Angle of arrival estimation based of generalized correlation coefficients are generally limited in accuracy by the maximum allowable delay samples based on sensor arrangement topology and sampling frequency. MUSIC based approaches are also limiting in their computational requirement due to their exhaustive search over the angle of arrival parameter, making them impracticable in online systems. A compromise is the use of root-MUSIC algorithm which eliminates the need for the search over the possible angles of arrival. The case for the two sensors case is even more compelling since exact close forms can be extracted.

Consider a far field acoustic signal impinging $2$ microphones with separation distance $d$ at an angle of $\theta^{\circ}$. The signal at microphone $i$, $x_i$ , can be denoted as

$x_i(t) = s(t - \tau_i) + \nu_i(t), i \in \{1, 2\}$

where $\tau_i =\frac{d}{c} \sin{\theta}$ is the delay of the desired signal at microphone $i$, $s(t)$ is the source signal, $\nu_i(t)$ is noise and $c$ is the speed of acoustic signals. Both $s(t)$ and $\nu (t)$ are   zero mean ergodic processes. We will like to estimate the angle of arrival and beamform $s(t)$.  The setup is as shown in Figure 1.

Two microphones with pairwise distance of $d$

The sample covariance matrix for the two signals is given as:

$\mathcal{R(\omega)} = \begin{bmatrix}|X_1(\omega)|^2 & X_1(\omega) X_2^*(\omega) \\ X_2(\omega) X_1^*(\omega) & |X_2(\omega)|^2 \end{bmatrix}$

Eigen decomposition of the sample co-variance matrix will delineate the signal and noise subspace. The noise subspace is a $2 \times 1$ subspace, denoted $\mathbf{E}_N(\omega) = \begin{bmatrix} \mathcal{E}_1(\omega) \\ \mathcal{E}_2(\omega) \end{bmatrix}$
The steering vector can simply be denoted as

$a(\theta)= \begin{bmatrix}1 \\ z^{-1} \end{bmatrix}$

The root music manifold is then defined as:

$f(\theta,\omega) = z^{-1}\left(\mathcal{E}_1(\omega) \mathcal{E}_2^*(\omega) +(\mathcal{E}_1(\omega) \mathcal{E}_1^*(\omega)+\mathcal{E}_2(\omega) \mathcal{E}_2^*(\omega) )z+\mathcal{E}_2(\omega) \mathcal{E}_1^*(\omega) z^2\right)$

The solution of the equation above is a one of the roots of a second order polynomial. The root close to the unit circle is the desired solution and the angle can be evaluated straight away using $z = e^{-j w \frac{d}{c} \sin{\theta}}$.

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