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Differential microphone arrays (DMAs) often encounter white noise amplification, especially at low frequencies. If the array geometry and the number of microphones are fixed, one can improve the white noise amplification problem by reducing the DMA order. With the existing differential beamforming methods, the DMA order can only be a positive integer number. Consequently, with a specified beampattern (or a kind of beampattern), reducing this order may easily lead to over compensation of the white noise gain (WNG) and too much reduction of the directivity factor (DF), which is not optimal. To deal with this problem, we present in this article a general approach to the design of DMAs with fractional orders. The major contributions of this article include but are not limited to: 1) we first define a directivity pattern that can achieve a continuous compromise between the pattern corresponding to the maximum DMA order and the omnidirectional pattern; 2) by approximating the beamformer's beampattern with the Jacobi-Anger expansion, we present a method to find the proper differential beamforming filter so that its beampattern matches closely the target directivity pattern of fractional orders; and 3) we show how to determine analytically the proper fractional order of the DMA with a given target beampattern when either the value of the DF or WNG is specified, which is useful in practice to achieve the desired beampattern and spatial gain while maintaining the robustness of the DMA system.
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