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Secure transmission in one-bit cell-free massive MIMO system with multiple non-colluding eavesdroppers

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Abstract

This paper investigates the secrecy performance of the one-bit cell-free massive multiple-input multiple-output system in the presence of multiple non-colluding eavesdroppers (Eves), considering that access points (APs) are deployed with one-bit analog-to-digital converters (ADCs) and digital-to-analog converters (DACs), or only one of them, or ideally infinite-resolution ones. Based on Bussgang decomposition theory and the introduction of maximum ratio transmission, artificial noise (AN) technologies, we derive closed-form expressions for achievable rates of legitimate users and upper-bound expressions for ergodic leakage information rates of Eves. We use the achievable secrecy rate to evaluate network security performance. Through theoretical analysis and numerical results under the four cases at APs , whether the resolution of ADCs or DACs is one-bit or not, we evaluate the effect of the number of APs or Eves, the number of APs’ or Eves’ antennas, the downlink transmit power, and the power fraction into signal symbols or AN symbols on the secrecy performance, respectively.

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Authors and Affiliations

  1. Graduate School, Army Engineering University of PLA, Nanjing, 210000, China

    Xiaoyu Wang, Yuanyuan Gao, Xianyu Zhang, Nan Sha, Mingxi Guo, Guozhen Zang & Na Li

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Appendices

Appendix

Appendix A

Above all, due to the fact that there is no correlation between the estimated channel and the error of the estimated channel by MMSE algorithm, the channel estimation error can be obtained as \({\tilde{\mathbf{g}}}_{mk}={\mathbf{g}}_{mk}-{\hat{\mathbf{g}}}_{mk}\), \({\tilde{\mathbf{g}}}_{mk}\sim {\mathcal{CN}}(0,\left( {{\beta }_{mk}}-{{\alpha }_{mk}} \right) {{I}_{{{N}_{A}}}})\). According to the formula (21), we handle the normalization constants into the specific forms based on MRT and R-AN technologies, which can be given by

$$\begin{aligned} \begin{aligned} \mu _{m}=\hbox{E}\left\{ tr({{{\mathbf{W}}}_{m}}{\mathbf{W}}_{m}^{H}) \right\} ={{N}_{A}}\sum _{k=1}^{K}{{{\alpha }_{mk}}}, \end{aligned} \end{aligned}$$
(51)
$$\begin{aligned} \begin{aligned} {{\nu }_{m}}=\hbox{E}\left\{ tr({{{\mathbf{S}}}_{m}}{\mathbf{S}}_{m}^{H}) \right\} ={{N}_{A}}, \end{aligned} \end{aligned}$$
(52)

And then, the terms of formula (28) can be calculated by

$$\begin{aligned} \left| D{{S}_{k}} \right|&=\hbox{E}\left\{ \sqrt{\frac{2\theta {{\rho }_{d}}}{\pi }}\sum _{m=1}^{M}{\mu _{m}^{-1/2}{\mathbf{g}}_{mk}^{T}{{\varvec{\omega }}_{mk}}} \right\} \\ &=\sqrt{\frac{2\theta {{\rho }_{d}}}{\pi }}\sum _{m=1}^{M}{\mu _{m}^{-1/2}\hbox{E}\left\{ {\mathbf{g}}_{mk}^{T}{{\varvec{\omega }}_{mk}} \right\} } \\ &=\sqrt{\frac{2\theta {{\rho }_{d}}}{\pi }}\sum _{m=1}^{M}{\mu _{m}^{-1/2}\hbox{E}\left\{ {\hat{\mathbf{g}}}_{mk}^{H}{\hat{\mathbf{g}}}_{mk} \right\} } \\ &=\sqrt{\frac{2{{N}_{A}}\theta {{\rho }_{d}}}{\pi }}\sum _{m=1}^{M}{{{\left( \sum _{k=1}^{K}{{{\alpha }_{mk}}} \right) }^{-1/2}}{{\alpha }_{mk}}}, \end{aligned}$$
(53)
$$\begin{aligned}&E\left\{ {{\left| B{{U}_{k}} \right| }^{2}} \right\} =E\left\{{{\left| \sqrt{\frac{2\theta {{\rho }_{d}}}{\pi }}\sum \limits_{m=1}^{M}{\mu _{m}^{-1/2}{\mathbf{g}}_{mk}^{T}{\varvec{\omega}}_{mk}}-D{{S}_{k}} \right| }^{2}} \right\} \\ &\quad=\frac{2\theta {{\rho }_{d}}}{\pi }\hbox{E}\left\{ {{\left| \sum\limits _{m=1}^{M}{\mu_{m}^{-1/2}{\mathbf{g}}_{mk}^{H}\varvec{{\hat{g}}}_{mk}}-\sum\limits _{m=1}^{M}{\mu _{m}^{-1/2}\hbox{E}\left\{{\hat{\varvec{g}}}_{mk}^{H}{\hat{\varvec{g}}}_{mk} \right\} }\right| }^{2}} \right\} \\ &\quad =\frac{2\theta {{\rho}_{d}}}{\pi }\hbox{E}\left\{ {{\left| \sum \limits _{m=1}^{M}{\mu_{m}^{-1/2}\left({\hat{\varvec{g}}}_{mk}^{H}+{\tilde{\varvec{g}}}_{mk}^{H} \right){\hat{\varvec{g}}}_{mk}} \right| }^{2}} \right\} \\ &\qquad-\frac{2\theta {{\rho }_{d}}}{\pi }{\left( \sum \limits_{m=1}^{M}{{{N}_{A}}\mu _{m}^{-1/2}{{\alpha }_{mk}}} \right) }^{2}\\ &\quad =\frac{2\theta {{\rho }_{d}}}{\pi }\sum \limits_{m=1}^{M}{\mu _{m}^{-1}\hbox{E}\left\{ {{\left|{\hat{\varvec{g}}}_{mk}^{H}{\hat{\varvec{g}}}_{mk} \right| }^{2}}\right\} } \\ &\qquad +\frac{4\theta {{\rho }_{d}}}{\pi}\hbox{E}\left\{ \sum \limits _{m=1}^{M}{\sum \limits _{n\ne m}{\mu_{m}^{-1/2}\mu _{n}^{-{\scriptstyle{}^{1}/{}_{2}}}{\hat{\varvec{g}}}_{mk}^{H}{\hat{\varvec{g}}}_{mk}}{\hat{\varvec{g}}}_{nk}^{H}{\hat{\varvec{g}}}_{nk}}\right\} \\ &\qquad +\frac{2\theta {{\rho }_{d}}}{\pi }\sum\limits _{m=1}^{M}{\mu _{m}^{-1}\hbox{E}\left\{ {{\left|{\tilde{\varvec{g}}}_{mk}^{H}{\hat{\varvec{g}}}_{mk} \right| }^{2}}\right\} }-\frac{2\theta {{\rho }_{d}}}{\pi }{{\left( \sum \limits_{m=1}^{M}{{{N}_{A}}\mu _{m}^{-{\scriptstyle {}^{1}/{}_{2}}}{{\alpha}_{mk}}} \right) }^{2}} \\ &\quad =\frac{2\theta {{\rho}_{d}}}{\pi }\sum \limits _{m=1}^{M}{N_{A}^{2}\mu _{m}^{-1}\alpha_{mk}^{2}+}\frac{4\theta {{\rho }_{d}}}{\pi }\sum \limits_{m=1}^{M}{\sum \limits _{n\ne m}{N_{A}^{2}\mu _{m}^{-1/2}\mu_{n}^{-1/2}{{\alpha }_{mk}}{{\alpha }_{nk}}}} \\ &\qquad+\frac{2\theta {{\rho }_{d}}}{\pi }\sum \limits _{m=1}^{M}{\mu_{m}^{-1}{{N}_{A}}\alpha _{mk}^{2}+}\frac{2\theta {{\rho }_{d}}}{\pi}\sum \limits _{m=1}^{M}{{{N}_{A}}\mu _{m}^{-1}{{\alpha}_{mk}}\left( {{\beta }_{mk}}-{{\alpha }_{mk}} \right) } \\&\qquad -\frac{2\theta {{\rho }_{d}}}{\pi }{{\left( \sum \limits_{m=1}^{M}{{{N}_{A}}\mu _{m}^{-1/2}{{\alpha }_{mk}}} \right) }^{2}}\\ &\quad =\frac{2\theta {{\rho }_{d}}}{\pi }\sum \limits_{m=1}^{M}{{{\left( \sum \limits _{{k}'=1}^{K}{{{\alpha }_{m{k}'}}}\right) }^{-1}}{{\alpha }_{mk}}{{\beta }_{mk}}}, \end{aligned}$$
(54)
$$\begin{aligned} \hbox{E}\left\{ {{\left| U{{I}_{k{k}'}} \right| }^{2}} \right\}&=\frac{2\theta {{\rho }_{d}}}{\pi }\hbox{E}\left\{ {{\left| \sum _{m=1}^{M}{\mu _{m}^{-{1}/{2}}{\mathbf{g}}_{mk}^{T}{{\varvec{\omega }}_{m{k}'}}} \right| }^{2}} \right\} \\ &\quad =\frac{2\theta {{\rho }_{d}}}{\pi }\hbox{E}\left\{ {{\left| \sum _{m=1}^{M}{\mu _{m}^{-{1}/{2}}{\mathbf{g}}_{mk}^{H}{\hat{\mathbf{g}}}_{m{k}'}} \right| }^{2}} \right\} \\ &\quad =\frac{2\theta {{\rho }_{d}}}{\pi }\sum _{m=1}^{M}{\mu _{m}^{-1}\hbox{E}\left\{ {{\left| {\mathbf{g}}_{mk}^{H}{\hat{\mathbf{g}}}_{m{k}'} \right| }^{2}} \right\} } \\ &\quad =\frac{2\theta {{\rho }_{d}}}{\pi }\sum _{m=1}^{M}{{{\left( \sum _{{{k}'}'=1}^{K}{{{\alpha }_{m{{k}'}'}}} \right) }^{-1}}{{\alpha }_{m{k}'}}{{\beta }_{mk}}}, \end{aligned}$$
(55)
$$\begin{aligned} \hbox{E}\left\{ {{\left| A{{N}_{k}} \right| }^{2}} \right\}&=\frac{2\left( 1-\theta \right) {{\rho }_{d}}}{\pi }\hbox{E}\left\{ {{\left| \sum _{m=1}^{M}{\nu _{m}^{-{1}/{2}}{\mathbf{g}}_{mk}^{T}{{{\mathbf{S}}}_{m}}{{{\mathbf{n}}}_{m}}} \right| }^{2}} \right\} \\ &\quad =\frac{2\left( 1-\theta \right) {{\rho }_{d}}}{\pi {{N}_{A}}}\sum _{m=1}^{M}{\hbox{E}\left\{ {{\left| {\mathbf{g}}_{mk}^{T}{{{\mathbf{S}}}_{m}}{{{\mathbf{n}}}_{m}} \right| }^{2}} \right\} } \\ &\quad =\frac{2\left( 1-\theta \right) {{\rho }_{d}}}{\pi {{N}_{A}}}\sum _{m=1}^{M}{\hbox{E}\left\{ {\mathbf{g}}_{mk}^{H}{{{\mathbf{S}}}_{m}}{{{\mathbf{n}}}_{m}}{\mathbf{n}}_{m}^{H}{\mathbf{S}}_{m}^{H}{\mathbf{g}}_{mk} \right\} } \\ &\quad =\frac{2\left( 1-\theta \right) {{\rho }_{d}}}{\pi {{N}_{A}}}\sum _{m=1}^{M}{\hbox{E}\left\{ tr\left( {\mathbf{g}}_{mk}{\mathbf{g}}_{mk}^{H} \right) \right\} } \\ &\quad =\frac{2\left( 1-\theta \right) {{\rho }_{d}}}{\pi }\sum _{m=1}^{M}{{{\beta }_{mk}}}, \end{aligned}$$
(56)
$$\begin{aligned} \hbox{E}\left\{ {{\left| Q{{N}_{k}} \right| }^{2}} \right\}&=\frac{{{\rho }_{d}}}{{{N}_{A}}}\hbox{E}\left\{ {{\left| \sum _{m=1}^{M}{{\mathbf{g}}_{mk}^{T}{{{\mathbf{q}}}_{d,m}}} \right| }^{2}} \right\} \\ &\quad =\frac{{{\rho }_{d}}}{{{N}_{A}}}\sum _{m=1}^{M}{\hbox{E}\left\{ {{\left| {\mathbf{g}}_{mk}^{T}{{{\mathbf{q}}}_{d,m}} \right| }^{2}} \right\} } \\ &\quad =\frac{{{\rho }_{d}}}{{{N}_{A}}}\sum _{m=1}^{M}{\hbox{E}\left\{ {\mathbf{g}}_{mk}^{H}{{{\mathbf{q}}}_{d,m}}{\mathbf{q}}{{_{d,m}^{H}}}{\mathbf{g}}_{mk} \right\} } \\ &\quad =\frac{{{\rho }_{d}}}{{{N}_{A}}}\sum _{m=1}^{M}{\hbox{E}\left\{ tr\left( {\mathbf{g}}_{mk}{\mathbf{g}}_{mk}^{H}{{{\mathbf{q}}}_{d,m}}{\mathbf{q}}{{_{d,m}^{H}}} \right) \right\} } \\ &\quad =\left( 1-\frac{2}{\pi } \right) {{\rho }_{d}}\sum _{m=1}^{M}{{{\beta }_{mk}}}, \end{aligned}$$
(57)

Plugging these results into the formula (29) and (30), the achievable rates of legitimate users in the case of one-bit ADCs and DACs can be obtained. And the derivations of the other three cases can be acquired in the same way, which will not be repeated here.

Appendix B

According to the formula (39), we can get

$$\begin{aligned} \hbox{E}\left\{ SIN{{R}_{Ei}} \right\} =\hbox{E}\left\{ {{\mathbf{D}}}{{\mathbf{S}}}_{Ei}^{H}{{\mathbf{\Theta }}^{-1}}{\mathbf{D}}{{{\mathbf{S}}}_{Ei}} \right\} , \end{aligned}$$
(58)

where

$$\begin{aligned} \Theta&=\sum _{{k}'\ne k}^{K}{\hbox{E}\left\{ {{\mathbf{U}}}{{\mathbf{I}}}_{Ei,k{k}'}{{\mathbf{U}}}{{\mathbf{I}}}_{Ei,k{k}'}^{H} \right\} }+\hbox{E}\left\{ {{\mathbf{A}}}{{\mathbf{N}}}_{Ei}{{\mathbf{A}}}{{\mathbf{N}}}_{Ei}^{H} \right\} \\ &\quad +\hbox{E}\left\{ {{\mathbf{Q}}}{{\mathbf{N}}}_{Ei}{{\mathbf{Q}}}{{\mathbf{N}}}_{Ei}^{H} \right\} +\sigma _{d}^{2}{{{\mathbf{I}}}_{{{N}_{E}}}}, \end{aligned}$$
(59)

And each term is deduced as follows:

$$\begin{aligned}&\hbox{E}\left\{ {{\mathbf{D}}}{{\mathbf{S}}}_{Ei}^{H}{\mathbf{D}}{{{\mathbf{S}}}_{Ei}} \right\} \\ &\quad =\frac{2\theta {{\rho }_{d}}}{\pi }\sum _{m=1}^{M}{\mu _{m}^{-1}tr\left( \hbox{E}\left\{ {{{\varvec{{\hat{g}}}}}_{m{{k}_{0}}}}{\hat{\mathbf{g}}}_{m{{k}_{0}}}^{H}{\mathbf{G}}_{mEi}{\mathbf{G}}_{mEi}^{H} \right\} \right) } \\ &\quad =\frac{2\theta {{\rho }_{d}}}{\pi }\sum _{m=1}^{M}{\mu _{m}^{-1}tr\left( \hbox{E}\left\{ {{{\varvec{{\hat{g}}}}}_{m{{k}_{0}}}}{\hat{\mathbf{g}}}_{m{{k}_{0}}}^{H}{\mathbf{G}}_{mEi}{\mathbf{G}}_{mEi}^{H} \right\} \right) } \\ &\quad =\frac{2\theta {{\rho }_{d}}}{\pi }\sum _{m=1}^{M}{\mu _{m}^{-1}tr\left( \hbox{E}\left\{ {{{\varvec{{\hat{g}}}}}_{m{{k}_{0}}}}{\hat{\mathbf{g}}}_{m{{k}_{0}}}^{H} \right\} \hbox{E}\left\{ {\mathbf{G}}_{mEi}{\mathbf{G}}_{mEi}^{H} \right\} \right) } \\ &\quad =\frac{2\theta {{\rho }_{d}}}{\pi }\sum _{m=1}^{M}{\mu _{m}^{-1}tr\left( {{N}_{E}}{{\beta }_{mEi}}{{\alpha }_{m{{k}_{0}}}}{{{\mathbf{I}}}_{{{N}_{A}}}} \right) } \\ &\quad =\frac{2{{N}_{E}}\theta {{\rho }_{d}}}{\pi }\sum _{m=1}^{M}{{{\left( \sum _{{k}'=1}^{K}{{{\alpha }_{m{k}'}}} \right) }^{-1}}{{\beta }_{mEi}}{{\alpha }_{m{{k}_{0}}}}}, \end{aligned}$$
(60)
$$\begin{aligned}&\hbox{E}\left\{ {{\mathbf{U}}}{{\mathbf{I}}}_{Ei,k{k}'}{{\mathbf{U}}}{{\mathbf{I}}}_{Ei,k{k}'}^{H} \right\} \\ &\quad =\frac{2\theta {{\rho }_{d}}}{\pi }\hbox{E}\left\{ \sum _{m=1}^{M}{\mu _{m}^{-1}{\mathbf{G}}_{mEi}^{H}{{{\varvec{{\hat{g}}}}}_{m{k}'}}\varvec{{\hat{g}}}{{_{mk}^{H}}_{m{k}'}}{\mathbf{G}}_{mEi}} \right\} \\ &\quad =\frac{2\theta {{\rho }_{d}}}{\pi }\sum _{m=1}^{M}{\mu _{m}^{-1}\hbox{E}\left\{ {\mathbf{G}}_{mEi}^{H}\hbox{E}\left\{ {{{\varvec{{\hat{g}}}}}_{m{k}'}}\varvec{{\hat{g}}}{{_{mk}^{H}}_{m{k}'}} \right\} {\mathbf{G}}_{mEi} \right\} } \\ &\quad =\frac{2\theta {{\rho }_{d}}}{\pi }\sum _{m=1}^{M}{\mu _{m}^{-1}{{\alpha }_{m{k}'}}\hbox{E}\left\{ {\mathbf{G}}_{mEi}^{H}{{{\mathbf{I}}}_{{{N}_{A}}}}{\mathbf{G}}_{mEi} \right\} } \\ &\quad =\frac{2\theta {{\rho }_{d}}}{\pi }\sum _{m=1}^{M}{\mu _{m}^{-1}{{N}_{A}}{{\beta }_{mEi}}{{\alpha }_{m{k}'}}}{{{\mathbf{I}}}_{{{N}_{E}}}} \\ &\quad =\frac{2\theta {{\rho }_{d}}}{\pi }\sum _{m=1}^{M}{{{\left( \sum _{{{k}'}'=1}^{K}{{{\alpha }_{m{{k}'}'}}} \right) }^{-1}}{{\beta }_{mEi}}{{\alpha }_{m{k}'}}}{{{\mathbf{I}}}_{{{N}_{E}}}}, \end{aligned}$$
(61)
$$\begin{aligned}&\hbox{E}\left\{ {{\mathbf{A}}}{{\mathbf{N}}}_{Ei}{{\mathbf{A}}}{{\mathbf{N}}}_{Ei}^{H} \right\} \\ &\quad =\frac{2\left( 1-\theta \right) {{\rho }_{d}}}{\pi }\sum _{m=1}^{M}{\hbox{E}\left\{ {\mathbf{G}}_{mEi}^{H}{{{\mathbf{S}}}_{m}}{{{\mathbf{n}}}_{m}}{{\left( {{{\mathbf{S}}}_{m}}{{{\mathbf{n}}}_{m}} \right) }^{H}}{\mathbf{G}}_{mEi} \right\} } \\ &\quad =\frac{2\left( 1-\theta \right) {{\rho }_{d}}}{\pi }\sum _{m=1}^{M}{\hbox{E}\left\{ {\mathbf{G}}_{mEi}^{H}\hbox{E}\left\{ {{{\mathbf{S}}}_{m}}{{{\mathbf{n}}}_{m}}{{\left( {{{\mathbf{S}}}_{m}}{{{\mathbf{n}}}_{m}} \right) }^{H}} \right\} {\mathbf{G}}_{mEi} \right\} } \\ &\quad =\frac{2{{N}_{A}}\left( 1-\theta \right) {{\rho }_{d}}}{\pi }\sum _{m=1}^{M}{{{\beta }_{mEi}}}{{{\mathbf{I}}}_{{{N}_{E}}}}, \end{aligned}$$
(62)
$$\begin{aligned}\hbox{E}\left\{ {{\mathbf{Q}}}{{\mathbf{N}}}_{Ei}{{\mathbf{Q}}}{{\mathbf{N}}}_{Ei}^{H} \right\} &=\frac{{{\rho }_{d}}}{{{N}_{A}}}\sum _{m=1}^{M}{\hbox{E}\left\{ {\mathbf{G}}_{mEi}^{H}{{{\mathbf{q}}}_{d,m}}{\mathbf{q}}_{d,m}^{H}{\mathbf{G}}_{mEi} \right\} } \\ &=\frac{{{\rho }_{d}}}{{{N}_{A}}}\sum _{m=1}^{M}{\hbox{E}\left\{ {\mathbf{G}}_{mEi}^{H}\hbox{E}\left\{ {{{\mathbf{q}}}_{d,m}}{\mathbf{q}}_{d,m}^{H} \right\} {\mathbf{G}}_{mEi} \right\} } \\ &=\left( 1-\frac{2}{\pi } \right) {{\rho }_{d}}\sum _{m=1}^{M}{{{\beta }_{mEi}}}{{{\mathbf{I}}}_{{{N}_{E}}}}, \end{aligned}$$
(63)

Substituting the above results into the formulas (39) and (58), we can obtain the Theorem 2.

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Wang, X., Gao, Y., Zhang, X. et al. Secure transmission in one-bit cell-free massive MIMO system with multiple non-colluding eavesdroppers. Wireless Netw 28, 2951–2966 (2022). https://doi.org/10.1007/s11276-022-03012-x

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