SCIENCE CHINA Information Sciences, Volume 64 , Issue 9 : 192303(2021) https://doi.org/10.1007/s11432-019-2919-5

## Covert communication with beamforming over MISO channels in the finite blocklength regime

• AcceptedMay 13, 2020
• PublishedAug 18, 2021
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### Acknowledgment

This work was supported by National Natural Science Foundation of China (Grant No. 61871264).

### Supplement

Appendix

łemma If $\tilde{{\boldsymbol~g}}$ is subject to $\mathcal{CN}(0,\delta^2_{{\boldsymbol~g}}~{\boldsymbol~I})$ and ${\boldsymbol~w}$ is a constant vector with the same dimension, $\tilde{{\boldsymbol~g}}^{\rm~H}{\boldsymbol~w}$ is a zero-mean complex circularly symmetric Gaussian random variable with variance $\|{\boldsymbol~w}\|^2\delta_{{\boldsymbol~g}}^2$.

proof Since $\tilde{{\boldsymbol~g}}$ follows $\mathcal{CN}(0,\delta^2_{{\boldsymbol~g}}~{\boldsymbol~I})$, the complex conjugate $\tilde{{\boldsymbol~g}}_i^*~(i=1,\ldots,n)$ are zero-mean complex circularly symmetric Gaussian with variance $\delta_{{\boldsymbol~g}}^2$ and independent of each other. $\tilde{{\boldsymbol~g}}^{\rm~H}{\boldsymbol~w}~=~\sum_{i=1}^n~\tilde{{\boldsymbol~g}}_i^*{\boldsymbol~w}_i$ is a linear combination of these $\tilde{{\boldsymbol~g}}_i^*$. The $i$th term in the right side of the equation is zero-mean complex Gaussian with variance $|{\boldsymbol~w}_i|^2\delta_{{\boldsymbol~g}}^2$. Hence, the sum of them is zero-mean complex circularly symmetric Gaussian with variance $\|{\boldsymbol~w}\|^2\delta_{{\boldsymbol~g}}^2$.

corollary With the same condition as Lemma lemma2, $~|\tilde{{\boldsymbol~g}}^{\rm~H}{\boldsymbol~w}~|$ is subject to Rayleigh distribution with probability density function: $$f(x) = \frac{x}{\|{\boldsymbol w}\|^2\delta_{{\boldsymbol g}}^2}{\rm e}^{-\frac{x^2}{2\|{\boldsymbol w}\|^2\delta_{{\boldsymbol g}}^2}}, x \geq 0.$$ $~|\tilde{{\boldsymbol~g}}^{\rm~H}{\boldsymbol~w}~|^2$ is subject to chi-squared distribution with two degrees of freedom, i.e., exponential distribution with pdf $$g(x) = \frac{1}{2\|{\boldsymbol w}\|^2\delta_{{\boldsymbol g}}^2}{\rm e}^{-\frac{x}{2\|{\boldsymbol w}\|^2\delta_{{\boldsymbol g}}^2}}, x \geq 0.$$

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• Figure 1

MISO channel model of covert communication in Subsection sect. 2.1.

• Figure 2

The model to compute the random variable $~|{\boldsymbol~g}^{\rm~H}{\boldsymbol~w}|^2$.

• Figure 3

(Color online) Comparison of the bounds with different beamforming strategies. (a) $\delta~=~0.2$; (b) $n~=~1000$.

• Figure 4

(Color online) (a) Available region of the pair ($P$, $\delta$) for beamforming. (b) Available regions for beamforming when $\delta~$ varies from $0.1$ to $1$.

• Figure 5

(Color online) Available regions for beamforming when (a) $\kappa~$ varies from $0$ to $0.9$ and (b) $\alpha~$ varies from $0.1$ to $0.9$.

• Figure 6

(Color online) Comparison of different beamforming strategies. (a) $(\delta,\kappa)=~(0.5,0.2)$; (b) $n~=~1000$ and $\kappa~=~0.2$.

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