Theoretical Analysis of Maclaurin Expansion Based Linear Differential Microphone Arrays and Improved Solutions

Linear differential microphone arrays (LDMAs) are becoming popular due to their potentially high directional gain and frequency-invariant beampattern. By increasing the number of microphones, the Maclaurin expansion-based LDMAs address the inherently poor robustness problem of the conventional LDMA at low frequencies. However, this method encounters severe beampattern distortion and the deep nulls problem in the white noise gain (WNG) and the directivity factor (DF) at high frequencies as the number of microphones increases. In this paper, we reveal that the severe beampattern distortion is attributed to the deviation term of the synthesized beampattern while the deep nulls problem in the WNG and the DF is attributed to the violation of the distortionless constraint in the desired direction. We then propose two new design methods to avoid the degraded performance of LDMAs. Compared to the Maclaurin series expansion-based method, the first method additionally imposes the distortionless constraint in the desired direction, and the deep nulls problem in the WNG and the DF can be avoided. The second method explicitly requires the response of the higher order spatial directivity pattern in the deviation term to be zero, and thus the beampattern distortion can be avoided. By choosing the frequency-wise parameter that determines the number of the considered higher order spatial directivity patterns, the second method enables a good trade-off between the WNG and the beampattern distortion. Simulations exemplify the superiority of the proposed method against existing methods in terms of the robustness and the beampattern distortion.