# Achievable Rate Maximization for Multi-Relay AF Cooperative SWIPT Systems with a Nonlinear EH Model
### Motivation and Contributions  
Except that a few studies (e.g., [14; 20; 21], etc.) considered multiple relays, the aforementioned literature on SWIPT relay systems focused on a single relay model. However, the power harvested at a single relay is typically small and a single relay may not provide satisfactory cooperative quality for the transmission [20]. Moreover, most of the works on SWIPT relay systems used linear EH model in the system design or performance analysis, while bringing the benefit of easy and tractable analysis, the use of conventional linear EH model for system design in the SWIPT systems may lead to the mismatches of resource allocation and thus resulting in the performance degradation, since the linear EH model is usually inaccurate and less practical in modeling the realistic nonlinear EH implementations [25; 26]. Motivated by the above observations, in this paper, we consider the nonlinear EH model for the SWIPT cooperative communication systems with multiple relays. The piece-wise linear model rather than the conventional linear EH model is used to capture the nonlinear input-output behavior for the energy harvester. Both the TS and PS schemes are considered for the receivers of the relay nodes. We maximize the achievable information rate for the SWIPT multi-relay systems by optimizing the PS or TS ratios for the relay nodes when PS or TS schemes are applied to the EH receivers, respectively. The contributions of the paper are summarized as follows. 1. For the SWIPT multi-relay cooperative AF systems with TS and PS schemes, we formulate the achievable rate maximization problems as convex and non-convex optimization problems, respectively. For the PS scheme, we transform the non-convex optimization problem into a convex one by transforming the objective function from maximizing the received signal-to-noise ratio (SNR) into minimizing the reciprocal of the received SNR at the destination. To obtain the optimal PS ratios and the maximum achievable rate for the system with PS receiver, the proximal gradient method is used to solve the convex optimization problem that is non-smooth due to the use of the piece-wise linear EH model. 2. By comparing the performance of the system with linear and nonlinear EH models, the impact of the resource allocation mismatch caused by using the linear EH model in modeling the realistic nonlinear EH circuit is investigated. We also analyze the asymptotic maximum achievable rates at low and high input SNRs for both the TS and PS schemes. It is demonstrated that for the TS scheme, the use of the traditional linear EH model leads to significant resource allocation mismatch and performance degradation, which becomes more serious as the input SNR increases, while for the PS scheme with the optimal PS ratios, the nonlinear EH model obtains only inconspicuous performance gain over the traditional linear EH model, especially in low and high input SNRs region, implying that the use of linear EH model in modeling the realistic nonlinear EH circuit does not bring significant resource allocation mismatch for the PS scheme. 3. We analyze the effect of the relays' position and the channel parameters on the system performance. It is shown that farther locations of the relay nodes to the source node bring better achievable rate performance for both of the proposed PS and TS schemes. It is found that for given channel parameters, the performance gap between the linear and nonlinear EH models for the TS scheme is larger than that for the PS scheme, implying that the TS scheme is more susceptible to the resource mismatch than the PS scheme. Meanwhile, the PS scheme is more susceptible to the variation of the relays' location and the channel parameters than the TS scheme. The remainder of this paper is organized as follows. In Section 2, the system model is described and the end-to-end SNRs are derived for both the TS and PS receiver. Section 3 provides the optimal problem formulation and solution and Section 4 presents the numerical results and discussion.