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Communication Under Normed Uncertainties

Capacity of MIMO Gaussian Channels with H ∞ Normed Channel Uncertainties. Communication Under Normed Uncertainties. S. Z. Denic C. D. Charalambous

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Communication Under Normed Uncertainties

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  1. Capacity of MIMO Gaussian Channels with H∞ Normed Channel Uncertainties Communication Under Normed Uncertainties S. Z. Denic C. D. Charalambous School of Information Technology Department of Electrical and and Engineering Computer Engineering University of Ottawa, Ottawa, Canada University of Cyprus, Nicosia, Cyprus S. M. Djouadi O. Milenkovic Department of Electrical and Electrical and Computer Engineering Computer Engineering Department University of Tennessee, Knoxville, USA University of Colorado, Boulder, USA

  2. Overview • Importance of Uncertainty in Communications • Review of Maximin Capacity; Recent results • Main Result • Examples • Conclusion and Future Work

  3. Example: Wireless Network hoping SISO MIMO

  4. Motivation MIMO communication schemes should be used to increase the capacity of newly emerged technologies such as • Wireless communication networks • 4G • Ad – hoc networks • Sensor networks • Teleoperation

  5. Importance of Uncertainty • Channel measurement errors • Network operating conditions • Channel modeling • Communication in presence of jamming • Sensor networks • Teleoperations

  6. MIMO Channel Model • Discrete time channel model n, uncertain x y + H uncertain + +

  7. Review of Maximin Capacity • The uncertainty can be modeled as the set of possible channels – compound channel • R is achievable rate if there exists a sequence of codes (M=2nR,n) such that the probability of error tends to zero as n tends to infinity uniformly over the set of channels (uncertainty set) • The operational capacity C is the supremum of all achievable rates

  8. Review of Maximin Capacity • Example: compound DMC • This result is due to Blackwell et. al. [6]. Also look at Csiszar [8], and Wolfowitz [21] • Blachman [5], and Dobrushin [12] were first to apply game theoretic approach in computing the channel capacity with mutual information as a pay-off function for discrete channels

  9. Recent Work • Raleigh, Cioffi, “Spatio-temporal coding for wireless communication”, Tran. On Communications, vol. 46, no. 3, pp. 357-366, March, 1998. • Vishwanath, Boyd, Goldsmith, ”Worst-case capacity of Gaussian vector channels,” Proceedings of 2003 CWIT, 2003. • Palomar, Cioffi, Lagunas, ”Uniform power allocation in MIMO channels: a game theoretic approach,” Tran. on Info. Theory, vol. 49, no. 7, pp. 1707-1727, July, 2003. • Yoo, Yoon, Goldsmith, ”MIMO Capacity with Channel Uncertainty: Does Feedback Help?”, IEEE GlobeCom 2004, Dallas, Texas, Dec. 2004. • Jafar, Goldsmith, ”Transmitter optimization and optimality of beamforming for multiple antenna systems with imperfect feedback”, IEEE Trans. on Wireless Comm., July 2004, vol. 3, no. 4, pp. 1165-1175.

  10. Uncertainty H∞ Modeling + + • Uncertainty models: additive and multiplicative

  11. Uncertainty H∞ Modeling Im Re

  12. Uncertainty H∞ Modeling • The uncertainty set represents the ball in frequency domain • Uncertainty set

  13. Uncertainty H∞ Modeling • The reasons for frequency domain uncertainty description • The proposed model is low-structured, as opposed to other parametric models found in the current literature • It is difficult to convert H∞ frequency domain uncertainty into uncertainty description in time domain using another norm • The frequency domain models are closer in the physical sense to real channels

  14. Uncertainty H∞ Modeling • The advantages of frequency domain uncertainty description (cont.) • The parameters of uncertainty models can be extracted from practical measurements • The evaluation of the channel capacity loss, and the optimal transmission strategy come naturally from the capacity solution

  15. Capacity of MIMO Channels Subject to Normed Uncertainties • The mutual information • Robust channel capacity

  16. Capacity of MIMO Channels Subject to Normed Uncertainties • Capacity formula • σi are singular values of Hnom, n is the rank of Hnom, μ is related to Lagrange multiplier

  17. Example • Channel uncertainty modeling

  18. Example • Channel uncertainty : magnitudes of • The is determined from Bode plot θ[rad/s]

  19. Example • The channel capacity versus uncertainty

  20. Capacity of MIMO Channels Subject to Normed Uncertainties • Optimal transmission where σi >|w| θ[rad/s] θ[rad/s]

  21. Example • Channel uncertainty θ[rad/s]

  22. Example • Channel uncertainty θ[rad/s]

  23. Capacity of MIMO Channels Subject to Normed Uncertainties • The beam-forming is optimal for high channel uncertainties • The beam-forming is optimal for small values of the transmitted power

  24. Future work • Generalization to ergodic capacity and capacity versus outage • Generalization to multi-terminal networks

  25. References [1] Ahlswede, R., “The capacity of a channel with arbitrary varying Gaussian channel probability functions”, Trans. 6th Prague Conf. Information Theory, Statistical Decision Functions, and Random Processes, pp. 13-31, Sept. 1971. [2] Baker, C. R., Chao, I.-F., “Information capacity of channels with partially unknown noise. I. Finite dimensional channels”, SIAM J. Appl. Math., vol. 56, no. 3, pp. 946-963, June 1996.

  26. References [4] Biglieri, E., Proakis, J., Shamai, S., “Fading channels: information-theoretic and communications aspects,” IEEE Transactions on Information Theory, vol. 44, no. 6, pp. 2619-2692, October, 1998. [5] Blachman, N. M., “Communication as a game”, IRE Wescon 1957 Conference Record, vol. 2, pp. 61-66, 1957. [6] Blackwell, D., Breiman, L., Thomasian, A. J., “The capacity of a class of channels”, Ann. Math. Stat., vol. 30, pp. 1229-1241, 1959. [7] Charalambous, C. D., Denic, S. Z., Djouadi, S. M. "Robust Capacity of White Gaussian Noise Channels with Uncertainty", accepted for 43th IEEE Conference on Decision and Control.

  27. References [8] Csiszar, I., Korner, J., Information theory: Coding theorems for discrete memoryless systems. New York: Academic Press, 1981. [9] Csiszar, I., Narayan P., “Capacity of the Gaussian arbitrary varying channels”, IEEE Transactions on Information Theory, vol. 37, no. 1, pp. 18-26, Jan., 1991. [10] Denic, S. Z., Charalambous, C. D., Djouadi, S.M., “Capacity of Gaussian channels with noise uncertainty”, Proceedings of IEEE CCECE 2004, Canada. [11] Denic, S.Z., Charalambous, C.D., Djouadi, S.M., “Robust capacity for additive colored Gaussian uncertain channels,” preprint.

  28. References [12] Dobrushin, L. “Optimal information transmission through a channel with unknown parameters”, Radiotekhnika i Electronika, vol. 4, pp. 1951-1956, 1959. [13] Doyle, J.C., Francis, B.A., Tannenbaum, A.R., Feedback control theory, New York: McMillan Publishing Company, 1992. [14] Forys, L.J., Varaiya, P.P., “The -capacity of classes of unknown channels,” Information and control, vol. 44, pp. 376-406, 1969. [15] Gallager, G.R., Information theory and reliable communication. New York: Wiley, 1968.

  29. References [16] Hughes, B., Narayan P., “Gaussian arbitrary varying channels”, IEEE Transactions on Information Theory, vol. 33, no. 2, pp. 267-284, Mar., 1987. [17] Hughes, B., Narayan P., “The capacity of vector Gaussian arbitrary varying channel”, IEEE Transactions on Information Theory, vol. 34, no. 5, pp. 995-1003, Sep., 1988.  [18] Lapidoth, A., Narayan, P., “Reliable communication under channel uncertainty,” IEEE Transactions on Information Theory, vol. 44, no. 6, pp. 2148-2177, October, 1998. [19] Medard, M., “Channel uncertainty in communications,” IEEE Information Theory Society Newsletters, vol. 53, no. 2, p. 1, pp. 10-12, June, 2003.

  30. References [20] Root, W.L., Varaiya, P.P., “Capacity of classes of Gaussian channels,” SIAM J. Appl. Math., vol. 16, no. 6, pp. 1350-1353, November, 1968. [21] Wolfowitz, Coding Theorems of Information Theory, Springer – Verlang, Belin Heildelberg, 1978. [22] McElice, R. J., “Communications in the presence of jamming – An information theoretic approach, in Secure Digital Communications, G. Longo, ed., Springer-Verlang, New York, 1983, pp. 127-166. [23] Diggavi, S. N., Cover, T. M., “The worst additive noise under a covariance constraint”, IEEE Transactions on Information Theory, vol. 47, no. 7, pp. 3072-3081, November, 2001.

  31. References [24] Vishwanath, S., Boyd, S., Goldsmith, A., “Worst-case capacity of Gaussian vector channels”, Proceedings of 2003 Canadian Workshop on Information Theory. [25] Shannon, C.E., “Mathematical theory of communication”, Bell Sys. Tech. J., vol. 27, pp. 379-423, pp. 623-656,July, Oct, 1948

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