First order adaptive differential mic array beamforming

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 $N$ constraints for N microphones. For two microphones, the constrainst are used to form:

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

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 a design parameter to shape the beam in the direction of the source signal. The optimal value for $\beta$ can be derived using a second cost function $J(w) = |y(w)|^2$ . It can be easily shown that the optimal value of $\beta$ will satisfy $\beta_{opt} = \frac{2 \mathbb{R}e\{z_b(w) z_f(w)^*\}}{|z_f(w)|^2}$
Figure 2 below illustrates these 4 spatial beams.

Figure 2: Different spatial patterns

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First order adaptive differential microphone array beamforming

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