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


In this work, we investigate the spectral efficiency (SE) and energy efficiency (EE) for a massive multiple-input multiple-output multi-pair two-way amplify-and-forward relaying system, where multi-pair users exchange information via a relay station equipped with large scale antennas. We assume that imperfect channel state information is available and maximum-ratio combining/maximum-ratio transmission beamforming is adopted at the relay station.Considering constant or scaled transmit power of pilot sequences, we quantify the asymptotic SE and EE under general power scaling schemes, in which the transmit power at each user and relay station can both be scaled down, as the number of relay antennas tends to infinity. In addition, a closed-form expression of the SE has been obtained approximately. Our results show that by using massive relay antennas, the transmit power at each user and the relay station can be scaled down, with a non-vanishing signal to interference and noise ratio (SINR). Finally, simulation results confirm the validity of our analysis.


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.


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