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# High resolution far field direction of arrival using three microphones

Recent advances in technology has led to requirements for finding direction of arrivals of acoustic disturbances in large open areas such as a car park or a stadium. Activities such as a car crash, an altercation or a discharge of a fire arm requires narrowed view of search to shorten the resolution time. In this regard, we presents  a solution to direction of arrival estimate of sound sources with a full circle view and attaining a resolution of less than $7^{\circ}$ for sources that are at distances of up to $100m$ away. Without loss of generality, our solution utilizes three non co-linear microphones but the approach easily scales to more than three microphones. Suppose $F_s \ge \frac{c}{d}$ , where $F_s$ is the sampling rate, $c$ is the speed of sound in free space, and $d$ is the microphone pairwise separation. This assumption ensures there is at least one full sample delay possible between pairwise microphones. Suppose also, for the purpose of this article that  $F_s = 16kHz$ with the separation distance between the three microphones  $d = 42.5 ~mm$. The orientation of the three microphones is as shown in Figure 1 below.

Figure 1: Three microphone array topology.

It can be shown that the direction of arrival for a far field acoustic source impinging on the triangular array topology in Figure 1 will obey: $\theta = \arctan{\left( \frac{\sqrt{3}(\tau_{1,2} -\tau_{1,3})}{\tau_{1,2} +\tau_{1,3}+2 \tau_{2,3}}\right)}$ where $\tau_{i,j}$ is the delay between microphones $i$ and $j$, $\theta$ is the angle to be estimated, and $d$ is the pairwise microphone distances apart with the signs of the numerator and denominator determining the quadrant in which the angle is located. The resolved angles will be as shown in Figure 2 below.

Figure 2:  Direction of arrival resolution.

The maximum resolution deviation falls within $\pm 7^{\circ}$. This bound can be further reduced if inter sample delays are employed for the pairwise microphone signal delays.

VOCAL Technologies offers custom designed solutions for beamforming with a direction of arrival estimation, 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!