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  • Research Article
  • Open Access

A General Theory for SIR Balancing

EURASIP Journal on Wireless Communications and Networking20062006:060681

  • Received: 12 May 2005
  • Accepted: 19 January 2006
  • Published:


We study the problem of maximizing the minimum signal-to-interference ratio (SIR) in a multiuser system with an adaptive receive strategy. The interference of each user is modelled by an axiomatic framework, which reflects the interaction between the propagation channel, the power allocation, and the receive strategy used for interference mitigation. Assuming that there is a one-to-one mapping between the QoS and the signal-to-interference ratio (SIR), the feasible QoS region is completely characterized by the max-min SIR balancing problem. In the first part of the paper, we derive fundamental properties of this problem for the most general case, when interference is modelled with an axiomatic framework. In the second part, we show more specific properties for interference functions based on a nonnegative coupling matrix. The principal aim of this paper is to provide a deeper understanding of the interaction between power allocation and interference mitigation strategies. We show how the proposed axiomatic approach is related to the matrix-based theory.


  • Information System
  • General Theory
  • Deep Understanding
  • Specific Property
  • System Application


Authors’ Affiliations

Heinrich Hertz Chair for Mobile Communications, Faculty of Electrical Engineering and Computer Science, Technical University of Berlin, Berlin, 10587, Germany
Fraunhofer German-Sino Lab for Mobile Communications (MCI), Einsteinufer 37, Berlin, 10587, Germany
Fraunhofer Institute for Telecommunications, Heinrich-Hertz-Institut (HHI), Einsteinufer 37, Berlin, 10587, Germany


  1. Verdu S: Multiuser Detection. Cambridge University Press, Cambridge, UK; 1998.MATHGoogle Scholar
  2. Boche H, Stańczak S: Log-convexity of the minimum total power in CDMA systems with certain quality-of-service guaranteed. IEEE Transactions on Information Theory 2005,51(1):374–381. 10.1109/TIT.2004.839530View ArticleMathSciNetMATHGoogle Scholar
  3. Boche H, Stańczak S: Convexity of some feasible QoS regions and asymptotic behavior of the minimum total power in CDMA systems. IEEE Transactions on Communications 2004,52(12):2190–2197. 10.1109/TCOMM.2004.838725View ArticleGoogle Scholar
  4. Zander J, Kim S-L: Radio Resource Management for Wireless Networks. Artech House, Boston, Mass, USA; 2001.Google Scholar
  5. Meyerhoff HJ: Method for computing the optimum power balance in multibeam satellites. COMSAT Technical Review 1974,4(1):139–146.Google Scholar
  6. Aein JM: Power balancing in systems employing frequency reuse. COMSAT Technical Review 1973,3(2):277–299.Google Scholar
  7. Boche H, Stańczak S: The infeasible SIR region is not a convex set. Proceedings of IEEE International Symposium on Information Theory (ISIT '05), September 2005, Adelaide, Australia 695–699.Google Scholar
  8. Yates RD: A framework for uplink power control in cellular radio systems. IEEE Journal on Selected Areas in Communications 1995,13(7):1341–1347. 10.1109/49.414651MathSciNetView ArticleGoogle Scholar
  9. Yates RD, Ching-Yao H: Integrated power control and base station assignment. IEEE Transactions on Vehicular Technology 1995,44(3):638–644. 10.1109/25.406632View ArticleGoogle Scholar
  10. Hanly SV: An algorithm for combined cell-site selection and power control to maximize cellular spread spectrum capacity. IEEE Journal on Selected Areas in Communications 1995,13(7):1332–1340. 10.1109/49.414650View ArticleGoogle Scholar
  11. Farsakh C, Nossek JA: Spatial covariance based downlink beamforming in an SDMA mobile radio system. IEEE Transactions on Communications 1998,46(11):1497–1506. 10.1109/26.729394View ArticleGoogle Scholar
  12. Rashid-Farrokhi F, Tassiulas L, Liu KJR: Joint optimal power control and beamforming in wireless networks using antenna arrays. IEEE Transactions on Communications 1998,46(10):1313–1324. 10.1109/26.725309View ArticleGoogle Scholar
  13. Visotsky E, Madhow U: Optimum beamforming using transmit antenna arrays. Proceedings of 49th IEEE Vehicular Technology Conference (VTC '99), May 1999, Houston, Tex, USA 1: 851–856.Google Scholar
  14. Bengtsson M, Ottersten B: Optimal and suboptimal transmit beamforming. In Handbook of Antennas in Wireless Communications. CRC Press, Boca Raton, Fla, USA; 2001. chapter 18Google Scholar
  15. Schubert M, Boche H: Solution of the multi-user downlink beamforming problem with individual SINR constraints. IEEE Transactions on Vehicular Technology 2004,53(1):18-28.View ArticleGoogle Scholar
  16. Wiesel A, Eldar YC, Shamai S: Linear precoding via conic optimization for fixed MIMO receivers. IEEE Transactions on Signal Processing 2006,54(1):161–176.View ArticleGoogle Scholar
  17. Gerlach D, Paulraj A: Base station transmitting antenna arrays for multipath environments. Signal Processing 1996,54(1):59–73. 10.1016/0165-1684(96)00093-XView ArticleMATHGoogle Scholar
  18. Montalbano G, Slock DTM: Matched filter bound optimization for multiuser downlink transmit beamforming. Proceedings of IEEE International Conference on Universal Personal Communications (ICUPC '98), October 1998, Florence, ItalyGoogle Scholar
  19. Schubert M, Boche H: A unifying theory for uplink and downlink multi-user beamforming. Proceedings of IEEE International Zurich Seminar on Broadband Communications, February 2002, Zurich, Switzerland 27–1-27–6.Google Scholar
  20. Boche H, Schubert M: Resource allocation for multi-antenna multi-user systems. Proceedings of IEEE International Conference on Communications (ICC '05), May 2005, Seoul, South Korea 2: 855–859.Google Scholar
  21. Boche H, Schubert M: Duality theory for uplink downlink multiuser beamforming. In Smart Antennas—State-of-the-Art, EURASIP Book Series. Hindawi, New York, NY, USA; 2006.Google Scholar
  22. Gantmacher FR: The Theory of Matrices, Vol. 2. Chelsea, New York, NY, USA; 1959.MATHGoogle Scholar
  23. Boche H, Schubert M: On the structure of the unconstrained multiuser QoS region. to appear in IEEE Transactions on Signal ProcessingGoogle Scholar
  24. Schubert M, Boche H: A generic approach to QoS-based transceiver optimization. to appear in IEEE Transactions on CommunicationsGoogle Scholar
  25. Schubert M, Boche H: Comparison of-norm and-norm optimization criteria for SIR-balanced multi-user beamforming. Signal Processing 2004,84(2):367–378. 10.1016/j.sigpro.2003.10.022View ArticleMATHGoogle Scholar
  26. Zander J: Performance of optimum transmitter power control in cellular radio systems. IEEE Transactions on Vehicular Technology 1992,41(1):57–62. 10.1109/25.120145View ArticleGoogle Scholar
  27. Wielandt H: Unzerlegbare, nicht negative Matrizen. Mathematische Zeitschrift 1950, (52):642–648. and Mathematische Werke/Mathematical Works, Vol. 2, 100–106 de Gruyter, Berlin, 1996Google Scholar
  28. Seneta E: Non-Negative Matrices and Markov Chains. Springer, New York, NY, USA; 1981.View ArticleMATHGoogle Scholar
  29. Boche H, Wiczanowski M, Stańczak S: Unifying view on min-max-fairness, max-min fairness, and utility optimization in cellular networks. in preparation, 2006Google Scholar


© H. Boche and M. Schubert 2006

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.