We show an approach to beamforming an acoustic source using a circular microphone topology. Consider the circular array illustrated in Figure 1 below and suppose a beamforming solution is desired.

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

Define the angle of arrival to be the impinging angle at microphone $m_{8}$ as $\theta$, using the line joining microphones $m_1$ and $m_8$ as reference. By choosing every consecutive pair of microphones together with the center microphone, 7 triples are possible. Each triple can be said to have an effective look direction width of $\frac{2\pi}{7}$. Thus anytime the signal of interest falls withing this range of any triplet, we chose these triplet to form the beam. Then by averaging the received signals at three microphones, it can be easily shown to that;

$Y(w) = \frac{1}{3} \sum\limits_{q} X_{q} (w) = X_0 (w) H(w) + \frac{1}{3}\sum\limits_{q} N_q(w)$

where

$X_{q} (w) = X_0 e^{-jw\tau_{8,q}} + N_{q} (w)$

Here, $\tau_{8,q}$ is the delay between microphones $8$ and $q$, $H(w)$ is the effective “sum filter”.

$H(w)$ can be synthesized as:

$H(w) = \frac{2}{3}\cos{\left( w \frac{d}{c} \sin{\frac{\psi}{2}} \sin{\theta} \right)}+ \frac{1}{3} e^{-jw \frac{d}{c} \cos{\frac{\psi}{2}}\cos{\theta}}$

Suppose we sum the signals at the three microphones, then we can use the above filter to form a distortionless response for the desired signal. An approximate inverse filter can be used with minimal distortion across all frequencies. Define such a filter as

$H^{-1}(w) = \frac{3}{ 2\cos{\left( w \frac{d}{c} \sin{\frac{\psi}{2}} \sin{\theta} \right)}+ e^{jw \frac{d}{c} \cos{\frac{\psi}{2}}\cos{\theta}} }$

The effective magnitude response of the above approach, for each triplet, is as shown in Figure 2 below:

Figure 2: Effective magnitude response for desired signal and noise.

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