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.

An Mth-order linear differential microphone array (LDMA) is conventionally designed by using a linear array of M+1 closely-spaced microphones. It is known that the conventional LDMAs suffer from the white noise amplification problem which may cause significant performance degradation in the presence of sensor imperfections. In order to resolve this, a more advanced solution has been proposed to employ more than M+1 microphones in the design of the LDMAs, which is shown to be effective against the white noise amplification and, hence, the resultant LDMAs are known as the robust LDMAs. Previous studies have shown that sensor imperfections can lead to dramatic performance degradation or even failure of the LDMAs due to the presence of the mainlobe orientation reversal (MOR) phenomenon. Therefore, avoiding the occurrence of the detrimental MOR phenomenon can serve as a minimum condition for the design of robust LDMAs in the presence of sensor imperfections. However, it is not yet quite clear how the sensor imperfections, such as microphone gain and phase mismatches, affect the robust LDMAs. Particularly in practical design, it will be of interest to know what the requirement is on the microphone imperfection tolerance for a given number of microphones, or alternatively, how many microphones should be used with a given microphone imperfection tolerance to guarantee no occurrence of the MOR phenomenon, which remains to be addressed. Motivated by the above, in this paper we first give an in-depth analysis of the impact of microphone imperfections on the mainlobe orientation of the robust LDMAs, and reveal how the MOR phenomenon occurs with the robust LDMAs. Then based on our theoretical foundation, the microphone imperfection tolerance analysis is performed, which provides a useful guidance when incorporating the influence of sensor imperfections into the practical design of robust LDMAs. Extensive numerical results are also presented to validate the effectiveness of our analysis.

Due to the increased sensitivity to microphone mismatches, lower-order differential microphone arrays (DMAs) are usually employed in practice, especially the first and second-order DMAs. It is known that the mainlobe orientation of the typical first- and second-order DMAs is both along the fixed endfire direction. This may be no longer true, however, in the presence of microphone mismatchs. This paper studies the fundamental problem of how microphone mismatches affect mainlobe orientation of the first- and second-order DMAs. Some insights into the effects of microphone mismatches on the mainlobe orientation of the two types of DMAs are revealed. In addition, the property on mainlobe orientation of the second-order DMA under ideal condition, which is yet to be known, is also studied. Moreover, tolerance analysis of microphone mismatches to ensure correct mainlobe orientation of the first- and second-order DMAs are performed in order to offer a useful guidance for practical design. Numerical examples are shown to verify the theoretical findings.