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 $\beta$.

VOCAL Technologies offers custom designed solutions for beamforming with a robust voice activity detector, acoustic echo cancellation and noise suppression. Our custom implementations of such systems are meant to deliver optimum performance for your specific beamforming task. Contact us today to discuss your solution!