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 $3$ constraints which are as shown below: $z_f(w) = S(w) \begin{bmatrix} 1 & e^{-jw\frac{d}{c}\cos{\theta}} & e^{-jw\frac{2d}{c}\cos{\theta}}\end{bmatrix} \begin{bmatrix} 1 & e^{-jw\frac{d}{c}} & e^{-jw\frac{2d}{c}} \\1 & e^{jw\frac{d}{c}} & e^{jw\frac{2d}{c}}\\1 & e^{-jw\frac{d}{c}\cos{\gamma}} & e^{-jw\frac{2d}{c}\cos{\gamma}}\end{bmatrix} ^{-1} \begin{bmatrix} 1\\0\\0\end{bmatrix}$

and $z_b(w) =S(w) \begin{bmatrix} 1 & e^{-jw\frac{d}{c}\cos{\theta}} & e^{-jw\frac{2d}{c}\cos{\theta}}\end{bmatrix} \begin{bmatrix} 1 & e^{-jw\frac{d}{c}} & e^{-jw\frac{2d}{c}} \\1 & e^{jw\frac{d}{c}} & e^{jw\frac{2d}{c}}\\1 & e^{-jw\frac{d}{c}\cos{\gamma}} & e^{-jw\frac{2d}{c}\cos{\gamma}}\end{bmatrix} ^{-1} \begin{bmatrix} 0\\1\\0\end{bmatrix}$

where $\theta$ is the true signal direction and $\gamma$ 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 $\gamma$, an optimal parameter $\beta$ is desired which will steer the beam towards the desired signal direction $\theta$ such that: $y(w) = z_b(w) - \beta z_f(w)$

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 $\beta_{opt} = \frac{2 \mathbb{R}e\{z_b(w) z_f(w)^*\}}{|z_f(w)|^2}$
An adaptive algorithm such as gradient descent can be employed over multiple temporal frames to find the optimal $\beta$.

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