Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Optimal beamforming via interior point methods

  • 97 Accesses

  • 3 Citations


We show that two antenna array pattern synthesis problems can be expressed as convex optimization problems. The first one deals with a symmetric planar array with real weights, which can be expressed as a linear program. The second one concerns a broadband acoustic array, which becomes a convex quadratically constrained quadratic program. Because these two problems are convex, they can be (numerically) solved with great efficiency by recently developed interior-point methods. Thanks to the efficiency of the interior point methods, we also built a computer-aided design tool for the first problem.

This is a preview of subscription content, log in to check access.


  1. 1.

    L.S. Lasdon, J. Plummer, B. Buehler, and A.D. Warren, “Optimal design of efficient acoustic antenna arrays,”Mathematical Programming, Vol. 39, No. 2, pp. 131–155, 1987.

  2. 2.

    O.M. Bucci, D. D'Elia, G. Mazzarella, and G. Panatiello, “Antenna pattern synthesis: A new general approach,”Proceedings of the IEEE, Vol. 82, pp. 358–371, March 1994.

  3. 3.

    L. Vandenberghe and S. Boyd, “Semidefinite programming,”SIAM Review, Vol. 38, March 1996.

  4. 4.

    N. Karmarkar, “A new polynomial-time algorithm for linear programming,”Combinatorica, Vol 4, No. 4, pp. 373–395, 1984.

  5. 5.

    G.W. McMahon, B. Hubley, and A. Mohammed, “Design of optimum directional arrays using linear programming techniques,”J. Acoust. Soc. Am., vol. 51, pp. 304–309, April 1972.

  6. 6.

    G.L. Wilson, “The design of antenna arrays with tapered sidelobe heights,”IEEE Trans. Antennas Propag., Vol. AP-26, pp. 345–347, March 1978.

  7. 7.

    F. Lorenzelli, A. Wang, D. Korompis, R. Hudson, and K. Yao, “Optimization and performance of broadband microphones array,”SPIE Proceedings on Advanced Signal Processing Algorithms, Vol. 2563, pp. 158–169, 1995.

  8. 8.

    M.H. Er and A. Cantoni,Control and Dynamic Systems, Academic Press, Inc., in Chapter, “Techniques in robust broadband beamforming,” pp. 321–386, 1992.

  9. 9.

    I. Thng, A. Cantoni, and Y.H. Leung, “Derivative constrained optimum broadband antenna arrays,”IEEE Transactions on Signal Processing, Vol. 41, pp. 2376–2388, July 1993.

  10. 10.

    J.B. Hiriart-Uruty and C. Lemaréchal,Convex Analysis and Minimization Algorithms, Springer-Verlag, 1993.

  11. 11.

    H. Lebret, “Antenna pattern synthesis through convex optimization,”Advanced Signal Processing Algorithms, F.T. Luk (Ed.),Proc. SPIE, Vol. 2563, pp. 182–192, 1995.

  12. 12.

    H. Lebret, “Synthèse de diagrammes de réseaux d'antennes par optimisation convexe,” Ph.D. Thesis, Université de Rennes I, Nov. 1994.

  13. 13.

    C.C. Gonzaga, “Path-following methods for linear programming,”SIAM Review, Vol. 34, pp. 167–224, June 1992.

  14. 14.

    Y. Nesterov and A. Nemirovsky, “Interior-point polynomial methods in convex programming,” ofStudies in Applied Mathematics. Philadelphia, PA: SIAM, Vol. 13, 1994.

  15. 15.

    I. Integrated Systems,Xmath Basics, Santa Clara, California, 1992.

Download references

Author information

Additional information

Research supported in part by Délégation Générale à l'Armement and by a fellowship from Thomson CSF.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Lebret, H. Optimal beamforming via interior point methods. J VLSI Sign Process Syst Sign Image Video Technol 14, 29–41 (1996).

Download citation


  • Antenna Array
  • Interior Point Method
  • Convex Optimization Problem
  • Beam Pattern
  • Normalization Constraint