Optimal design of distributed microphone array

Abstract

This thesis concentrates on the study of distributed broadband beamforming system and source localization problem. The main contributions of this thesis consist of the following four parts. 1. For the design of distributed broadband beamforming system, each microphone is equipped with wireless communications capability. It is designed such that the error between the actual response and the desired response is minimized which is then formulated as a minimax optimization problem. Since we find that the performance of the optimized designs is very sensitive to the perturbations in microphone locations, we first use sensor network technology which solved by semi-definite programming method to estimate the microphone locations, and then incorporate it into the design process. We propose a suitable robust formulation as a remedy to regain the performance. The minimax optimization problem is transformed into a semi-definite programming problem so that interior point algorithms can be applied. We illustrate the proposed method by several designs and demonstrate that this approach is essential to regain accuracy in the optimized designs. 2. The broadband beamforming design problem is formulated as a non-strictly convex semi-infinite programming problem. The approach to solve it is that adding a small perturbation quadratic function to the objective function to make it strictly convex. We demonstrate that the solution of the per-turbation semi-infinite programming problem approximates the solution of the original problem as the perturbation going to 0. The new exchange algorithm is applied successfully to the filter design problem. 3. We present a new method to solve the source localization problem with time-difference information. Fist we formulate a mixed SDP-SOCP relaxation model and then state how to obtain the exact solution from the solutions of the mixed SDP-SOCP relaxation model and the second order polynomial equation. The estimator properties for the true source location under noises is proposed. We also give bi-level method to solve the source localization problem that formulated only as a semi-definite programming. Then a mixed SDP-SOCP relaxation model for source localization combined with sensor network localization problem is studied, also we give some statistical analyses for it. Many illustrated examples demonstrate those approaches can be applied successfully and some comparisons are presented. 4. We obtain a representation for the solution of the mixed SDP-SOCP model and the characterization such that the mixed SDP-SOCP model has an exact relaxation in two-dimensional case. We derive the geometry of the localizable region for the proposed mixed model. The characterization shows that the source localization with some time-difference information can be solved exactly by the mixed SDP-SOCP relaxation model in a larger region than the triangle region determined by three points.