There are certain applications where the assumtion that the source of the impinging sound source is far field does not hold. The analytical expressions for far field source localization will thus be laden with large margins of errors. There are certain applications where the assumtion that the source of the impinging sound source is far field does not hold. The analytical expressions for far field source localization will thus be laden with large margins of errors. In such situations, a different solution is required. Suppose there is a near field signal impinging on the circular array microphones as shown in as shown in Figure 1 below:

Circular microphone array topology for 8 microphones

Also, suppose the direction of arrival, DOA, of the impinging signal is desired. It is clear that the solutions that assumes that the signal impinges on all microphones at the same angle does not hold. There are $7!$ time delay of arrivals that can be estimated using pairwise correlates. However, for real time operating systems, RTOSs, we can not utilize all $7!$ possible delay estimates between paired microphones without depleting considerably computing resources.  A reduction by a factor of $6!$ is possible by using only $7$ time delay of arrivals. It can be shown that by pivoting on a single microphone, the DOA should obey:

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

Equation (1) can be compressed in matrix form to:

A least squares approach can then be used to efficiently compute the DOAs to a resolution of below $7^{\circ}$ using $8$ microphones.

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