Optimal Array Pattern Synthesis With Desired Magnitude Response

Letting Euclidean norm be the performance parameter, the task of finding the best approximation of a complex function in a finite dimension subspace leads to a convex optimization problem that can be easily solved by the least-squares method. However, this procedure leads to a sub-optimal solution in applications that have no phase requirements on the approximated function. In this case, semidefinite programming has been used to obtain optimal magnitude responses. In this work, this non-convex optimization problem is dealt with by using an iterative method based on the least-squares, which is illustrated on directivity synthesis by spherical loudspeaker arrays. Usually, instead of synthesize directly the desired pattern, the strategy adopted is to reproduce its truncated spherical harmonic representation. The truncation order is determined by the number of drivers of the spherical array. It is shown that truncation error and signal powers can be significantly reduced if phase error is neglected, providing potential means to improve directivity synthesis for applications requiring only magnitude response.45994604Kassakian, P., Magnitude least-squares fitting via semidefinite programming with applications to beamforming and multidimensional filter design (2005) Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, 3, pp. iii/53-iii/56. , Philadelphia, USA, 18-23 MarchWarusfel, O., Misdariis, N., Directivity synthesis with a 3D array of loudspeakers: Application for stage performance (2001) Proceedings of the COST G-6 Conference on Digital Audio Effects (DAFX-01), , Limerick, Ireland, 6-8 DecemberKassakian, P., Wessel, D., Characterization of spherical loudspeaker arrays (2004) Proceedings of the 117th Audio Engineering Society Convention, , San Francisco, USA, 28-31 October, Paper Number 6283Zotter, F., Sontacchi, A., Höldrich, R., Modeling a spherical loudspeaker system as multipole source (2007) Proceedings of the 33rd German Annual Conference on Acoustics, , Stuttgart, Germany, 19-22 MarchAvizienis, R., Freed, A., Kassakian, P., Wessel, D., A compact 120 independent element spherical loudspeaker array with programmable radiation patterns (2006) Proceedings of the 120th Audio Engineering Society Convention, , Paris, France, Paper Number 6783Boyd, S., Vandenberghe, L., (2006) Convex Optimization, p. 716. , Ed. Cambridge University Press, Cambridge, United KingdomKazhdan, M., Funkhouser, T., Rusinkiewicz, D., Rotation invariant spherical harmonic representation of 3D shape descriptors (2003) Proceedings of the Eurographics Symposium on Geometry Processing, pp. 156-164. , Aachen, GermanyKassakian, P., Wessel, D., Design of low-order filters for radiation synthesis (2003) Proceedings of the 115th Audio Engineering Society Convention, , New York, USA, 10-13 October, Paper Number 5925Wang, F., Balakrishnan, V., Zhou, P.Y., Optimal array pattern synthesis using semidefinite programming (2003) IEEE Transactions on Signal Processing, 51 (5), pp. 1172-118