Wireless Networks

, Volume 14, Issue 5, pp 573–590 | Cite as

On the transport capacity of Gaussian multiple access and broadcast channels

  • G. A. Gupta
  • S. ToumpisEmail author
  • J. Sayir
  • R. R. Müller


We study the transport capacity of the Gaussian multiple access channel (MAC), which consists of multiple transmitters and a single receiver, and the Gaussian broadcast channel (BC), which consists of a single transmitter and multiple receivers. The transport capacity is defined as the sum, over all transmitters (for the MAC) or receivers (for the BC), of the product of the data rate with a reward r(x) which is a function of the distance x that the data travels.

In the case of the MAC, assuming that the sum of the transmit powers is upper bounded, we calculate in closed form the optimal power allocation among the transmitters, that maximizes the transport capacity, using Karush-Kuhn-Tucker (KKT) conditions. We also derive asymptotic expressions for the optimal power allocation, that hold as the number of transmitters approaches infinity, using the most-rapid-approach method of the calculus of variations. In the case of the BC, we calculate in closed form the optimal allocation of the transmit power among the signals to the different receivers, both for a finite number of receivers and for the case of asymptotically many receivers, using our results for the MAC together with duality arguments. Our results can be used to gain intuition and develop good design principles in a variety of settings. For example, they apply to the uplink and downlink channel of cellular networks, and also to sensor networks which consist of multiple sensors that communicate with a single central station.


Broadcast channel Multiple access channel Power allocation Transport capacity Wireless networks 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    S. Toumpis, R. Müller and J. Sayir, On the transport capacity of a multiple access gaussian channel, in: Proc. IEEE Inter. Workshop on Wireless Ad Hoc Networks, Oulu, Finland (May–June 2004).Google Scholar
  2. 2.
    G.A. Gupta, S. Toumpis, J. Sayir and R.R. Müller, On the transport capacity of gaussian multiple access and broadcast channels, in: Proc. IEEE WiOpt, Riva del Garda, Trentino, Italy (April 2005).Google Scholar
  3. 3.
    G.A. Gupta, S. Toumpis, J. Sayir and R.R. Müller, Transport capacity of gaussian multiple access and broadcast channels with a large number of nodes, in: Proc. IEEE ISIT, Adelaide, Australia (September 2005).Google Scholar
  4. 4.
    H. Takagi and L. Kleinrock, Optimal transmission ranges for randomly distributed packet radio networks, IEEE Transactions on Communication 32(3) (March 1984) 246–257.CrossRefGoogle Scholar
  5. 5.
    R. Nelson and L. Kleinrock, The spatial capacity of a slotted ALOHA multihop packet radio network with capture, IEEE Transactions on Communication 32(6) (June 1984) 684–694.CrossRefGoogle Scholar
  6. 6.
    P. Gupta and P.R. Kumar, The capacity of wireless networks, IEEE Transactions on Information Theory 46(2) (March 2000) 388–404.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    A. Reznik and S. Verdú, On the transport capacity of a broadcast Gaussian channel, Communications in Information and Systems 2(2) (December 2002) 183–216.zbMATHGoogle Scholar
  8. 8.
    T. Cover and J. Thomas, Elements of Information Theory, 1st edn. (Wiley Interscience, New York, 1991).zbMATHGoogle Scholar
  9. 9.
    N. Jindal, S. Vishwanath and A.J. Goldsmith, On the duality of gaussian multiple-access and broadcast channels, IEEE Transactions on Information Theory 50(5) (May 2004) 768–783.CrossRefMathSciNetGoogle Scholar
  10. 10.
    D.N.C. Tse and S.V. Hanly, Multiaccess fading channels—Part I: Polymatroid structure, optimal resource allocation and throughput capacities, IEEE Transactions on Information Theory 44(7) (November 1998) 2796–2815.zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    S.V. Hanly and D.N.C. Tse, Multiaccess fading channels—Part II: Delay-limited capacities, IEEE Transactions on Information Theory 44(7) (November 1998) 2816–2831.zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    S. Boyd and L. Vandenberghe, Convex Optimization, 1st edn. (Cambridge University Press, 2004).Google Scholar
  13. 13.
    R.R. Müller, Power and bandwidth efficiency of multiuser systems with random spreading, Ph.D. dissertation, University of Erlangen-Nuremberg (March 1999).Google Scholar
  14. 14.
    S. Vishwanath, S.A. Jafar and A.J. Goldsmith, Optimum power and rate allocation strategies for multiple access fading channels, in: Proc. Spring IEEE VTC, Rhodes, Greece (May 2001). Vol. 4, pp. 2888–2892.Google Scholar
  15. 15.
    R. Müller, G. Caire and R. Knopp, Multiuser diversity in delay-limited cellular wideband systems, in: Proc. IEEE Information Theory Workshop, Rotorua, New Zealand (August–September 2005).Google Scholar
  16. 16.
    T.S. Rappaport, Wireless Communications: Principles and Practice, 1st edn. (Prentice Hall, Upper Saddle River, NJ, 1996).Google Scholar
  17. 17.
    D. Hughes-Hartogs, The capacity of a degraded spectral Gaussian broadcast channel, Ph.D. dissertation, Stanford University (July 1975).Google Scholar
  18. 18.
    L. Li and A.J. Goldsmith, Capacity and optimal resource allocation for fading broadcast channels—Part I: Ergodic capacity, IEEE Transactions on Information Theory 47(3) (March 2001) 1083.zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    R. Knopp and P.A. Humblet, Information capacity and power control in single cell multiuser communications, in: Proc. IEEE ICC, Seattle, WA (June 1995).Google Scholar
  20. 20.
    L. Euler, De serie Lambertina plurimisque eius insignibus proprieatibus, Opera Mathematica 1. 6 (1921) 350–369 (original data 1779).Google Scholar
  21. 21.
    R.M. Corless, G.H. Gonnet, D.E.G. Hare, D.J. Jeffrey and D.E. Knuth, On the Lambert W function, Advances in Computational Mathematics 5 (1996) 329–359.zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    F.Y.M. Wan, Introduction to the Calculus of Variations and its Applications (Chapman & Hall Mathematics, 1995).Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2006

Authors and Affiliations

  • G. A. Gupta
    • 1
  • S. Toumpis
    • 2
    Email author
  • J. Sayir
    • 3
  • R. R. Müller
    • 4
  1. 1.Asia Pacific Business Analytics Unit of CitibankGurgaonIndia
  2. 2.Department of Electrical and Computer Engineering of the University of CyprusNicosiaCyprus
  3. 3.Telecommunications Research Center Vienna (ftw.)ViennaAustria
  4. 4.Norwegian University of Science and TechnologyTrondheimNorway

Personalised recommendations