Energy Efficiency Fairness of Active Reconfigurable Intelligent Surfaces-Aided Cell-Free Network

Reconfigurable intelligent surface (RIS) has been widely recognized as a promising technique for future wireless communications. Benefiting from its ability to improve channel capacity with low cost and power consumption, the concept of RIS-aided cell-free network is proposed to break the bottleneck of current network architecture. However, the multiplicative fading effect (MFE) introduced by RIS prevents RIS-aided cell-free network from higher network capacity. To further increase the capacity of the RIS-aided cell-free network, in this paper, we propose the concept of active RIS-aided cell-free network, where the RIS is replaced by active RIS in traditional RIS-aided cell-free network. Moreover, we analyze the proposed network by investigating its energy-efficiency fairness (EEF). To elaborate, we formulate the EEF maximization problem in the active RIS-aided cell-free network, and propose a joint beamforming and resource allocation (JBRA) algorithm by exploiting alternating optimization (AO) and fractional programming (FP). Simulations results verify the proposed active RIS-aided cell-free network as well as the JBRA algorithm as an energy-efficient design for cell-free network with large network capacity.


I. INTRODUCTION
Reconfigurable intelligent surface (RIS) has been widely recognized as a promising technique for future wireless communications. Specifically, RIS is a large-scale reflective antenna array composed of massive elements, each of which can be manipulated with a controllable additive phase shift on the incident electromagnetic waves. Benefiting from the characteristics of low cost and power consumption, RIS can overcome blockage and provide high array gains [1], [2], [3].
One of the most representative usage of RIS in future wireless communications is the RIS-aided cell-free network where multiple RISs are employed for improving the network capacity [4]. This network architecture provides an energyefficient solution to the inter-cell interferences, which is the bottleneck of boosting the network capacity in current The associate editor coordinating the review of this manuscript and approving it for publication was Mauro Fadda . architecture [5]. However, employing RIS in cell-free network cannot always provide significant gains in network capacity. This is because the multiplicative fading effect (MFE) introduced by RIS, that the pathloss of the transmitter-RIS-receiver link is the product of that of the transmitter-RIS and RIS-receiver links, which results in limited gain of the reflection link [6]. Therefore, only when the direct link is weak enough can the employment of RIS be efficient, and this brings about the new challenge for further increasing the capacity of the RIS-aided cell-free network.
To overcome the challenge of MFE, fortunately, an improved hardware named active RIS is proposed, where a power amplifier (PA) is additionally integrated in each element of RIS [6], [7]. This PA is responsible for amplifying the reflected signals apart from controlling their phase shifts, which increases the array gains of RIS and provably changes the multiplicative fading back to the additive fading [8]. Several works have been done revolving around active RIS, and have revealed the benefits of replacing RIS with active RIS. For example, in [9], the optimal placement strategy is investigated for active RIS. In [10], a sub-connected architecture of active RIS is proposed for power saving. In [11], the possible ability of active RIS for secure transmission is discussed.
To further increase the capacity of the RIS-aided cell-free network, in this paper, we propose the concept of active RIS-aided cell-free network, where the RIS is replaced by its active counterpart in traditional RIS-aided cell-free network. Moreover, we analyze the active RIS-aided cell-free network by investigating its energy-efficiency fairness (EEF). To elaborate, we formulate the EEF maximization problem in the active RIS-aided cell-free network, where the energy efficiency of the worst user is maximized. Furthermore, to solve this non-convex problem with decoupled variables, we propose a joint beamforming and resource allocation (JBRA) algorithm by exploiting alternating optimization (AO) and fractional programming (FP), which gives a sub-optimal design for the active RIS-aided cell-free network. Simulations results verify the proposed active RIS-aided cell-free network as well as the JBRA algorithm as an energy-efficient design for cell-free network with large network capacity.
A. RELATED WORKS RIS has been regarded as a powerful tool in wireless communications. There are extensive works studying the RIS-aided network and RIS beamforming designs.
In [12], the rate splitting multiple access is considered in the uplink network. The constraints of minimum quality-ofservice (QoS) is applied, and a sum-rate maximization problem is formulated and solved. In [13], the cognitive satellite terrestrial network and a secure transmission maximization problem is considered. A interference threshold is considered which introduces larger non-convexity of the problem. The authors first approximate the objective function to a feasible form, and then use AO to solve different optimizations with respect to the resources. In [14], the cooperation of active RIS and passive RIS is studied. The constraints of active and passive RIS beamforming are applied to solve the rate maximization problem. The practical parameters are considered to approximate the problem into a convex form. In [15], the self-sustainability of RIS is discussed, where the wireless energy harvesting capability is taken into account. One novelty of this work is the constraint on RIS that the total consumed power should not exceed the total harvested power. The framework of solving this optimization is also based on variables decoupling, where all the resource allocation parameters are separately solved. In [16], the case of discrete RIS phase shift optimizations is solved in the MIMO system. Instead of modeling the problem as a continuous optimization problem, this work formulate the beamfoming optimization within a finite solution space. A low complexity approach is proposed for efficiently finding the optimal solution in the discrete feasible set. There are also recent works focusing on multi-RIS design, where the method of decoupling the beamforming optimization into several subproblems and solving them one by one is what most works preferred [17], [18], [19].
To our best knowledge, the EEF problem in the active RIS-aided cell-free network has not been delivered before. In this paper, we develop the beamforming optimization of this problem, and apply the mainstream AO method to solve this problem. It is possible the our work can be extended to more general scenarios or the complexity of the optimization can be reduced by referring to previous works [12], [13], [14], [15], [16], [17], [18], [19].
Organization: The remainder of this paper is organized as follows. In Section II, the network architecture and system model of active RIS-aided cell-free network is introduced, and the EEF maximization problem is formulated. In Section III, to solve the formulated problem, the proposed JBRA algorithm is developed. In Section IV, simulation results are shown to demonstrate the proposed active RIS-aided cell-free network as well as the JBRA algorithm as an energy-efficient design. Finally, in Section V, conclusions and discussions are presented to summarize this paper.
Notation: C and R + denote the set of complex and positive real numbers, respectively; {L} represents the set of integers {1, · · · , L}; [·] * , [·] T and [·] H denote the conjugate, transpose and conjugate transpose of a matrix, respectively; ∥·∥ denotes the Frobenius norm of a matrix; diag(·) sends a vector to a corresponding diagonal matrix; [·] + is the operation max{0, ·}; ℜ{·} denotes the real part of complex numbers; CN (µ, ) denotes the multivariate complex Gaussian distribution with the mean µ and the variance ; I represents the identity matrix.

II. SYSTEM MODEL
In this section, we propose the network architecture of active RIS-aided cell-free network at first. Then, we develop the signal model to analyze the performance of the network. Based on this signal model, the EEF maximization problem is formulated.

A. ACTIVE RIS-AIDED CELL-FREE NETWORK
The concept of active RIS-aided cell-free network is proposed to increase the network capacity by overcoming the shortages of the inter-cell interferences of cell-centric network and the MFE of RIS. To elaborate, in the active RIS-aided cell-free network, multiple base stations (BSs) work cooperatively with multiple active RISs to serve multiple users in the network. Usually, to intelligently control the network, central processing units (CPUs) are employed, to which the BSs and the RISs are connected. Moreover, multiple orthogonal carriers are available for transmissions, which generally satisfies the rule of orthogonal frequency division multiple access (OFDMA). Without generality, in this paper we consider the uplink transmission. It can be seen that the design of this active RIS-aided cell-free network includes the design of the beamforming at the users and the RISs, the combining at the BSs, the transmit power allocation at the users, and the subcarrier allocation of the resources, which should be a joint design of both the beamforming and the resource allocation.

B. SIGNAL MODEL
As shown in Fig. 1, let us consider an active RIS-aided cell-free network consists of P subcarriers, where B BSs communicate with K users with the aid of Q active RISs. Assume that each BS is equipped with M antennas, each user is equipped with a single antenna, and each active RIS has N elements. Specifically, an active RIS has a beamforming matrix ∈ C N ×N that can be written as and a ∈ C N ×1 denotes the phase shifted and the power amplified on each RIS element, respectively.
Let us denote the symbol transmitted by user k on subcarrier p as s k,p , which are assumed to be independent normalized variables. Then, the signal received by BS b, which is transmitted by user k on subcarrier p, can be represented as where h b,k,p ∈ C M ×1 , G b,q,p ∈ C N ×M , and f q,k,p ∈ C N ×1 denote the channel spanning from user k to user b, from user k to active RIS q, and from active RIS q to BS b. Additionally, W k,p denotes the allocated transmit power for user k on subcarrier p, and n b,p ∼ CN 0, σ 2 n I and z q,p ∼ CN 0, σ 2 z I denotes the additive white Gaussian noise (AWGN) at the receiver BS b and the dynamic noise introduced by the active RIS q, respectively. Let us also denote as the equivalent channel for the transmission from user k to BS b on subcarrier p. At the receiver side, the BS would perform the combining operation, represented by a vector v b,p ∈ C M ×1 to combine signals received by multiple antennas. Therefore, the combined signal received by BS b on subcarrier p is Here, we also denote v = [v b,p ] and ψ = [ψ q ] for notation simplicity. To focus on the beamforming and resource allocation, in this paper, we take the general assumption that the channel state information (CSI) is accurately acquired beforehand, i.e., channel matrices h b,k,p , G b,q,p , and f q,k,p are known, which is usually achieved by channel estimation. Based on the above signal model, we obtain: • the sum rate R k for user k, represented by where ρ k,p ∈ {0, 1} indicates if subcarrier p is available to user k, and SNR k,p is the receiving signal to noise ratio (SNR) given by • the overall power consumption W ′ k for user k, represented by where W k is the transmit power of user k, and ξ k and W k,0 denote the energy conversion efficiency and the static power, respectively. Thus, we can model the energy efficiency η k of user k as which characterize how much information is transmitted with unit energy. Note that, this energy efficiency is a function with respect to the subcarrier assignment ρ, the power allocation W , the combining v, and the active RIS beamforming ψ.
In this paper, we aim to maximize the worst user's energy efficiency in the network, which is the so-called EEF maximization problem.

C. PROBLEM FORMULATION
According to the signal model, the EEF maximization problem can be formulated as To elaborate, we aim at maximizing the worst user's energy efficiency, i.e., min k η k = R k /W ′ k . The user's energy efficiency is a very important metric that characterize the performance of a network and has always been a topic in communications. However, energy efficiency is not only a problem about the entire system but also a problem about individual users. It can happen that the network energy efficiency improves at the cost of a severe drop for the energy efficiency of individual links. This can lead to the unfairness among links in terms of energy efficiency. So in this work, the max-min problem of the energy efficiency is set as the objective. We want to maximize the work user's energy efficiency to achieve the fairness. To formulate this problem practically, we consider the following constraints: C 1 requires each user's transmit rate to be at least R min k ; C 2 requires each user's overall power consumption to be at least W max k ; C 3 and C 4 are the constraints for active RISs' beamforming, which requires the phase-shift to be normalized and the power amplification to be positive, respectively; C 5 and C 6 are the constraints of OFDMA, which indicates that every subcarrier can be assigned to at most one user.
Note that the formulated problem (9) is a highly coupled and non-convex problem, which makes finding the optimal solution to be challenging. It is also worth noting that the subcarrier assignment ρ is a boolean variable, which makes the optimization a NP-hard mixed-integer quadratic programming (MIQP) problem [26]. In the following section, we begin solving this problem by proposing a joint system design.

III. JOINT BEAMFORMING AND RESOURCE ALLOCATION (JBRA) ALGORITHM
To solve the non-convex problem (9), in this section, we propose a JBRA algorithm by exploiting AO and FP.
power allocation W , the combining v, and the active RIS beamforming ψ with other variables fixed. Once the objective function (9a) converges, we obtain the optimal solution for the above variables. Note that the AO guarantees the monotonous increase of the objective function, it can only ensure the local optimality.

B. TRANSFORMATION
To tackle the highly coupled objective function (9a) in the fractional form, we first perform a transformation step that sends it to a subtractive form [20]. Specifically, the optimal energy efficiency η opt of problem (9) satisfies This implies that we can iteratively finds the optimal energy efficiency η opt by solving max ρ,W ,v,ψ min k R k − η opt W ′ k s.t. C 1 , C 2 , C 3 , C 4 , C 5 , C 6 .
Moreover, we relax ρ k,p ∈ {0, 1} to ρ k,p ∈ [0, 1] to circumvent the mixed-integer problem. We also introduce a new variable ϕ to transform this max-min problem to an easier maximization problem, which is formulated as max ρ,W ,v,ψ,ϕ ϕ s.t. C 1 , C 2 , C 3 , C 4 , C 6 , For ease of notation, let
The Lagrangian multipliers can be updated by conventional methods like subgradient projection, while the update of the subcarrier assignment ρ, the power allocation W , the combining v, the active RIS beamforming ψ, and the introduced variable ϕ is discussed in the following subsection.

C. OPTIMAL BEAMFORMING AND RESOURCE ALLOCATION
As mentioned in Subsection III-A, each variable is optimized with the other variables fixed. The detailed solution to the optimization is provided as follows.

1) OPTIMAL SUBCARRIER ASSIGNMENT
Let us denote s k,p = ρ k,p W k,p , and we only extract the terms with respect to the subcarrier assignment ρ k,p and power allocation W k,p , i.e., max ρ,W where We exploit Karush-Kuhn-Tucker (KKT) conditions, which specifies that Then, substituting (18)

2) OPTIMAL POWER ALLOCATION
Similarly, by using the KKT conditions with respect to ρ k,p , we derive from (18) that wherẽ In practice, we notice constraints and (9g) that requires, for given k, there exists at most one ρ k,p = 1 among all p ∈ {P}. Therefore, we modify our result as

3) OPTIMAL COMBINING
We also extract the terms with respect to the combining v and the active RIS beamforming ψ first. The subproblem is formulated as where L 2 (w, θ) The logarithm and fractional part make (25) very challenging to tackle. Inspired by the Lagrangian dual transform [20], we introduce the auxiliary variables µ, ν ∈ C K ×P to address this non-convexity. Specifically, we reformulate problem (24) as max v,ψ,µ,ν where By setting ∂L 3 /∂µ k,p and ∂L 3 /∂ν k,p to 0, we obtain the optimal values for these auxiliary variables as With the derived optimal auxiliary variables, we can calculate the optimal combining v p by setting

4) OPTIMAL ACTIVE RIS BEAMFORMING
For active RIS beamforming, let us first define where c k,p and d k,p are known variables defined for notation simplicity. Substituting (31) into (26), we obtain the subproblem with respect to active RIS beamforming as where L 4 (ψ)  VOLUME 11, 2023 which can simplify (34) as a quadratic form where Observe that problem (33) is a quadratic constraint quadratic programming (QCQP) problem in convex optimization. Thus, we can obtain the optimal solution ψ opt by existing methods like alternating direction method of multipliers (ADMM).

5) OPTIMAL INTRODUCED VARIABLE
Finally, for the selection of optimal introduced variable ϕ, the problem is simplified as Therefore, we can readily conclude that Note that there exists various algorithms for solving optimizations in RIS-aided system. To tackle the nonconvexity introduced by RIS, for example, [21] use a penalty term to force the function close to the feasible region of the constrained problem. In [22], the authors apply the majorization-minimization (MM) algorithm, where an upperbound function is calculated and an optimal solution is given in the feasible region iteratively. Another representative solution is [23], where the feasible set of RIS beamforming is first relaxed to a convex hull, and the obtained solution is then projected to the unit circle. Since this paper mainly focus on the construction of RIS-aided cell-free network as well as the EEF maximization, we just use a straightforward approach to solve the RIS optimizations. More advanced approached can be applied for future works.

D. ALGORITHM ANALYSIS 1) CONVERGENCE AND OPTIMALITY
For the proposed JBRA algorithm, we adopt the assumption that the phase-shifts of the RIS elements can be continuously controlled. Under this assumption, if the variables are alternately updated based on (20), (23), (28a), (28b), (30), (33), and (39) in Algorithm 1, which is the optimal solution of the corresponding subproblem with all the other variables fixed, the objective function can increase monotonically, which provides the strict convergence of the JBRA algorithm. However, since the JBRA algorithm is based on AO, the global optimality cannot be guaranteed, while the local optimality can be guaranteed.

2) COMPLEXITY
The computational complexity of Algorithm 1 mainly comes from the computation of the parameter updates (20), (23), (28a), (28b), (30), (33), and (39). We have summarized these complexities in Table 1, where I ψ refers to the number of iterations to solve (33) [4]. We can readily find that the computation complexities for W , ρ, µ, ν, and ϕ are the same, which are much smaller than that of v and ψ. Therefore, by assuming that the number of RIS elements and the number of BSs are usually large [24], [25], we can approximate the overall complexity as I 0 ( where I 0 refers to the number of iterations for (15) to converge. Generally, this complexity can be further reduced by applying some tricks in the computation, which is left as a good topic for future investigation.

IV. SIMULATION RESULTS
In this section, we use simulation results to further demonstrate the performance of the proposed active RIS-aided cellfree network and the corresponding JBRA algorithm.
As the simulation setup, we consider a scenario that 4 BSs (located at (±10 m, ±10 m)) with 4 RISs (located at (±8 m, ±8 m)) cooperatively serve 4 users (randomly located at a circle centered at (0 m, 0 m) with radius 4 m). We assume the numbers of antennas employed by BS and RIS are 16 and 48, respectively. Assume furthermore that the noise power is  σ 2 n = σ 2 z = 10 −6 W and the number of subcarriers is P = 32. For the channel model, the BS-RIS channel is assumed as a line-of-sight (LoS) channel, while other channels are assumed as the Rayleigh channels. Other related parameters can be found in 3GPP propagation environments [22].
First, we use numerical results in Fig. 2 to show the convergence of the proposed algorithm. In the proposed algorithm, there are two loops of iterative algorithm that need to take convergence into consideration. The outer loop is the iterative update of η by solving problem (11), which is marked as the blue line in Fig. 2. This blue line shows that the update of η would converge within 8 iterations. The inner loop is the AO in Algorithm 1, where the value of objective function is marked as the orange line in Fig. 2. This orange line shows that the optimization converges quickly within 6 iterations.
Intuitively, we use Table. 2 to show why it is important to have the energy efficiency maximization. We record the single user's energy efficiency in Table. 2, and the second and third columns are the energy efficiency that is/isn't optimized using the EEF settings. We can read that although in the ''not optimized EEF'' case, user 4 has a very large energy efficiency, it sacrifices the other users' energy efficiency. While in the ''optimized EEF'' case, the energy efficiency tends to allocate evenly, with all the users sharing a large energy efficiency.
Moreover, we investigate the performance of the worst user's energy efficiency versus the number of BS antennas in Fig. 3. One can observe that, under the same number of BS antennas, the network without RIS performs similar  to the RIS-aided cell-free network with random beamforming of RIS, while the worst user's energy efficiency has a 14.7% increment if the RIS has the optimal beamforming. This indicates the significance of RIS beamforming. For comparison, we also show the case of active RIS-aided cellfree network with random or optimal beamforming of active RIS. Observe that there is a 13.8% improvement from the random beamforming case to the optimal beamforming case, which demonstrate the effectiveness of our proposed active RIS-aided cell-free network and the JBRA algorithm. Overall, we can see that our proposed scheme with active RIS and optimal beamforming can increase the worst user's energy efficiency by over 50%. This phenomenon also provides a good demonstration for active RIS from the aspect of capacity VOLUME 11, 2023 We also exhibit a similar result in Fig. 4 which depicts the relationship between the number of active RIS elements and the worst user's energy efficiency. We can make the same conclusion that the beamforming of RIS or active RIS is essential to the network's performance. Fig. 5 discuss the relationship between the total transmit power and the worst user's energy efficiency. In our setting we assume the transmit power of BS and active RIS is equal. We can tell that the performance is dominant by the transmit power when it is small. While, when the transmit power is high (>10 dBW in our setting), the performance is dominant by other resources in the network. Combining the results in Fig. 3 and Fig. 4, possible efficient ways to further improve the performance when it is dominant by the transmit power include increasing the number of BS antennas or the number of active RIS elements. Considering the cost and power of active RIS, it is more cost-and energy-efficient to employ larger number of RIS elements in this case. The results coincide with that in Fig. 3, which provides good illustration to the superiority of our proposed active RIS-aided cell-free network as well as the JBRA algorithm.

V. CONCLUSION
In this paper, we have proposed the concept of active RIS-aided cell-free network to take a further step to increase the capacity of the RIS-aided cell-free network. In this active RIS-aided cell-free network, multiple active RISs assist the transmission from multiple BSs to multiple users. Moreover, we have analyzed the proposed network by investigating its EEF. Specifically, we have formulated the EEF maximization problem in the active RIS-aided cell-free network, and have proposed a JBRA algorithm that alternatingly solves the optimal solution of the subcarrier assignment, the power allocation, the combining, and the active RIS beamforming. Simulations results have shown that our proposed active RIS-aided cell-free network as well as the JBRA algorithm as an energy-efficient design for cell-free network with large network capacity.