I Introduction
Energy efficiency has emerged as a key performance indicator in the design of wireless communication systems, due to operational, economic, and environmental concerns [1]. Device-to-device (D2D) communication has been proposed as a promising technology to increase energy efficiency (EE), which allows new peer-to-peer communication service by leveraging the proximity and reuse gains [2].
Reconfigurable intelligent surface (RIS) is designed to improve the propagation environment and enhance wireless communications, and has attracted much attention recently [3, 4, 5]. Specifically, RIS is a meta-surface with a large number of passive reflecting elements. Each element can induce a phase shift to the incident signal independently. All elements cooperatively perform the reflect beamforming. Unlike the conventional amplify-and-forward (AF) relay, the RIS forwards the incident signals using passive reflection beamforming, which reduces the energy consumption of the system. These characteristics make the RIS technology attractive from an energy efficiency standpoint [5].
In this letter, we propose a RIS-assisted D2D communication network with a full-frequency reuse to support D2D users, which is assisted by some RISs. The EE maximization problem is formulated to optimize the RIS phase shifts and the transmit powers under phase-shift, maximum power and minimum transmission rate constraints. We divide the non-convex problem into two subproblems, which are passive beamforming and power control. The two subproblems can be optimized alternately. We first rewrite the passive beamforming at RIS based on the Lagrangian dual transform [6], and solve this subproblem based on the fractional programming [6]. Then, we transform the power control subproblem in a fractional form into an equivalent optimization problem in a subtractive form, which is solvable by the Dinkelbach method [7]. The two subproblems are solved iteratively until convergence, yielding a suboptimal solution for the joint optimization problem. Numerical results have verified the effectiveness of the proposed algorithm, which can significantly improve the energy efficiency of the D2D network.
Ii System Model
In this letter, we consider the RIS-aided D2D communication network with full-frequency reuse to support D2D users. Without loss of generality, we assume that there are D2D links in the network, assisted by geographically separated RISs, where the -th RIS, denoted by , has
elements. We assume that the power of the signals reflected by the RIS for two or more times can be ignored due to significant path loss. Besides, we assume that all the channels experience quasi-static flat-fading, and the channel state information (CSI) can be estimated by using the existing channel estimation methods, such as in
[8].Denote by the channel coefficient from transmitter to receiver
. The channel coefficient vectors from transmitter
to and from receiver to are denoted by and , respectively. Let be the total number of reflecting elements. The channel coefficient vectors from transmitter to RISs and from receiever to RISs are denoted by and , respectively. A simple scenario with one RIS and two D2D pairs is shown in Fig. 1.The phase-shift matrix of the RISs is denoted by a diagonal matrix , where , and .
Denote by the transmit data symbol to user . The received signal of receiver is given by
(1) |
where is the additive white Gaussian noise (AWGN) at the -th receiver. Denote by as the transmit power of D2D link . The signal-to-interference-plus-noise ratio (SINR) of link is expressed as
(2) |
Then, the achievable sum rate of the system is given by
(3) |
The total power consumption contains D2D users’ transmit power , circuit power and the RISs’ power. The RISs’ power consumption depends on the type and the resolution of its individual elements [5]. Elements with -bit resolution can perform -bit phase shifting on the impinging signal. The power consumption of the RISs with reflecting elements can be written as
(4) |
where denotes the power consumption of each element having -bit resolution. Thus, the toal power consumption can be expressed as
(5) |
We define the ratio between the achievable sum rate and the total power consumption as the energy efficiency (EE). The energy efficiency is maximized by jointly optimizing the transmit power and the phase-shift matirx , subject to RIS’s phase-shift, maximum power and minimum transmission rate constraints. Thus, the problem can be formulated as
(6a) | ||||
s.t. | (6b) | |||
(6c) | ||||
(6d) |
It is difficult to obtain the optimal solution to this problem due to its nonconvex objective function and constant-modulus constraints. In the next section, an efficient algorithm is proposed to obtain a suboptimal solution to this problem.
Iii Energy Efficiency Maximization
In this section, we adress the energy efficiency maximization problem of the RIS-aided D2D system. We divide the problem (6) into two subproblems, i.e., the passive beamforming and the power control, respectively. Then, the power control and passive beamforming can be optimized alternately.
Iii-a Optimizing for Given
For a fixed , the problem (6) reduces to
s.t. | (7) |
We apply the the Lagrangian dual transform [9] to (III-A), with the new objective function given by
(8) | ||||
where refers to a set of auxiliary variables. We propose to optimize and alternately. For a fixed , the optimal is
(9) |
Then, for a fixed , optimizing is reduced to
s.t. | (10) |
The problem (III-A) can be solved by using the FP method. Define and . Using the quadratic transform proposed in [6], is reformulated as
(11) | ||||
We optimize and alternately. The optimal can be obtained by letting , yielding
(12) |
Then the remaining problem is to optimize for a given . By simplifying (11), the optimization problem for is represented as
s.t. | (13) |
where
(14) |
(15) |
and is a constant. Since for all and are positive-definite matrices, is a positive-definite matrix and is a quadratic concave functions of . Problem (III-A) is non-convex and inhomogeneous due to the non-convexity of the quadratic constraints and the unit modulus constraints. Letting and , and dropping the rank-one constraint of , problem (III-A) can be rewritten as
(16a) | ||||
s.t. | ||||
(16b) | ||||
(16c) | ||||
(16d) |
where,
, .
Iii-B Optimizing for Given
For a fixed , the problem (6) reduces to
s.t. | (17) |
Problem (III-B) is a non-concave FP problem. From [7], we define a new optimization problem as
s.t. | (18) |
where is a non-negative parameter, and
(19) |
is continuous and strictly monotonically decreasing in and has a unique root . The optimal solution of problem (III-B) is the same as that of problem (III-B) with , where can be obtained by using the Dinkelbach method [7].
The key of the method is to solve problem (III-B) for a given . To be specific, (19) can be rewritten as
(20) |
where
(21) | ||||
(22) |
Clearly, and are concave. Hence, the objective function of (III-B) is the difference of two concave functions. Constraints in (6b) are also the difference of two concave functions, which are
(23) |
where
(24) |
(25) |
Iv Numerical Result
In this section, numerical results are provided to demonstrate the validity of the proposed algorithm. We consider a RIS-aided D2D network with multiple single-antenna D2D users, which are assumed randomly and uniformly placed in the rectangular. And there are four RISs in the rectangular. The distance between D2D users is 20-40m. The baseband channels of the RIS-user link, user-RIS link, and user-user link are modeled as Rician fading channels with Rician factor . The background noise at the receivers is dBm. We set the path-loss constant , the path-loss exponent . Circuit power at each user dBm. RIS reflection efficiency . All presented results are obtained by averaging over 1000 independent channel realizations.
We compare the performance of the proposed algorithm with two baselines. Baseline 1 is the energy efficiency optimized by power control without the aid of RIS. Baseline 2 is the energy efficiency optimized by optimal power control with the aid of RISs. We adopt the Branch and Bound method to obtain the optimal power control [12]. Fig. 2 illustrates the energy efficiency of different power parameters with respect to the size of of RIS. As we can see, significant performance gains are achieved by joint power control and RISs’ passive beamforming optimization. Typical power consumption values of each RIS element are 1.5, 4.5, 6, and 7.8 mW for 3-, 4-, 5-, and 6-bit resolution [13]. As the number of quantized bits increases, the energy efficiency decreases gradually. Although high-bit RIS will improve the spectral efficiency of the network, its high power consumption leads to the decrease of energy efficiency.
Particularly, EE performance increases as increases. However, for a large , EE starts decreasing. This is because for a large , RISs’ improvement in spectrum efficiency is no longer enough to compensate for the decrease in system energy efficiency caused by high power consumption. So, there exists an optimal number of RIS elements. Besides, the energy efficiency tends to decrease firstly and then increase. This is because when is too small, there will be little improvement in spectral efficiency with the assistance of the RISs. However, the power consumption of the RISs results in the decrease of energy efficiency.
The effect of the different values of and EE performances versus in dBm is depicted in Fig. 3. For the cases where the design problems turned out to be infeasible, EE is set to zero, corresponding to a communication failure. Fig. 3(a) compares the EE with the different minimum rate . When , EE increases to a fixed value as the power limit increases. For the higher , EE will increase to a lower fixed value. Fig. 3(b) compares the failure rate with the different minimum rate . Similarly, for the higher
, the failure rate will reduce to a higher fixed value. At this moment, increasing
does not improve efficiency or reduce failure rate.V Conclusion
In this letter, the RIS technique was applied to enhance the energy efficiency of D2D communication network. To maximize the energy efficiency, we proposed a joint power control and RISs’ passive beamforming optimization algorithm for obtaining the high-quality suboptimal solution. Simulation results demonstrated that the assistance of the RISs is beneficial to substantially improve the energy efficiency of the D2D communication network.
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