SCIENCE CHINA Information Sciences, Volume 60 , Issue 10 : 100303(2017) https://doi.org/10.1007/s11432-017-9194-y

## Pilot reuse and power control of D2D underlaying massive MIMO systems for energy efficiency optimization

• AcceptedJul 27, 2017
• PublishedSep 1, 2017
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### Abstract

It is predicted that there will be billions of machine type communication (MTC) devices to be deployed in near future. This will certainly cause severe access congestion and system overload which is one of the major challenges for the proper operation of 5G networks. Adopting device-to-device (D2D) communications into massive multiple-input multiple-output (MIMO) systems has been considered as a potential solution to alleviate the overload of MTC devices by offloading the MTC traffic onto D2D links. This work proposes a novel pilot reuse (PR) and power control (PC) for energy efficiency (EE) optimization of the uplink D2D underlaying massive MIMO cellular systems. Although the use of large scale antenna array at the base station (BS) can eliminate most of the D2D-to-Cellular interference, the Cellular-to-D2D interference and the channel estimation error caused by PR will remain significant. Motivated by this, and in order to reduce the channel estimation error, in this paper a novel heuristic PR optimum pilot reuse scheme is proposed for D2D transmitters (D2DTs) selection. By taking into account the interference among users as well as the overall power consumption, the overall system EE is maximized through power optimization while maintaining the quality-of-service (QoS) provisions for both cellular users (CUEs) and D2D pairs. The power optimization problem is modeled as a non-cooperative game and, as such, a distributed iterative power control algorithm which optimizes users' power sequentially is proposed. Various performance evaluation results obtained by means of computer simulations have shown that the proposed PR scheme and PC algorithm can significantly increase the overall system EE.

### Acknowledgment

This work was supported by National Natural Science Foundation of China (Grant Nos. 61371109, 61671278), National Science Foundation for Excellent Young Scholars of China (Grant No. 61622111), and Shandong Provincial Natural Science Foundation for Young Scholars of China (Grant No. ZR2017QF008).

### Supplement

Appendix

Proof of Theorem 1

From (10), the SINR at the $k$th CUE can be rewritten as $$\mathrm{{SINR}}_k^c=\frac{P_k^c\beta_k^c|\frac{\widehat{\boldsymbol{h}}_k^* \widehat{\boldsymbol{h}}_k}{N}|^2}{\frac{\eta_1^c+\eta_2^c+\eta_3^c}{N^2} +\|\frac{\widehat{\boldsymbol{h}}_k^*}{N}\|^2N_0}. \tag{38}$$ Based on the favorable propagation condition [12] in massive MIMO systems and the distribution of $\widehat{\boldsymbol{h}}_k$ in (5a), when the number of antennas at BS is much larger than the number of users, i.e., $N\gg~K+M$, we have $$\begin{split} \left|\frac{\widehat{\boldsymbol{h}}_k^*\widehat{\boldsymbol{h}}_k}{N}\right|^2\approx \left(\delta_k^c\right)^2. \end{split} \tag{39}$$ Then, from (4), it can be observed that the estimation of channel of the $k$th CUE, i.e., $\widehat{\boldsymbol{h}}_k$, is only affected by the D2D pairs who reuse its pilot, so $\widehat{\boldsymbol{h}}_k$ is independent of $\widehat{\boldsymbol{h}}_i$, $\forall~i\in\mathcal{K}\backslash~k$, and $\mathbf{g}_j$, $\forall~j\in\mathcal{K}\backslash\mathcal{D}_k$. Next, according to (5a) and (11a), we have $$\left|\frac{\widehat{\boldsymbol{h}}_k^*\widehat{\boldsymbol{h}}_i}{N}\right|^2\approx\frac{\delta_k^c\delta_i^c}{N}, \forall i\in\mathcal{K}\backslash k \tag{40}$$ and $$\left|\frac{\widehat{\boldsymbol{h}}_k^*\mathbf{g}_j}{N}\right|^2\approx\frac{\delta_k^c}{N}, \forall j \in\mathcal{K}\backslash\mathcal{D}_k. \tag{41}$$ Similarly, because $\widehat{\boldsymbol{h}}_k$ is independent of $\widetilde{\boldsymbol{h}}_i$, $\forall~i\in\mathcal{K}$, according to (eq5b) and (eq11b), we get $$\frac{\widehat{\boldsymbol{h}}_k^*\mathbb{E}\{\|\widetilde{\boldsymbol{h}}_i\widetilde{\boldsymbol{h}}_i^*\|\}\widehat{\boldsymbol{h}}_k}{N^2}\approx\frac{\delta_k^c\varepsilon_i^c}{N}, \forall i\in\mathcal{K}. \tag{42}$$ Moreover, we can obtain that $\mathbf{g}_j\left(\forall~j\in\mathcal{D}_k\right)$ is not independent of $\widehat{\boldsymbol{h}}_k$ from (4). As a result, we have \begin{align} \left|\frac{\widehat{\boldsymbol{h}}_k^*\mathbf{g}_j}{N}\right|^2 &=\frac{\delta_k^c|\sqrt{P_p^c\beta_k^c}\boldsymbol{h}_k^*\mathbf{g}_j+\sum_{i\in\mathcal{D}_k} \sqrt{P_p^d\beta_i^d} \mathbf{g}_i^*\mathbf{g}_j+(\boldsymbol{W}\boldsymbol{\phi}_k)^*\mathbf{g}_j |^2}{N^2(N_0+P_p^c\beta_k^c +P_p^d\sum_{i\in\mathcal{D}_k}\beta_i^d)} \\ &\approx\frac{\delta_k^c(\frac{1}{N}P_p^c\beta_k^c+\frac{1}{N}\sum_{i\in\mathcal{D}_k\backslash j}P_p^d\beta_i^d+P_p^d\beta_j^d+\frac{1}{N}N_0)}{N_0+P_p^c\beta_k^c +P_p^d\sum_{i\in\mathcal{D}_k}\beta_i^d} \\ &=\frac{\delta_k^c(\frac{1}{N}N_0+\frac{1}{N}P_p^c\beta_k^c+\frac{1}{N}\sum_{i\in\mathcal{D}_k}P_p^d\beta_i^d +\frac{N-1}{N}P_p^d\beta_j^d)}{N_0+P_p^c\beta_k^c +P_p^d\sum_{i\in\mathcal{D}_k}\beta_i^d} \\ &\approx \frac{\delta_k^c}{N}+\frac{(\delta_k^c)^2P_p^d\beta_j^d}{P_p^c\beta_k^c}. \tag{43} \end{align} By substituting (39)–(43) into (38), we can obtain $$\begin{split} \rm{{SINR}}\it{_k^c}&=\frac{P_k^c\beta_k^c|\frac{\widehat{\boldsymbol{h}}_k^*\widehat{\boldsymbol{h}}_k}{N}|^2}{\frac{\eta_1^c+\eta_2^c+\eta_3^c}{N^2}+\|\frac{\widehat{\boldsymbol{h}}_k^*}{N}\|^2N_0} \approx\frac{(\delta_k^c)^2P_k^c\beta_k^c}{\delta_k^c(\mu_1^c+\mu_2^c+\mu_3^c)} =\frac{P_k^c\beta_k^c\delta_k^c}{\mu_1^c+\mu_2^c+\mu_3^c} \triangleq\widetilde{\rm{SINR}}_k^c. \end{split} \tag{44}$$ Therefore, the SINR at the $k$th CUE given in (10) can be deterministically approximated by (12).

Proof of Theorem 2

A Nash equilibrium exists if the utility function is continuous and quasi-concave, and the set of strategies is a nonempty compact convex subset of a Euclidean space 1). For the $k$th CUE $\left(\forall~k\in\mathcal{K}\right)$, denote the sublevel set of the objective function given by (22) which, for convenience, is repeated below $$U_k^c(P_k^c, \boldsymbol{P}_{-k}^c)=\frac{R_k^c}{P_k^c+P_{cir}} =\frac{\log_2(1+\frac{\delta_k^cP_k^c\beta_k^c}{\frac{1}{N}P_k^c\beta_k^c\varepsilon_k^c+\chi_k^c})}{P_k^c+P_{cir}} \tag{45}$$ as $S_{\alpha}=\{P_k^c\geq0\;|\;U_k^c(P_k^c,~\boldsymbol{P}_{-k}^c)\geq\alpha\}$. $U_k^c(P_k^c,~\boldsymbol{P}_{-k}^c)$ is quasi-concave if $S_{\alpha}$ is convex for $\alpha\in\mathbf{R}$ 2). If $\alpha\leq~0$, it is obvious that $S_{\alpha}$ is convex. For $\alpha~>~0$, since $U_k^c(P_k^c,~\boldsymbol{P}_{-k}^c)=\frac{R_k^c}{P_k^c+P_{cir}}\geq\alpha$, $S_{\alpha}$ can be rewritten as $S_{\alpha}=\{P_k^c\geq0\;|\;R_k^c-\alpha(P_k^c+P_{cir})\geq~0\}$. Then, let's consider $R_k^c=\log_2(1+\frac{\delta_k^cP_k^c\beta_k^c}{\frac{1}{N}P_k^c\beta_k^c\varepsilon_k^c+\chi_k^c})$, for all $P_k^c\geq~0$, the second-order derivation of which can be expressed as $$\begin{split} \frac{\partial^2R_k^c}{\partial \left(P_k^c\right)^2}=-\frac{\left(\beta_k^c \delta_k^c+\frac{1}{N}\beta_k^c\varepsilon_k^c\right)\left(\frac{2}{N}P_k^c\beta_k^c\varepsilon_k^c +\chi_k^c\right)+\frac{1}{N}\beta_k^c\varepsilon_k^c\chi_k^c}{\left(\delta_k^cP_k^c\beta_k^c+\frac{1}{N}P_k^c\beta_k^c\varepsilon_k^c+\chi_k^c\right)^2} \cdot\frac{\beta_k^c\delta_k^c\chi_k^c\log_2{e}}{\left(\frac{1}{N}P_k^c\beta_k^c\varepsilon_k^c+\chi_k^c\right)^2} < 0. \end{split} \tag{46}$$ So, $R_k^c$ is concave in $P_k^c\geq0$ given the power strategies of other users. Since $P_k^c+P_{cir}$ is an affine function of $P_k^c$, $R_k^c-\alpha(P_k^c+P_{cir})$ is a concave function of $P_k^c(P_k^c\geq0)$. Therefore, $S_{\alpha}$ is also convex when $\alpha>~0$, i.e., $S_{\alpha}$ is convex for $\alpha\in\mathbf{R}$, so that $U_k^c(P_k^c,~\boldsymbol{P}_{-k}^c)$ is continuous and quasi-concave. The power strategy $P_k^c$ is a nonempty, compact, and convex subset of the Euclidean space $\mathbb{R}$. Similarly, for the $m$th D2D pair $(\forall~m\in\mathcal{M})$, $U_m^d(P_m^d,~\boldsymbol{P}_{-m}^d)$ can be proved to be continuous and quasi-concave and the power strategy $P_m^d$ is also a nonempty, compact, and convex subset of the Euclidean space $\mathbb{R}$. Hence, a Nash equilibrium exists in the noncooperaive game.

If the power strategy $P_k^{c*}$ obtained by using Algorithm 2 is not the Nash equilibrium, the $k$th CUE can choose the Nash equilibrium $\widehat{P}_k^c$ $(\widehat{P}_k^c\neq~P_k^{c*})$ to obtain the maximum EE. However, since in Algorithm 2 $P_k^{c*}$ has converged, then we must have $P_k^{c*}=\widehat{P}_k^c$, which contradicts with the assumption. Therefore, $P_k^{c*}$ is part of the Nash equilibrium. A similar proof holds for $P_m^{d*}$. It thus can be concluded that the power strategy set of users $\{P_k^{c*},P_m^{d*}\;|\;k\in\mathcal{K},~m\in\mathcal{M}\}$ obtained by using Algorithm 2 is the Nash equilibrium.

Osborne M J, Rubinstein A. A Course in Game Theory. Cambridge: MIT Press, 1994. 11–29.

Boyd S, Vandenberghe L. Convex Optimization. New York: Cambridge University Press, 2004. 95–103.

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