A new beampattern synthesis formulation is proposed to compute the minimum number of array sensors required. In order to satisfy all the prescribed specifications of the beampattern, the proposed method imposes linear matrix inequality (LMI) constraints on the beampattern as developed by Davidson et al., which remove the need to discretize the beampattern region. As the proposed formulation is quasi-convex, an iterative procedure is used to decompose it into a systematic sequence of convex feasibility problems, in order to find the minimum number of sensors. The proposed method guarantees convergence if the globally optimal solution lies in the search interval, which is easily ensured at the start of the search.
Siew Eng Nai,
Zhu Liang Yu,
"Beampattern Synthesis with Linear Matrix Inequalities Using Minimal Array Sensors," Progress In Electromagnetics Research M,
Vol. 9, 165-176, 2009. doi:10.2528/PIERM09092804
1. Van Trees, H. L., Optimum Array Processing, Part IV of Detection, Estimation and Modulation Theory, John Wiley and Sons, New York, 2002.
2. Qu, Y., G. S. Liao, S. Q. Zhu, and X. Y. Liu, "Pattern synthesis of planar antenna array via convex optimization for airborne forward looking radar," Progress In Electromagnetics Research, Vol. 84, 1-10, 2008. doi:10.2528/PIER08060301
3. Mouhamadou, M., P. Vaudon, and M. Rammal, "Smart antenna array patterns synthesis: Null steering and multi-user beamforming by phase control," Progress In Electromagnetics Research, Vol. 60, 95-106, 2006. doi:10.2528/PIER05112801
4. Guo, B., Y. Wang, J. Li, P. Stoica, and R. Wu, "Microwave imaging via adaptive beamforming methods for breast cancer detection," PIERS Online, Vol. 1, No. 3, 350-353, 2005.
5. Davidson, T. N., Z. Q. Luo, and J. F. Sturm, "Linear matrix inequality formulation of spectral mask constraints with applications to FIR filter design ," IEEE Trans. Signal Process., Vol. 50, 2702-2715, Nov. 2002.
6. Davidson, T. N., Z. Q. Luo, and K. M. Wong, "Design of orthogonal pulse shapes for communications via semidefinite programming," IEEE Trans. Signal Process., Vol. 48, 1433-1445, May 2000. doi:10.1109/78.839988
7. Woodward, P. M. and J. D. Lawson, "The theoretical precision with which an arbitrary radiation pattern may be obtained from a source of finite size," J. IEE, Vol. 95, 363-370, Sep. 1948.
8. Wu, S.-P., S. P. Boyd, and L. Vandenberghe, "FIR filter design via spectral factorization and convex optimization," Appl. Computational Contr., Signal and Commun., Vol. 1, 215-245, B. N. Datta, Birkhauser, Boston, MA, 1997.
9. Nai, S. E., W. Ser, Z. L. Yu, and S. Rahardja, "A robust adaptive beamforming framework with beampattern shaping constraints," IEEE Trans. Antennas Propag., Vol. 57, 2198-2203, Jul. 2009.
10. Hoang, H. G., H. D. Tuan, and B.-N. Vo, "Low-dimensional SDP formulation for large antenna array synthesis," IEEE Trans. Antennas Propag., Vol. 55, 1716-1725, Jun. 2007. doi:10.1109/TAP.2007.898573
11. Sturm, J. F., "Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones," Optimization Methods Softw., Vol. 11-12, 625-653, 1999. doi:10.1080/10556789908805766
12. Yu, Z. L., W. Ser, and M. H. Er, "Robust adaptive beamformers with linear matrix inequality constraints," Proc. IEEE Int. Symp. Circuits Syst. (ISCAS'08), 3214-3217, Seattle, WA, May 2008.
13. Roh, T. and L. Vandenberghe, "Discrete transforms, semidefinite programming, and sum-of-squares representations of nonnegative polynomials," Soc. Ind. Appl. Math. (SIAM) J. Optimization, Vol. 16, 939-364, 2006.
14. Dumitrescu, B., Positive Trigonometric Polynomials and Signal Processing Applications, Springer, 2007.
15. Boyd, S. P. and L. Vandenberghe, Convex Optimization, Cambridge Univ. Press, United Kingdom, 2004.