LMS implementation of first order differential beamformer

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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LMS Implementation of First Order Adaptive Differential Beamforming

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