SCIENCE CHINA Information Sciences, Volume 60 , Issue 10 : 102303(2017) https://doi.org/10.1007/s11432-016-9007-2

Spectral and energy efficiency analysis for massive MIMO multi-pair two-way relaying networks under generalized power scaling

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  • ReceivedOct 21, 2016
  • AcceptedNov 28, 2016
  • PublishedMar 13, 2017



This work was supported by National Natural Science Foundation of China (Grant Nos. 61301111, 61472343) and China Postdoctoral Science Foundation (Grant No. 2014M56074).



Proof of Theorem sect. 3

According to the law of large numbers, we obtain \begin{align} \frac{1}{N}\hat{{\boldsymbol g}}^{\rm T}_k{\boldsymbol e}_k\xrightarrow[N\to\infty]{\rm a.s.}0, \frac{1}{N}{{\boldsymbol e}}^{\rm T}_k{\boldsymbol e}_{k'}\xrightarrow[N\to\infty]{\rm a.s.}0. \tag{35} \end{align} As such, we have \begin{align} \mathrm{P_A}\xrightarrow[N\to\infty]{\rm a.s.}0, \mathrm{P_{II}}\xrightarrow[N\to\infty]{\rm a.s.}0, \mathrm{P_{SI}}\xrightarrow[N\to\infty]{\rm a.s.}0, \mathrm{and} \mathrm{P_N}\xrightarrow[N\to\infty]{\rm a.s.}\sigma_n^2\|{\hat{\boldsymbol g}}_{k'}^{\rm T}{\boldsymbol F}\|^2 +\sigma_n^2. \tag{36} \end{align} Utilizing 36, we have \begin{align} \gamma_{k'}\xrightarrow[N\to\infty]{\rm a.s.}\frac{P_{\rm U}|\hat{{\boldsymbol g}}_{k'}^{\rm T}{\boldsymbol F}{\hat{\boldsymbol g}}_k|^2}{\sigma_n^2\|{\hat{\boldsymbol g}}_{k'}^{\rm T}{\boldsymbol F}\|^2 +\sigma_n^2}. \tag{37} \end{align} Rewriting $\hat{{\boldsymbol~g}}_k~\sim~\mathcal{CN}(0,\frac{~P_{\rm~P}\eta_k^2}{~P_{\rm~P}\eta_k+1}\mathbf{1}_N)$ due to 6, we obtain \begin{align} \hat{{\boldsymbol g}}_{k'}^{\rm T}{\boldsymbol F} \xrightarrow[N\to\infty]{\rm a.s.}\beta N\big(\eta_{k'}-\sigma_{e_{k'}}^2\big){\hat{\boldsymbol g}}_{k}^{\rm H}, \tag{38} \end{align} and \begin{align} \hat{{\boldsymbol g}}_{k'}^{\rm T}{\boldsymbol F}{\hat{\boldsymbol g}}_i \xrightarrow[N\to\infty]{\rm a.s.}\beta N^2\big(\eta_{k'}-\sigma_{e_{k'}}^2\big)\left(\eta_i-\sigma_{e_i}^2\right)\delta _{ki}. \tag{39} \end{align} Additionally, using the property $\mathrm{Tr}\left(\mathbf{AB}\right)=\mathrm{Tr}\left(\mathbf{BA}\right)$ and rewriting $[\hat{\mathbf{\Lambda}}]_{kk}=(\eta_k-\sigma_{e_k}^2)$ due to 6, $\mathrm{Tr}({\boldsymbol~Z}_1)$ and $\mathrm{Tr}({\boldsymbol~Z}_2)$ in 9 are given by \begin{align} \mathrm{Tr}\left({\boldsymbol Z}_1\right) &\xrightarrow[N\to\infty]{\rm a.s.} \mathrm{Tr}\left(N\hat{{\boldsymbol G}}^{\rm T}\hat{{\boldsymbol G}}^\ast{\boldsymbol P}\hat{{\boldsymbol G}}^{\rm H}\hat{{\boldsymbol G}}{\boldsymbol P}\right) \xrightarrow[N\to\infty]{\rm a.s.}\mathrm{Tr}\left(N\hat{\mathbf{\Lambda}}{\boldsymbol P}\hat{\mathbf{\Lambda}}{\boldsymbol P}\right) \\ &=2\sum\limits_{i=1}^K\big(\eta_{2i-1}-\sigma_{e_{2i-1}}^2\big)\left(\eta_{2i}-\sigma_{e_{2i}}^2\right)N^2= \varphi_1N^2, \tag{40} \end{align} \begin{align} \mathrm{Tr}\left({\boldsymbol Z}_2\right) &\xrightarrow[N\to\infty]{\rm a.s.} \mathrm{Tr}\left(N^2\hat{{\boldsymbol G}}^{\rm T}\hat{{\boldsymbol G}}^\ast{\boldsymbol P}\hat{{\boldsymbol G}}^{\rm H}\hat{{\boldsymbol G}}\hat{{\boldsymbol G}}^{\rm H}\hat{{\boldsymbol G}}{\boldsymbol P}\right) \xrightarrow[N\to\infty]{\rm a.s.} \mathrm{Tr}\left(N^2\hat{\mathbf{\Lambda}}{\boldsymbol P}\hat{\mathbf{\Lambda}}^2{\boldsymbol P}\right) \\ &=\sum\limits_{i=1}^K\big(\eta_{2i-1}-\sigma_{e_{2i-1}}^2\big)\left(\eta_{2i}-\sigma_{e_{2i}}^2\right)\left[\big(\eta_{2i-1}-\sigma_{e_{2i-1}}^2\big)+\left(\eta_{2i}-\sigma_{e_{2i}}^2\right)\right] N^3=\varphi_2N^3. \tag{41} \end{align} Therefore, we get \begin{align} \beta^2 \xrightarrow[N\to\infty]{\rm a.s.} \frac{P_{\rm R}}{P_{\rm U}\varphi_2N^3+\varphi_1\sigma_n^2N^2}. \tag{42} \end{align} Substituting 38 and 39 into 37, we have \begin{align} \gamma_{k'}\xrightarrow[N\to\infty]{\rm a.s.}\frac{P_{\rm U}\beta^2 N^4\big(\eta_{k'}-\sigma_{e_{k'}}^2\big)^2(\eta_{k}-\sigma_{e_{k}}^2)^2}{\sigma_n^2\beta^2 N^3\big(\eta_{k'}-\sigma_{e_{k'}}^2\big)^2(\eta_{k}-\sigma_{e_{k}}^2)+\sigma_n^2}. \tag{43} \end{align} Let $P_{\rm~U}={~E_{\rm~U}}/{~N^{a}},P_{\rm~R}={~E_{\rm~R}}/{~N^b}$, $a,b\geq0$, substituting 42 into 43, Theorem sect. 3.2 can be deduced.

Proof of Theorem 4.1

Recalling that the SINR at the $k'$th user, we have \begin{align} {\rm E}\left\{\left[\gamma_{k'}\right]^{-1}\right\}={\rm E}\left\{\frac{\mathrm{P_A}+\mathrm{P_N}+\mathrm{P_{SI}}+\mathrm{P_{II}}}{P_{\rm U}|\hat{{\boldsymbol g}}_{k'}^{\rm T}{\boldsymbol F}{\hat{\boldsymbol g}}_k|^2}\right\}. \tag{44} \end{align} Due to 4, one can obtain \begin{equation} \hat{{\boldsymbol g}}_{k'}^{\rm T}{\boldsymbol F} \xrightarrow[N\to\infty]{\rm a.s.} \beta\hat{{\boldsymbol g}}_{k'}^{\rm T}\hat{{\boldsymbol g}}_{k'}^\ast\mathbf{1}_{k'}{\boldsymbol P}\hat{{\boldsymbol G}}^{\rm H} =\beta\|\hat{{\boldsymbol g}}_{k'}\|^2\hat{{\boldsymbol g}}_k^{\rm H}, \tag{45}\end{equation} and \begin{align} \hat{{\boldsymbol g}}_{k'}^{\rm T}{\boldsymbol F}{\hat{\boldsymbol g}}_i \xrightarrow[N\to\infty]{\rm a.s.} \beta\|\hat{{\boldsymbol g}}_{k'}\|^2\|\hat{{\boldsymbol g}}_{i}\|^2\delta_{ki}. \tag{46} \end{align} Consequently, recalling 36 and substituting 45 and 46 into 44 we have \begin{align} {\rm E}\{\left[\gamma_{k'}\right]^{-1}\}&\xrightarrow[N\to\infty]{\rm a.s.}{\rm E}\left\{\frac{\sigma_n^2\beta^2\|\hat{{\boldsymbol g}}_{k'}\|^4\|\hat{{\boldsymbol g}}_k\|^2+\sigma_n^2}{P_{\rm U}\beta^2\|\hat{{\boldsymbol g}}_{k'}\|^4\|\hat{{\boldsymbol g}}_{k}\|^4}\right\} \\ &=\frac{\sigma_n^2}{P_{\rm U}}{\rm E}\left\{\frac{1}{\|\hat{{\boldsymbol g}}_k\|^2}\right\} +\frac{\sigma_n^2}{P_{\rm U}\beta^2} {\rm E}\left\{\frac{1}{\|\hat{{\boldsymbol g}}_{k}\|^4}\right\}\cdot{\rm E}\left\{\frac{1}{\|\hat{{\boldsymbol g}}_{k'}\|^4}\right\}. \tag{47} \end{align} Utilizing the properties of Wishart matrix, we obtain \begin{align} {\rm E}\left\{\frac{1}{\|\hat{{\boldsymbol g}}_{k}\|^2}\right\}=\frac{1}{\left(N-1\right)(\eta_k-\sigma_{e_{k}}^2)}\approx\frac{1}{N(\eta_k-\sigma_{e_{k}}^2)}, \tag{48} \end{align} and \begin{align} {\rm E}\left\{\frac{1}{\|\hat{{\boldsymbol g}}_{k}\|^4}\right\}\cdot{\rm E}\left\{\frac{1}{\|\hat{{\boldsymbol g}}_{k'}\|^4}\right\} &=\frac{1}{\left(N-1\right)^2\left(N-2\right)^2(\eta_k-\sigma_{e_{k}}^2)^2\big(\eta_{k'}-\sigma_{e_{k'}}^2\big)^2} \\ &\approx\frac{1}{N^4(\eta_k-\sigma_{e_{k}}^2)^2\big(\eta_{k'}-\sigma_{e_{k'}}^2\big)^2}. \tag{49} \end{align} Substituting 48, 49 and 42 into 47, we have \begin{align} {\rm E}\{\left[\gamma_{k'}\right]^{-1}\}&\xrightarrow[N\to\infty]{\rm a.s.}\frac{\sigma_n^2}{P_{\rm U}}\frac{1}{\left(N-1\right)(\eta_k-\sigma_{e_{k}}^2)} +\frac{\sigma_n^2}{P_{\rm U}\beta^2}\frac{1}{\left(N-1\right)^2\left(N-2\right)^2(\eta_k-\sigma_{e_{k}}^2)^2(\eta_{k'}-\sigma_{e_{k'}}^2)^2} \\ &\approx\frac{1}{N}\cdot\frac{\sigma_n^2}{P_{\rm U}}\frac{1}{(\eta_k-\sigma_{e_{k}}^2)} +\frac{1}{N^4}\cdot\frac{1}{\beta^2}\cdot\frac{\sigma_n^2}{P_{\rm U}}\cdot\frac{1}{(\eta_k-\sigma_{e_{k}}^2)^2\big(\eta_{k'}-\sigma_{e_{k'}}^2\big)^2} \\ &\xrightarrow[N\to\infty]{\rm a.s.}\frac{\sigma_n^2}{NP_{\rm U}}\left\{\frac{1}{(\eta_k-\sigma_{e_{k}}^2)} +\frac{({P_{\rm U}\varphi_2N^3+\varphi_1\sigma_n^2N^2})}{P_{\rm R}(\eta_k-\sigma_{e_{k}}^2)^2\big(\eta_{k'}-\sigma_{e_{k'}}^2\big)^2N^3}\right\} \\ &=\frac{\sigma_n^2}{NP_{\rm U}(\eta_k-\sigma_{e_{k}}^2)}\left\{1 +\frac{{P_{\rm U}\varphi_2+\varphi_1\sigma_n^2/N}}{P_{\rm R}(\eta_k-\sigma_{e_{k}}^2)(\eta_{k'}-\sigma_{e_{k'}}^2)^2}\right\}. \tag{50} \end{align} Substituting 50 into 32 and using 13, Theorem 4.1 can be deduced.


[1] Wang D, Zhang Y, Wei H. An overview of transmission theory and techniques of large-scale antenna systems for 5G wireless communications. Sci China Inf Sci, 2016, 59: 081301 CrossRef Google Scholar

[2] Ma Z, Zhang Z Q, Ding Z G, et al. Key techniques for 5G wireless communications: network architecture, physical layer, and MAC layer perspectives. Sci China Inf Sci, 2015, 58: 041301. Google Scholar

[3] Larsson E, Edfors O, Tufvesson F, et al. Massive MIMO for next generation wireless systems. IEEE Commun Mag, 2014, 52: 186--195. Google Scholar

[4] Yang A, Xing C W, Fei Z S, et al. Performance analysis for uplink massive MIMO systems with a large and random number of UEs. Sci China Inf Sci, 2016, 59: 022312. Google Scholar

[5] Zhang Q, Jin S, Wong K K. Power Scaling of Uplink Massive MIMO Systems With Arbitrary-Rank Channel Means. IEEE J Sel Top Signal Process, 2014, 8: 966-981 CrossRef ADS arXiv Google Scholar

[6] Hien Quoc Ngo , Larsson E G, Marzetta T L. Energy and spectral efficiency of very large multiuser MIMO systems. IEEE Trans Commun, 2013, 61: 1436-1449 CrossRef Google Scholar

[7] Han B, Zhao S, Yang B. et al. Historical PMI based multi-user scheduling for FDD massive MIMO systems. In: Proceedings of the IEEE Vehicular Technology Conference (VTC), Nanjing. 2016. 1--5. Google Scholar

[8] Yang J, Fan P, Duong T Q. Exact performance of two-way AF relaying in Nakagami-m fading environment. IEEE Trans Wireless Commun, 2011, 10: 980-987 CrossRef Google Scholar

[9] Zhang H, Xing H, Cheng J. Secure Resource Allocation for OFDMA Two-Way Relay Wireless Sensor Networks Without and With Cooperative Jamming. IEEE Trans Ind Inf, 2016, 12: 1714-1725 CrossRef Google Scholar

[10] Multi-Pair Two-Way Amplify-and-Forward Relaying with Very Large Number of Relay Antennas. IEEE Trans Wireless Commun, 2014, 13: 2636-2645 CrossRef Google Scholar

[11] Jin S, Liang X, Wong K K, et al. Ergodic rate analysis for multipair massive MIMO two-way relay networks. IEEE Trans Wirel Commun, 2014, 14: 1480--1491. Google Scholar

[12] Dai Y, Dong X. Power Allocation for Multi-Pair Massive MIMO Two-Way AF Relaying With Linear Processing. IEEE Trans Wireless Commun, 2016, 15: 5932-5946 CrossRef Google Scholar

[13] Ngo H Q, Suraweera H A, Matthaiou M. Multipair Full-Duplex Relaying With Massive Arrays and Linear Processing. IEEE J Select Areas Commun, 2014, 32: 1721-1737 CrossRef Google Scholar

[14] Zhang Z S, Wang X, Zhang C Y, et al. Massive MIMO technology and challenges (in Chinese). Sci Sin Inform, 2015, 45: 1095--1110. Google Scholar

[15] Xue Y, Zhang J, Gao X. Resource allocation for pilot-assisted massive MIMO transmission. Sci China Inf Sci, 2017, 60: 042302 CrossRef Google Scholar

[16] Khansefid A, Hlaing M. Asymptotically optimal power allocation for massive MIMO uplink. In: Proceedings of the IEEE International Conference on Signal and Information Processing (GlobalSIP), Atlanta, 2014. 627--631. Google Scholar

[17] Dong Y, Qiu L. Closed-form ASE of downlink massive MIMO system with uplink and downlink training. In: Proceedings of the IEEE International Conference on Wireless Communications Signal Processing (WCSP), Nanjing, 2015. 1--5. Google Scholar

[18] Yang A, He Z, Xing C. The Role of Large-Scale Fading in Uplink Massive MIMO Systems. IEEE Trans Veh Technol, 2016, 65: 477-483 CrossRef Google Scholar

[19] Yang J, Wang H Y, Ding J, et al. Spectral and energy efficiency for massive MIMO multi-pair two-way relay networks with ZFR/ZFT and imperfect CSI. In: Proceedings of Asia-Pacific Conference on Communications (APCC), Kyoto, 2015. 47--51. Google Scholar

[20] Kong C, Zhong C, Papazafeiropoulos A K. Sum-Rate and Power Scaling of Massive MIMO Systems With Channel Aging. IEEE Trans Commun, 2015, 63: 4879-4893 CrossRef Google Scholar

[21] Silva S, Amarasuriya G, Tellambura C, et al. Massive MIMO two-way relay networks with channel imperfections. In: Proceedings of the IEEE International Conference on Communications (ICC), Kuala Lumpur, 2016. 1--7. Google Scholar

[22] Xing C, Ma S, Zhou Y. Matrix-Monotonic Optimization for MIMO Systems. IEEE Trans Signal Process, 2015, 63: 334-348 CrossRef ADS Google Scholar

[23] Cramér H. Random Variables and Probability Distributions. Cambridge: Cambridge University Press, 1970. Google Scholar


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