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Physical layer security for space–time-block-coded MIMO system in Rician fading in the presence of imperfect feedback

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Abstract

We investigate physical layer security (PLY) of space–time-block-coded (STBC)–multi-input–multi-output (MIMO) wiretap channel in the presence of imperfect channel state information (ICSI) over Rician fading channel. Secrecy for data transmission to legitimate user in the presence of eavesdropper is metered in terms of average secrecy capacity (ASC), probability of non-zero secrecy rate (PNZSR) and secrecy outage probability (SOP). We derive the closed form expression for security parameter by using the probability distribution function and cumulative distribution function of output signal to noise ratio (SNR). Simulation results are presented to validate our analytical results and demonstrate the impact of ICSI on the secrecy performance of STBC–MIMO system.

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Authors and Affiliations

  1. Electronics and Communication Engineering Department, National Institute of Technology, Kurukshetra, Haryana, 136119, India

    Rajiv Kumar & Sudakar Singh Chauhan

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  1. Rajiv Kumar
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  2. Sudakar Singh Chauhan
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Correspondence to Rajiv Kumar.

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Appendices

Appendix 1

Secrecy capacity is described as:

$$ C_{s} = \left[ {C_{B} - C_{E} } \right]^{ + } , $$
(14)

where \(C_{B} = \log \left( {1 + x_{b} } \right)\;{\text{and}}\;C_{E} = \log \left( {1 + x_{e} } \right).\) The ASC can be written as

$$ \overline{C}_{s} = \int_{0}^{\infty } {\left[ {\int_{0}^{\infty } {C_{s} f_{{\gamma_{E} }} \left( {x_{e} } \right)dx_{e} } } \right]} f_{{\gamma_{B} }} \left( {x_{b} } \right)dx_{b} . $$
(15)

Let us consider, \(\Delta_{1} = \int_{0}^{\infty } {C_{s} f_{{\gamma_{E} }} \left( {x_{e} } \right)dx_{e} } .\) By putting the value of \(C_{s}\) into this equation we get

$$ \Delta_{1} = \int_{0}^{{x_{b} }} {\left[ {\log_{2} \left( {1 + x_{b} } \right) - \log_{2} \left( {1 + x_{e} } \right)} \right]} f_{{\gamma_{E} }} \left( {x_{e} } \right)dx_{e} . $$
(16)

By doing integration by parts method, and utilizing algebra on Eq. (16), we get equation as

$$ \begin{aligned} \Delta_{1} & = \log_{2} \left( {1 + x_{b} } \right)F_{{\gamma_{E} }} \left( {x_{b} } \right) - \left[ {\log_{2} \left( {1 + x_{b} } \right)F_{{\gamma_{E} }} \left( {x_{b} } \right) - \frac{1}{\ln 2}\int_{0}^{{x_{b} }} {\frac{{F_{{\gamma_{E} }} \left( {x_{e} } \right)}}{{1 + x_{e} }}dx_{e} } } \right] \\ & = \frac{1}{\ln 2}\int_{0}^{{x_{b} }} {\frac{{F_{{\gamma_{E} }} \left( {x_{e} } \right)}}{{1 + x_{e} }}dx_{e} } . \\ \end{aligned} $$
(17)

By putting the value of \(\Delta_{1}\) into Eq. (15), we get final expression for ASC as shown below:

$$ \overline{C}_{s} = \frac{1}{\ln 2}\int_{0}^{\infty } {\frac{{F_{{\gamma_{E} }} \left( {x_{e} } \right)}}{{1 + x_{e} }}\left( {1 - F_{{\gamma_{B} }} \left( {x_{b} } \right)} \right)} dx_{b} . $$
(18)

By putting the value of CDF \(F_{{\gamma_{E} }} \left( {x_{e} } \right)\) and CDF \(F_{{\gamma_{B} }} \left( {x_{b} } \right)\) into Eq. (18), and by using [27, 9.211.4], we expressed the ASC as shown below in Eq. (19):

$$ \begin{aligned} & \overline{C} _{s} = - \frac{1}{{\ln 2}}\left( {\frac{{L_{A} r_{e} \left( {1 + \sigma _{e}^{2} \overline{\gamma } _{e} } \right)}}{{c_{e}^{2} \overline{\gamma } _{e} }}} \right)^{{\left( {\frac{{L_{A} L_{E} + 1}}{2}} \right)}} e^{{\left( { - \frac{{L_{A} L_{E} K_{e} }}{{c_{e}^{2} }}} \right)}} \left( {\frac{{c_{e}^{2} }}{{L_{A} L_{E} K_{e} }}} \right)^{{\left( {\frac{{L_{A} L_{E} - 1}}{2}} \right)}} \\ & \quad \quad \times \sum\limits_{{n = 0}}^{\infty } {\frac{{\left( { - 1} \right)^{n} }}{{n{!}}}\left( {\frac{{r_{e} L_{E} L_{A}^{2} K_{e} \left( {1 + \sigma _{e}^{2} \overline{\gamma } _{e} } \right)}}{{c_{e}^{4} \overline{\gamma } _{e} }}} \right)} ^{{\left( {n + \frac{{L_{A} L_{E} - 1}}{2}} \right)}} \sum\limits_{{l = 0}}^{{n + L_{A} L_{E} - 1}} {\left( {n + L_{A} L_{E} - 1} \right){!}\left( {\frac{{c_{e}^{2} \overline{\gamma } _{e} }}{{L_{A} r_{e} \left( {1 + \sigma _{e}^{2} \overline{\gamma } _{e} } \right)}}} \right)} ^{{n + L_{A} L_{E} - l}} \\ & \quad \Psi \left( {l + 1,\;l + 1;\;\frac{{L_{A} r_{e} \left( {1 + \sigma _{e}^{2} \overline{\gamma } _{e} } \right)}}{{c_{e}^{2} \overline{\gamma } _{e} }}} \right) - \frac{1}{{\ln 2}}\left( {\frac{{L_{A} r_{e} \left( {1 + \sigma _{e}^{2} \overline{\gamma } _{e} } \right)}}{{c_{e}^{2} \overline{\gamma } _{e} }}} \right)^{{\left( {\frac{{L_{A} L_{E} + 1}}{2}} \right)}} e^{{\left( { - \frac{{L_{A} L_{E} K_{e} }}{{c_{e}^{2} }}} \right)}} \left( {\frac{{c_{e}^{2} }}{{L_{A} L_{E} K_{e} }}} \right)^{{\left( {\frac{{L_{A} L_{E} - 1}}{2}} \right)}} \\ & \quad \quad \times \sum\limits_{{n = 0}}^{\infty } {\frac{{\left( { - 1} \right)^{n} }}{{n{!}}}\left( {\frac{{r_{e} L_{E} L_{A}^{2} K_{e} \left( {1 + \sigma _{e}^{2} \overline{\gamma } _{e} } \right)}}{{c_{e}^{4} \overline{\gamma } _{e} }}} \right)} ^{{\left( {n + \frac{{L_{A} L_{E} - 1}}{2}} \right)}} \left( {\frac{{L_{A} r_{b} \left( {1 + \sigma _{b}^{2} \overline{\gamma } _{b} } \right)}}{{c_{b}^{2} \overline{\gamma } _{b} }}} \right)^{{\left( {\frac{{L_{A} L_{B} + 1}}{2}} \right)}} e^{{\left( { - \frac{{L_{A} L_{B} K_{b} }}{{c_{b}^{2} }}} \right)}} \left( {\frac{{c_{b}^{2} }}{{L_{A} L_{B} K_{e} }}} \right)^{{\left( {\frac{{L_{A} L_{B} - 1}}{2}} \right)}} \\ & \quad \quad \times \sum\limits_{{m = 0}}^{\infty } {\frac{{\left( { - 1} \right)^{m} }}{{m{!}}}\left( {\frac{{r_{b} L_{B} L_{A}^{2} K_{b} \left( {1 + \sigma _{b}^{2} \overline{\gamma } _{b} } \right)}}{{c_{b}^{4} \overline{\gamma } _{b} }}} \right)} ^{{\left( {m + \frac{{L_{A} L_{B} - 1}}{2}} \right)}} \left[ \begin{gathered} - \left( {\frac{{c_{b}^{2} \overline{\gamma } _{b} }}{{L_{A} r_{b} \left( {1 + \sigma _{b}^{2} \overline{\gamma } _{b} } \right)}}} \right)^{{m + L_{A} L_{B} }} \left( {m + L_{A} L_{B} - 1} \right){!} \hfill \\ \sum\limits_{{l = 0}}^{{n + L_{A} L_{E} - 1}} {\left( {n + L_{A} L_{E} - 1} \right){!}\left( {\frac{{c_{e}^{2} \overline{\gamma } _{e} }}{{L_{A} r_{e} \left( {1 + \sigma _{e}^{2} \overline{\gamma } _{e} } \right)}}} \right)} ^{{n + L_{A} L_{E} - l}} \Psi \left( {l + 1,\;l + 1;\;\frac{{L_{A} r_{e} \left( {1 + \sigma _{e}^{2} \overline{\gamma } _{e} } \right)}}{{c_{e}^{2} \overline{\gamma } _{e} }}} \right) \hfill \\ \end{gathered} \right] \\ & \quad \quad \times - \frac{1}{{\ln 2}}\left( {\frac{{L_{A} r_{e} \left( {1 + \sigma _{e}^{2} \overline{\gamma } _{e} } \right)}}{{c_{e}^{2} \overline{\gamma } _{e} }}} \right)^{{\left( {\frac{{L_{A} L_{E} + 1}}{2}} \right)}} e^{{\left( { - \frac{{L_{A} L_{E} K_{e} }}{{c_{e}^{2} }}} \right)}} \left( {\frac{{c_{e}^{2} }}{{L_{A} L_{E} K_{e} }}} \right)^{{\left( {\frac{{L_{A} L_{E} - 1}}{2}} \right)}} \\ & \quad \quad \times \sum\limits_{{n = 0}}^{\infty } {\frac{{\left( { - 1} \right)^{n} }}{{n{!}}}\left( {\frac{{r_{e} L_{E} L_{A}^{2} K_{e} \left( {1 + \sigma _{e}^{2} \overline{\gamma } _{e} } \right)}}{{c_{e}^{4} \overline{\gamma } _{e} }}} \right)} ^{{\left( {n + \frac{{L_{A} L_{E} - 1}}{2}} \right)}} \left( {\frac{{L_{A} r_{b} \left( {1 + \sigma _{b}^{2} \overline{\gamma } _{b} } \right)}}{{c_{b}^{2} \overline{\gamma } _{b} }}} \right)^{{\left( {\frac{{L_{A} L_{B} + 1}}{2}} \right)}} e^{{\left( { - \frac{{L_{A} L_{B} K_{b} }}{{c_{b}^{2} }}} \right)}} \left( {\frac{{c_{b}^{2} }}{{L_{A} L_{B} K_{e} }}} \right)^{{\left( {\frac{{L_{A} L_{B} - 1}}{2}} \right)}} \\ & \quad \quad \times \sum\limits_{{m = 0}}^{\infty } {\frac{{\left( { - 1} \right)^{m} }}{{m{!}}}\left( {\frac{{r_{b} L_{B} L_{A}^{2} K_{b} \left( {1 + \sigma _{b}^{2} \overline{\gamma } _{b} } \right)}}{{c_{b}^{4} \overline{\gamma } _{b} }}} \right)} ^{{\left( {m + \frac{{L_{A} L_{B} - 1}}{2}} \right)}} \left[ \begin{gathered} - \sum\limits_{{k = 0}}^{{m + L_{A} L_{B} - 1}} {\left( {m + L_{A} L_{B} - 1} \right){!}\left( {\frac{{c_{b}^{2} \overline{\gamma } _{b} }}{{L_{A} r_{b} \left( {1 + \sigma _{b}^{2} \overline{\gamma } _{b} } \right)}}} \right)} ^{{m + L_{A} L_{B} - k}} \left( {n + L_{A} L_{E} - 1} \right){!} \times \hfill \\ \left( {\frac{{c_{e}^{2} \overline{\gamma } _{e} }}{{L_{A} r_{e} \left( {1 + \sigma _{e}^{2} \overline{\gamma } _{e} } \right)}}} \right)^{{n + L_{A} L_{E} }} \Psi \left( {k + 1,\;k + 1;\,\frac{{L_{A} r_{b} \left( {1 + \sigma _{b}^{2} \overline{\gamma } _{b} } \right)}}{{c_{b}^{2} \overline{\gamma } _{b} }}} \right) \hfill \\ \end{gathered} \right] \\ & \quad \quad - \frac{1}{{\ln 2}}\left( {\frac{{L_{A} r_{e} \left( {1 + \sigma _{e}^{2} \overline{\gamma } _{e} } \right)}}{{c_{e}^{2} \overline{\gamma } _{e} }}} \right)^{{\left( {\frac{{L_{A} L_{E} + 1}}{2}} \right)}} e^{{\left( { - \frac{{L_{A} L_{E} K_{e} }}{{c_{e}^{2} }}} \right)}} \left( {\frac{{c_{e}^{2} }}{{L_{A} L_{E} K_{e} }}} \right)^{{\left( {\frac{{L_{A} L_{E} - 1}}{2}} \right)}} \sum\limits_{{n = 0}}^{\infty } {\frac{{\left( { - 1} \right)^{n} }}{{n{!}}}\left( {\frac{{r_{e} L_{E} L_{A}^{2} K_{e} \left( {1 + \sigma _{e}^{2} \overline{\gamma } _{e} } \right)}}{{c_{e}^{4} \overline{\gamma } _{e} }}} \right)} ^{{\left( {n + \frac{{L_{A} L_{E} - 1}}{2}} \right)}} \\ & \quad \left( {\frac{{L_{A} r_{b} \left( {1 + \sigma _{b}^{2} \overline{\gamma } _{b} } \right)}}{{c_{b}^{2} \overline{\gamma } _{b} }}} \right)^{{\left( {\frac{{L_{A} L_{B} + 1}}{2}} \right)}} e^{{\left( { - \frac{{L_{A} L_{B} K_{b} }}{{c_{b}^{2} }}} \right)}} \left( {\frac{{c_{b}^{2} }}{{L_{A} L_{B} K_{e} }}} \right)^{{\left( {\frac{{L_{A} L_{B} - 1}}{2}} \right)}} \sum\limits_{{m = 0}}^{\infty } {\frac{{\left( { - 1} \right)^{m} }}{{m{!}}}\left( {\frac{{r_{b} L_{B} L_{A}^{2} K_{b} \left( {1 + \sigma _{b}^{2} \overline{\gamma } _{b} } \right)}}{{c_{b}^{4} \overline{\gamma } _{b} }}} \right)} ^{{\left( {m + \frac{{L_{A} L_{B} - 1}}{2}} \right)}} \\ & \quad \left[ \begin{gathered} \sum\limits_{{k = 0}}^{{m + L_{A} L_{B} - 1}} {\frac{{\left( {m + L_{A} L_{B} - 1} \right){!}}}{{k{!}}}\left( {\frac{{c_{b}^{2} \overline{\gamma } _{b} }}{{L_{A} r_{b} \left( {1 + \sigma _{b}^{2} \overline{\gamma } _{b} } \right)}}} \right)^{{m + L_{A} L_{B} - k}} } \sum\limits_{{l = 0}}^{{n + L_{A} L_{E} - 1}} {\frac{{\left( {n + L_{A} L_{E} - 1} \right){!}}}{{l{!}}}\left( {\frac{{c_{e}^{2} \overline{\gamma } _{e} }}{{L_{A} r_{e} \left( {1 + \sigma _{e}^{2} \overline{\gamma } _{e} } \right)}}} \right)^{{n + L_{A} L_{E} - l}} } \hfill \\ \left( {k + l} \right){!}\Psi \left( {k + l + 1,\;k + l + 1;\;\frac{{L_{A} r_{b} \left( {1 + \sigma _{b}^{2} \overline{\gamma } _{b} } \right)}}{{c_{b}^{2} \overline{\gamma } _{b} }} + \frac{{L_{A} r_{e} \left( {1 + \sigma _{e}^{2} \overline{\gamma } _{e} } \right)}}{{c_{e}^{2} \overline{\gamma } _{e} }}} \right) \hfill \\ \end{gathered} \right]. \\ \end{aligned} $$
(19)

Appendix 2

$$ \Pr \left( {C_{S} > 0} \right) = \int_{0}^{\infty } {f_{{\gamma_{B} }} \left( x \right)F_{{\gamma_{E} }} \left( x \right)} dx. $$
(20)

By putting the value of PDF of output SNR at Bob \(f_{{\gamma_{B} }} \left( x \right)\) and CDF of output SNR at Eve \(F_{{\gamma_{E} }} \left( x \right)\) into Eq. (20) and using results of [27, 3.351.3], the PNZSR can be expressed as shown below:

$$ \begin{aligned} & \Pr \left( {C_{S} > 0} \right) = \left( {\frac{{L_{A} r_{b} \left( {1 + \sigma_{b}^{2} \overline{\gamma }_{b} } \right)}}{{c_{b}^{2} \overline{\gamma }_{b} }}} \right)^{{\left( {\frac{{L_{A} L_{B} + 1}}{2}} \right)}} e^{{\left( { - \frac{{L_{A} L_{B} K_{b} }}{{c_{b}^{2} }}} \right)}} \left( {\frac{{c_{b}^{2} }}{{L_{A} L_{B} K_{b} }}} \right)^{{\left( {\frac{{L_{A} L_{B} - 1}}{2}} \right)}} \\ & \quad \times \sum\limits_{m = 0}^{\infty } {\frac{{\left( { - 1} \right)^{m} }}{{m{!}\left( {m + L_{A} L_{B} - 1} \right){!}}}\left( {\frac{{r_{b} L_{B} L_{A}^{2} K_{b} \left( {1 + \sigma_{b}^{2} \overline{\gamma }_{b} } \right)}}{{c_{b}^{4} \overline{\gamma }_{b} }}} \right)}^{{\left( {m + \frac{{L_{A} L_{B} - 1}}{2}} \right)}} \left( {\frac{{L_{A} r_{e} \left( {1 + \sigma_{e}^{2} \overline{\gamma }_{e} } \right)}}{{c_{e}^{2} \overline{\gamma }_{e} }}} \right)^{{\left( {\frac{{L_{A} L_{E} + 1}}{2}} \right)}} \\ & \quad \times e^{{\left( { - \frac{{L_{A} L_{E} K_{e} }}{{c_{e}^{2} }}} \right)}} \left( {\frac{{c_{e}^{2} }}{{L_{A} L_{E} K_{e} }}} \right)^{{\left( {\frac{{L_{A} L_{E} - 1}}{2}} \right)}} \sum\limits_{n = 0}^{\infty } {\frac{{\left( { - 1} \right)^{n} }}{{n{!}\left( {n + L_{A} L_{E} - 1} \right){!}}}\left( {\frac{{r_{e} L_{E} L_{A}^{2} K_{e} \left( {1 + \sigma_{e}^{2} \overline{\gamma }_{e} } \right)}}{{c_{e}^{4} \overline{\gamma }_{e} }}} \right)}^{{\left( {n + \frac{{L_{A} L_{E} - 1}}{2}} \right)}} \\ & \quad \times \left[ {\left( {n + L_{A} L_{E} - 1} \right){!}\left( {\frac{{c_{e}^{2} \overline{\gamma }_{e} }}{{L_{A} r_{e} \left( {1 + \sigma_{e}^{2} \overline{\gamma }_{e} } \right)}}} \right)^{{n + L_{A} L_{E} }} \left( {m + L_{A} L_{B} - 1} \right){!}\left( {\frac{{L_{A} r_{b} \left( {1 + \sigma_{b}^{2} \overline{\gamma }_{b} } \right)}}{{c_{b}^{2} \overline{\gamma }_{b} }}} \right)^{{ - \left( {m + L_{A} L_{B} } \right)}} } \right] \\ & \quad - \left( {\frac{{L_{A} r_{b} \left( {1 + \sigma_{b}^{2} \overline{\gamma }_{b} } \right)}}{{c_{b}^{2} \overline{\gamma }_{b} }}} \right)^{{\left( {\frac{{L_{A} L_{B} + 1}}{2}} \right)}} e^{{\left( { - \frac{{L_{A} L_{B} K_{b} }}{{c_{b}^{2} }}} \right)}} \left( {\frac{{c_{b}^{2} }}{{L_{A} L_{B} K_{b} }}} \right)^{{\left( {\frac{{L_{A} L_{B} - 1}}{2}} \right)}} \\ & \quad \times \sum\limits_{m = 0}^{\infty } {\frac{{\left( { - 1} \right)^{m} }}{{m{!}\left( {m + L_{A} L_{B} - 1} \right){!}}}\left( {\frac{{r_{b} L_{B} L_{A}^{2} K_{b} \left( {1 + \sigma_{b}^{2} \overline{\gamma }_{b} } \right)}}{{c_{b}^{4} \overline{\gamma }_{b} }}} \right)}^{{\left( {m + \frac{{L_{A} L_{B} - 1}}{2}} \right)}} \left( {\frac{{L_{A} r_{e} \left( {1 + \sigma_{e}^{2} \overline{\gamma }_{e} } \right)}}{{c_{e}^{2} \overline{\gamma }_{e} }}} \right)^{{\left( {\frac{{L_{A} L_{E} + 1}}{2}} \right)}} \\ & \quad \times e^{{\left( { - \frac{{L_{A} L_{E} K_{e} }}{{c_{e}^{2} }}} \right)}} \left( {\frac{{c_{e}^{2} }}{{L_{A} L_{E} K_{e} }}} \right)^{{\left( {\frac{{L_{A} L_{E} - 1}}{2}} \right)}} \sum\limits_{n = 0}^{\infty } {\frac{{\left( { - 1} \right)^{n} }}{{n{!}\left( {n + L_{A} L_{E} - 1} \right){!}}}\left( {\frac{{r_{e} L_{E} L_{A}^{2} K_{e} \left( {1 + \sigma_{e}^{2} \overline{\gamma }_{e} } \right)}}{{c_{e}^{4} \overline{\gamma }_{e} }}} \right)}^{{\left( {n + \frac{{L_{A} L_{E} - 1}}{2}} \right)}} \\ & \quad \times \left[ \begin{gathered} \sum\limits_{l = 0}^{{n + L_{A} L_{E} - 1}} {\frac{{\left( {n + L_{A} L_{E} - 1} \right){!}}}{{l{!}}}} \left( {\frac{{c_{e}^{2} \overline{\gamma }_{e} }}{{L_{A} r_{e} \left( {1 + \sigma_{e}^{2} \overline{\gamma }_{e} } \right)}}} \right)^{{n + L_{A} L_{E} - l}} \left( {m + l + L_{A} L_{B} - 1} \right){!} \hfill \\ \times \left( {\frac{{L_{A} r_{b} \left( {1 + \sigma_{b}^{2} \overline{\gamma }_{b} } \right)}}{{c_{b}^{2} \overline{\gamma }_{b} }} + \frac{{L_{A} r_{e} \left( {1 + \sigma_{e}^{2} \overline{\gamma }_{e} } \right)}}{{c_{e}^{2} \overline{\gamma }_{e} }}} \right)^{{ - \left( {m + l + L_{A} L_{B} } \right)}} \hfill \\ \end{gathered} \right]. \\ \end{aligned} $$
(21)

Appendix 3

$$ P_{out} \left( {R_{S} } \right) = \Pr (R_{S} < C_{S} ) = = \int_{0}^{\infty } {F_{{\gamma_{B} }} \left( {2^{{R_{S} }} \left( {1 + x} \right) - 1} \right)f_{{\gamma_{e} }} } \left( x \right)dx. $$
(22)

By putting the value of PDF of output SNR at Eve \(f_{{\gamma_{e} }} \left( x \right)\) and CDF of output SNR at Bob with some modification \(F_{{\gamma_{B} }} \left( {2^{{R_{S} }} \left( {1 + x} \right) - 1} \right)\) into (22), we get final expression for SOP, which is shown in Eq. (23).

$$ \begin{aligned} & P_{out} \left( {R_{S} } \right) = \left( {\frac{{L_{A} r_{b} \left( {1 + \sigma_{b}^{2} \overline{\gamma }_{b} } \right)}}{{c_{b}^{2} \overline{\gamma }_{b} }}} \right)^{{\left( {\frac{{L_{A} L_{B} + 1}}{2}} \right)}} e^{{\left( { - \frac{{L_{A} L_{B} K_{b} }}{{c_{b}^{2} }}} \right)}} \left( {\frac{{c_{b}^{2} }}{{L_{A} L_{B} K_{b} }}} \right)^{{\left( {\frac{{L_{A} L_{B} - 1}}{2}} \right)}} \\ & \quad \times \sum\limits_{m = 0}^{\infty } {\frac{{\left( { - 1} \right)^{m} }}{{m{!}\left( {m + L_{A} L_{B} - 1} \right){!}}}\left( {\frac{{r_{b} L_{B} L_{A}^{2} K_{b} \left( {1 + \sigma_{b}^{2} \overline{\gamma }_{b} } \right)}}{{c_{b}^{4} \overline{\gamma }_{b} }}} \right)}^{{\left( {m + \frac{{L_{A} L_{B} - 1}}{2}} \right)}} \left( {\frac{{L_{A} r_{e} \left( {1 + \sigma_{e}^{2} \overline{\gamma }_{e} } \right)}}{{c_{e}^{2} \overline{\gamma }_{e} }}} \right)^{{\left( {\frac{{L_{A} L_{E} + 1}}{2}} \right)}} \\ & \quad \times e^{{\left( { - \frac{{L_{A} L_{E} K_{e} }}{{c_{e}^{2} }}} \right)}} \left( {\frac{{c_{e}^{2} }}{{L_{A} L_{E} K_{e} }}} \right)^{{\left( {\frac{{L_{A} L_{E} - 1}}{2}} \right)}} \sum\limits_{n = 0}^{\infty } {\frac{{\left( { - 1} \right)^{n} }}{{n{!}\left( {n + L_{A} L_{E} - 1} \right){!}}}\left( {\frac{{r_{e} L_{E} L_{A}^{2} K_{e} \left( {1 + \sigma_{e}^{2} \overline{\gamma }_{e} } \right)}}{{c_{e}^{4} \overline{\gamma }_{e} }}} \right)}^{{\left( {n + \frac{{L_{A} L_{E} - 1}}{2}} \right)}} \\ & \quad \times \left[ {\left( {m + L_{A} L_{B} - 1} \right)!\left( {\frac{{\overline{\gamma }_{b} c_{b}^{2} }}{{L_{A} r_{b} \left( {1 + \sigma_{b}^{2} \overline{\gamma }_{b} } \right)}}} \right)^{{m + L_{A} L_{B} }} \left( {n + L_{A} L_{E} - 1} \right)!\left( {\frac{{L_{A} r_{e} \left( {1 + \sigma_{e}^{2} \overline{\gamma }_{e} } \right)}}{{c_{e}^{2} \overline{\gamma }_{e} }}} \right)^{{ - \left( {n + L_{A} L_{E} } \right)}} } \right] \\ & \quad - \left( {\frac{{L_{A} r_{b} \left( {1 + \sigma_{b}^{2} \overline{\gamma }_{b} } \right)}}{{c_{b}^{2} \overline{\gamma }_{b} }}} \right)^{{\left( {\frac{{L_{A} L_{B} + 1}}{2}} \right)}} e^{{\left( { - \frac{{L_{A} L_{B} K_{b} }}{{c_{b}^{2} }}} \right)}} \left( {\frac{{c_{b}^{2} }}{{L_{A} L_{B} K_{b} }}} \right)^{{\left( {\frac{{L_{A} L_{B} - 1}}{2}} \right)}} \\ & \quad \times \sum\limits_{m = 0}^{\infty } {\frac{{\left( { - 1} \right)^{m} }}{{m{!}\left( {m + L_{A} L_{B} - 1} \right){!}}}\left( {\frac{{r_{b} L_{B} L_{A}^{2} K_{b} \left( {1 + \sigma_{b}^{2} \overline{\gamma }_{b} } \right)}}{{c_{b}^{4} \overline{\gamma }_{b} }}} \right)}^{{\left( {m + \frac{{L_{A} L_{B} - 1}}{2}} \right)}} \left( {\frac{{L_{A} r_{e} \left( {1 + \sigma_{e}^{2} \overline{\gamma }_{e} } \right)}}{{c_{e}^{2} \overline{\gamma }_{e} }}} \right)^{{\left( {\frac{{L_{A} L_{E} + 1}}{2}} \right)}} \\ & \quad \times e^{{\left( { - \frac{{L_{A} L_{E} K_{e} }}{{c_{e}^{2} }}} \right)}} \left( {\frac{{c_{e}^{2} }}{{L_{A} L_{E} K_{e} }}} \right)^{{\left( {\frac{{L_{A} L_{E} - 1}}{2}} \right)}} \sum\limits_{n = 0}^{\infty } {\frac{{\left( { - 1} \right)^{n} }}{{n{!}\left( {n + L_{A} L_{E} - 1} \right){!}}}\left( {\frac{{r_{e} L_{E} L_{A}^{2} K_{e} \left( {1 + \sigma_{e}^{2} \overline{\gamma }_{e} } \right)}}{{c_{e}^{4} \overline{\gamma }_{e} }}} \right)}^{{\left( {n + \frac{{L_{A} L_{E} - 1}}{2}} \right)}} \\ & \quad \times \left[ \begin{gathered} \sum\limits_{k = 0}^{{\left( {m + L_{A} L_{B} - 1} \right)}} {\frac{{\left( {m + L_{A} L_{B} - 1} \right)!}}{k!}} \left( {\frac{{\overline{\gamma }_{b} c_{b}^{2} }}{{L_{A} r_{b} \left( {1 + \sigma_{b}^{2} \overline{\gamma }_{b} } \right)}}} \right)^{{m + L_{A} L_{B} - k}} e^{{ - \left( {\frac{{L_{A} r_{b} \left( {1 + \sigma_{b}^{2} \overline{\gamma }_{b} } \right)}}{{c_{b}^{2} \overline{\gamma }_{b} }}} \right)\left( {2^{{R_{S} }} - 1} \right)}} \hfill \\ \times \sum\limits_{i = 0}^{k} {\left( {\begin{array}{*{20}c} k \\ i \\ \end{array} } \right)\left( {2^{{R_{S} }} - 1} \right)^{k - i} \left( {2^{{R_{S} }} } \right)^{i} \left( {i + n + L_{A} L_{E} - 1} \right)!\left( {\frac{{L_{A} r_{e} \left( {1 + \sigma_{e}^{2} \overline{\gamma }_{e} } \right)}}{{c_{e}^{2} \overline{\gamma }_{e} }} + \frac{{L_{A} r_{b} \left( {1 + \sigma_{b}^{2} \overline{\gamma }_{b} } \right)}}{{c_{b}^{2} \overline{\gamma }_{b} }}} \right)^{{ - \left( {n + L_{A} L_{E} + i} \right)}} } \hfill \\ \end{gathered} \right]. \\ \end{aligned} $$
(23)

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Kumar, R., Chauhan, S.S. Physical layer security for space–time-block-coded MIMO system in Rician fading in the presence of imperfect feedback. Wireless Netw 26, 4239–4247 (2020). https://doi.org/10.1007/s11276-020-02327-x

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