Energy efficiency analysis and optimization for reconfigurable intelligent surface aided DF relay cooperation with minimum-rate guarantee

Abstract

We consider an energy-efficient reconfigurable intelligent surface (RIS)-aided decode-and-forward relay cooperation scheme with minimum-rate guarantee. Although the emerging RIS has a similar role as the traditional relay, RIS and relay are essentially different and can complement each other. Firstly, we derive the upper bounds on the energy efficiency (EE) of the considered scheme over Rayleigh fading channels for given transmit powers at the source and relay. Secondly, we investigate the EE optimization problem with minimum-rate guarantee in two scenarios with fixed and upper-bounded total transmit powers. In the fixed power scenario, the phase shifts at two time slots are optimized based on the channel state information (CSI), and then the EE optimization problem is reformulated to an equivalent optimal power allocation problem with minimum-rate guarantee, which can be solved by convex optimization techniques. In the upper-bounded power scenario, the corresponding non-convex EE optimization problem is solved by the proposed method using fractional programming and generalized Dinkelbach’s algorithm. Finally, illustrative simulation results demonstrate the superiorities of the considered scheme as compared with the benchmark schemes and reveal the effects of various factors on its performance. Simulation results also show the good robustness of the considered scheme against imperfect CSI and discrete phase shifts of RIS.

Appendices

Appendix A

Lemma 1

For any ${\tilde{\textbf{P}}} \in S$ and ${\xi _{\tilde{\textbf{P}}}} = \mathop {\min }\limits _{1 \le k \le K} \frac{{{f_k}\left( {\tilde{\textbf{P}}} \right) }}{{{g_k}\left( {\tilde{\textbf{P}}} \right) }}$, we have $F\left( {{\xi _{\tilde{\textbf{P}}}}} \right) \ge 0$, with equality if and only if ${\tilde{\textbf{P}}} = \arg \mathop {\max }\limits _{{\textbf{P}} \in S} \left\{ \mathop {\min }\limits _{1 \le k \le K} \left\{ {{f_k}\left( {\textbf{P}} \right) - {\xi _{\tilde{\textbf{P}}}}{g_k}\left( {\textbf{P}} \right) } \right\} \right\}$.

Proof

For any ${\tilde{\textbf{P}}} \in S$ and ${\xi _{\tilde{\textbf{P}}}} = \mathop {\min }\limits _{1 \le k \le K} \frac{{{f_k}\left( {\tilde{\textbf{P}}} \right) }}{{{g_k}\left( {\tilde{\textbf{P}}} \right) }}$, Lemma 1 can be obtained as follows:

$$\begin{aligned}&F\left( {{\xi _{\tilde{\textbf{P}}}}} \right) = \mathop {\max }\limits _{{\textbf{P}} \in S} \left\{ {\mathop {\min }\limits _{1 \le k \le K} \left\{ {{f_k}\left( {\textbf{P}} \right) - {\xi _{\tilde{\textbf{P}}}}{g_k}\left( {\textbf{P}} \right) } \right\} } \right\} \nonumber \\&= \mathop {\max }\limits _{{\textbf{P}} \in S} \left\{ {\mathop {\min }\limits _{1 \le k \le K} \left\{ {{f_k}\left( {\textbf{P}} \right) \mathrm{{-}} \mathop {\min }\limits _{1 \le k \le K} \frac{{{f_k}\left( {\tilde{\textbf{P}}} \right) }}{{{g_k}\left( {\tilde{\textbf{P}}} \right) }}{g_k}\left( {\textbf{P}} \right) } \right\} } \right\} \nonumber \\&\mathop {\ge } \limits ^a \mathop {\min }\limits _{1 \le k \le K} \left\{ {{f_k}\left( {\tilde{\textbf{P}}} \right) - \mathop {\min }\limits _{1 \le k \le K} \frac{{{f_k}\left( {\tilde{\textbf{P}}} \right) }}{{{g_k}\left( {\tilde{\textbf{P}}} \right) }}{g_k}\left( {\tilde{\textbf{P}}} \right) } \right\} \nonumber \\&= {f_{{k^*}}}\left( {\tilde{\textbf{P}}} \right) - \frac{{{f_{{k^*}}}\left( {\tilde{\textbf{P}}} \right) }}{{{g_{{k^*}}}\left( {\tilde{\textbf{P}}} \right) }}{g_{{k^*}}}\left( {\tilde{\textbf{P}}} \right) = 0. \end{aligned}$$

(A1)

(a: The equation holds if and only if ${\tilde{\textbf{P}}} = \arg \mathop {\max }\limits _{{\textbf{P}} \in S}\bigg \{ \mathop {\min }\limits _{1 \le k \le K} \left\{ {{f_k}\left( {\textbf{P}} \right) - {\xi _{\tilde{\textbf{P}}}}{g_k}\left( {\textbf{P}} \right) } \right\} \bigg \}$.) $\square$

Appendix B

Lemma 2

$F\left( \xi \right)$ is strictly monotonic decreasing and has a unique root.

Proof

Supposing that ${\xi _2} > {\xi _1}$, and ${\textbf{P}}_2^* = \arg \mathop {\max }\limits _{{\textbf{P}} \in S} \left\{ \mathop {\min }\limits _{1 \le k \le K} \left\{ {{f_k}\left( {\textbf{P}} \right) - {\xi _2}{g_k}\left( {\textbf{P}} \right) } \right\} \right\}$, then we can obtain the assertion that $F\left( \xi \right)$ is strictly monotonic decreasing as follows:

$$\begin{aligned} \begin{aligned} F\left( {\xi {}_2} \right)&= \mathop {\min }\limits _{1 \le k \le K} \left\{ {{f_k}\left( {{\textbf{P}}_2^*} \right) - {\xi _2}{g_k}\left( {{\textbf{P}}_2^*} \right) } \right\} \\&< \mathop {\min }\limits _{1 \le k \le K} \left\{ {{f_k}\left( {{\textbf{P}}_2^*} \right) - {\xi _1}{g_k}\left( {{\textbf{P}}_2^*} \right) } \right\} \\&\le \mathop {\max }\limits _{{\textbf{P}} \in S} \left\{ {\mathop {\min }\limits _{1 \le k \le K} \left\{ {{f_k}\left( {\textbf{P}} \right) - {\xi _1}{g_k}\left( {\textbf{P}} \right) } \right\} } \right\} \\&= F\left( {{\xi _1}} \right) . \end{aligned} \end{aligned}$$

(B1)

Besides, it can be seen that $\mathop {\lim }\limits _{\xi \rightarrow - \infty } F\left( \xi \right) = + \infty$, and $\mathop {\lim }\limits _{\xi \rightarrow + \infty } F\left( \xi \right) = - \infty$, thus $F\left( \xi \right)$ has a unique root. $\square$