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 allow for precise source localization.

source localization crcular microphone array

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}


\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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