logo

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

More info
  • ReceivedDec 3, 2019
  • AcceptedMay 13, 2020
  • PublishedAug 18, 2021

Abstract


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 $$g(x) = \frac{1}{2\|{\boldsymbol w}\|^2\delta_{{\boldsymbol g}}^2}{\rm e}^{-\frac{x}{2\|{\boldsymbol w}\|^2\delta_{{\boldsymbol g}}^2}}, x \geq 0.$$


References

[1] Shafiee S, Liu N, Ulukus S. Towards the Secrecy Capacity of the Gaussian MIMO Wire-Tap Channel: The 2-2-1 Channel. IEEE Trans Inform Theor, 2009, 55: 4033-4039 CrossRef Google Scholar

[2] Khisti A, Wornell G W. Secure Transmission With Multiple Antennas I: The MISOME Wiretap Channel. IEEE Trans Inform Theor, 2010, 56: 3088-3104 CrossRef Google Scholar

[3] Khisti A, Wornell G W. Secure Transmission With Multiple Antennas-Part II: The MIMOME Wiretap Channel. IEEE Trans Inform Theor, 2010, 56: 5515-5532 CrossRef Google Scholar

[4] Zhang L, Wu G, Li S Q. Capacity bounds of transmit beamforming over MISO time-varying channels with imperfect feedback. Sci China Inf Sci, 2010, 53: 1417-1430 CrossRef Google Scholar

[5] Tie Liu , Shamai S. A Note on the Secrecy Capacity of the Multiple-Antenna Wiretap Channel. IEEE Trans Inform Theor, 2009, 55: 2547-2553 CrossRef Google Scholar

[6] Gerbracht S, Scheunert C, Jorswieck E A. Secrecy Outage in MISO Systems With Partial Channel Information. IEEE TransInformForensic Secur, 2012, 7: 704-716 CrossRef Google Scholar

[7] Rezki Z, Khisti A, Alouini M S. On the secrecy capacity of the MISO wiretap channel under imperfect channel estimation. In: Proceedings of IEEE Global Communications Conference(GLOBECOM'2014), Austin, 2014. 1602--1607. Google Scholar

[8] Zhou X, Rezki Z, Alomair B. Achievable Rates of Secure Transmission in Gaussian MISO Channel With Imperfect Main Channel Estimation. IEEE Trans Wireless Commun, 2016, 15: 4470-4485 CrossRef Google Scholar

[9] Bash B A, Goeckel D, Towsley D. Limits of Reliable Communication with Low Probability of Detection on AWGN Channels. IEEE J Sel Areas Commun, 2013, 31: 1921-1930 CrossRef Google Scholar

[10] Wang L, Wornell G W, Zheng L. Fundamental Limits of Communication With Low Probability of Detection. IEEE Trans Inform Theor, 2016, 62: 3493-3503 CrossRef Google Scholar

[11] Bloch M R. Covert Communication Over Noisy Channels: A Resolvability Perspective. IEEE Trans Inform Theor, 2016, 62: 2334-2354 CrossRef Google Scholar

[12] Abdelaziz A, Koksal C E. Fundamental limits of covert communication over MIMO AWGN channel. In: Proceedings of 2017 IEEE Conference on Communications and Network Security (CNS) Las Vegas, 2017. 1--9. Google Scholar

[13] Lu W, Xu Z, Gong Y. A novel covert communication system based on symmetric ${\alpha}$-stable distribution. Sci Sin-Inf, 2017, 47: 374-384 CrossRef Google Scholar

[14] Zhao H J, Ahn G J, Hu H X. 常规隐信道下可证明安全隐写术的有效构造方法. Sci Sin-Inf, 2013, 43: 657-669 CrossRef Google Scholar

[15] Lee S, Baxley R J, Weitnauer M A. Achieving Undetectable Communication. IEEE J Sel Top Signal Process, 2015, 9: 1195-1205 CrossRef ADS Google Scholar

[16] Che P H, Bakshi M, Jaggi S. Reliable deniable communication: hiding messages in noise. In: Proceedings of IEEE International Symposium on Information Theory (ISIT2013), Istanbul, 2013. 2945--2949. Google Scholar

[17] He B, Yan S, Zhou X. On Covert Communication With Noise Uncertainty. IEEE Commun Lett, 2017, 21: 941-944 CrossRef Google Scholar

[18] Shahzad K, Zhou X, Yan S. Covert communication in fading channels under channel uncertainty. In: Proceedings of IEEE 85th Vehicular Technology Conference (VTC Spring), Sydney, 2017. 1--5. Google Scholar

[19] Sobers T V, Bash B A, Goeckel D, et al. Covert communication with the help of an uninformed jammer achieves positive rat. In: Proceedings of the 49th Asilomar Conference on Signals, Systems and Computers, Pacific Grove, 2015. 625--629. Google Scholar

[20] Sobers T V, Bash B A, Goeckel D, et al. Covert communication with the help of an uninformed jammer achieves positive rate. In: Proceedings of the 49th Asilomar Conference on Signals, Systems and Computers, Pacific Grove, 2015. 625--629. Google Scholar

[21] Soltani R, Goeckel D, Towsley D. Covert Wireless Communication With Artificial Noise Generation. IEEE Trans Wireless Commun, 2018, 17: 7252-7267 CrossRef Google Scholar

[22] Li J, Petropulu A P. On Ergodic Secrecy Rate for Gaussian MISO Wiretap Channels. IEEE Trans Wireless Commun, 2011, 10: 1176-1187 CrossRef Google Scholar

[23] Tahmasbi M, Bloch M R. Second-order asymptotics of covert communications over noisy channels. In: Proceedings of IEEE International Symposium on Information Theory (ISIT), Barcelona, 2016. 2224--2228. Google Scholar

[24] Tahmasbi M, Bloch M R. First- and Second-Order Asymptotics in Covert Communication. IEEE Trans Inform Theor, 2019, 65: 2190-2212 CrossRef Google Scholar

[25] Tang H, Wang J, Zheng Y R. Covert communication with extremely low power under finite block length over slow fading. In: Proceedings of IEEE Conference on Computer Communications Workshops (INFOCOM WKSHPS), Honolulu, 2018. 657--661. Google Scholar

[26] Yu X, Wei S, Luo Y. One-shot achievability and converse bounds of Gaussian random coding in AWGN channels under covert constraint. In: Proceedings of the 57th Annual Allerton Conference on Communication, Control, and Computing (Allerton), Monticello, 2019. Google Scholar

[27] Yu X, Wei S, Luo Y. Finite blocklength analysis of Gaussian random coding in AWGN channels under covert constraints. Submitted to IEEE Transactions on Information Forensics & Security in Nov. 2019. Google Scholar

[28] Polyanskiy Y, Poor H V, Verdu S. Channel Coding Rate in the Finite Blocklength Regime. IEEE Trans Inform Theor, 2010, 56: 2307-2359 CrossRef Google Scholar

[29] Yan S, Cong Y, Hanly S V. Gaussian Signalling for Covert Communications. IEEE Trans Wireless Commun, 2019, 18: 3542-3553 CrossRef Google Scholar

[30] Telatar I E. Capacity of multi-antenna Gaussian channels. Internal Tech. Memo., AT&T-Bell Laboratories, 1995. Google Scholar

[31] Yu X, Wei S, Luo Y. Finite blocklength analysis of Gaussian random coding in AWGN channels under covert constraints. 2019,. arXiv Google Scholar

[32] Tse D, Viswanath P. Fundamentals of Wireless Communication. Cambridge: Cambridge University Press, 2005. Google Scholar

qqqq

Contact and support