formulating a long-term average total power cost minimization problem in a time-varying system under imperfect channel conditions. The problem was transformed into a weighted sum rate maximization problem for each slot using the Lyapunov optimization technique. To combat the channel uncertainty constraint, approximation techniques were employed to design robust sparse beamforming. To meet the QoS requirements of users, a convex–concave procedure and first-order Taylor expansion were used to transform the QoS constraints into a convex form. By applying DFP and alternating optimization, a locally optimal point was obtained. The proposed algorithm can minimize the average power consumption of the network while satisfying the QoS requirements for each user. The simulation results demonstrate that our proposed algorithm converges rapidly and outperforms the traditional WMMSE algorithm and illustrate the tradeoff between the average network energy consumption and the queue latency. An improved case considering finite-capacity backhaul is expected for future research.