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Conventional delay and sum beamformers are additive in nature. Another approach to forming a super directive beam in a specific direction is by the use of differential beamformers, distinctive from additive beamforming. The number of microphones available limits the order of the differential microphone array (DMA) beamforming. For N microphones, an N-1 order differential microphone beamforming is realized. Thus, for a first order DMA beamforming, two microphones are required. An advantage of using DMA beamforming is that the amplitude response tends to be flat.

Consider a two microphones array as shown in Figure 1:


Two Microphone Array

Figure 1: Two microphone array

The DMA beamforming is an optimization problem to meet N constraints for N microphones. For two microphones, the constrainst are:

h_1 + h_2 e^{-jw\frac{d}{c}\cos{\theta}} =1


h_1 + h_2 e^{-jw\frac{d}{c}\cos{\beta}} =0

where \theta is the desired beam direction and \beta is a desired null direction. Given \theta, the choice of \beta will determine whether the spatial beam will be a dipole, cardiod, hypercardiod or supercardiod. Figure 2 below illustrates these 4 spatial beams.

Different spatial patterns

Figure 2: Different spatial patterns

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