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Centered circular array direction of arrival estimation using least squares

The use of microphone arrays to estimate the direction of arrival (DOA) of sound sources is widespread. Most algorithms however are tailored for linear array topologies. A limitation of linear arrays is that they can only resolve a half plane with folding spacial ambiguities. We now consider the case of a centered circular array topology for estimating the DOA. Consider a far field speech impinging N+1 microphones arranged in a centered circular topology and suppose it is desired to estimate the DOA. This is depicted in Figure 1 below:

Centered circular array direction of arrival estimation using least squares

N+1 microphones in centered circular array

 

The signal at each microphone obeys:

x_i(t) = h_i(t)*s(t - \tau_i) + \nu(t)

where \tau_i is the delay from the source and * denotes convolution. Its is easy to verify that the time difference of arrivals (TDOA) will obey:

\tau_j - \tau_0 = \tau_{0,j} = \frac{d}{c} \cos{\left((j-1)\psi - \theta\right)}, j \ge 1 \tau_j - \tau_1 = \tau_{1,j} = \frac{d}{c} \sin{\left((j-1)\frac{\psi}{2} - \theta\right)}, j \ge 2

where d is as shown on Figure 1 and \theta is the DOA with respect to the ordinate. The angle \psi = \frac{2\pi}{N}. For N+1 microphones, we get \frac{N(N+1)}{2} unique tuples from which the TDOA can be estimated. We can however reduce it to use only the two equations above, making 2N-1 tuples. This will lead to a system of equations given as:

\underbrace{\begin{bmatrix}\tau_{0,1}\\\tau_{0,2}\\\vdots\\\tau_{0,N}\\\tau_{1,2}\\\vdots\\\tau_{1,N}\end{bmatrix}}_{Y} = \underbrace{ \frac{d}{c}\begin{bmatrix}0 & 1\\\sin{\psi} & \cos{\psi}\\\vdots & \vdots\\\sin{\left((N-1)\psi\right)} & \cos{\left((N-1)\psi\right)}\\-\cos{\left(\frac{\psi}{2}\right)} & \sin{\left(\frac{\psi}{2}\right)}\\\vdots\\-\cos{\left((N-1)\frac{\psi}{2}\right)} & \sin{\left((N-1)\frac{\psi}{2}\right)}\end{bmatrix}}_{A} \begin{bmatrix}\sin{\theta}\\\cos{\theta}\end{bmatrix}

Then the least squares solution becomes

\begin{bmatrix}\sin{\theta}\\\cos{\theta}\end{bmatrix} = (A^T A)^{-1} A^T Y

It should be noted that (A^T A)^{-1} A^T can be precomputed once and then only 2N-1 additions and multiplications are required to estimate \sin{\theta} and \cos{\theta} after the time delay tuples have been estimated. Given \sin{\theta} and \cos{\theta}, the unique \theta can be readily estimated without any spatial aliasing. The choice of sampling Frequency and/or d, together with the number of microphones can be used to determine the resolution of the returned DOA. Figure 2 below illustrates and example with real speech, 8 microphones at a sampling frequency of 16kHz and d = 0.024mm. The true DOA is 0 or 360 degrees.

 

centered circular array angle estimate
[TOP] Speech signal [BOTTOM] Estimate of the true angle of 0 or 360 degrees.

It can be seen that in the DOA estimate is pretty accurate. This technique can be easily extended to estimate multiple speakers if the speakers can be separated in the time-frequency domain with some binning to isolate specific DOAs.

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