LMS implementation of first order adaptive differential microphone array beamforming. In the implementation of first order adaptive differential microphone, the difference between two spatial beams are taking using a scaling parameter $\beta$. The choice of the optimal $\beta$ 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 $\beta$.
Consider  a two  microphone array as shown in Figure 1: Figure 1: Two microphone array

The constraints for beamforming are: $z_f(w) = \frac{S(w)}{e^{-jw\frac{d}{c}\cos{\gamma}}-e^{-jw\frac{d}{c}\cos{\theta}} } \begin{bmatrix} 1 & e^{-jw\frac{d}{c}\cos{\theta}} \end{bmatrix} \begin{bmatrix} e^{-jw\frac{d}{c}\cos{\gamma}} \\ -1 \end{bmatrix}$

and $z_b(w) = \frac{S(w) }{e^{-jw\frac{d}{c}\cos{\gamma}}-e^{-jw\frac{d}{c}\cos{\theta}} } \begin{bmatrix} 1 & e^{-jw\frac{d}{c}\cos{\theta}} \end{bmatrix} \begin{bmatrix} -1 \\ e^{-jw\frac{d}{c}\cos{\gamma}} \end{bmatrix}$

where $\theta$ is the desired beam direction and $\gamma$ is a desired null direction. Given $\theta$ and $\gamma$, the choice of $\beta$ will determine whether the spatial beam will be a dipole, cardiod, hypercardiod or supercardiod. The final solution is given as $y(w) = z_b(w) - \beta z_f(w)$

where $\beta$ is the design parameter of concern. Consider a cost  function $J(w) = |y(w)|^2$. It can be easily shown that a gradient descent optimization of beta $\beta$ will satisfy $\beta [k+1] = \beta[k] - \mu \frac{\partial{J(w)}}{\partial{\beta}}|_{\beta = \beta[k]}$where $\frac{\partial{J(w)}}{\partial{\beta}} = -2 \mathbf{R}e \{z_f(w)^* y(w) \}$
A time average can be deployed to optimize the estimate of $\beta$. The descent algorithm then becomes: $\beta_t [k+1] = \beta_t[k] +2 \mu \mathbf{R}e \{z_{f_t}(w)^* y_t(w) \}$

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