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Multiuser Two-Way Relaying with Large-Scale Antenna Arrays and Energy Harvesting

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Abstract

This paper considers an energy-harvesting multiuser cellular two-way relay channel (EH-cTWRC), where multiple energy-harvesting users exchange information with a base station (BS) via a relay station (RS). Both the BS and the RS are powerful infrastructures equipped with a very large antenna array, while the energy-constrained users are equipped with a single antenna and harvest energy from the RS. We propose a three-phase harvest-decode-and-forward (HDF) relaying protocol for the considered EH-cTWRC. In the first phase, the RS transfers energy to the mobile users. With the harvested power and their inherent power supply, the users exchange information with the BS via the RS in the second and third phases using a low-complexity antenna-selection based decode-and-forward relaying scheme. We analyze performance of the proposed HDF scheme, and derive closed-form asymptotic expressions for the signal-to-interference-plus-noise ratio and the achievable sum-rate when the numbers of antennas at the BS and the RS become large. We investigate the optimal power allocation at the RS to maximize the achievable sum-rate. A low-complexity suboptimal power allocation scheme with closed-form expression is also proposed. Both analytical and numerical results show that large antenna arrays can improve the spectral efficiency and wireless power transfer efficiency significantly.

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Acknowledgments

The work in this paper was supported by the Natural Science Foundation of China under Grant No. 61401400, the public welfare project of Zhejiang Province under Grant No. 2016C33036, the Natural Science Foundation of Zhejiang Province under Grant No. LY14F020011 and LY17F010002, and the science and technology project for public wellbeing of Ningbo City under Grant No. 2016C51039.

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Correspondence to Zhaoxi Fang.

Appendix: Proof of Proposition 2

Appendix: Proof of Proposition 2

First, consider the received SINRs at the RS. For the received SINR \(\gamma _{BR,k}\), the variance of the noise \({\varvec{w}}_{RR,k}^H{\varvec{z}}_R\) in (11) is given by

$$\begin{aligned}&{\mathbb{V}}\text{ar}\left[ {\varvec{w}}_{RR,k}^H{\varvec{z}}_R \right] = \sigma ^2 {\mathbb{E}} \Vert {\varvec{w}}_{RR,k} \Vert ^2 \\&\quad = \sigma ^2 {\mathbb{E}} \left[ \left( \left( {\varvec{H}}_{SR}^H {\varvec{H}}_{SR}\right) ^{-1} \right) _{kk} \right] \\&\quad = \sigma ^2 {\mathbb{E}} \left[ \left( \left( \tilde{{\varvec{H}}}_{SR}^H \tilde{{\varvec{H}}}_{SR}\right) ^{-1} \left( {\varvec{\Lambda }}_{SR}^H {\varvec{\Lambda }}_{SR}\right) ^{-1} \right) _{kk} \right] \\&\quad = \frac{\sigma ^2}{{\ell }_{B}} {\mathbb{E}} \left[ \left( \left( \tilde{{\varvec{H}}}_{SR}^H \tilde{{\varvec{H}}}_{SR}\right) ^{-1} \right) _{kk} \right] \\&\quad = \frac{\sigma ^2}{K {\ell }_{B}} {\mathbb{E}} \left[ \text{tr} \left( \left( \tilde{{\varvec{H}}}_{SR}^H \tilde{{\varvec{H}}}_{SR}\right) ^{-1} \right) \right] \\&\quad = \frac{\sigma ^2}{ (N_R-K) {\ell }_{B}}, \end{aligned}$$
(36)

\(k= 1,\ldots ,K\), where in the last step we used the identity \({\mathbb{E}} \left[ \text{tr} ({\varvec{X}}^{-1}) \right] = \frac{M}{N-M}\), when \({\varvec{X}}\) is a \(M \times M\) central Wishart matrix with N degrees of freedom [18]. Substituting (36) into (11), we have the desired result in (26a).

Similarly, the variance of the noise \({\varvec{w}}_{RR,K+k}^H{\varvec{z}}_R\) in (12) is given by

$$\begin{aligned} {\mathbb{V}}\text{ar}\left[ {\varvec{w}}_{RR,K+k}^H{\varvec{z}}_R \right] = \frac{\sigma ^2}{ (N_R-K) {\ell }_{U}}, \quad k= 1,\ldots ,K. \end{aligned}$$
(37)

Substituting (37) into (12), we obtain the result in (26b).

Now consider the received SINRs at the BS and the users. With large antenna arrays at the BS and the RS, from (14), the power normalizing factor \(\alpha _{RI}\) can be determined by Fang et al. [17]

$$\begin{aligned} \alpha _{RI}&= {} \sqrt{\frac{P_{RI}}{K \ell _{U} {\mathbb{E}} \left\{ \text{tr} \left[ \left( {\varvec{H}}_{RU} {\varvec{H}}_{RU}^{ H} \right) ^{-1} \right] \right\} }} \\&= {} \sqrt{\frac{(N_R-K) {\ell }_{U} P_{RI}}{K}}. \end{aligned}$$
(38)

For the received SINRs at the BS, the variance of the noise term \({\varvec{w}}_{B,k}^H {\varvec{z}}_B\) in (18) is given by

$$\begin{aligned}{\mathbb{V}}\text{ar}\left[ {\varvec{w}}_{B,k}^H {\varvec{z}}_B \right] &= \sigma ^2 {\mathbb{E}} \left[ \Vert {\varvec{w}}_{B,k} \Vert ^2 \right] \\&= \sigma ^2 {\mathbb{E}} \left\{ \left[ \left( \left( {\varvec{H}}_{RB} {\varvec{H}}_{RU}^{\dag } \right) ^H \left( {\varvec{H}}_{RB} {\varvec{H}}_{RU}^{\dag } \right) \right) ^{-1} \right] _{k,k} \right\} \\&= \sigma ^2 {\mathbb{E}} \left\{ \left[ \left( {\varvec{H}}_{RU}^{+H} {\varvec{H}}_{RB}^H {\varvec{H}}_{RB} {\varvec{H}}_{RU}^{\dag } \right) ^{-1} \right] _{k,k} \right\} \\&= \frac{\sigma ^2 }{\ell _{B} N_B} {\mathbb{E}} \left\{ \left[ \left( {\varvec{H}}_{RU}^{+H} {\varvec{H}}_{RU}^{\dag } \right) ^{-1} \right] _{k,k} \right\} \\&= \frac{\sigma ^2 }{\ell _{B} N_B} {\mathbb{E}} \left\{ \left[ {\varvec{H}}_{RU} {\varvec{H}}_{RU}^H \right] _{k,k} \right\} \\&= \frac{\ell _U N_R \sigma ^2 }{\ell _{B} N_B}. \end{aligned}$$
(39)

Substituting (38) and (39) into (18), we have the result in (26c). Similarly, we can obtain the result in (26d) by substituting (38) into (19).

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Fang, Z., Liang, F., Li, J. et al. Multiuser Two-Way Relaying with Large-Scale Antenna Arrays and Energy Harvesting. Wireless Pers Commun 95, 1299–1315 (2017). https://doi.org/10.1007/s11277-016-3830-7

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