Differential beamforming using two microphones, and hence first order, has two different implementations. The first involves a delay of one microphone signal then taking the difference between the resultant and the second signal. An alternate to this approach is the so-called adaptive differential microphone array beamforming which forms two cardioids and combines them for the required spatial beam pattern.
Consider a two microphone array as shown in Figure 1:
Figure 1: Two microphone array
The DMA beamforming is an optimization problem to meet constraints for N microphones. For two microphones, the constrainst are used to form:
where is the desired beam direction and is a desired null direction. Given and , the choice of will determine whether the spatial beam will be a dipole, cardiod, hypercardiod or supercardiod. The final solution is given as
where is a design parameter to shape the beam in the direction of the source signal. The optimal value for can be derived using a second cost function . It can be easily shown that the optimal value of will satisfy
Figure 2 below illustrates these 4 spatial beams.
Figure 2: Different spatial patterns
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