SCIENTIA SINICA Informationis, Volume 51 , Issue 8 : 1360(2021) https://doi.org/10.1360/SSI-2020-0287

Threat region development of covert wireless communication based on 3D beamforming

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  • ReceivedSep 15, 2020
  • AcceptedDec 14, 2020
  • PublishedAug 2, 2021


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  • Figure 1

    (Color online) System model of downlink covert wireless communication based on 3D beamforming

  • Figure 8

    (Color online) Thermogram of the maximum covert throughput and the covert threat region. (a) Matched group; (b) ${K_b}=~{K_w}=~30$; (c) $\Delta~d/\lambda~=~1.5$; (d) ULA with 32 antennas; (e) $\sigma~_n^2~=~-~20~{\rm{~dB}}$; (f) $\rho~=~1.5$; (g) $\delta~=~0.001$;protect łinebreak (h) $\varepsilon~=~0.01$

  • Figure 9

    (Color online) Thermogram of the optimal transmission power and the maximum covert throughput. (a) Covert threat region based on Algorithm 1; (b) error of ${P^~*~}$; (c) error of ${\eta~^~*~}$; (d) error ratio of ${\eta~^~*~}$

  • Table 1   Simulation parameters
    Channel model Antenna model at Alice Antenna model at Bob and Willie
    Rician fading with ${K_b}~=~{K_w}~=~3$ $L~\times~M~=~8~\times~4$, $T~=~30~{\rm{~m}}$, $\Delta~d/\lambda~~=~1$ Single antenna
    Beamforming mode Service radius of base station Path loss exponent
    MRT $r~=~100~{\rm{~m}}$ $\alpha~~=~3$
    Position of Alice Position of Bob Position of Willie
    (0, 0) (30, 30) (20, 25)
    Noise level Noise uncertainty level Maximum transmission power requirement
    $\sigma~_b^2~=~\sigma~_n^2~=~~-~30~{\rm{~dB}}$ $\rho~~=~1.1$ ${P_{\max~}}~=~10~{\rm{~W}}$
    Covertness requirement Reliability requirement Maximum covert throughput requirement
    $\varepsilon~~=~0.1$ $\delta~~=~0.01$ ${\eta~_l}~=~0.1~{\rm{~bit/s/Hz}}$

    Algorithm 1 Search algorithm for maximum covert throughput

    Require:Covertness requirement $\varepsilon~$, reliability requirement $\delta~$, and maximum transmit power requirement ${P_{\max~}}$.

    Output:Maximum covert throughput ${\eta~^~*~}$, optimal transmit power ${P^~*~}$, and optimal target rate ${R^~*~}$.

    Transform the covertness constraint into an equality, and solve the integral equation ${\bar~\xi~^~*~}(P)~=~1~-~\varepsilon~$ for ${P^\Delta~}$ through binary search. Then, obtain ${P^~*~}~=~\min~\{~{P^\Delta~},{P_{\max~}}\}~$ under the maximum transmit power constraint;

    Transform the reliability constraint into an equality, and obtain ${R^\Delta~}$ by solving the integral equation ${p_{{\rm{out~}}}}({P^~*~},R)~=~\delta~$;

    Set the searching range as $R~\in~(0,{R^\Delta~}]$, and obtain ${R^~*~}$ and ${\eta~^~*~}$ through one-side exhaustive search.


    Algorithm 2 Lightweight search algorithm for covert threat region

    Require:Covertness requirement $\varepsilon~$, reliability requirement $\delta~$, maximum transmit power requirement ${P_{\max~}}$, and covert throughput requirement ${\eta~_l}$.

    Output:Covert threat region ${{\mathop{\rm~Area}\nolimits}~_D}\left(~{x,y}~\right)$.

    Calculate the fitting factor $\mu~$ through image method. Transform the fitting covertness constraint (28) into an equality and compute ${P^\Delta~}~=~\frac{{LM\sigma~_n^2\gamma~_w^{\rm~th}\left(~{{K_b}~+~1}~\right)\left(~{{K_w}~+~1}~\right)}}{{\mu~d_{aw}^{~-~\alpha~}\left(~{{K_b}{K_w}{g_{bw}}~+~LM\left(~{{K_b}~+~{K_w}~+~1}~\right)}~\right)}}$. Then, obtain ${P^~*~}~=~\min~\{~{P^\Delta~},{P_{\max~}}\}~$ under the maximum transmit power constraint (24d);

    Compute the numerical solution of the maximum point ${R_{\max~}}$ for $\eta$, and obtain the corresponding $p_{\rm~out}^{\max~}$. Next, obtian ${\eta~^~*~}~=~\left(~{1~-~p_{\rm~out}^{\max~}}~\right){R_{\max~}}$ directly when $\delta~~\ge~p_{\rm~out}^{\max~}$. Otherwise, when $\delta~~<~p_{\rm~out}^{\max~}$, transform the reliability constraint (24c) into an equality and obtain ${R^\Delta~}$ by solving the integral equation ${p_{{\rm{out~}}}}({P^~*~},R)~=~\delta~$, as well as ${\eta~^~*~}~=~\left(~{1~-~\delta~}~\right){R^\Delta~}$;

    Traverse the whole system service area with the position of Willie, and calculate ${\eta~^~*~}(x,y)$. At last, obtain ${{\mathop{\rm~Area}\nolimits}~_D}\left(~{x,y}~\right)$ with the comparison of ${\eta~_l}$.


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