In the implementation of first order adaptive differential microphone, the difference between two spatial beams are taking using a scaling parameter . The choice of the optimal is not exact in the presence of large amplitude noise or low signal to noise ratios. We present a LMS approach to estimating the optimal .
Consider a two microphone array as shown in Figure 1:
Figure 1: Two microphone array
The constraints for beamforming are:
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 the design parameter of concern. Consider a cost function . It can be easily shown that a gradient descent optimization of beta will satisfy where
A time average can be deployed to optimize the estimate of . The descent algorithm then becomes:
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