Theoretical Lower Bounds on the Performance of the First-Order Differential Microphone Arrays With Sensor Imperfections

The first-order differential microphone arrays (DMAs) offer the appealing properties such as compact size, and frequency-invariant response, and thus have found a wide range of applications. However, they are known to be sensitive to sensor imperfections, i.e., microphone gain errors, phase errors and self-noise. In this paper, we study the analysis and design of the first-order DMAs in the presence of sensor imperfections. Particularly, the theoretical lower bounds on the performance of the first-order DMAs with sensor imperfections are derived in closed form by using the theory of interval analysis, including the lower bounds on directivity factor (DF) and front-to-back ratio (FBR), which are two essential performance measures in DMAs design. Unlike the existing related works, we have considered a more general and realistic scenario when all the above-mentioned three kinds of sensor imperfections coexist. Based on the derived performance lower bounds, a method is presented to design the first-order DMA with its performance exceeding the worst-case optimum performance. Moreover, an analysis is presented, which takes into account the sensor imperfections in the design of the first-order DMAs so the optimum worst-case performance is guaranteed. Extensive simulation results are also shown to demonstrate the effectiveness of our theoretical findings.