logo

SCIENCE CHINA Information Sciences, Volume 61, Issue 4: 042303(2018) https://doi.org/10.1007/s11432-016-9126-1

Robust MMSE precoding for massive MIMO transmission with hardware mismatch

More info
  • ReceivedOct 20, 2016
  • AcceptedMay 5, 2017
  • PublishedNov 20, 2017

Abstract

Due to hardware mismatch, the channel reciprocity of time-division duplex massive multiple-input multiple-output system is impaired.Under this condition, there exist several different approaches for base station (BS) to obtain downlink (DL) channel information based on the minimum mean-square-error (MMSE) estimation method.In this paper, we show that with the hardware mismatch parameters BS can obtain the same DL channel information via these different approaches.As the obtained DL channel information is usually imperfect,we propose a precoding technique based on the criterion that minimizes the mean-square-error (MSE) of signal detection at the user terminals (UTs).The proposed precoding is robust to the channel estimation error and significantly improves the system performance compared to the conventional regularized zero-forcing precoding.Furthermore, we derive an asymptotic approximation of the ergodic sum rate for the proposed precoding using the large dimensional random matrix theory,which is tight as the number of antennas both at the BS and UT approach infinity with a fixed non-zero and finite ratio.This approximation can provide a reliable sum rate prediction at a much lower computation cost than Monte Carlo simulations.Simulation results show that the approximation is accurate even for a realistic system dimension.


Acknowledgment

This work was supported by National Natural Science Foundation of China (Grant Nos. 61320106003, 61471113, 61521061, 61631018), National High Technology Research and Development Program of China (863) (Grant Nos. 2015AA01A701, 2014AA01A704), National Science and Technology Major Project of China (Grant No. 2014ZX03003006-003), and Huawei Cooperation Project.


Supplement

Appendix

Proof of Proposition 1

Firstly, we obtain the DL channel estimate via these three approaches in turn. For simplicity, we treat RF circuit gains as known constants at the BS during the channel estimation phase and the concerned data transmission interval [8].

Approach 1. Base on (4), the MMSE estimate of ${\boldsymbol~h}_{\mathrm{u}}$ is given by [27] \begin{eqnarray} \widehat{{\boldsymbol h}}_u = {\boldsymbol C}_{{\boldsymbol h}_{\mathrm{u}}{\boldsymbol y}_{\mathrm{t}}}{\boldsymbol C}^{-1}_{{\boldsymbol y}_{\mathrm{t}}{\boldsymbol y}_{\mathrm{t}}}{\boldsymbol y}_{\mathrm{t}} =\left({\boldsymbol R}_{\mathrm{b}} \otimes {\boldsymbol T}_{\mathrm{u}}\right) {\boldsymbol C}_{{\boldsymbol v}{\boldsymbol v}} \left({\boldsymbol R}_{\mathrm{b}} \otimes {\boldsymbol T}_{\mathrm{u}}\right)^{\rm H} {\boldsymbol C}^{-1}_{{\boldsymbol y}_{\mathrm{t}}{\boldsymbol y}_{\mathrm{t}}}{\boldsymbol y}_{\mathrm{t}} . \tag{38} \end{eqnarray}

According to the orthogonality principle of MMSE estimation [27], the estimation error ${\boldsymbol~e}_{\mathrm{u}}~=~{\boldsymbol~h}_{\mathrm{u}}-~\widehat{{\boldsymbol~h}}_{\mathrm{u}}$ is uncorrelated with ${\boldsymbol~h}_{\mathrm{u}}$. Furthermore, the covariance matrix of the estimation error is given by \begin{equation} {\boldsymbol C}_{{\boldsymbol e}_{\mathrm{u}}{\boldsymbol e}_{\mathrm{u}}} = \left({\boldsymbol R}_{\mathrm{b}} \otimes {\boldsymbol T}_{\mathrm{u}}\right) {\boldsymbol C}_{{\boldsymbol vv}} \left({\boldsymbol R}_{\mathrm{b}} \otimes {\boldsymbol T}_{\mathrm{u}}\right)^{\rm H} - \left({\boldsymbol R}_{\mathrm{b}} \otimes {\boldsymbol T}_{\mathrm{u}}\right) {\boldsymbol C}_{{\boldsymbol vv}} \left({\boldsymbol R}_{\mathrm{b}} \otimes {\boldsymbol T}_{\mathrm{u}}\right)^{\rm H}{\boldsymbol C}^{-1}_{{\boldsymbol y}_{\mathrm{t}}{\boldsymbol y}_{\mathrm{t}}}\left({\boldsymbol R}_{\mathrm{b}} \otimes {\boldsymbol T}_{\mathrm{u}}\right) {\boldsymbol C}_{{\boldsymbol vv}} \left({\boldsymbol R}_{\mathrm{b}} \otimes {\boldsymbol T}_{\mathrm{u}}\right)^{\rm H}. \tag{39}\end{equation}

Then we obtain the DL channel estimate $\widehat{{\boldsymbol~h}}_{\mathrm{d}1}$: \begin{eqnarray} \widehat{{\boldsymbol h}}_{\mathrm{d}1} &=& \left({\boldsymbol T}_{\mathrm{b}} {\boldsymbol R}^{-1}_{\mathrm{b}} \otimes {\boldsymbol R}_{\mathrm{u}}{\boldsymbol T}^{-1}_{\mathrm{u}}\right) \widehat{{\boldsymbol h}}_{\mathrm{u}} \\ & = &\left({\boldsymbol T}_{\mathrm{b}} \otimes {\boldsymbol R}_{\mathrm{u}}\right) {\boldsymbol C}_{{\boldsymbol vv}} \left({\boldsymbol R}_{\mathrm{b}} \otimes {\boldsymbol T}_{\mathrm{u}}\right)^{\rm H} {\boldsymbol C}^{-1}_{{\boldsymbol y}_{\mathrm{t}}{\boldsymbol y}_{\mathrm{t}}}{\boldsymbol y}_{\mathrm{t}}. \tag{40} \end{eqnarray}

The estimation error ${\boldsymbol~e}_{\mathrm{d1}}$, which is expressed as ${\boldsymbol~e}_{\mathrm{d1}}~=~({\boldsymbol~T}_{\mathrm{b}}~{\boldsymbol~R}^{-1}_{\mathrm{b}}~\otimes~{\boldsymbol~R}_{\mathrm{u}}{\boldsymbol~T}^{-1}_{\mathrm{u}})~{\boldsymbol~e}_{\mathrm{u}}$, is uncorrelated with ${\boldsymbol~h}_{\mathrm{d}}$ since ${\boldsymbol~e}_{\mathrm{u}}$ is independent on ${\boldsymbol~h}_{\mathrm{u}}$. The corresponding estimation error covariance matrix is given by \begin{eqnarray} {\boldsymbol C}_{{\boldsymbol e}_1{\boldsymbol e}_1} &=& \left({\boldsymbol T}_{\mathrm{b}} {\boldsymbol R}^{-1}_{\mathrm{b}} \otimes {\boldsymbol R}_{\mathrm{u}}{\boldsymbol T}^{-1}_{\mathrm{u}}\right) {\boldsymbol C}_{{\boldsymbol e}_{\mathrm{u}}{\boldsymbol e}_{\mathrm{u}}} \left({\boldsymbol T}_{\mathrm{b}} {\boldsymbol R}^{-1}_{\mathrm{b}} \otimes {\boldsymbol R}_{\mathrm{u}}{\boldsymbol T}^{-1}_{\mathrm{u}}\right)^{\rm H} \\ &=& \left({\boldsymbol T}_{\mathrm{b}} \otimes {\boldsymbol R}_{\mathrm{u}}\right) {\boldsymbol C}_{{\boldsymbol vv}} \left({\boldsymbol T}_{\mathrm{b}} \otimes {\boldsymbol R}_{\mathrm{u}}\right)^{\rm H} - \left({\boldsymbol T}_{\mathrm{b}} \otimes {\boldsymbol R}_{\mathrm{u}}\right) {\boldsymbol C}_{{\boldsymbol vv}} \left({\boldsymbol R}_{\mathrm{b}} \otimes {\boldsymbol T}_{\mathrm{u}}\right)^{\rm H}{\boldsymbol C}^{-1}_{{\boldsymbol y}_{\mathrm{t}}{\boldsymbol y}_{\mathrm{t}}}\left({\boldsymbol R}_{\mathrm{b}} \otimes {\boldsymbol T}_{\mathrm{u}}\right) {\boldsymbol C}_{{\boldsymbol vv}} \left({\boldsymbol T}_{\mathrm{b}} \otimes {\boldsymbol R}_{\mathrm{u}}\right)^{\rm H}. \tag{41} \end{eqnarray}

Approach 2. Base on (5), the MMSE estimate of ${\boldsymbol~v}_{\mathrm{w}}$ is given by [27] \begin{eqnarray} \widehat{{\boldsymbol v}}_{\rm w} = {\boldsymbol C}_{{\boldsymbol v}{\boldsymbol y}_{\mathrm{t}}}{\boldsymbol C}^{-1}_{{\boldsymbol y}_{\mathrm{t}}{\boldsymbol y}_{\mathrm{t}}}{\boldsymbol y}_{\mathrm{t}} = {\boldsymbol C}_{{\boldsymbol v}{\boldsymbol v}} \left({\boldsymbol R}_{\mathrm{b}} \otimes {\boldsymbol T}_{\mathrm{u}}\right)^{\rm H} {\boldsymbol C}^{-1}_{{\boldsymbol y}_{\mathrm{t}}{\boldsymbol y}_{\mathrm{t}}}{\boldsymbol y}_{\mathrm{t}}. \tag{42} \end{eqnarray}

The covariance matrix of the estimation error ${\boldsymbol~e}_{\mathrm{w}}~=~{\boldsymbol~v}_{\mathrm{w}}~-~\widehat{{\boldsymbol~v}}_{\mathrm{w}}$ is given by \begin{equation} {\boldsymbol C}_{{\boldsymbol e}_{\mathrm{w}}{\boldsymbol e}_{\mathrm{w}}} ={\boldsymbol C}_{{\boldsymbol v}{\boldsymbol v}} - {\boldsymbol C}_{{\boldsymbol v}{\boldsymbol v}} \left({\boldsymbol R}_{\mathrm{b}} \otimes {\boldsymbol T}_{\mathrm{u}}\right)^{\rm H}{\boldsymbol C}^{-1}_{{\boldsymbol y}_{\mathrm{t}}{\boldsymbol y}_{\mathrm{t}}}\left({\boldsymbol R}_{\mathrm{b}} \otimes {\boldsymbol T}_{\mathrm{u}}\right) {\boldsymbol C}_{{\boldsymbol v}{\boldsymbol v}}. \tag{43}\end{equation}

Then we obtain the DL channel estimate $\widehat{{\boldsymbol~h}}_{\mathrm{d}2}$: \begin{eqnarray} \widehat{{\boldsymbol h}}_{\mathrm{d}2} = \left({\boldsymbol T}_{\mathrm{b}} \otimes {\boldsymbol R}_{\mathrm{u}}\right) \widehat{{\boldsymbol v}}_{\mathrm{w}} = \left({\boldsymbol T}_{\mathrm{b}} \otimes {\boldsymbol R}_{\mathrm{u}}\right) {\boldsymbol C}_{{\boldsymbol v}{\boldsymbol v}} \left({\boldsymbol R}_{\mathrm{b}} \otimes {\boldsymbol T}_{\mathrm{u}}\right)^{\rm H} {\boldsymbol C}^{-1}_{{\boldsymbol y}_{\mathrm{t}}{\boldsymbol y}_{\mathrm{t}}}{\boldsymbol y}_{\mathrm{t}}. \tag{44} \end{eqnarray}

The estimation error ${\boldsymbol~e}_{\mathrm{d2}}$, which is expressed as ${\boldsymbol~e}_{\mathrm{d2}}=\left({\boldsymbol~T}_{\mathrm{b}}~\otimes~{\boldsymbol~R}_{\mathrm{u}}\right){\boldsymbol~e}_{\mathrm{w}}$, is uncorrelated with ${\boldsymbol~h}_{\mathrm{d2}}$ for the same reason as ${\boldsymbol~e}_{\mathrm{d1}}$. The covariance matrix is give by \begin{eqnarray} {\boldsymbol C}_{{\boldsymbol e}_2{\boldsymbol e}_2} &=& \left({\boldsymbol T}_{\mathrm{b}} \otimes {\boldsymbol R}_{\mathrm{u}}\right){\boldsymbol C}_{{\boldsymbol e}_{\mathrm{w}}{\boldsymbol e}_{\mathrm{w}}}\left({\boldsymbol T}_{\mathrm{b}} \otimes {\boldsymbol R}_{\mathrm{u}}\right)^{\rm H} \\ &=& \left({\boldsymbol T}_{\mathrm{b}} \otimes {\boldsymbol R}_{\mathrm{u}}\right) {\boldsymbol C}_{{\boldsymbol v}{\boldsymbol v}} \left({\boldsymbol T}_{\mathrm{b}} \otimes {\boldsymbol R}_{\mathrm{u}}\right)^{\rm H} - \left({\boldsymbol T}_{\mathrm{b}} \otimes {\boldsymbol R}_{\mathrm{u}}\right) {\boldsymbol C}_{{\boldsymbol v}{\boldsymbol v}} \left({\boldsymbol R}_{\mathrm{b}} \otimes {\boldsymbol T}_{\mathrm{u}}\right)^{\rm H}{\boldsymbol C}^{-1}_{{\boldsymbol y}_{\mathrm{t}}{\boldsymbol y}_{\mathrm{t}}}\left({\boldsymbol R}_{\mathrm{b}} \otimes {\boldsymbol T}_{\mathrm{u}}\right) {\boldsymbol C}_{{\boldsymbol v}{\boldsymbol v}} \left({\boldsymbol T}_{\mathrm{b}} \otimes {\boldsymbol R}_{\mathrm{u}}\right)^{\rm H}. \tag{45} \end{eqnarray}

Approach 3. Based on (6), the MMSE estimate of ${\boldsymbol~h}_{\mathrm{d}}$ is given by [27] \begin{eqnarray} \widehat{{\boldsymbol h}}_{\mathrm{d}} = {\boldsymbol C}_{{\boldsymbol h}_{\mathrm{d}}{\boldsymbol y}_{\mathrm{t}}}{\boldsymbol C}^{-1}_{{\boldsymbol y}_{\mathrm{t}}{\boldsymbol y}_{\mathrm{t}}}{\boldsymbol y}_{\mathrm{t}} = \left({\boldsymbol T}_{\mathrm{b}} \otimes {\boldsymbol R}_{\mathrm{u}}\right) {\boldsymbol C}_{{\boldsymbol v}{\boldsymbol v}} \left({\boldsymbol R}_{\mathrm{b}} \otimes {\boldsymbol T}_{\mathrm{u}}\right)^{\rm H} {\boldsymbol C}^{-1}_{{\boldsymbol y}_{\mathrm{t}}{\boldsymbol y}_{\mathrm{t}}}{\boldsymbol y}_{\mathrm{t}}. \tag{46} \end{eqnarray}

The covariance matrix of the estimation error ${\boldsymbol~e}_{\mathrm{d}3}~=~{\boldsymbol~h}_{\mathrm{d}}-\widehat{{\boldsymbol~h}}_{\mathrm{d}}$ is given by \begin{equation} {\boldsymbol C}_{{\boldsymbol e}_3{\boldsymbol e}_3} = \left({\boldsymbol T}_{\mathrm{b}} \otimes {\boldsymbol R}_{\mathrm{u}}\right) {\boldsymbol C}_{{\boldsymbol v}{\boldsymbol v}} \left({\boldsymbol T}_{\mathrm{b}} \otimes {\boldsymbol R}_{\mathrm{u}}\right)^{\rm H} - \left({\boldsymbol T}_{\mathrm{b}} \otimes {\boldsymbol R}_{\mathrm{u}}\right) {\boldsymbol C}_{{\boldsymbol v}{\boldsymbol v}} \left({\boldsymbol R}_{\mathrm{b}} \otimes {\boldsymbol T}_{\mathrm{u}}\right)^{\rm H}{\boldsymbol C}^{-1}_{{\boldsymbol y}_{\mathrm{t}}{\boldsymbol y}_{\mathrm{t}}}\left({\boldsymbol R}_{\mathrm{b}} \otimes {\boldsymbol T}_{\mathrm{u}}\right) {\boldsymbol C}_{{\boldsymbol v}{\boldsymbol v}} \left({\boldsymbol T}_{\mathrm{b}} \otimes {\boldsymbol R}_{\mathrm{u}}\right)^{\rm H}. \tag{47}\end{equation}

Comparing (40), (44) and (46), it can be seen that $\widehat{{\boldsymbol~h}}_{\mathrm{d}1}$, $\widehat{{\boldsymbol~h}}_{\mathrm{d}2}$ and $\widehat{{\boldsymbol~h}}_{\mathrm{d}3}$ are the same. Moreover, we can see that the covariance of the estimation error is also the same by comparing (41), (45), and (47). This concludes the proof.

Proof of Proposition 2

First we simplify the MSE expression as \begin{eqnarray} \epsilon &=& {\rm E}\left\{ \| \alpha{\boldsymbol y}_{\mathrm{d}} - {\boldsymbol s} \|^2_2 \right\} \\ &=& {\rm E}\left\{\left\| \left[\left({\boldsymbol R}_{\mathrm{u}} \widehat{{\boldsymbol V}}_{\mathrm{d}} {\boldsymbol T}_{\mathrm{b}} + {\boldsymbol R}_{\mathrm{u}} {\boldsymbol E}_\mathrm{v} {\boldsymbol T}_{\mathrm{b}}\right) {\boldsymbol P} + {\boldsymbol I}_K \right]{\boldsymbol s} \right\|^2_2\right\} +\alpha^2K\sigma^2_{\mathrm{d}} \\ &=& \mathrm{tr}\left\{ \alpha^2{\boldsymbol P}^{\rm H} \left(\widehat{{\boldsymbol H}}_{\mathrm{d}}^{\rm H}\widehat{{\boldsymbol H}}_{\mathrm{d}} + {\boldsymbol C}_E\right){\boldsymbol P}\right\} -\mathrm{tr}\left\{\alpha \widehat{{\boldsymbol H}}_{\mathrm{d}} {\boldsymbol P}\right\} - \mathrm{tr}\left\{\alpha{\boldsymbol P}^{\rm H} \widehat{{\boldsymbol H}}_{\mathrm{d}}^{\rm H}\right\} + \left(\alpha^2\sigma^2_{\mathrm{d}} + 1\right)K , \tag{48} \end{eqnarray} where ${\boldsymbol~C}_{\rm~E}~=~{\rm~E}~\{~\left({\boldsymbol~R}_{\mathrm{u}}~~{\boldsymbol~E}_\mathrm{v}~{\boldsymbol~T}_{\mathrm{b}}\right)^{\rm~H}~\left(~{\boldsymbol~R}_{\mathrm{u}}~{\boldsymbol~E}_\mathrm{v}{\boldsymbol~T}_{\mathrm{b}}\right)\}$ can be obtained from (47). Furthermore, it can be seen from Proposition 1 that the covariance of the estimation error is abe to be obtained only with the relative calibration matrices.

The objective function in (48) is non-convex with respect to $({\boldsymbol~P},~\alpha)$, but there exist a global minimum Lemmas 4, 8. The global optimal solution must satisfy the Karush-Kunhn-Tucker (KKT) necessary condition 1). We will derive all the solutions that satisfy the KKT conditions, and then identify the optimal solution.

The Lagrangian associated with the problem (14) can be formulated as \begin{equation} \mathcal{L}\left({\boldsymbol P}, \alpha, \lambda\right) = \epsilon + \eta\left(\mathrm{tr}\{{\boldsymbol P} {\boldsymbol P}^{\rm H} \}-P\right) , \tag{49}\end{equation} where $\eta$ is the Lagrange multiplier associated with the inequality constraint.

The KKT conditions can be obtained as 2) \begin{eqnarray}& & \frac {\partial}{\partial{\boldsymbol P}}\mathcal{L}({\boldsymbol P}, \alpha, \lambda) = \alpha^2 \left(\widehat{{\boldsymbol H}}_{\mathrm{d}}^{\rm H}\widehat{{\boldsymbol H}}_{\mathrm{d}} + {\boldsymbol C}_{\rm E}\right)^{\rm T} {\boldsymbol P}^* - \alpha \left(\widehat{{\boldsymbol H}}_{\mathrm{d}}\right)^{\rm T} + \eta {\boldsymbol P}^* =0 , \tag{50} \\ & &\frac {\partial}{\partial\alpha}\mathcal{L}({\boldsymbol P}, \alpha, \lambda)= 2\alpha\mathrm{tr}\left\{{\boldsymbol P}^{\rm H} \left(\widehat{{\boldsymbol H}}_{\mathrm{d}}^{\rm H}\widehat{{\boldsymbol H}}_{\mathrm{d}} + {\boldsymbol C}_E\right) {\boldsymbol P}\right\} -\mathrm{tr}\left\{\widehat{{\boldsymbol H}}_{\mathrm{d}} {\boldsymbol P}\right\} -\mathrm{tr}\left\{{\boldsymbol P}^{\rm H} \widehat{{\boldsymbol H}}_{\mathrm{d}}^{\rm H}\right\} +2\alpha \sigma^2_{\mathrm{d}} K= 0, \tag{51} \\ & & \eta \geq0, \tag{52} \\ & & \mathrm{tr}\left\{ {\boldsymbol P} {\boldsymbol P}^{\rm H}\right\}\leq P , \tag{53} \\ & & \eta\left(\mathrm{tr}\{ {\boldsymbol P} {\boldsymbol P}^{\rm H}\}-P\right)=0. \tag{54} \end{eqnarray}

An obvious solution is $(\alpha~=~0,~{\boldsymbol~P}~=~0,~\eta~=~0)$, and the corresponding MSE $\epsilon$ equals $K$. In the case of $\alpha~\neq~0$, from (50) we can obtain \begin{equation} \alpha {\boldsymbol P}^{\rm H} \bigg(\widehat{{\boldsymbol H}}_{\mathrm{d}}^{\rm H}\widehat{{\boldsymbol H}}_{\mathrm{d}} + {\boldsymbol C}_{\rm E} + \frac {\eta}{\alpha^2}{\boldsymbol I}_N \bigg) =\widehat{{\boldsymbol H}}_{\mathrm{d}}. \tag{55}\end{equation}

Combining (55) with (51), we obtain \begin{equation} \eta\mathrm{tr}\left\{{\boldsymbol P}{\boldsymbol P}^{\rm H}\right\} = \alpha^2 \sigma^2_{\mathrm{d}} K. \tag{56}\end{equation}

Substituting (56) into (54), yields \begin{equation} \eta = \frac {\alpha^2K\sigma^2_{\mathrm{d}}}{P} > 0, \mathrm{tr}\left\{{\boldsymbol P}{\boldsymbol P}^{\rm H}\right\} = P. \tag{57}\end{equation}

Then we derive the precoding matrix \begin{equation} {\boldsymbol P} =\frac {1}{\alpha}\left(\widehat{{\boldsymbol H}}_{\mathrm{d}}^{\rm H}\widehat{{\boldsymbol H}}_{\mathrm{d}} + {\boldsymbol C}_{\rm E} + \frac {K}{\rho_{\mathrm{d}}}{\boldsymbol I}_N \right)^{-1}\widehat{{\boldsymbol H}}_{\mathrm{d}}^{\rm H}, \tag{58}\end{equation} and $\alpha$ is chosen to satisfy $\mathrm{tr}\left\{{\boldsymbol~P}{\boldsymbol~P}^{\rm~H}\right\}~=~P$, given by \begin{equation} \alpha = \sqrt{\frac {\mathrm{tr}\left\{\widehat{{\boldsymbol H}}_{\mathrm{d}} \left(\widehat{{\boldsymbol H}}_{\mathrm{d}}^{\rm H}\widehat{{\boldsymbol H}}_{\mathrm{d}} + {\boldsymbol C}_{\rm E} + \frac {K}{\rho_{\mathrm{d}}}{\boldsymbol I}_N\right)^{-2}\widehat{{\boldsymbol H}}_{\mathrm{d}}^{\rm H}\right\}}{P}}. \tag{59}\end{equation}

Substituting (58) into (48), the corresponding MSE is expressed as \begin{equation} \epsilon = \mathrm{tr}\left\{{\boldsymbol I}_K - \widehat{{\boldsymbol H}}_{\mathrm{d}} \left(\widehat{{\boldsymbol H}}_{\mathrm{d}}^{\rm H}\widehat{{\boldsymbol H}}_{\mathrm{d}} + {\boldsymbol C}_{\rm E} + \frac {K}{\rho_{\mathrm{d}}}{\boldsymbol I}_N\right)^{-1}\widehat{{\boldsymbol H}}_{\mathrm{d}}^{\rm H}\right\}. \tag{60}\end{equation}

Since ${\boldsymbol~C}_{\rm~E}$ is Hermitian, the second part in the brace is also Hermitian. Then the MSE $\epsilon$ can be described as $\epsilon~=~K~-~X$, where $X$ is some positive real number, so it is smaller than $K$. Thus, Eq. (58) is the optimal precoding matrix. This concludes the proof.

Boyd S, Vandenberghe L. Convex Optimization. Cambridge: Cambridge University Press, 2004.

Hjørungnes A. Complex-valued Matrix Derivatives with Applications in Signal Processing and Communications. Cambridge: Cambridge University Press, 2011.

Proof of Proposition 3

The SINR in (17) consists of tree terms: (1) the desired signal power $|~\left({\boldsymbol~T}_{\mathrm{b}}~{\boldsymbol~v}_k~r_{\mathrm{u}k}~\right)^{\rm~T}~\widehat{{\boldsymbol~W}}~\left({\boldsymbol~T}_{\mathrm{b}}~\widehat{{\boldsymbol~v}}_k~r_{\mathrm{u}k}~\right)^*|^2$; (2) the interference power $\left({\boldsymbol~T}_{\mathrm{b}}~{\boldsymbol~v}_k~r_{\mathrm{u}k}~\right)^{\rm~T}~\widehat{{\boldsymbol~W}}~\left({\boldsymbol~T}_{\mathrm{b}}~\widehat{{\boldsymbol~v}}_j~r_{\mathrm{u}k}~\right)^*$; (3) the term $\xi$. We will derive a deterministic equivalent for each term, and then they together constitute the final expression for $\gamma^\circ_{k}$.

Let us show that the spectral norms of some matrices are uniformly bounded. We take the proof for ${\boldsymbol~T}^*_{\mathrm{b}}~\boldsymbol\varPhi^{\rm~T}_k~{\boldsymbol~T}_{\mathrm{b}}~\widehat{{\boldsymbol~C}}$ where $\widehat{{\boldsymbol~C}}~=~(\frac{1}{N}\widehat{{\boldsymbol~H}}^{\rm~H}_{\mathrm{d}}~\widehat{{\boldsymbol~H}}_{\mathrm{d}}+~\frac{1}{N}{\boldsymbol~C}_{\rm~E}~+~\frac{1}{c\left(N,K\right)\rho_{\mathrm{d}}}{\boldsymbol~I}_N)^{-1}$ ($c\left(N,K\right)=\frac{N}{K}$) as an example. The proof for the other terms is similar, under the condition in Proposition 3 that $c\left(N,K\right)\leq~c$ for all $N$.

The eigenvalues of $\widehat{{\boldsymbol~C}}$ can be written as \begin{equation}\delta_k = \frac{1}{f(N)_k+\frac{1}{c\left(N,K\right)\rho_{\mathrm{d}}}}, \tag{61}\end{equation} where $f(N)_k~\geq~0$ is the eigenvalue of the non-negative definite part $\frac{1}{N}\widehat{{\boldsymbol~H}}^{\rm~H}_{\mathrm{d}}~\widehat{{\boldsymbol~H}}_{\mathrm{d}}+~\frac{1}{N}{\boldsymbol~C}_{\rm~E}~$. Thus $0~<~\delta_k~\leq~c\rho_{\mathrm{d}}$ is upper bounded for all $N$, which means that the spectral norm of $\widehat{{\boldsymbol~C}}$ is uniformly bounded.

The spectral norm of ${\boldsymbol~T}^*_{\mathrm{b}}~\boldsymbol\varPhi^{\rm~T}_k~{\boldsymbol~T}_{\mathrm{b}}~\widehat{{\boldsymbol~C}}$ can be written as \begin{equation}\|{\boldsymbol T}^*_{\mathrm{b}} \boldsymbol\varPhi^{\rm T}_k {\boldsymbol T}_{\mathrm{b}} \widehat{{\boldsymbol C}}\| \leq \|{\boldsymbol T}^*_{\mathrm{b}} \| \| \boldsymbol\varPhi^{\rm T}_k \| \|{\boldsymbol T}_{\mathrm{b}}\| \| \widehat{{\boldsymbol C}}\|. \tag{62}\end{equation}

Since the amplitudes of RF circuit gains are identically and uniformly distributed around $1$, and we have assumed that all correlation matrices $\boldsymbol{\varPhi}_k$ have uniformly bounded spectral norm on $N$, we can obtain that $\|{\boldsymbol~T}^*_{\mathrm{b}}~\boldsymbol\varPhi^{\rm~T}_k~{\boldsymbol~T}_{\mathrm{b}}~\widehat{{\boldsymbol~C}}\|$ is uniformly bounded.

Now we will derive the deterministic equivalent for each term.

(1) Deterministic equivalent for $\left({\boldsymbol~T}_{\mathrm{b}}~{\boldsymbol~v}_k~r_{\mathrm{u}k}~\right)^{\rm~T}~\widehat{{\boldsymbol~W}}~\left({\boldsymbol~T}_{\mathrm{b}}~\widehat{{\boldsymbol~v}}_k~r_{\mathrm{u}k}~\right)^*$.

By applying Lemma 1 of 20 (matrix inversion lemma) we have \begin{eqnarray}\left({\boldsymbol T}_{\mathrm{b}} {\boldsymbol v}_k r_{\mathrm{u}k} \right)^{\rm T} {\widehat{{\boldsymbol W}} }\left({\boldsymbol T}_{\mathrm{b}} \widehat{{\boldsymbol v}}_k r_{\mathrm{u}k} \right)^* &=& \frac {\left({\boldsymbol T}_{\mathrm{b}} {\boldsymbol v}_k r_{\mathrm{u}k} \right)^{\rm T} {\widehat{{\boldsymbol W}}_{[k]}}\left({\boldsymbol T}_{\mathrm{b}} \widehat{{\boldsymbol v}}_k r_{\mathrm{u}k} \right)^*}{1+\left({\boldsymbol T}_{\mathrm{b}} \widehat{{\boldsymbol v}}_k r_{\mathrm{u}k} \right)^{\rm T} {\widehat{{\boldsymbol W}}_{[k]}}\left({\boldsymbol T}_{\mathrm{b}} \widehat{{\boldsymbol v}}_k r_{\mathrm{u}k} \right)^*} \tag{63} \\ &=& \frac { \left|r_{\mathrm{u}k}\right|^2{\boldsymbol z}^{\rm T}_k\left(\boldsymbol{\varPhi}^{\frac{1}{2}}_k\right)^{\rm T}{\boldsymbol T}_{\mathrm{b}} {\widehat{{\boldsymbol C}}_{[k]}} {\boldsymbol T}^*_{\mathrm{b}} \left(\boldsymbol{\varPhi}^{\frac{1}{2}}_k\right)^* {\boldsymbol z}^*_k}{1+\left|r_{\mathrm{u}k}\right|^2 \left(\boldsymbol{\varPhi}^{1/2}_k{\boldsymbol z}_k - \boldsymbol{\varPhi}^{1/2}_{\mathrm{e}_k} {\boldsymbol e}_{\mathrm{v}k} \right)^{\rm T}{\boldsymbol T}_{\mathrm{b}} {\widehat{{\boldsymbol C}}_{[k]}} {\boldsymbol T}^*_{\mathrm{b}}\left(\boldsymbol{\varPhi}^{1/2}_k{\boldsymbol z}_k - \boldsymbol{\varPhi}^{1/2}_{\mathrm{e}_k} {\boldsymbol e}_{\mathrm{v}k} \right)^*} \\ & & + \frac { \left|r_{\mathrm{u}k}\right|^2{\boldsymbol z}^{\rm T}_k\left(\boldsymbol{\varPhi}^{\frac{1}{2}}_k\right)^{\rm T}{\boldsymbol T}_{\mathrm{b}} {\widehat{{\boldsymbol C}}_{[k]}} {\boldsymbol T}^*_{\mathrm{b}} \left(\boldsymbol{\varPhi}^{\frac{1}{2}}_{{\boldsymbol e}_{\mathrm{v}k}}\right)^* {\boldsymbol e}^*_k}{1+\left|r_{\mathrm{u}k}\right|^2\left(\boldsymbol{\varPhi}^{1/2}_k{\boldsymbol z}_k - \boldsymbol{\varPhi}^{1/2}_{\mathrm{e}_k} {\boldsymbol e}_{\mathrm{v}k} \right)^{\rm T}{\boldsymbol T}_{\mathrm{b}} {\widehat{{\boldsymbol C}}_{[k]}} {\boldsymbol T}^*_{\mathrm{b}} \left(\boldsymbol{\varPhi}^{1/2}_k{\boldsymbol z}_k - \boldsymbol{\varPhi}^{1/2}_{\mathrm{e}_k} {\boldsymbol e}_{\mathrm{v}k} \right)^*}, \tag{64} \end{eqnarray} where $\widehat{{\boldsymbol~W}}_{[k]}=[\sum_{j\neq~k}\left({\boldsymbol~T}_{\mathrm{b}}~{\boldsymbol~v}_j~r_{\mathrm{u}j}~\right)^*\left({\boldsymbol~T}_{\mathrm{b}}~{\boldsymbol~v}_j~r_{\mathrm{u}j}~\right)^{\rm~T}+~{\boldsymbol~C}_{\rm~E}~+~\frac~{K}{\rho_{\mathrm{d}}}{\boldsymbol~I}_N]^{-1}$ and $\widehat{{\boldsymbol~C}}_{[k]}~=~N\widehat{{\boldsymbol~W}}_{[k]}$. Eq. (64) is due to substituting (9) and $\widehat{{\boldsymbol~v}}_k~=~\sqrt{N}(\boldsymbol{\varPhi}^{1/2}_k{\boldsymbol~z}_k~-~\boldsymbol{\varPhi}^{1/2}_{\mathrm{e}_k}~{\boldsymbol~e}_{\mathrm{v}k}~)$ into (63).

By applying Lemma 14.2 of [34] and Lemma 5 of [32], \begin{eqnarray}& & {\boldsymbol z}^{\rm T}_k\left(\boldsymbol{\varPhi}^{\frac{1}{2}}_k\right)^{\rm T}{\boldsymbol T}_{\mathrm{b}} {\widehat{{\boldsymbol C}}_{[k]}} {\boldsymbol T}^*_{\mathrm{b}} \left(\boldsymbol{\varPhi}^{\frac{1}{2}}_k\right)^* {\boldsymbol z}^*_k -\frac{1}{N}\mathrm{tr}\left\{{\boldsymbol T}^*_{\mathrm{b}} \boldsymbol\varPhi^{\rm T}_k {\boldsymbol T}_{\mathrm{b}} \widehat{{\boldsymbol C}}_{[k]}\right\} \xrightarrow[N\rightarrow\infty]{\rm a.s.}0 , \tag{65} \\ & & {\boldsymbol e}^{\rm T}_k\left(\boldsymbol{\varPhi}^{\frac{1}{2}}_{\mathrm{e}_k}\right)^{\rm T}{\boldsymbol T}_{\mathrm{b}} {\widehat{{\boldsymbol C}}_{[k]}} {\boldsymbol T}^*_{\mathrm{b}} \left(\boldsymbol{\varPhi}^{\frac{1}{2}}_{\mathrm{e}_k}\right)^* {\boldsymbol e}^*_k -\frac{1}{N}\mathrm{tr}\left\{{\boldsymbol T}^*_{\mathrm{b}} \boldsymbol\varPhi^{\rm T}_{\mathrm{e}_k} {\boldsymbol T}_{\mathrm{b}} \widehat{{\boldsymbol C}}_{[k]}\right\} \xrightarrow[N\rightarrow\infty]{\rm a.s.}0 , \tag{66} \\ & & {{\boldsymbol z}}^{\rm T}_k\left(\boldsymbol{\varPhi}^{\frac{1}{2}}_k\right)^{\rm T}{\boldsymbol T}_{\mathrm{b}} {\widehat{{\boldsymbol C}}_{[k]}} {\boldsymbol T}^*_{\mathrm{b}} \left(\boldsymbol{\varPhi}^{\frac{1}{2}}_k\right)^* {{\boldsymbol e}}^*_k \xrightarrow[N\rightarrow\infty]{\rm a.s.}0. \tag{67} \end{eqnarray}

Furthermore, we have \begin{equation}\left(\boldsymbol{\varPhi}^{1/2}_k{\boldsymbol z}_k - \boldsymbol{\varPhi}^{1/2}_{\mathrm{e}_k} {\boldsymbol e}_{\mathrm{v}k} \right)^{\rm T}{\boldsymbol T}_{\mathrm{b}} {\widehat{{\boldsymbol C}}_{[k]}} {\boldsymbol T}^*_{\mathrm{b}}\left(\boldsymbol{\varPhi}^{1/2}_k{\boldsymbol z}_k - \boldsymbol{\varPhi}^{1/2}_{\mathrm{e}_k} {\boldsymbol e}_{\mathrm{v}k} \right)^* - \frac{1}{N}\mathrm{tr}\left\{ {\boldsymbol T}^*_{\mathrm{b}} \boldsymbol{\varPhi}^{\rm T}_{\mathrm{eq}_k} {\boldsymbol T}_{\mathrm{b}}\widehat{{\boldsymbol C}}_{[k]}\right\}\xrightarrow[N\rightarrow\infty]{\rm a.s.}0, \tag{68}\end{equation} where $\boldsymbol\varPhi_{\mathrm{eq}_k}~=~\boldsymbol\varPhi_{k}~+\boldsymbol\varPhi_{\mathrm{e}_k}$.

With Lemma 14.3 of [34] we have \begin{eqnarray}& & \frac{1}{N}\mathrm{tr}\left\{{\boldsymbol T}^*_{\mathrm{b}} \boldsymbol\varPhi^{\rm T}_k {\boldsymbol T}_{\mathrm{b}} \widehat{{\boldsymbol C}}_{[k]}\right\}-\frac{1}{N}\mathrm{tr}\left\{{\boldsymbol T}^*_{\mathrm{b}} \boldsymbol\varPhi^{\rm T}_k {\boldsymbol T}_{\mathrm{b}} \widehat{{\boldsymbol C}}\right\}\xrightarrow[N\rightarrow\infty]{\rm a.s.}0, \tag{69} \\ & & \frac{1}{N}\mathrm{tr}\left\{{\boldsymbol T}^*_{\mathrm{b}} \boldsymbol\varPhi^{\rm T}_{\mathrm{eq}_k} {\boldsymbol T}_{\mathrm{b}} \widehat{{\boldsymbol C}}_{[k]}\right\}-\frac{1}{N}\mathrm{tr}\left\{{\boldsymbol T}^*_{\mathrm{b}} \boldsymbol\varPhi^{\rm T}_{\mathrm{eq}_k} {\boldsymbol T}_{\mathrm{b}} \widehat{{\boldsymbol C}}\right\}\xrightarrow[N\rightarrow\infty]{\rm a.s.}0. \tag{70} \end{eqnarray}

By applying Theorem 1 of [33], we obtain \begin{eqnarray}& & \frac{1}{N}\mathrm{tr}\left\{{\boldsymbol T}^*_{\mathrm{b}} \boldsymbol\varPhi^{\rm T}_k {\boldsymbol T}_{\mathrm{b}}\widehat{{\boldsymbol C}}\right\} - \frac{1}{N}\mathrm{tr}\left\{ {\boldsymbol T}^*_{\mathrm{b}} \boldsymbol\varPhi^{\rm T}_k {\boldsymbol T}_{\mathrm{b}}\boldsymbol{\varPsi} \right\}\xrightarrow[N\rightarrow\infty]{\rm a.s.}0, \tag{71} \\ & & \frac{1}{N}\mathrm{tr}\left\{{\boldsymbol T}^*_{\mathrm{b}} \boldsymbol\varPhi^{\rm T}_{\mathrm{eq}_k} {\boldsymbol T}_{\mathrm{b}}\widehat{{\boldsymbol C}}\right\} - \frac{1}{N}\mathrm{tr}\left\{ {\boldsymbol T}^*_{\mathrm{b}} \boldsymbol\varPhi^{\rm T}_{\mathrm{eq}_k} {\boldsymbol T}_{\mathrm{b}}\boldsymbol{\varPsi} \right\}\xrightarrow[N\rightarrow\infty]{\rm a.s.}0, \tag{72} \end{eqnarray} where $\boldsymbol{\varPsi}$ is defined in (26). Then we have \begin{eqnarray}& & {\boldsymbol z}^{\rm T}_k\left(\boldsymbol{\varPhi}^{\frac{1}{2}}_k\right)^{\rm T}{\boldsymbol T}_{\mathrm{b}} {\widehat{{\boldsymbol C}}_{[k]}} {\boldsymbol T}^*_{\mathrm{b}} \left(\boldsymbol{\varPhi}^{\frac{1}{2}}_k\right)^* {\boldsymbol z}^*_k - \frac{1}{N}\mathrm{tr}\left\{ {\boldsymbol T}^*_{\mathrm{b}} \boldsymbol\varPhi^{\rm T}_k {\boldsymbol T}_{\mathrm{b}}\boldsymbol{\varPsi} \right\}\xrightarrow[N\rightarrow\infty]{\rm a.s.}0, \tag{73} \\ & & \widehat{{\boldsymbol g}}^{\rm T}_k{\boldsymbol T}_{\mathrm{b}} {\widehat{{\boldsymbol C}}_{[k]}} {\boldsymbol T}^*_{\mathrm{b}}\widehat{{\boldsymbol g}}^*_k - \frac{1}{N}\mathrm{tr}\left\{ {\boldsymbol T}^*_{\mathrm{b}} \boldsymbol{\varPhi}^{\rm T}_{\mathrm{eq}_k} {\boldsymbol T}_{\mathrm{b}}\boldsymbol{\varPsi} \right\}\xrightarrow[N\rightarrow\infty]{\rm a.s.}0, \tag{74} \end{eqnarray} where $\widehat{{\boldsymbol~g}}_k~=~\boldsymbol{\varPhi}^{1/2}_k{\boldsymbol~z}_k~+~\boldsymbol{\varPhi}^{1/2}_{\mathrm{e}_k}~{\boldsymbol~e}_{\mathrm{v}k}$.

Now we obtain the deterministic equivalent for desired signal power, \begin{equation} \left({\boldsymbol T}_{\mathrm{b}} {\boldsymbol v}_k r_{\mathrm{u}k} \right)^{\rm T} \widehat{{\boldsymbol W}} \left({\boldsymbol T}_{\mathrm{b}} \widehat{{\boldsymbol v}}_k r_{\mathrm{u}k} \right)^* - \frac{\left|r_{\mathrm{u}k}\right|^2 m^\circ_k}{1 + \left|r_{\mathrm{u}k}\right|^2 m^\circ_{\mathrm{eq}_k}}\xrightarrow[N\rightarrow\infty]{\rm a.s.} 0, \tag{75}\end{equation} where $~m^\circ_k~=~\frac{1}{N}~\mathrm{tr}\left\{{\boldsymbol~T}^*_{\mathrm{b}}~\boldsymbol\varPhi^{\rm~T}_k~{\boldsymbol~T}_{\mathrm{b}}~\boldsymbol{\varPsi}~\right\}$ and $~m^\circ_{\mathrm{eq}_k}~=~\frac{1}{N}~\mathrm{tr}\{{\boldsymbol~T}^*_{\mathrm{b}}~\boldsymbol{\varPhi}^{\rm~T}_{\mathrm{eq}_k}~{\boldsymbol~T}_{\mathrm{b}}~\boldsymbol{\varPsi}~\}$.

(2) Deterministic equivalent for $|\left({\boldsymbol~T}_{\mathrm{b}}{\boldsymbol~v}_kr_{uk}~\right)^{\rm~T}~\widehat{{\boldsymbol~W}}~\left({\boldsymbol~T}_{\mathrm{b}}\widehat{{\boldsymbol~v}}_j~r_{\mathrm{u}j}\right)^*~|^2$.

The undesired signal power can be rewritten as \begin{eqnarray}\left|\left({\boldsymbol T}_{\mathrm{b}} {\boldsymbol v}_k r_{\mathrm{u}k} \right)^{\rm T} \widehat{{\boldsymbol W}} \left({\boldsymbol T}_{\mathrm{b}} \widehat{{\boldsymbol v}}_j r_{\mathrm{u}j} \right)^*\right |^2 & & = \frac{\left({\boldsymbol T}_{\mathrm{b}}\widehat{{\boldsymbol v}}_j r_{\mathrm{u}j} \right)^{\rm T} \widehat{{\boldsymbol W}}_{[j]} \left({\boldsymbol T}_{\mathrm{b}} {\boldsymbol v}_k r_{\mathrm{u}k} \right)^*\left({\boldsymbol T}_{\mathrm{b}} {\boldsymbol v}_k r_{\mathrm{u}k} \right)^{\rm T} \widehat{{\boldsymbol W}}_{[j]} \left({\boldsymbol T}_{\mathrm{b}} \widehat{{\boldsymbol v}}_j r_{\mathrm{u}j} \right)^*} {\left[1+\left|r_{\mathrm{u}j}\right|^2\widehat{{\boldsymbol g}}^{\rm T}_k{\boldsymbol T}_{\mathrm{b}} {\widehat{{\boldsymbol C}}_{[j]}} {\boldsymbol T}^*_{\mathrm{b}} \widehat{{\boldsymbol g}}^*_k\right]^2} \tag{76} \\ & & \xrightarrow[N\rightarrow\infty]{\rm a.s.} \frac{\left|r_{\mathrm{u}j}\right|^2\left|r_{\mathrm{u}k}\right|^2} {N(1+\left|r_{\mathrm{u}j}\right|^2 m^\circ_{\mathrm{eq}_j})^2} {{\boldsymbol z}^{\rm T}_k\left(\boldsymbol{\varPhi}^{\frac{1}{2}}_k\right)^{\rm T}{\boldsymbol T}_{\mathrm{b}} \widehat{{\boldsymbol C}}_{[j]} {\boldsymbol T}^*_{\mathrm{b}}\boldsymbol{\varPhi}^{\rm T}_{\mathrm{eq}_j} {\boldsymbol T}_{\mathrm{b}}\widehat{{\boldsymbol C}}_{[j]}{\boldsymbol T}^*_{\mathrm{b}} \left(\boldsymbol{\varPhi}^{\frac{1}{2}}_k\right)^* {\boldsymbol z}^*_k}, \tag{77} \end{eqnarray} where Eq. (76) is due to applying Lemma 1 of 20 (matrix inversion lemma) for term $\widehat{{\boldsymbol~W}}$; Eq. (77) is obtained by applying (74) for the denominator.

By applying Lemma 2 of 20 for term $\widehat{{\boldsymbol~C}}_{[j]}$, we can write \begin{eqnarray}& & {{\boldsymbol z}^{\rm T}_k\left(\boldsymbol{\varPhi}^{\frac{1}{2}}_k\right)^{\rm T}{\boldsymbol T}_{\mathrm{b}} \widehat{{\boldsymbol C}}_{[j]} {\boldsymbol T}^*_{\mathrm{b}} \boldsymbol{\varPhi}^{\rm T}_{\mathrm{eq}_j} {\boldsymbol T}_{\mathrm{b}}\widehat{{\boldsymbol C}}_{[j]}{\boldsymbol T}^*_{\mathrm{b}} \left(\boldsymbol{\varPhi}^{\frac{1}{2}}_k\right)^* {\boldsymbol z}^*_k} \\ & & = {{\boldsymbol z}^{\rm T}_k\left(\boldsymbol{\varPhi}^{\frac{1}{2}}_k\right)^{\rm T}{\boldsymbol T}_{\mathrm{b}} \widehat{{\boldsymbol C}}_{[jk]} {\boldsymbol T}^*_{\mathrm{b}} \boldsymbol{\varPhi}^{\rm T}_{\mathrm{eq}_j} {\boldsymbol T}_{\mathrm{b}}\widehat{{\boldsymbol C}}_{[jk]}{\boldsymbol T}^*_{\mathrm{b}} \left(\boldsymbol{\varPhi}^{\frac{1}{2}}_k\right)^* {\boldsymbol z}^*_k} \\ & & + \frac{\left|r_{\mathrm{u}k}\right|^4\left|{\boldsymbol z}^{\rm T}_k\left(\boldsymbol{\varPhi}^{\frac{1}{2}}_k\right)^{\rm T}{\boldsymbol T}_{\mathrm{b}} \widehat{{\boldsymbol C}}_{[jk]}{\boldsymbol T}^*_{\mathrm{b}} \left(\boldsymbol{\varPhi}^{1/2}_k{\boldsymbol z}_k - \boldsymbol{\varPhi}^{1/2}_{\mathrm{e}_k} {\boldsymbol e}_{\mathrm{v}k} \right)^*\right|^2 \widehat{{\boldsymbol g}}^{\rm T}_k{\boldsymbol T}_{\mathrm{b}} \widehat{{\boldsymbol C}}_{[jk]} {\boldsymbol T}^*_{\mathrm{b}} \boldsymbol{\varPhi}^{\rm T}_{\mathrm{eq}_j} {\boldsymbol T}_{\mathrm{b}}\widehat{{\boldsymbol C}}_{[jk]}{\boldsymbol T}^*_{\mathrm{b}} \widehat{{\boldsymbol g}}^*_k} {{\left[1+\left|r_{\mathrm{u}k}\right|^2\widehat{{\boldsymbol g}}_k^{\rm T}{\boldsymbol T}_{\mathrm{b}} \widehat{{\boldsymbol C}}_{[jk]}{\boldsymbol T}^*_{\mathrm{b}}\widehat{{\boldsymbol g}}_k^*\right]^2}} \\ & & - 2\Re\left\{\frac {\left|r_{\mathrm{u}k}\right|^2 {\boldsymbol z}^{\rm T}_k\left(\boldsymbol{\varPhi}^{\frac{1}{2}}_k\right)^{\rm T}{\boldsymbol T}_{\mathrm{b}} \widehat{{\boldsymbol C}}_{[jk]}{\boldsymbol T}^*_{\mathrm{b}} \widehat{{\boldsymbol g}}^*_k\widehat{{\boldsymbol g}}^{\rm T}_k{\boldsymbol T}_{\mathrm{b}} \widehat{{\boldsymbol C}}_{[jk]} {\boldsymbol T}^*_{\mathrm{b}} \boldsymbol{\varPhi}^{\rm T}_{\mathrm{eq}_j} {\boldsymbol T}_{\mathrm{b}}\widehat{{\boldsymbol C}}_{[jk]}{\boldsymbol T}^*_{\mathrm{b}} \left(\boldsymbol{\varPhi}^{\frac{1}{2}}_k\right)^* {\boldsymbol z}^*_k} {1+\left|r_{\mathrm{u}k}\right|^2\widehat{{\boldsymbol g}}_k^{\rm T}{\boldsymbol T}_{\mathrm{b}} \widehat{{\boldsymbol C}}_{[jk]}{\boldsymbol T}^*_{\mathrm{b}} \widehat{{\boldsymbol g}}_k^*} \right\}. \tag{78} \end{eqnarray}

By applying Lemma 14.2 of [34] and Theorem 1 of [33] as steps for deriving (65) similarly, we can obtain \begin{eqnarray}& & {\boldsymbol z}^{\rm T}_k\left(\boldsymbol{\varPhi}^{\frac{1}{2}}_k\right)^{\rm T}{\boldsymbol T}_{\mathrm{b}} \widehat{{\boldsymbol C}}_{[jk]} {\boldsymbol T}^*_{\mathrm{b}} \boldsymbol{\varPhi}^{\rm T}_{\mathrm{eq}_j} {\boldsymbol T}_{\mathrm{b}}\widehat{{\boldsymbol C}}_{[jk]}{\boldsymbol T}^*_{\mathrm{b}} \left(\boldsymbol{\varPhi}^{\frac{1}{2}}_k\right)^* {\boldsymbol z}^*_k - \frac{1}{N} \mathrm{tr} \left\{{\boldsymbol T}^*_{\mathrm{b}} \boldsymbol{\varPhi}^{\rm T}_k {\boldsymbol T}_{\mathrm{b}}\widehat{{\boldsymbol C}} {\boldsymbol T}^*_{\mathrm{b}} \boldsymbol{\varPhi}^{\rm T}_{\mathrm{eq}_j} {\boldsymbol T}_{\mathrm{b}}\widehat{{\boldsymbol C}}\right\}\xrightarrow[N\rightarrow\infty]{\rm a.s.}0, \tag{79} \\ & & \widehat{{\boldsymbol g}}^{\rm T}_k{\boldsymbol T}_{\mathrm{b}} \widehat{{\boldsymbol C}}_{[jk]} {\boldsymbol T}^*_{\mathrm{b}} \boldsymbol{\varPhi}^{\rm T}_{\mathrm{eq}_j} {\boldsymbol T}_{\mathrm{b}}\widehat{{\boldsymbol C}}_{[jk]}{\boldsymbol T}^*_{\mathrm{b}} \widehat{{\boldsymbol g}}^*_k - \frac{1}{N} \mathrm{tr} \left\{{\boldsymbol T}^*_{\mathrm{b}} \boldsymbol{\varPhi}^{\rm T}_{\mathrm{eq}_k} {\boldsymbol T}_{\mathrm{b}}\widehat{{\boldsymbol C}} {\boldsymbol T}^*_{\mathrm{b}} \boldsymbol{\varPhi}^{\rm T}_{\mathrm{eq}_j} {\boldsymbol T}_{\mathrm{b}}\widehat{{\boldsymbol C}}\right\}\xrightarrow[N\rightarrow\infty]{\rm a.s.}0, \tag{80} \\ & & {\boldsymbol z}^{\rm T}_k\left(\boldsymbol{\varPhi}^{\frac{1}{2}}_k\right)^{\rm T}{\boldsymbol T}_{\mathrm{b}} \widehat{{\boldsymbol C}}_{[jk]} {\boldsymbol T}^*_{\mathrm{b}} \boldsymbol{\varPhi}^{\rm T}_{\mathrm{eq}_j} {\boldsymbol T}_{\mathrm{b}}\widehat{{\boldsymbol C}}_{[jk]}{\boldsymbol T}^*_{\mathrm{b}} \widehat{{\boldsymbol g}}^*_k - \frac{1}{N} \mathrm{tr} \left\{{\boldsymbol T}^*_{\mathrm{b}} \boldsymbol{\varPhi}^{\rm T}_k {\boldsymbol T}_{\mathrm{b}}\widehat{{\boldsymbol C}} {\boldsymbol T}^*_{\mathrm{b}} \boldsymbol{\varPhi}^{\rm T}_{\mathrm{eq}_j} {\boldsymbol T}_{\mathrm{b}}\widehat{{\boldsymbol C}}\right\}\xrightarrow[N\rightarrow\infty]{\rm a.s.}0, \tag{81} \end{eqnarray} where Eq. (81) is similar as the numerator in (75).

By applying Theorem 2 of [33] for which ${\boldsymbol~Q}~=~{\boldsymbol~T}^*_{\mathrm{b}}~\boldsymbol{\varPhi}^{\rm~T}_{j}~{\boldsymbol~T}_{\mathrm{b}}$ and ${\boldsymbol~D}~=~{\boldsymbol~T}^*_{\mathrm{b}}~\boldsymbol{\varPhi}^{\rm~T}_{\mathrm{eq}_j}~{\boldsymbol~T}_{\mathrm{b}}$, we have \begin{equation} \frac{1}{N} \mathrm{tr} \left\{{\boldsymbol T}^*_{\mathrm{b}} \boldsymbol{\varPhi}^{\rm T}_k {\boldsymbol T}_{\mathrm{b}}\widehat{{\boldsymbol C}} {\boldsymbol T}^*_{\mathrm{b}}\boldsymbol{\varPhi}^{\rm T}_{\mathrm{eq}_j} {\boldsymbol T}_{\mathrm{b}}\widehat{{\boldsymbol C}}\right\}- \frac{1}{N} \mathrm{tr} \left\{{\boldsymbol T}^*_{\mathrm{b}} \boldsymbol{\varPhi}^{\rm T}_k {\boldsymbol T}_{\mathrm{b}}\boldsymbol{\varPsi}^{\prime}_{\mathrm{T}j}\right\}\xrightarrow[N\rightarrow\infty]{\rm a.s.}0, \tag{82}\end{equation} where $\boldsymbol{\varPsi}^{\prime}_{\mathrm{T}j}$ is given by (31). Similarly, we have \begin{equation} \frac{1}{N} \mathrm{tr} \left\{{\boldsymbol T}^*_{\mathrm{b}} \boldsymbol{\varPhi}^{\rm T}_{\mathrm{eq}_k} {\boldsymbol T}_{\mathrm{b}}\widehat{{\boldsymbol C}} {\boldsymbol T}^*_{\mathrm{b}}\boldsymbol{\varPhi}^{\rm T}_{\mathrm{eq}_j} {\boldsymbol T}_{\mathrm{b}}\widehat{{\boldsymbol C}}\right\}- \frac{1}{N} \mathrm{tr} \left\{{\boldsymbol T}^*_{\mathrm{b}} \boldsymbol{\varPhi}^{\rm T}_{\mathrm{eq}_k} {\boldsymbol T}_{\mathrm{b}}\boldsymbol{\varPsi}^{\prime}_{\mathrm{T}j}\right\}\xrightarrow[N\rightarrow\infty]{\rm a.s.}0. \tag{83}\end{equation}

By substituting (67), (73), (79)–(83) into (78), then we obtain the deterministic equivalent for undesired signal power, \begin{equation}\left|\left({\boldsymbol T}_{\mathrm{b}}{\boldsymbol v}_kr_{uk} \right)^{\rm T} \widehat{{\boldsymbol W}} \left({\boldsymbol T}_{\mathrm{b}}\widehat{{\boldsymbol v}}_j r_{\mathrm{u}j}\right)^*\right |^2 \xrightarrow[N\rightarrow\infty]{\rm a.s.} \frac{\left|r_{\mathrm{u}j}\right|^2\left|r_{\mathrm{u}k}\right|^2} {N\left(1+\left|r_{\mathrm{u}j}\right|^2 m^\circ_{\mathrm{eq}_j}\right)^2} \left[\mu^\circ_{\mathrm{T}j} + \frac{\left|r_{\mathrm{u}k}\right|^4\left|m^\circ_k\right|^2\mu^\circ_{\mathrm{Te}j}}{\left(1+\left|r_{\mathrm{u}k}\right|^2m^\circ_{\mathrm{eq}_k}\right)^2} -\frac{2\left|r_{\mathrm{u}k}\right|^2\Re\left\{m^\circ_k\mu^\circ_{\mathrm{T}j}\right\}}{\left(1+\left|r_{\mathrm{u}k}\right|^2m^\circ_{\mathrm{eq}_k}\right)}\right], \tag{84}\end{equation} where $\mu^\circ_{\mathrm{T}j}$ is given by (28) and $\mu^\circ_{\mathrm{Te}j}$ is given by (29).

(3) Deterministic equivalent for $\xi$.

The term $\xi$ can be rewritten as \begin{eqnarray}\xi &=& \sum^K_{k=1} \left({\boldsymbol T}_{\mathrm{b}} \widehat{{\boldsymbol v}}_k r_{\mathrm{u}k} \right)^{\rm T} \widehat{{\boldsymbol W}}^{2} \left({\boldsymbol T}_{\mathrm{b}} \widehat{{\boldsymbol v}}_k r_{\mathrm{u}k} \right)^* =\frac{1}{N} \sum^{K}_{k=1} \frac{\left|r_{\mathrm{u}k}\right|^2\widehat{{\boldsymbol g}}^{\rm T}_k{\boldsymbol T}_{\mathrm{b}} \widehat{{\boldsymbol C}}^{2}_{[k]}{\boldsymbol T}^*_{\mathrm{b}} \widehat{{\boldsymbol g}}^*_k} {\left[1 + \left|r_{\mathrm{u}k}\right|^2\widehat{{\boldsymbol g}}^{\rm T}_k{\boldsymbol T}_{\mathrm{b}} \widehat{{\boldsymbol C}}_{[k]}{\boldsymbol T}^*_{\mathrm{b}} \widehat{{\boldsymbol g}}^*_k \right]^2}, \tag{85} \end{eqnarray} where Eq. (85) is due to matrix inversion lemma.

By applying Theorem 2 of [33] as the steps for deriving (82) for which ${\boldsymbol~Q}~=~{\boldsymbol~T}^*_{\mathrm{b}}~\boldsymbol{\varPhi}^{\rm~T}_{\mathrm{eq}_k}~{\boldsymbol~T}_{\mathrm{b}}$ and ${\boldsymbol~D}={\boldsymbol~I}_N$, we have \begin{equation} \widehat{{\boldsymbol g}}^{\rm T}_k{\boldsymbol T}_{\mathrm{b}} \widehat{{\boldsymbol C}}^{2}_{[k]}{\boldsymbol T}^*_{\mathrm{b}} \widehat{{\boldsymbol g}}^*_k - \frac{1}{N} \mathrm{tr} \left\{{\boldsymbol T}^*_{\mathrm{b}}\boldsymbol{\varPhi}^{\rm T}_{\mathrm{eq}_k} {\boldsymbol T}_{\mathrm{b}}\boldsymbol{\varPsi}^{\prime}_I\right\}\xrightarrow[N\rightarrow\infty]{\rm a.s.}0, \tag{86}\end{equation} where $\boldsymbol{\varPsi}^{\prime}_I$ is given by (31).

By substituting (74), (86) into (85), we obtain the deterministic equivalent for $\xi$, \begin{equation}\xi - \frac{1}{N} \sum^{K}_{k = 1}\frac{\left|r_{\mathrm{u}k}\right|^2\mu^\circ_{\mathrm{I}k}}{\left[1+\left|r_{\mathrm{u}k}\right|^2 m^\circ_{\mathrm{eq}_k}\right]^2}\xrightarrow[N\rightarrow\infty]{\rm a.s.}0, \tag{87}\end{equation} where $\mu^\circ_{\mathrm{I}k}$ is given by (30).

Until this point, we have obtained all the deterministic equivalents of the three terms. Combining the results, the deterministic equivalent of the SINR of UT $k$ is given by (20).


References

[1] Marzetta T L. Noncooperative Cellular Wireless with Unlimited Numbers of Base Station Antennas. IEEE Trans Wireless Commun, 2010, 9: 3590-3600 CrossRef Google Scholar

[2] Rusek F, Persson D, Lau B K, et al. Scaling up MIMO: opportunities and challenges with very large arrays. IEEE Signal Process Mag, 2013, 30: 40--60. Google Scholar

[3] Larsson E G, Edfors O, Tufvesson F, et al. Massive MIMO for next generation wireless systems. IEEE Commun Mag, 2014, 2: 186--195. Google Scholar

[4] Gao X, Edfors O, Fredrik R, et al. Linear pre-coding performance in measured very-large MIMO channels. In: Proceedings of IEEE Vehicular Technology Conference (VTC Fall), San Francisco, 2011. 1--5. Google Scholar

[5] Marzetta T L. How much training is required for multiuser MIMO. In: Proceedings of the 40th Asilomar Conference on Signals, Systems and Computers, Pacific Grove, 2006. 359--363. Google Scholar

[6] 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

[7] Schenk T. RF Imperfections in High-rate Wireless Systems: Impact and Digital Compensation. Berlin: Springer, 2008. Google Scholar

[8] Zhang W, Ren H, Pan C. Large-Scale Antenna Systems With UL/DL Hardware Mismatch: Achievable Rates Analysis and Calibration. IEEE Trans Commun, 2015, 63: 1216-1229 CrossRef Google Scholar

[9] Bjornson E, Hoydis J, Kountouris M. Massive MIMO Systems With Non-Ideal Hardware: Energy Efficiency, Estimation, and Capacity Limits. IEEE Trans Inform Theor, 2014, 60: 7112-7139 CrossRef Google Scholar

[10] Bourdoux A. Non-reciprocal transceivers in OFDM/SDMA systems: impact and mitigation. In: Proceedings of Radio and Wireless Conference, Boston, 2003. 183--186. Google Scholar

[11] Guillaud M, Slock D T M, Knopp R. A practical method for wireless channel reciprocity exploitation through relative calibration. In: Proceedings of the 8th International Symposium on Signal Processing and Its Applications, Sydney, 2005. 403--406. Google Scholar

[12] Shi J, Luo Q L, You L M. An efficient method for enhancing TDD over the air reciprocity calibration. In: Proceedings of IEEE Wireless Communications and Networking Conference, Cancun, 2011. 339--344. Google Scholar

[13] Han S Q, Yang C Y, Wang G, et al. Coordinated multipoint transmission strategies for TDD systems with non-ideal channel reciprocity. IEEE Trans Commun, 2012, 10: 4256--4270. Google Scholar

[14] Guillaud M, Kaltenberger F. Towards practical channel reciprocity exploitation: relative calibration in the presence of frequency offset. In: Proceedings of IEEE Wireless Communications and Networking Conference, Shanghai, 2013. 2525--2530. Google Scholar

[15] Rogalin R, Bursalioglu O Y, Papadopoulos H, et al. Scalable synchronization and reciprocity calibration for distributed multiuser MIMO. IEEE Trans Wirel Commun, 2014, 4: 1815--1831. Google Scholar

[16] Su L Y, Yang C Y, Wang G, et al. Retrieving channel reciprocity for coordinated multi-point transmission with joint processing. IEEE Trans Commun, 2014, 5: 1541--1553. Google Scholar

[17] Shepard C, Yu H, Anand N, et al. Argos: practical many-antenna base stations. In: Proceedings of the 18th Annual International Conference on Mobile Computing and Networking, Istanburl, 2012. 53--64. Google Scholar

[18] Kaltenberger F. Relative channel reciprocity calibration in MIMO/TDD systems. In: Proceedings of IEEE Future Network and Mobile Summit, Florence, 2010. 1--10. Google Scholar

[19] Yang H, Marzetta T L. Performance of Conjugate and Zero-Forcing Beamforming in Large-Scale Antenna Systems. IEEE J Sel Areas Commun, 2013, 31: 172-179 CrossRef Google Scholar

[20] Peel C B, Hochwald B M, Swindlehurst A L. A Vector-Perturbation Technique for Near-Capacity Multiantenna Multiuser Communication-Part I: Channel Inversion and Regularization. IEEE Trans Commun, 2005, 53: 195-202 CrossRef Google Scholar

[21] Joham M, Utschick W, Nossek J A. Linear transmit processing in MIMO communications systems. IEEE Trans Signal Process, 2005, 53: 2700-2712 CrossRef ADS Google Scholar

[22] Costa M. Writing on dirty paper (Corresp.). IEEE Trans Inform Theor, 1983, 29: 439-441 CrossRef Google Scholar

[23] Hochwald B M, Peel C B, Swindlehurst A L. A Vector-Perturbation Technique for Near-Capacity Multiantenna Multiuser Communication- Part II: Perturbation. IEEE Trans Commun, 2005, 53: 537-544 CrossRef Google Scholar

[24] Björnson E, Larsson E G, Marzetta T L. Massive MIMO: ten myths and one critical question. IEEE Commun Mag, 2016, 54: 114--123. Google Scholar

[25] Wei H, Wang D M, Wang J Z, et al. Impact of RF mismatches on the performance of massive MIMO systems with ZF precoding. Sci China Inf Sci, 2016, 59: 022302. Google Scholar

[26] Meng X, Jiang B, Gao X Q. Efficient co-channel interference suppression in MIMO-OFDM systems. Sci China Inf Sci, 2015, 58: 022301. Google Scholar

[27] Kay S M. Fundamentals of Statistical Signal Processing: Estimation Theory. Englewood Cliffs: Prentice Hall, 1993. Google Scholar

[28] You L, Gao X, Xia X G. Pilot Reuse for Massive MIMO Transmission over Spatially Correlated Rayleigh Fading Channels. IEEE Trans Wireless Commun, 2015, 14: 3352-3366 CrossRef Google Scholar

[29] Kermoal J P, Schumacher L, Pedersen I K, et al. A stochastic MIMO radio channel model with experimental validation. IEEE Trans Commun, 2002, 20: 1211--1226. Google Scholar

[30] Sayeed A M. Deconstructing multiantenna fading channels. IEEE Trans Signal Process, 2002, 50: 2563-2579 CrossRef ADS Google Scholar

[31] Weichselberger W, Herdin M, Ozcelik H. A stochastic MIMO channel model with joint correlation of both link ends. IEEE Trans Wireless Commun, 2006, 5: 90-100 CrossRef Google Scholar

[32] Wagner S, Couillet R, Debbah M. Large System Analysis of Linear Precoding in Correlated MISO Broadcast Channels Under Limited Feedback. IEEE Trans Inform Theor, 2012, 58: 4509-4537 CrossRef Google Scholar

[33] Hoydis J, Brink S T, Debbah M. Massive MIMO in the UL/DL of celluar networks: how many antennas do we need? IEEE J Sel Areas Commun, 2013, 31: 160--171. Google Scholar

[34] Couillet R, Debbah M. Random Matrix Methods for Wireless Communications. Cambridge: Cambridge University Press, 2011. Google Scholar

[35] Silverstein J W, Bai Z D. On the Empirical Distribution of Eigenvalues of a Class of Large Dimensional Random Matrices. J Multivariate Anal, 1995, 54: 175-192 CrossRef Google Scholar

[36] Jacks W C, Cox D C. Microwave Mobile Communications. New York: Wiley, 1994. Google Scholar

  • Figure 4

    (Color online) Comparison of the ergodic sum rates obtained by Monte Carlo simulations and the asymptotic approximation, results are shown versus the SNR with $N~=~128$ and $K~=~32$.

Copyright 2020 Science China Press Co., Ltd. 《中国科学》杂志社有限责任公司 版权所有

京ICP备18024590号-1       京公网安备11010102003388号