Design of Minimax Broadband Beamformers that are Robust to Microphone Gain, Phase, and Position Errors

Broadband beamformers with small-size microphone arrays are known to be highly sensitive to microphone imperfections. A new method for the design of minimax broadband beamformers that are robust to microphone gain, phase, and position errors is proposed. In the method, the maximum variations in the microphone errors are used in formulating a convex optimization problem where the worst-case passband error is minimized under the constraint that the worst-case stopband error is below a prescribed level. To include the microphone imperfections in the optimization problem, we developed a suitable model that incorporates the variations due to the microphone errors and at the same time is efficient to compute. An important advantage of the proposed method is the availability of corresponding worst-case passband- and stopband-error bounds for the beamformer that has been designed; a second advantage is that it does not require the probability distributions of microphone errors. We then describe a two-phase method where the proposed method is used in the first phase to derive the passband and stopband error constraints for solving an optimization problem in the second phase where the white noise gain (WNG) of the beamformer is maximized. In our experiments, we compare beamformers designed using the proposed method, the two-phase method and a modified version of a competing method. Experimental results show that beamformers designed using the proposed method have much better performance than those of the modified competing method and comparable performance with those of the two-phase method; however, unlike the two-phase method, the proposed method provides the additional guarantee that the errors will always lie within the worst-case error bounds.