In this paper, we propose a convex optimization method for the design of 3-D High Order Ambisonics (3-D HOA) encoding matrices using spherical microphone arrays, which offers the possibility to impose spatial stop-bands in the directivity patterns of all the spherical harmonics while keeping the transformed audio channels still compatible with the 3-D HOA reproduction sound format. Using the proposed convex optimization method, the globally optimal encoding matrices can be obtained, and the suitable trade-off among several design factors, e.g. response distortions in the obtained spherical harmonic, the dynamic range of matrix coefficients, i.e., the system robustness, and frequencies, can be analyzed and illustrated. The proposed convex optimization is formulated as a form of second-order cone programming which can be efficiently solved. Numerical results validate the proposed method. This method can be easily generalized to the 2-D HOA cases.
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