It is desired that a fixed beamformer should maintain the frequency-invariant beampattern and achieve the high white noise gain (WNG), i.e., high robustness against the mismatch in practice. However, existing methods for the design of concentric circular differential microphone arrays (CCDMAs) cannot achieve a compromise between the high robustness and the frequency-invariant beampattern. To address this problem, a new analytical expression for the synthesized beampattern of CCDMAs is derived without any truncation error. Then CCDMAs are designed by matching mode coefficients of the approximated synthesized beampattern to that of the target differential beampattern, where an adjustable truncation order is utilized to enable a trade-off between the robustness and the beampattern distortion. A simple and effective procedure is presented to determine the frequency-wise truncation order. The proposed method reduces to three existing methods, i.e., the Jacobi-Anger method, the improved Jacobi-Anger method, and the minimum mean square error-based method, for a fixed truncation order, which in turn establishes a close connection with these methods and provides a unified view on the design of the CCDMAs. The superiority of the proposed method in terms of robustness and beampattern distortion is demonstrated through computer simulations.
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