Resource allocation for full-duplex MIMO relaying system with self-energy recycling

Abstract

Self-energy recycling cooperative communication, in which the relay nodes simultaneously harvest energy from the source and their own transmitted signal, has been demonstrated to significantly increase the total harvested energy. In this paper, we present an optimization framework for the resource allocation and relay selection with the objective of maximizing the sum-throughput in afull-duplex multi-relay multi-user multiple-input-multiple-output network by incorporating self-energy recycling at the relays, for the first time. The formulated problem is a mixed integer non-linear programming problem, which is difficult to solve for the global optimal solution in polynomial-time. As a solution strategy, we decompose the optimization problem into two sub-problems: resource allocation problem and relay selection problem. First, we solve the resource allocation problem for a predetermined relay selection by using Taylor series approximation and convex optimization techniques. For the relay selection problem, we propose a polynomial-time sub-optimal algorithm based on the idea of iteratively selecting the relay offering the maximum throughput at each time. Through simulations, we demonstrate that the proposed dynamic time FD-MIMO system increases the throughput by 30% as compared to the conventional static time protocol system, and by a factor of 2.4 over the system without self-energy recycling.

Appendix

Proof

Taking the derivative of Lagrangian function L w.r.t $\tau _1^{m_k,k}$ and $\tau _2^{m_k,k}$ yields

$$\begin{aligned}{} & {} \frac{\partial L}{\partial \tau _1^{m_k,k}} = \frac{1-\sigma _k}{T} \bigg ( \sum _{n=1}^{N} \mathrm{log_2}\bigg (1+{p_{s,n}^{m_k,k} \gamma _{s,n}^{m_k,k}}\bigg )\bigg ) - \nu _k = 0, \end{aligned}$$

(27)

$$\begin{aligned}{} & {} \frac{\partial L}{\partial \tau _2^{m_k,k}} = \frac{1+\sigma _k}{T} \bigg ( \sum _{n=1}^{N} \mathrm{log_2}\bigg (1+{p_{r,n}^{m_k,k} \gamma _{r,n}^{m_k,k}}\bigg ) \bigg ) - \nu _k = 0, \end{aligned}$$

(28)

We know that the optimization problem is maximized when

$$\begin{aligned} \begin{aligned}&\frac{\tau _1^{m_k,k}}{T} \sum _{n=1}^{N} \mathrm{log_2}\bigg (1+{p_{s,n}^{m_k,k} \gamma _{s,n}^{m_k,k}}\bigg ) \\&=\frac{\tau _2^{m_k,k}}{T} \sum _{n=1}^{N} \mathrm{log_2}\bigg (1+{p_{r,n}^{m_k,k} \gamma _{r,n}^{m_k,k}}\bigg ), \end{aligned} \end{aligned}$$

(29)

Therefore, the following equality holds

$$\begin{aligned} \frac{\tau _1^{m_k,k}}{1-\sigma _k} = \frac{\tau _2^{m_k,k}}{1+\sigma _k}, \end{aligned}$$

(30)

Substituting constraint (17d) into Eq. (30) gives the optimal solution of $\tau _1^{m_k,k}$ and $\tau _2^{m_k,k}$, as presented in Theorem 1.

Similarly, taking the derivative of L w.r.t $p_{s,n}^{m_k,k}$ and $p_{r,n}^{m_k,k}$ is given as

$$\begin{aligned}{} & {} \frac{\partial L}{\partial p_{s,n}^{m_k,k}} = \frac{(1-\sigma _k)\tau _1^{m_k,k}\gamma _{s,n}^{m_k,k} }{T ln(2)\bigg (\gamma _{s,n}^{m_k,k} p_{s,n}^{m_k,k}+1\bigg )} \nonumber \\{} & {} - \mu _k + \lambda _{k}\bigg (\triangledown _{p_{s,n}^{m_k,k}} P_{m_k}\bigg (\varvec{p_s^{(j)}}, \varvec{p_m^{(j)}}\bigg )\bigg ) = 0, \end{aligned}$$

(31)

$$\begin{aligned}{} & {} \frac{\partial L}{\partial p_{r,n}^{m_k,k}} = \frac{(1+\sigma _k)\tau _2^{m_k,k}\gamma _{r,n}^{m_k,k}}{T ln(2)\bigg (\gamma _{r,n}^{m_k,k} p_{r,n}^{m_k,k}+1\bigg )} \nonumber \\{} & {} - \lambda _{k} + \lambda _{k}\bigg (\triangledown _{p_{r,n}^{m_k,k}} P_{m_k}\bigg (\varvec{p_s^{(j)}}, \varvec{p_m^{(j)}}\bigg )\bigg )= 0, \end{aligned}$$

(32)

Putting the values of $\tau _1^{m_k,k}$ and $\tau _2^{m_k,k}$ from Eqs. (19) and (20) give

$$\begin{aligned}{} & {} \frac{\partial L}{\partial p_{s,n}^{m_k,k}} = \frac{(1-\sigma _k)(1-\sigma _k)T\gamma _{s,n}^{m_k,k} }{2 T ln(2)\bigg (\gamma _{s,n}^{m_k,k} p_{s,n}^{m_k,k}+1\bigg )} \nonumber \\{} & {} - \mu _k + \lambda _{k}\bigg (\triangledown _{p_{s,n}^{m_k,k}}P_{m_k}\bigg (\varvec{p_s^{(j)}}, \varvec{p_m^{(j)}}\bigg )\bigg )] = 0, \end{aligned}$$

(33)

$$\begin{aligned}{} & {} \frac{\partial L}{\partial p_{r,n}^{m_k,k}} = \frac{(1+\sigma _k)(1+\sigma _k)T\gamma _{r,n}^{m_k,k}}{2 T ln(2)(\gamma _{r,n}^{m_k,k} p_{r,n}^{m_k,k}+1)} \nonumber \\{} & {} - \lambda _{k} + \lambda _{k}\bigg (\triangledown _{p_{r,n}^{m_k,k}} P_{m_k}\bigg (\varvec{p_s^{(j)}}, \varvec{p_m^{(j)}}\bigg )\bigg )= 0, \end{aligned}$$

(34)

After some simple algebraic manipulations which are skipped for brevity, Eqs. (33) and (34) can be written as in Theorem 1. $\square$