Considerable works on adaptive schemes for transmit beamforming in distributed networks have emerged in the past years. In all these works, it was assumed that channels between all transmitters and the re- ceiver experience frequency flat slow-fading and a static environment was often considered. In practical environments, however, system uncertainties such as channels fluctuations, networks random node ad- dition and random node removal may rise and the aforementioned ideal assumptions may fail in these settings. Therefore, we focus on robust designs in this thesis and proposed a systematic analytical framework where stochastic stability is employed to demonstrate the tracking capability of the general adaptive schemes when channels are subject to fast variations in time. In addition, for time-varying channel and time-varying network topology, we defined a set of robustness criteria that can be used as comparison metrics for existing adaptive schemes. By utilizing the proposed analytical frameworks and metrics, we develop an bio-inspired scheme, BioRARSA2, that possess significantly superior ro- bustness with respect to environmental variations and system uncertainties. The improved robustness of the proposed algorithm is further validated through extensive numerical simulations.

Considerable works on adaptive schemes for transmit beamforming in distributed networks have emerged in the past years. In all these works, it was assumed that channels between all transmitters and the re- ceiver experience frequency flat slow-fading and a static environment was often considered. In practical environments, however, system uncertainties such as channels fluctuations, networks random node ad- dition and random node removal may rise and the aforementioned ideal assumptions may fail in these settings. Therefore, we focus on robust designs in this thesis and proposed a systematic analytical framework where stochastic stability is employed to demonstrate the tracking capability of the general adaptive schemes when channels are subject to fast variations in time. In addition, for time-varying channel and time-varying network topology, we defined a set of robustness criteria that can be used as comparison metrics for existing adaptive schemes. By utilizing the proposed analytical frameworks and metrics, we develop an bio-inspired scheme, BioRARSA2, that possess significantly superior ro- bustness with respect to environmental variations and system uncertainties. The improved robustness of the proposed algorithm is further validated through extensive numerical simulations.

[1] H.Ochiai,P.Mitran,H.Poor,andV.Tarokh,“Collaborativebeamformingfordistributedwireless ad hoc sensor networks,” IEEE Transactions on Signal Processing, vol. 53, no. 11, pp. 4110 – 4124, nov. 2005.
[2] R. Mudumbai, G. Barriac, and U. Madhow, “On the feasibility of distributed beamforming in wireless networks,” IEEE Transactions on Wireless Communications, vol. 6, no. 5, pp. 1754 – 1763, may 2007.
[3] R. Mudumbai, B. Wild, U. Madhow, and K. Ramch, “Distributed beamforming using 1 bit feed- back: From concept to realization,” in Allerton Conference on Communication, Control, and Computing, 2006.
[4] I. Thibault, G. Corazza, and L. Deambrogio, “Random, deterministic, and hybrid algorithms for distributed beamforming,” in Advanced satellite multimedia systems conference and the 11th sig- nal processing for space communications workshop, 2010.
[5] C.-S.Tseng,C.-C.Chen,andC.Lin,“Abio-inspiredrobustadaptiverandomsearchalgorithmfor distributed beamforming,” in IEEE International Conference on Communications, June 2011.
[6] S. Song and J. Thompson, “One-bit feedback algorithm with decreasing step size for distributed beamforming,” in 2010 Second UK-India-IDRC International Workshop on Cognitive Wireless Systems (UKIWCWS), Dec. 2010.
39

[7] P.Fertl,A.Hottinen,andG.Matz,“Perturbation-baseddistributedbeamformingforwirelessrelay networks,” in IEEE Global Telecommunications Conference, 2008.
[8] ——, “A multiplicative weight perturbation scheme for distributed beamforming in wireless relay networks with 1-bit feedback,” in IEEE International Conference on Acoustics, Speech and Signal Processing, 2009.
[9] I. Thibault, G. Corazza, and L. Deambrogio, “Phase synchronization algorithms for distributed beamforming with time varying channels in wireless sensor networks,” in 7th International Wire- less Communications and Mobile Computing Conference (IWCMC), Jul. 2011.
[10] J. Denis, C.-S. Tseng, C.-W. Lee, C.-Y. Tsai, and C. Lin, “Deterministic bisection search algo- rithm for distributed sensor/relay networks,” in Global Communications Conference (GLOBE- COM), 2012 IEEE, 2012, pp. 4851–4855.
[11] J. Bucklew and W. Sethares, “Convergence of a class of decentralized beamforming algorithms,” IEEE Transactions on Signal Processing, vol. 56, 2008.
[12] C. Lin, V. Veeravalli, and S. Meyn, “A random search framework for convergence analysis of dis- tributed beamforming with feedback,” IEEE Transactions on Information Theory, vol. 56, no. 12, pp. 6133 –6141, dec. 2010.
[13] C.-C. Chen and C. Lin, “Adaptive distributed beamforming for relay networks: Convergence analysis,” submitted to IEEE Transactions on Signal Processing, 2011.
[14] S. Song, J. Thompson, P.-J. Chung, and P. Grant, “Exploiting negative feedback information for one-bit feedback beamforming algorithm,” IEEE Transactions on Wireless Communications, vol. 11, no. 2, pp. 516 –525, Feb. 2012.
[15] H. J. Kushner, Stochastic Stability and Control. Academy Press Inc., 1967.
40

[16] H. Kushner, “On the construction of stochastic liapunov functions,” Automatic Control, IEEE Transactions on, vol. 10, no. 4, pp. 477–478, 1965.
[17] L. Weiss and E. F. Infante, “On the stability of systems defined over a finite time interval,” Proceedings of the National Academy of Sciences, vol. 54, no. 1, pp. 44–48, July 1965. [Online]. Available: http://www.pnas.org/content/54/1/44.short
[18] K. Passino, “Biomimicry of bacterial foraging for distributed optimization andcontrol,” IEEE Control Systems Magazine, vol. 22, 2002.
[19] H. Kushner, “Finite time stochastic stability and the analysis of tracking systems,” Automatic Control, IEEE Transactions on, vol. 11, no. 2, pp. 219–227, 1966.
[20] H. J. Kushner, Introduction to Stochastic Control. Holt, Rinehart and Winston, Inc., 1971.
[21] G. Schrack and M. Choit, “Optimized relative step size random searches,” Mathematical Pro-
gramming, vol. 10, 1976.
[22] D. Young and N. Beaulieu, “The generation of correlated rayleigh random variates by inverse discrete fourier transform,” IEEE Transactions on Communications, vol. 48, no. 7, pp. 1114 – 1127, Jul. 2000.