We show an approach to localizing an acoustic source using a circular microphone topology. Consider the circular array illustrated in Figure 1 below and suppose the angle of arriving signal,$\theta$, and the range $r$, are desired. Note that these two parameters, $(r,\theta)$, taken together localize the source precisely.

Figure 1: Circular microphone array topology for $8$ microphones, $\phi = \frac{2\pi}{7}$

Define the angle of arrival to be the impinging angle at microphone $m_{8}$ as $\theta$, using the line joing micrphones $m_1$ and $m_8$ as reference. Then the microphones can be easily shown to obey;

$dc^2 \left(\tau_{8,9-n}^2-\tau_{8,n}^2 \right) \cos{\left((n-1)\psi\right)} \sin{\theta} + d\left(c^2\tau_{8,9-n}^2 -c^2 \tau_{8,n} -2d^2\right) \sin{\left( (n-1) \psi\right)} \cos{\theta} = c\left(d^2 +c^2 \tau_{8,9-n}\tau_{8,n} \right) \left(\tau_{8,n} - \tau_{8,9-n} \right) , n \in \{2,\cdots,4\}$

where $\tau_{i,j}$ is the time difference of arrival between the signals at microphones $m_{i}$ and $m_{j}$.

The least squares solution to solve the angle of arrival, $\theta$, thus:

$\mathbf{A} \begin{bmatrix}\sin{\theta} \\ \cos{\theta} \end{bmatrix} = \mathbf{Y}$

where

$\mathbf{A}=d\begin{bmatrix}c^2 \left(\tau_{8,7}^2-\tau_{8,2}^2 \right) \cos{\left(\psi\right)} & \left(c^2\tau_{8,7}^2 -c^2 \tau_{8,2} -2d^2\right) \sin{\left(\psi\right)}\\c^2 \left(\tau_{8,6}^2-\tau_{8,3}^2 \right) \cos{\left(2\psi\right)} & \left(c^2\tau_{8,6}^2 -c^2 \tau_{8,3} -2d^2\right) \sin{\left( 2\psi\right)}\\c^2 \left(\tau_{8,5}^2-\tau_{8,4}^2 \right) \cos{\left(3\psi\right)} & \left(c^2\tau_{8,5}^2 -c^2 \tau_{8,4} -2d^2\right) \sin{\left(3 \psi\right)} \end{bmatrix}$

$\mathbf{Y}=c\begin{bmatrix} \left(d^2 +c^2 \tau_{8,7}\tau_{8,2} \right) \left(\tau_{8,2} - \tau_{8,7} \right)\\ \left(d^2 +c^2 \tau_{8,6}\tau_{8,3} \right) \left(\tau_{8,3} - \tau_{8,6} \right)\\ \left(d^2 +c^2 \tau_{8,5}\tau_{8,4} \right) \left(\tau_{8,4} - \tau_{8,5} \right) \end{bmatrix}$

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