User cooperation for enhanced throughput fairness in wireless powered communication networks

Abstract

This paper studies a novel user cooperation method in a wireless powered cooperative communication network (WPCN) in which a pair of distributed terminal users first harvest wireless energy broadcasted by one energy node and then use the harvested energy to transmit information to a destination node (DN). In particular, the two cooperating users exchange their independent information with each other so as to form a virtual antenna array and transmit jointly to the DN. By allowing the users to share their harvested energy to transmit each other’s information, the proposed method can effectively mitigate the inherent user unfairness problem in WPCN, where one user may suffer from very low data rate due to poor energy harvesting performance and high data transmission consumptions. Depending on the availability of channel state information at the transmitters, we consider the two users cooperating using either coherent or non-coherent data transmissions. In both cases, we derive the maximum common throughput achieved by the cooperation schemes through optimizing the time allocation on wireless energy transfer, user message exchange, and joint information transmissions in a fixed-length time slot. We also perform numerical analysis to study the impact of channel conditions on the system performance. By comparing with some existing benchmark schemes, our results demonstrate the effectiveness of the proposed user cooperation in a WPCN under different application scenarios.

Appendices

Appendix 1

Proof of Lemma 4.1

The transmit power of user X is $P_X=E_X/(t_2+t_4)$, we have from (4) that

$$\begin{aligned} R_X^{(2)}&= t_2\log _{2}\left( 1+\frac{E_Xh_{XY}}{(t_2+t_4)N_0}\right)\,\triangleq\,t_2\log _{2}\left( 1+ \frac{c_1}{t_2+c_2}\right) , \end{aligned}$$

(42)

where $c_1\,\triangleq\, E_Xh_{XY}/N_0$, $c_2\,\triangleq\,t_4$ are both constant. By taking the first and second order derivatives of $R_X^{(2)}$ in $t_2$, we have

$$\begin{aligned} \frac{dR_X^{(2)}}{\mathrm{d}t_2}&=\log _{2}\left( 1+ \frac{c_1}{t_2+c_2}\right) - \frac{c_1t_2}{\ln 2(t_2+c_3)(t_2+c_2)}, \end{aligned}$$

(43)

$$\begin{aligned} \frac{d^2R_X^{(2)}}{\mathrm{d}t_2^2}&= - \frac{c_1}{\ln 2} \frac{(c_2+c_3)t_2+2c_2c_3}{(t_2+c_3)(t_2+c_2)}, \end{aligned}$$

(44)

where $c_3\triangleq c_1+c_2$. Because $\frac{d^2R_X^{(2)}}{\mathrm{d}t_2^2}<0$ and $\lim \limits _{t_2 \rightarrow +\infty } \frac{\mathrm{d}R_X^{(2)}}{\mathrm{d}t_2} = 0$, we can infer that $\frac{\mathrm{d}R_X^{(2)}}{\mathrm{d}t_2} > 0$ when $t_2>0$, which leads to the proof of Lemma 4.1 that $R_X^{(2)}$ increases in $t_2 \in \left[ 0,T_0\right]$. Similarly, we have $R_Y^{(3)}$ deceases with $t_2 \in \left[ 0,T_0\right]$. $\square$

Appendix 2

Proof of Lemma 4.2

First of all, we show that both $t_2$ and $t_3$ decrease as $t_4$ increases. Otherwise, we assume without loss of generality that $t_2$ increases and $t_3$ decreases when $t_4$ become larger. We denote the updated values of $t_2$ and $t_3$ after $t_4$ becomes $\bar{t}_4=t_4+{\Delta }t_4$ as $\bar{t}_2=t_2+{\Delta } t_2$ and $\bar{t}_3=t_3-{\Delta } t_3$, respectively, where ${\Delta } t_2,{\Delta } t_3,{\Delta }t_4>0$, and ${\Delta } t_2+ {\Delta }t_4 -{\Delta } t_3 =0$. Besides, we denote the updated values of $R_X^{(2)}$ and $R_Y^{(3)}$ as $\bar{R}_X^{{(2)}}$ and $\bar{R}_Y^{{(3)}}$, respectively. It can be easily shown from Lemma 4.1 that $\bar{R}_X^{{(2)}}>\bar{R}_Y^{{(3)}}$ given $R_X^{(2)} = R_Y^{(3)}$. However, this contradicts with the necessary condition of an optimal solution that requires $\bar{R}_X^{{(2)}}=\bar{R}_Y^{{(3)}}$. Therefore, we reject our assumption and conclude that both $t_2$ and $t_3$ decrease as $t_4$ increases. Because $t_2+ t_3 +t_4 =T_1$, we can infer that $t_2+t_4 = T_1 -t_3$ increases with $t_4$, so does $t_3+t_4$. This, together with the result that $t_2$ (and $t_3$) decrease with $t_4$, leads to the proof that $R_X^{(2)}$ in (23) (and $R_Y^{(3)}$ in (24)) is a decreasing function with $t_4$.

Next, we show that $R_X^{(4)}$ in (25) increases with $t_4$. To see this, we let $\bar{R}_X^{(4)}$ denote the updated value of $R_X^{(4)}$ after $t_4$ increases to $\bar{t}_4 = t_4 + \Delta t_4$. First, we can infer from $\Delta t_4 = \Delta t_2 + \Delta t_3$ and $\Delta t_2,\Delta t_3>0$ that $0< \Delta t_3 \le \Delta t_4$ and $0<\Delta t_2\le \Delta t_4$ hold. Then, we have

$$\begin{aligned} \bar{R}_X^{(4)}&= \frac{t_4+\Delta t_4}{2}\log _{2}\left( 1+ \rho _3 \frac{t_1}{1-t_1 - t_3 + \Delta t_3 }+\,\rho _4 \frac{t_1}{1-t_1 - t_3 + \Delta t_2}\right) \\&\ge \frac{t_4+\Delta t_4}{2}\log _{2}\left( 1+ \rho _3 \frac{t_1}{1-t_1 - t_3 + \Delta t_4 }+ \rho _4 \frac{t_1}{1-t_1 - t_3 + \Delta t_4}\right) \\ &\ge \frac{t_4}{2}\log _{2}\left( 1+ \rho _3 \frac{t_1}{1-t_1 - t_3 }+ \rho _4 \frac{t_1}{1-t_1 - t_3 }\right) = R_X^{(4)}, \end{aligned}$$

(45)

where the first inequality holds because $0< \Delta t_3 \le \Delta t_4$ and $0<\Delta t_2\le \Delta t_4$, and the second inequality holds because $R_X^{(4)}$ increases monotonically with $t_4$ when $t_2$ and $t_3$ are fixed. This leads to the proof that $R_X^{(4)}$ increases with $t_4$. $\square$