Suppose there is a far field signal impinging on the circular array microphones as shown in as shown in Figure 1 below:

Figure 1: Circular microphone array topology for 8 microphones.

Also, suppose the direction of arrival, DOA, of the impinging signal is desired. There are $7!$ possible delay estimates between paired microphones, ignoring the simple sign changes of the delays. For RTOSs, that will be too many pairs to be computed. Rather, the delays with respect to the center microphone can be utilized to attained fine grained estimates of the DOA. The DOA should obey:

$\begin{bmatrix}0 &1\\ \sin{\psi} &\cos{\psi} \\ \sin{2\psi} &\cos{2\psi} \\ - \cos{3\psi} &\sin{3\psi} \\ - \sin{3\psi} &-\cos{3\psi} \\ \sin{2\psi} &- \cos{2\psi} \\ - \sin{\psi} &\cos{\psi}\end{bmatrix} \begin{bmatrix}\sin{\theta} \\ \cos{\theta} \end{bmatrix} = \frac{c}{d}\begin{bmatrix}\tau_1 \\ \tau_2 \\ \tau_3 \\ \tau_4 \\ \tau_5 \\ \tau_6 \\ \tau_7\end{bmatrix}$

where $\tau_i$ is the delay between microphone $8$ and microphone $i$ and $c$ is the speed of sound in free space. The solution of the above equation leads to:

$\theta = \arctan{\left(\frac{2+2\cos{\psi}^2 + 2\cos{2\psi}^2}{1+2\sin{\psi}^2 + 2\sin{2\psi}^2} \frac{(\tau_2 - \tau_7) \sin{\psi}+ (\tau_3 + \tau_6) \sin{2\psi}-\tau_4\cos{3\psi} -\tau_5\sin{3\psi}}{\tau_1 +(\tau_2 + \tau_7) \cos{\psi}+ (\tau_3 - \tau_6) \cos{2\psi}+\tau_4\sin{3\psi} -\tau_5\cos{3\psi}} \right)}$

The signs of the numerator and denominator terms of the argument of the arc tangent is used to discriminate the quadrant of the DOA. There are more than $1900$ unique angles between $0^{\circ}$ and $360^{\circ}$ that can be resolved using these seven TDOAs with a maximum error depicting uncovered angle of less than $\frac{1.7}{2}^{\circ}$. Figure 2 illustrates the granularity of this approach for DOA estimation. The TDOAs can be computed using fast microphone array topology constrained GCC-PHAT like algorithms.

Figure 2: Resolution of DOA estimation using 7 TDOAs.

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