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SCIENCE CHINA Information Sciences, Volume 61, Issue 12: 122302(2018) https://doi.org/10.1007/s11432-018-9591-8

Energy- and spectral-efficiency of zero-forcing beamforming in massive MIMO systems with imperfect reciprocity calibration: bound and optimization

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  • ReceivedFeb 12, 2018
  • AcceptedSep 7, 2018
  • PublishedNov 14, 2018

Abstract

In a time-division duplex (TDD) system with massive multiple input multiple output (MIMO), channel reciprocity calibration (RC) is generally required in order tocope with the reciprocity mismatch between the uplink and downlink channel state information. Currently, evaluating the achievable spectral efficiency (SE) and energy efficiency (EE) of TDD massive MIMO systems with imperfect RC (IRC) mainly relies on exhausting Monte Carlo simulations and it is infeasible to precisely and concisely quantify the achievable SE and EE with IRC. In this study, a novel method is presented for tightly bounding the achievable SE of massive MIMO systems with zero-forcing beamforming under IRC On the basis of the analytical results, we demonstrate key insights for practical system design with IRC in three aspects: the scaling rule for interference power, saturation region of the SE, and the bound on the SE loss. Finally, the trade-off between spectral and energy efficiencies in the presence of IRC isdetermined with algorithms developed to optimize SE (EE) under a constrained EE (SE) value. The loss of optimal total SE and EE due to IRC is also quantified, which shows that the loss of optimal EE is more sensitive to IRC in a typical range of transmit power values.


Acknowledgment

This work was supported by National Natural Science Foundation of China (Grant Nos. 61531009, 61471108, 61771107, 61701075), the National Major Projects (Grant No. 2016ZX03001009), the Fund from the China Scholarship Council (Grant No. 201706070084), and the Fundamental Research Funds for the Central Universities.


Supplement

Appendix

Applying singular value decomposition to ${\boldsymbol{H}}_{\rm{ul}}^{\rm~T}$ yields ${\boldsymbol{H}}_{\rm{ul}}^{\rm~T}~=~{\boldsymbol{\tilde~Y}}\left(~{{\bf{\Sigma~}},{\bf{0}}}~\right){{\boldsymbol{\tilde~X}}^{\rm~H}}$, where ${\bf{\Sigma~}}~=~{\rm{diag}}\left(~{{\chi~_1},~\ldots~,{\chi~_K}}~\right)$ is a diagonal matrix containing the non-negative real singular values of ${\boldsymbol{H}}_{\rm{ul}}^{\rm~T}$. ${\boldsymbol{\tilde~Y}}$ and ${\boldsymbol{\tilde~X}}$ denote the left and right singular matrices for the corresponding singular values. Substituting ${\boldsymbol{H}}_{\rm{ul}}^{\rm~T}$, we obtain ${\boldsymbol{H}}_{\rm{ul}}^{\rm~T}~=~{\boldsymbol{\tilde~Y}}\left(~{{\bf{\Sigma~}},{\bf{0}}}~\right){{\boldsymbol{\tilde~X}}^{\rm~H}}$ into (10) yields \begin{eqnarray}{} \begin{array}{l} E\left[ {{{\left( {{\boldsymbol{u}}{{\boldsymbol{u}}^{\rm H}}} \right)}_{k,k}}} \right] = \frac{1}{K}{\rm E}\left\{ {{\rm{tr}}\left[ {{{{\boldsymbol{\tilde X}}}^{\rm H}}{{\hat {\boldsymbol D}}}_{\rm{J}}^{\rm{n}}{\boldsymbol{\tilde X}}\left( {\begin{array}{*{20}{c}} {{\Sigma ^{ - 2}}}&{\bf{0}} \\ {\bf{0}}&{\bf{0}} \end{array}} \right){{{\boldsymbol{\tilde X}}}^{\rm H}}{{\left( {{{\hat {\boldsymbol D}}}_{\rm{J}}^{\rm{n}}} \right)}\!^{\rm H}}\!{\boldsymbol{\tilde X}}\left( {\begin{array}{*{20}{c}} {{\Sigma ^2}}&{\bf{0}} \\ {\bf{0}}&{\bf{0}} \end{array}} \right)} \right]} \right\}. \end{array} \tag{31} \end{eqnarray}

Define an $M\times~M$ matrix $${\boldsymbol{A}} = {{\boldsymbol{\tilde X}}^{\rm H}}{{\hat {\boldsymbol D}}}_{\rm{J}}^{\rm{n}}{\boldsymbol{\tilde X}}\left( {\begin{array}{*{20}{c}} {{\Sigma ^{ - 2}}}&{\bf{0}}\\ {\bf{0}}&{\bf{0}} \end{array}} \right){{\boldsymbol{\tilde X}}^{\rm H}}{\left( {{{\hat {\boldsymbol D}}}_{\rm{J}}^{\rm{n}}} \right)^{\rm H}{\boldsymbol{\tilde X}}}.$$ One should note that ${\boldsymbol{H}}_{\rm{ul}}^{\rm~T}{\left(~{{\boldsymbol{H}}_{\rm{ul}}^{\rm~T}}~\right)^{\rm~H}}\approx~K{\bf{I}}_K$ for massive MIMO with $M\rightarrow~\infty$ and small HM magnitudes, i.e., ${\chi^2~_k}\approx~K$ Thus, \begin{eqnarray}{} {\rm E}\left[ {{{\left( {{\boldsymbol{u}}{{\boldsymbol{u}}^{\rm H}}} \right)}_{k,k}}} \right] \approx \frac{1}{K}{\rm E}\left\{{\rm{tr}}\left[{\boldsymbol{A}}\left({\begin{array}{*{20}{c}} {{K{\bf{I}}_K}}&{\bf{0}} \\ {\bf{0}}&{\bf{0}} \end{array}}\right)\right]\right\}= \frac{1}{K} \times K{\rm E}\left\{ {\sum\limits_{i = 1}^K {{{\boldsymbol{A}}_{i,i}}} } \right\} ={\rm E}\left\{ {\sum\limits_{i = 1}^K {{{\boldsymbol{A}}_{i,i}}} } \right\}, \tag{32} \end{eqnarray} where ${{\boldsymbol{A}}_{i,i}}$ denotes the $i$th diagonal element of ${{\boldsymbol{A}}}$.

Again, from ${\chi^2~_k}\approx~K$ it is known that ${\rm{tr}}\left(~{\boldsymbol{A}}~\right)~\approx~{\delta~_{\rm{e}}}$. As the elements of ${{{\hat~{\boldsymbol~D}}}_{\rm{J}}^{\rm{n}}}$ are ergodic and permutations of ${\chi^2~_k}$ can be random with equal probability, ${{{\boldsymbol{A}}_{i,i}}}$ contributes equally to ${\rm{tr}}\left(~{\boldsymbol{A}}~\right)$. Hence, \begin{eqnarray}{} {\rm E}\left\{ {\sum\limits_{i = 1}^K {{{\boldsymbol{A}}_{i,i}}} } \right\}\approx \frac{K}{M}{\rm{tr}}\left( {\boldsymbol{A}} \right) = \frac{{{K}{\delta _{\rm{e}}}}}{M}, \tag{33} \end{eqnarray} since ${\boldsymbol{A}}$ has $M$ elements on the diagonal. By substituting (33) and (SVDC1-v) into (9), one obtains ${\rm~E}\left(~{\xi~_k}{{\overline~{\cal~I}}_k}~\right)~\approx~{\xi~_k}\frac{{K~-~1}}{M}{\delta~_{\rm{e}}}$, which concludes the proof.

From ${\overline~{\cal~N}~_k}~=~{{\left\|~{\boldsymbol{W}}~\right\|_F^2}~\mathord{\left/ ~{\vphantom~{{\left\|~{\boldsymbol{W}}~\right\|_F^2}~{{{\left|~{{\mu~_k}}~\right|}^2}}}}~\right. ~\kern-\nulldelimiterspace}~{{{\left|~{{\mu~_k}}~\right|}^2}}}$, we have \begin{eqnarray}{} {\rm E}\left( {\overline {\cal N} _k} \right) = {\rm E}\left[ {{{\left| {\frac{{d_{\rm{ut}}^k}}{{d_{\rm{ur}}^k}}} \right|}^{\rm{2}}}{\rm{tr}}\left( {{{{\hat {\boldsymbol D}}}_{\rm{J}}}{\boldsymbol{\tilde W}}{{{\boldsymbol{\tilde W}}}^{\rm H}}{{\hat {\boldsymbol D}}}_{\rm{J}}^{\rm H}} \right)} \right] = {\rm E}\left [{{{\left| {\frac{{d_{\rm{ut}}^k}}{{d_{\rm{ur}}^k}}} \right|}^{\rm{2}}}}{\rm{tr}}\left( {{{\hat {\boldsymbol D}}}_{\rm{J}}^{\rm H}{{{\hat {\boldsymbol D}}}_{\rm{J}}}{\boldsymbol{\tilde W}}{{{\boldsymbol{\tilde W}}}^{\rm H}}} \right)\right]. \tag{34} \end{eqnarray} One should note that ${\rm{tr}}(~{{{{\hat~{\boldsymbol~D}}}_{\rm{J}}}^{\rm~H}{{{{\hat~{\boldsymbol~D}}}_{\rm{J}}}}{\boldsymbol{\tilde~W}}{{{\boldsymbol{\tilde~W}}}^{\rm~H}}}~)$ is a weighted summation of diagonal elements of ${\boldsymbol{\tilde~W}}{{{\boldsymbol{\tilde~W}}}^{\rm~H}}$ with weighting factor given in ${{{{\hat~{\boldsymbol~D}}}}_{\rm{J}}}^{\rm~H}{{{{\hat~{\boldsymbol~D}}}}_{\rm{J}}}$. The complicated uplink channel model indicated by (1) makes obtaining the mathematical expectations of (34) demanding. In order to obtain explicit analytical results, we consider ${{\boldsymbol{H}}_{{\rm{ul}}}}~\approx~{\boldsymbol{H}}{{\boldsymbol{D}}_{{\rm{ut}}}}$ when calculating ${\rm~E}\left(~{\overline~{\cal~N}~_k}~\right)$ since the HM should be small due to the EVM requirements of modern transmitters. Thus, \begin{eqnarray}{} {\rm E}\left( {{{\overline N }_k}} \right)\!\!\approx\! {\rm E}\left[ {{{\left| {{{\left( {{{{{\hat {\boldsymbol D}}}}_{\rm J}}} \right)}_{k,k}}} \right|}^2}} \right]{\rm E}\left[ {{{\left| {\frac{{d_{{\rm{ut}}}^k}}{{d_{{\rm{ur}}}^k}}} \right|}^{\rm{2}}}{\rm{tr}}\left( {{\boldsymbol{\tilde W}}{{{\boldsymbol{\tilde W}}}^{\rm H}}} \right)} \right] = \frac{{\rm E}{\left| {{{\left( {{{{\hat{\boldsymbol D}}}_{\rm J}}} \right)}_{k,k}}} \right|^2}{{\rm E}}\left[ {{{\left| {\frac{{d_{\rm{ut}}^k}}{{d_{\rm{ur}}^k}}} \right|}^{\rm{2}}}\sum\nolimits_{k = 1}^K {{{\left| {\frac{1}{{d_{\rm{ut}}^k}}} \right|}^{\rm{2}}}} } \right]}{{\left( {M - K} \right)}}, \tag{35} \end{eqnarray} where Eq. (35) is obtained from the fact that the elements of ${{{\hat~{\boldsymbol~D}}}_{\rm~J}}$ are ergodic from [7] \begin{eqnarray} {{\rm E}_{\rm{H}}}\left[ {{\rm{tr}}\left( {{\boldsymbol{\tilde W}}{{{\boldsymbol{\tilde W}}}^{\rm H}}} \right)} \right] = {{\rm E}_{\rm{H}}}\left[ {{\rm{tr}}\left( {{{{\boldsymbol{\tilde W}}}^{\rm{H}}}{\boldsymbol{\tilde W}}} \right)} \right] = \frac{1}{{\left( {M - K} \right)}}\sum\limits_{k = 1}^K {{{\left| {\frac{1}{{d_{\rm{ut}}^k}}} \right|}^{\rm{2}}}} \tag{36} \end{eqnarray} with ${\rm~E}_{\rm{H}}$ denoting the expectation over small-scale fading in $\boldsymbol{H}$. It is worth mentioning that similar results can also be obtained for ${\rm~E}\left(~{\overline~{\cal~N}~_k}~\right)$ by following the approximation strategy [15,18] for calculating the average of ${{\left\|~{\boldsymbol{W}}~\right\|_F^2}}~$. The key of these approximations relies on the fact that HM has small values. The simulation results in this paper also show that the calculation in this appendix provides a good approximation.

Although deriving an explicit form of ${{{{\rm~E}_{{\beta~_k}}}}\left(~{R_k^{{\rm{LB}}}}~\right)}$ is usually infeasible, $f_{\rm{se}}$ can be numerically calculated using the Monte Carlo method: \begin{eqnarray}{} f_{\rm{se}} \approx \frac{1}{N}\sum\limits_{n = 1}^N {K \times R_n^{{\rm{LB}}}}, \tag{37} \end{eqnarray} where $\beta_n$ in $R_n^{{\rm{LB}}}$ denotes the $n$th path loss realization from the distribution of user locations.

\begin{eqnarray}{} \frac{{{\rm{\partial }}KR_k^{{\rm{LB}}}}}{{{\rm{\partial }}K}} &=& R_k^{{\rm{LB}}} + K\frac{{{{A}}\left( K \right)}}{{{{B}}\left( K \right)}} \tag{38} \\ &=& R_k^{{\rm{LB}}} + \frac{{{{\log }_2}e\left[ { - {\xi _k}{\delta _{\rm e}}{K^3} + 2{\xi _k}{\delta _{\rm e}}M{K^2} - {M^2}\left( {1 + {\xi _k}{\delta _{\rm e}}} \right)K} \right]}}{{{\xi _k}{\delta _{\rm e}}{K^3} - \left( {2{\xi _k}{\delta _{\rm e}}M + M + {\xi _k}{\delta _{\rm e}}} \right){K^2} + \left( {{\xi _k}{\delta _{\rm e}}{M^2} + {M^2} + 2{\xi _k}{\delta _{\rm e}}M} \right)K - {\xi _k}{\delta _{\rm e}}{M^2}}}. \end{eqnarray}

Hence, the convexity of $f_{\rm{se}}(K,E_s)$ can be observed from (37). In this appendix, we show that $\frac{{{\rm~{\partial~}~^2}KR_k^{{\rm{LB}}}}}{{{\rm{\partial~}~}K^2}}~<~0$ if $M\gg~K$. First, $\frac{{\partial~KR_k^{{\rm{LB}}}}}{{\partial~K}}$ is given at the top of the next page, where $\frac{{{\rm{\partial~}}R_k^{{\rm{LB}}}}}{{{\rm{\partial~}}K}}~=~\frac{{{{A}}\left(~K~\right)}}{{{{B}}\left(~K~\right)}}$. Thus, $\frac{{{\rm~{\partial~}~^2}KR_k^{{\rm{LB}}}}}{{{\rm{\partial~}~}K^2}}$ can be written as \begin{eqnarray} \frac{{{{\rm{d}}^{\rm{2}}}R_k^{{\rm{LB}}}}}{{{\rm{d}}{K^2}}} &=& \frac{{2A\left( K \right)B\left( K \right) \!+\! KA'\left( K \right)B\left( K \right) \!-\! KA\left( K \right)B'\left( K \right)}}{{{B^2}\left( K \right)}} \\ &=& \frac{{B\left( K \right)\left[ {A\left( K \right) \!+\! KA'\left( K \right)} \right] \!+\! A\left( K \right)\left[ {B\left( K \right) \!-\! KB'\left( K \right)} \right]}}{{{B^2}\left( K \right)}}, \tag{39} \end{eqnarray} where $A'\left(~K~\right)~=~\frac{{{\rm{\partial~}}A\left(~K~\right)}}{{{\rm{\partial~}}K}}$ and $B'\left(~K~\right)~=~\frac{{{\rm{\partial~}}B\left(~K~\right)}}{{{\rm{\partial~}}K}}$.

Note that this paper considers large-scale fading to be dominated by path loss, as was the case in [19]. Thus, the instantaneous value of $\xi_k$ is bounded. Based on the condition $M~\gg~K$, it is known that \begin{eqnarray} & &{\xi _k}{\delta _{\rm e}}{K^3} \ll \left( {2{\xi _k}{\delta _{\rm e}}M + M + {\xi _k}{\delta _{\rm e}}} \right){K^2}\ \left( {2{\xi _k}{\delta _{\rm e}}M + M + {\xi _k}{\delta _{\rm e}}} \right){K^2} \ll \left( {{\xi _k}{\delta _{\rm e}}{M^2} + {M^2} + 2{\xi _k}{\delta _{\rm e}}M} \right)K, \tag{40} \\ & &{\xi _k}{\delta _{\rm e}}{M^2} \ll \left( {{\xi _k}{\delta _{\rm e}}{M^2} + {M^2} + 2{\xi _k}{\delta _{\rm e}}M} \right)K, \ {\xi _k}{\delta _{\rm e}}{K^2} \ll 2{\xi _k}{\delta _{\rm e}}M{K} \rm{and} \ 2{\xi _k}{\delta _{\rm e}}M{K} \ll {M^2}\left( {1 + {\xi _k}{\delta _{\rm e}}} \right). \tag{41} \end{eqnarray}

Thus, \begin{align} {A\left( K \right) + KA'\left( K \right)}\approx& -M^2\left( 1+{\xi _k}{\delta _{\rm e}} \right), A\left( K \right) \approx - {M^2}\left( {1 + {\xi _k}{\delta _{\rm e}}} \right), \\ B\left( K \right) - KB'\left( K \right) = & - 2{\xi _k}{\delta _{\rm e}}{K^3} - {\xi _k}{\delta _{\rm e}}{M^2}+ \left( {2{\xi _k}{\delta _{\rm e}}M + M + {\xi _k}\delta } \right){K^2} {\rm and} B\left( K \right) \tag{42} \\ &\approx \left( {{\xi _k}{\delta _{\rm e}}{M^2} + {M^2} + 2{\xi _k}{\delta _{\rm e}}M} \right)K. \end{align} It is worth mentioning that the exact forms of ${A\left(~K~\right)}$ and ${B\left(~K~\right)}$ are used when calculating ${A'\left(~K~\right)}$ and ${B'\left(~K~\right)}$.

One can verify that ${B\left(~K~\right)\left[~{A\left(~K~\right)~+~KA'\left(~K~\right)}~\right]}+{A\left(~K~\right)\left[~{B\left(~K~\right)~-~KB'\left(~K~\right)}~\right]}<0$ conditioned on $M~\gg~K$. Hence, $\frac{{{{\rm{\partial~}}^{\rm{2}}}KR_k^{{\rm{LB}}}}}{{{\rm{\partial~}}{K^2}}}<0$, which implies $f_{\rm{se}}~\approx~\frac{1}{N}\sum\nolimits_{n~=~1}^N~{K~\times~R_n^{{\rm{LB}}}}$ is concave.

Similarly, the convexity of $\eta_{\rm{ee}}$ can be observed from $\eta_{\rm{ee}}~\approx~\frac{1}{{{E_s}}}\frac{1}{N}\sum\nolimits_{n~=~1}^N~{K~\times~R_n^{{\rm{LB}}}}$. By calculating $\frac{{\partial~\left(~{{{KR_n^{{\rm{LB}}}}~\mathord{\left/ ~{\vphantom~{{KR_n^{{\rm{LB}}}}~{{E_s}}}}~\right. ~\kern-\nulldelimiterspace}~{{E_s}}}}~\right)}}{{\partial~{E_s}}}$, we have \begin{eqnarray}{} \frac{{\partial \left( {{{R_n^{{\rm{LB}}}} \mathord{\left/ {\vphantom {{R_n^{{\rm{LB}}}} {{E_s}}}} \right. \kern-\nulldelimiterspace} {{E_s}}}} \right)}}{{\partial {E_s}}} \!\!=\!\! \frac{1}{{E_s^2}}\left[ {\frac{{\frac{K}{{M - K}} \times {{\log }_2}e}}{{\frac{{{\beta _n}{E_s}}}{{{\sigma ^2}}}\frac{{K - 1}}{M}{\delta _{\rm e}} \!+\! \frac{{K\left( {1 + {\delta _{\rm e}}} \right)}}{{M - K}}}} - R_n^{{\rm{LB}}}} \right]. \tag{43} \end{eqnarray} Letting $\frac{{\partial~\left(~{{{R_n^{{\rm{LB}}}}~\mathord{\left/ ~{\vphantom~{{R_n^{{\rm{LB}}}}~{{E_s}}}}~\right. ~\kern-\nulldelimiterspace}~{{E_s}}}}~\right)}}{{\partial~{E_s}}}<0$, i.e., \begin{eqnarray}{} {\log _2}\left[ {\frac{{\exp \left( {\frac{{\frac{K}{{M - K}}}}{{\frac{{{\beta _n}{E_s}}}{{{\sigma ^2}}}\frac{{K - 1}}{M}{\delta _{\rm e}} + \frac{K}{{M - K}}\left( {1 + {\delta _{\rm e}}} \right)}}} \right)}}{{1 + \frac{{{\beta _n}{E_s}}}{{{\sigma ^2}}}\frac{{K - 1}}{M}{\delta _{\rm e}} + \frac{K}{{M - K}}\left( {1 + {\delta _{\rm e}}} \right)}}} \right]<0, \tag{44} \end{eqnarray} we have $E_s~\in~{\cal~U}_{0}~=~\{~{~{{E_s}}~|{E_s}~>~\frac{{{\sigma~^2}}}{{{\beta~_n}}}~\times~{Z_0}}~\}$, where ${Z_0}~=~\frac{{MK}}{{2\left(~{K~-~1}~\right)\left(~{M~-~K}~\right)}}\frac{{\sqrt~{{{\left(~{1~+~{\delta~_{\rm~e}}}~\right)}^2}~+~\frac{{4{\delta~_{\rm~e}}\left(~{K~-~1}~\right)}}{M}}~-~\left(~{1~+~{\delta~_{\rm~e}}}~\right)}}{{{\delta~_{\rm~e}}}}$ and the Taylor Series ${e^x}~=~1~+~x~+~0.5{x^2}~+~o\left(~{{x^2}}~\right)$ is used to obtain the explicit solution to (44). One can verify that $Z_0$ is a monotonically decreasing function of $\delta_e$ for $\delta_{\rm{e}}>0$. Thus, $ {\cal~U}_{1}~=~\{~{~{{E_s}}~|{E_s}~>~\frac{{{\sigma~^2}}}{{{\beta~_n}}}{{~{{Z_0}}~|}_{{\delta~_{\rm~e}}~\to~0}}~=~\frac{{{\sigma~^2}}}{{{\beta~_n}}}\frac{M}{{(~{M~-~K}~)}}}~\}$ is a subset of ${\cal~U}_0$ since ${Z_0}~<~{~{\max~(~{{Z_0}}~)~=~{Z_0}}~|_{{\delta~_{\rm~e}}~\to~0}}$.

Note that this paper considers $\gamma_n~\gg~1$, which implies \begin{eqnarray}{} {\cal {U}}_{2} = \left\{ {\left. {{E_s}} \right|{E_s} \gg \frac{{{\sigma ^2}}}{{{\beta _n}}}\frac{M}{{M - K}}\frac{{M\left( {1 + {\delta _{\rm e}}} \right)}}{{M - \left( {K - 1} \right){\delta _{\rm e}}}}} \right\}. \tag{45} \end{eqnarray}

Hence, ${\cal~U}_{2}~\subseteq~{\cal~U}_{1}~\subseteq~{\cal~U}_{0}$, implying $\frac{{R_n^{{\rm{LB}}}}}{{{E_s}}}$ is decreasing with respect to $E_s$. As ${\cal~U}_{2}~\subseteq~{\cal~U}_{1}~\subseteq~{\cal~U}_{0}$ holds and is independent of $K$, $\frac{{R_n^{{\rm{LB}}}}}{{{E_s}}}$ is decreasing with respect to $E_s$, regardless of $K$. In fact, in a reasonable range of ${\frac{{{\beta~_n}{E_s}}}{{{\sigma~^2}}}}$ values, $M\gg~K$, ${E_s}~\in~{\cal~U}_1$ can always be satisfied without requiring (45).


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  • Figure 1

    (Color online) Average equivalent CSCG noise power, where $\sigma^2$ is normalized to 1 (0 dBm). (a) $\rho~=~0.2$ and $\delta_{\rm{e}}=~0.001$; (b) $\rho~=~0.03$ and $\delta_{\rm{e}}=~0.001$.

  • Figure 2

    (Color online) Average power of multi-user interference resulting from imperfect calibration, where the number of users is 2, 4, 8, and 10; $\rho~=~0.2$; $\delta_{\rm{e}}=~0.001$; $\xi_k=0~{\rm{dB}}$.

  • Figure 3

    (Color online) Total spectral efficiency versus the number of antennas, where $\rho~=~0.1$ and $f_{\rm{c}}=1.8~\rm{GHz}$.

  • Figure 4

    (Color online) Spectral efficiency loss from a single user side due to imperfect calibration, where $\xi_k$ is interpreted as the achievable SISO SNR, $1/\delta_{\rm{e}}$ is interpreted as the calibration SNR, $K$=10, and $M$=100.

  • Figure 5

    (Color online) Optimal EE (a) and Loss of EE (b) under a required SE value, where $2\leq~K\leq20$ and $0.01\leq~E_{{s}}\leq1$.

  • Figure 6

    (Color online) Optimal SE (a) and Loss of SE (b) under a required EE value, where $2\leq~K\leq20$ and $0.01\leq~E_{{s}}\leq1$.

  •   

    Algorithm 1 Optimal $E_{\rm{s}}^*$ and $K^*$ for P1

    ; $E_{{s}}^*=E_1$

    and $K^*=K_1$;

    Initialization $E_{\rm{1}}=E_{\rm{min}},E_{\rm{2}}=E_{\rm{max}}$, tolerance $\varepsilon>0$;

    Let $E_{{s}}=E_{\rm{2}}$ and compute $K_2$ and $\eta_{\rm{ee}}(K_2,E_2)$ by Algorithm 2; Let $E_{{s}}=E_{\rm{1}}$ and compute $K_1$ and $\eta_{\rm{ee}}(K_1,E_1)$ by Algorithm 2;

    if $\eta_{\rm{ee}}(K_2,E_2)<\eta_0$ then

    no solution and stop;

    end if

    if $\eta_{\rm{ee}}(K_1,E_1)\geq~\eta_0$ then

    return $E_{{s}}^*=E_2$ and $K^*=K_2$;

    end if

    repeat

    $E_{{m}}=(E_1+E_2)/2$; compute $K_{{m}}$ and $\eta_{\rm{ee}}(K_{{m}},E_{{m}})$ by Algorithm 2;

    if $\eta_{\rm{ee}}(K_{{m}},E_{{m}})>\eta_0$ then

    $E_1=E_{{m}}$ and $K_1~=~K_{{m}}$;

    else

    $E_2=E_{{m}}$ and $K_2~=~K_{{m}}$;

    end if

    until $|\eta_{\rm{ee}}(K_1,E_1)-\eta_0|<\varepsilon$

  •   

    Algorithm 2 Optimal number of active users

    Initialization $l=2,r=K_{\rm{max}},\Delta=1$;

    while $l+\Delta<r$ do

    $m=(l+r)/2,~mm=(m+r)/2$;

    compute $R(K=m)=\frac{1}{N}\sum\nolimits_{n~=~1}^N{KR_{n}^{\rm{LB}}}$ and $R(K=mm)=\frac{1}{N}\sum\nolimits_{n~=~1}^N{KR_{n}^{\rm{LB}}}$;

    if $R(K=m)\geq~R(K=mm)$ then

    $r=m$;

    else

    $l=m$;

    end if

    end while

    $K_1=\lfloor~l\rfloor$, $K_2=\lceil~l\rceil$, and $K_3~=~\lceil~r\rceil$;

    compute $R(K=K_i)=\frac{1}{N}\sum\nolimits_{n~=~1}^N{KR_{n}^{\rm{LB}}}$, $i~=1,2,3$;

    $f_{\rm{se}}(K_0,E_{{s}})=\max\{R(K=K_i)\}$, where $K_0={\rm{argmax}}\{R(K=K_i)\}$;

    $\eta_{\rm{ee}}(K_0,E_{{s}})=f_{\rm{se}}(K_0,E_{{s}})/E_{{s}}$;

    return $\eta_{\rm{ee}}(K_0,E_{{s}})$ and $K_0$;

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