Differential beamforming using three uniform linear array (ULA) microphones, and hence second order, has two different implementations. An alternate to the static approach with a fixed direction of arrival is the so-called adaptive differential microphone array beamforming which forms two cardioids and combines them for the required spatial beam pattern. Consider a three microphone array as shown in Figure 1:
Figure 1: Three ULA microphones
The DMA beamforming is an optimization problem to meet constraints which are as shown below:
where is the true signal direction and is an extra desired null direction. It should be noted that the inverted matrix has a high condition number and as such the pseudo-inverse should be used for all practical purposes. Given , an optimal parameter is desired which will steer the beam towards the desired signal direction such that:
The optimal value for can be derived using a second cost function . It can be easily shown that the optimal value of will satisfy
An adaptive algorithm such as gradient descent can be employed over multiple temporal frames to find the optimal .
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