本論文探討保密性波束形成技術在多輸入單輸出且傳送端僅具通道方向資訊的情況下的最佳性。在之前的成果中，已經發現如果輸入端的共變異矩陣的rank為1，那麼由Khisti等人所導出的保密性通道容量上界就可以被一個使用高斯波束形成技術的保密傳輸速率所達到，也就是說該保密傳輸速率為最大的速率。因此根據上述結果，我們想要從數學上和數值上兩個角度切入去驗證rank-1的輸入端的共變異矩陣對該上界來說是不是一個最佳解。在這篇論文中，我們從數學上證明了在一些特別的情況下，rank-1的輸入端共變異矩陣的確是一個最佳解。在一般性的情況下，我們則是使用數值解(也就是interior point algorithm)的方式去驗證，而這個演算法在目標問題是convex-concave問題時才能保證其解的最佳性。最後我們根據模擬結果畫出上界和下界的比較結果。

[1] A. D. Wyner, “The wiretap channel,” Bell Syst. Tech. J., vol. 54, pp. 1355–1387, 1975.
[2] S. K. Leung-Yan-Cheong and M. E. Hellman, “The Gaussian wiretap channel,” IEEE
Trans. Inf. Theory, vol. 24, no. 4, pp. 451–456, July 1978.
[3] I. Csisz´ar and J. Korner, “Broadcast channels with confidential messages,” IEEE Trans.
Inf. Theory, vol. 24, no. 3, pp. 339–348, 1978.
[4] T. Liu and S. Shamai, “A note on the secrecy capacity of the multiple-antenna wiretap
channel,” IEEE Trans. Inf. Theory, vol. 55, no. 6, pp. 2547–2553, JUNE 2009.
[5] H. V. P. R. Bustin, R. Liu and S. Shamai, “An MMSE approach to the secrecy capacity
of the MIMO Gaussian wiretap channel,” in Proc. IEEE Int. Symp. Information Theory,
July. 2009, pp. 2602–2606.
[6] A. Khisti and G. W. Wornell, “Secure transmission with multiple antennas-I: the MISOME
wiretap channel,” IEEE Trans. Inf. Theory, vol. 56, no. 7, pp. 3088–3104, Jul.
2010.
[7] ——, “Secure transmission with multiple antennas-II: the MIMOME wiretap channel,”
IEEE Trans. Inf. Theory, vol. 56, no. 11, pp. 5515–5532, Nov. 2010.
[8] F. Oggier and B. Hassibi, “The secrecy capacity of the MIMO wiretap channel,” IEEE
Trans. Inf. Theory, vol. 57, no. 8, pp. 4961–4972, Aug. 2011.
[9] S. Shafiee and S. Ulukus, “Achievable rates in Gaussian MISO channels with secrecy
constraints,” IEEE Trans. Inf. Theory, pp. 2466–2470, Jun. 2007.
[10] J. Li and A. Petropulu, “Optimal input covariance for achieving secrecy capacity in
Gaussian MIMO wiretap channels,” in Proc. IEEE Int. Workshop Information Forensics
and Security (WIFS), Tenerife, Spain, Mar. 2010, pp. 3362–3365.
[11] S. Shafiee and S. Ulukus, “Towards the secrecy capacity of the Gaussian MIMO wire-tap
channel: the 2-2-1 channel,” IEEE Trans. Inf. Theory, vol. 55, no. 9, pp. 4033–4039,
Sept. 2009.
[12] J. Li and A. Petropulu, “Optimality of beamforming for secrecy capacity MIMO wiretap
channels,” in 2012 IEEE International Workshop on Information Forensics and Security
(WIFS), Dec. 2012, pp. 276–281.
[13] S.-C. Lin, T.-H. Chang, Y.-L. Liang, Y.-W. P. Hong, and C.-Y. Chi, “On the impact
of quantized channel feedback in guaranteeing secrecy with artificial noise: The noise
leakage problem,” IEEE Trans. on Wireless Commun., vol. 10, no. 3, pp. 901–915, Mar.
2011.
[14] S.-C. Lin, L.-R. Chen, and Y.-W. P. Hong, “On the optimality of secrecy beamforming
in multiple-input single-output single-antenna eavesdropper scenarios with only channel
direction information at the transmitter.”
[15] P. Gopala, L. Lai, and H. El Gamal, “On the secrecy capacity of fading channels,” IEEE
Trans. Inf. Theory, vol. 54, no. 10, pp. 4687–4698, 2008.
[16] S. A. Jafar and A. J. Goldsmith, “Isotropic fading vector broadcast channels: The scalar
upperbound and loss in degrees of freedom,” IEEE Trans. Inform. Theory, vol. 51, pp.
848–857, 2005.
[17] A. N. Dimitri P. Bertsekas and A. E. Ozdaglar, Convex Analysis and Optmization.
Athena Scientific, 2003.
[18] R. Hunger, Analysis and Transceiver Design for the MIMO Broadcast Channel.
Springer, 2013.
[19] W. Yu and T. Lan, “Transmitter optimization for the multi-antenna downlink with
per-antenna power constraints,” IEEE Trans. Signal Process., vol. 55, no. 6, June 2007.
[20] H. C. P. H. Huh and G. Caire, “Multiuser MISO transmitter optimization for intercell
interference mitigation,” IEEE Trans. Signal Process., vol. 58, pp. 4272 – 4285, Aug.
2010.
[21] M. Payaro and D. P. Palomar, “Hessian and concavity of mutual information, differential
entropy, and entropy power in linear vector Gaussian channels,” IEEE Trans. Inf.
Theory, vol. 55, no. 8, Aug. 2009.
[22] A. Khisti, A. Tchamkerten, and G. W. Wornell, “Secure broadcasting over fading channels,”
IEEE Trans. Inf. Theory, vol. 54, no. 6, pp. 2453–2469, Jun. 2008.
[23] D. A. Harville, Matrix Algebra From a Statistician's Perspective, second edition ed.
Springer, 2008.
[24] R. A. Horn and C. R. Johnson, Matrix Analysis. Cambridge, UK: Cambridge University
Press, 1999.
[25] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge, UK: Cambridge
University Press, 2004.
[26] A. Hjørungnes, Complex-Valued Matrix Derivatives. Cambridge University Press,
2011. [Online]. Available: http://dx.doi.org/10.1017/CBO9780511921490