## 2 Methodologies  
### Sound Source Localization  
Unlike frequency-domain acoustic beamformers, which have undergone rapid development, few time-domain algorithms have been proposed in the past because of their high computational cost [37]. One of the most famous acoustic beamformers is the conventional DAS beamformer, which is considered effective and accurate for localizing moving, transient, and broadband sound sources based on various indicators, such as the peak value and root mean square (RMS) [20,38]. For an accurate localization performance, the spherical wave-front assumption of the sound wave propagation model is adopted so that the scanning region is meshed with certain horizontal and vertical coordinates. Figure 2 illustrates the spherical wavefronts from \(N\), point sound sources, incident on a linear array of \(M\), microphones. The sound signals received at the \(m\)th (\(m=1,\ 2,\ \ldots,\ M\)) microphone can be expressed as  
\[p_{m}(t) = x_{m}^{1}(t)+x_{m}^{2}(t)+\ldots+x_{m}^{k}(t)+\ldots+x_{m}^{N}(t) \tag{1}\] \[= \frac{1}{4\pi}\sum\limits_{n=1}^{N}r_{nm}^{-1}s_{n}\Big{(}t-\frac {r_{nm}}{c_{0}}\Big{)}+\sum\limits_{n=1}^{N}\Phi_{n}(t)\]  
where \(n=1,\ 2,\ \ldots,\ N\), and the total sound signals received by the microphone array, can be expressed as  
\[\mathbf{p}(t)=\big{[}\ p_{1}(t)\quad p_{2}(t)\quad\ldots\quad p_{m}(t)\quad \ldots\quad p_{M}(t)\ \big{]}^{T}, \tag{2}\]  
where \(x_{m}^{k}(t)=\frac{1}{4\pi}r_{km}^{-1}s_{k}\Big{(}t-\frac{r_{nm}}{c_{0}}\Big{)} +\Phi_{k}(t)\) is the signal from the \(k\)th sound source, \(\Phi_{k}(t)\) is the incoherent noise, \(\frac{1}{4\pi}r_{km}^{-1}\) represents the sound pressure attenuation factor due to the distance between the sound source and microphone, \(\frac{r_{nm}}{c_{0}}\) represents the time delay, and \(c_{0}\) is the speed of sound. With respect to a positional parameter \(r_{m}\), which is the distance between the scanning point and the \(m\)th microphone, an intentional compensating time delay \(\frac{r_{m}}{c_{0}}\) is applied to each measured signal, and the traditional DAS beamformer output can be derived by summing and averaging all the delayed signals, i.e.,  
\[b(r_{m},t)=\frac{1}{M}\sum\limits_{m=1}^{M}p_{m}\Big{(}t-\frac{r_{m}}{c}\Big{)}, \tag{3}\]  
and it is a function of time and space [20,37]. To indicate the spatial information of the dominant sound source in the scanning area only, an expectation operator, \(E\), is applied such that the corresponding beamforming power can be expressed as  
\[\beta(r_{m})=E\Big{[}|b(r_{m},t)|^{2}\Big{]}. \tag{4}\]  
To achieve a higher accurate localization performance, the MUSIC algorithm uses the eigen-decomposition of the correlation matrix of the measured sound signal to extract the signal and noise subspaces. After the eigen-decomposition of the correlation matrix, it can be observed that the eigenvectors of the noise subspace are orthogonal to the basis vectors of the signal subspace. Furthermore, the MUSIC beamforming output can be calculated by the inverse of a designed scan vector, which belongs to the subspace obtained by multiplying the signal subspace by the correlation matrix of the noise subspace [20,27,34]. Considering the measured, complex sound field described in Equation (2) of a specific frequency bin, \(\omega\), (2) can be expressed as  
\[\mathbf{p}(\omega)=\left[ \begin{array}{cccc}&p_{1}(\omega)&p_{2}(\omega)&\dots& p_{M}(\omega)\end{array} \right]^{T}, \tag{5}\]  
and the measured correlation matrix \(\mathbf{R}\) is  
\[\mathbf{R}(\omega)=E[\mathbf{p}(\omega)\mathbf{p}(\omega)^{H}], \tag{6}\]  
where each element represents the correlation between two microphone signals. The correlation matrix can be expressed in an eigen-decomposition form:  
\[\mathbf{R}=\mathbf{U}\mathbf{D}\mathbf{U}^{H}. \tag{7}\]  
Because the columns of \(\mathbf{U}\) are orthogonal, the correlation matrix can be separated into two parts:  
\[