# Robust DOA Estimation Using Multi-Scale Fusion Network with Attention Mask
## 3 Signal Analysis in the Spherical Harmonic Domain  
Consider a spherical microphone array (SMA) with the radius \(r\) consisting of \(L\) array elements. The center of the array coincides with the origin of the coordinate system and the position of the \(l\)-th microphone, which is \(r_{l}=r(cos\Phi_{l}sin\theta_{l},sin\Phi_{l}sin\theta_{l},cos\theta_{l})^{T}\) in Cartesian coordinates, where \(\theta_{l}\) and \(\Phi_{l}\) are the elevation and azimuth angle of the \(l\)-th microphone, respectively.  
Assuming that the elevation and azimuth of the \(D\) far-field sources are \(\theta_{d}\) and \(\varphi_{d}\), respectively, the model of the signals received by the SMA can be expressed as  
\[\mathbf{x}(t)=\mathbf{A}\mathbf{s}(t)+\mathbf{v}(t), \tag{1}\]  
where \(\mathbf{x}(t)\) is the received signal vector, \(\mathbf{A}=[a(\theta_{1},\varphi_{1}),\cdots,a(\theta_{D},\varphi_{D})]\) is the steering matrix, \(\mathbf{s}(t)\) is the source signal vector, \(\mathbf{v}(t)\) is the noise vector, and \(t\) is the time index.  
The steering matrix \(\mathbf{A}\) can be decomposed as follows:  
\[\mathbf{A}=\mathbf{Y}(\Omega)\mathbf{B}(kr)\mathbf{Y}^{H}(\Phi), \tag{2}\]where \(\mathbf{Y}(\Omega)=\big{[}\mathbf{y}(\theta_{1},\Phi_{1}),\cdots,\mathbf{y}(\theta_{L},\Phi_{L}) \big{]}^{T}\) is the spherical harmonic matrix corresponding to the microphone position, \(\mathbf{B}(kr)=diag\{b_{0}(kr),b_{1}(kr),b_{1}(kr),\cdots,b_{N}(kr)\}\) is the mode strength matrix, \(\mathbf{Y}(\Phi)=\big{[}\mathbf{y}(\theta_{1},\varphi_{1}),\cdots,\mathbf{y}(\theta_{D}, \varphi_{D})\big{]}^{T}\) is similar in structure to \(\mathbf{Y}(\Omega)\) but depends only on the DOA of the \(D\) sources, \([\cdot]^{T}\) represents the transpose, \([\cdot]^{H}\) is the conjugate transpose, and \(\mathbf{y}(\theta,\Phi)\) is the spherical harmonic vector [39].  
After STFT, the signal model in (1) can be rewritten as  
\[\mathbf{x}(\tau,k)=\mathbf{Y}(\Omega)\mathbf{B}(kr)\mathbf{Y}^{H}(\Phi)\mathbf{s}(\tau,k)+\mathbf{v}( \tau,k), \tag{3}\]  
where \(\tau\) is the time frame index, \(k=2\pi f/c\) is the wavenumber corresponding to the frequency \(f\), \(c\) is the speed of sound, \(\mathbf{x}(\tau,k),\mathbf{s}(\tau,k)\), and \(\mathbf{v}(\tau,k)\) are the short-term Fourier transforms corresponding to \(\mathbf{x}(t),\mathbf{s}(t)\) and \(\mathbf{v}(t)\) respectively.  
According to the orthogonality criterion for uniformly or approximately uniformly sampled spherical harmonic functions \(\mathbf{Y}^{H}(\Omega)\mathbf{Y}(\Omega)=\mathbf{I}\)[45], by left multiplying (3) by \(\mathbf{Y}^{H}(\Omega)\), the spherical harmonic coefficients are given as  
\[\mathbf{x}_{nm}(\tau,k)=\mathbf{B}(kr)\mathbf{Y}^{H}(\Phi)\mathbf{s}(\tau,k)+\mathbf{v}_{nm}(\tau,k), \tag{4}\]  
where \(\mathbf{I}\) is the identity matrix, \(\mathbf{x}_{nm}(\tau,k)=\mathbf{Y}^{H}(\Omega)\mathbf{x}(\tau,k)\) and \(\mathbf{v}_{nm}(\tau,k)=\mathbf{Y}^{H}(\Omega)\mathbf{v}(\tau,k)\).  
Further left multiplication of the model (4) by \(\mathbf{B}^{-1}(kr)\) leads to  
\[\hat{\mathbf{x}}_{nm}(\tau,k)=\mathbf{Y}^{H}(\Phi)\mathbf{s}(\tau,k)+\hat{\mathbf{v}}_{nm}( \tau,k), \tag{5}\]  
where \(\hat{\mathbf{x}}_{nm}(\tau,k)=\mathbf{B}^{-1}(kr)\mathbf{x}_{nm}(\tau,k)\) and \(\hat{\mathbf{v}}_{nm}(\tau,k)=\mathbf{B}^{-1}(kr)\mathbf{v}_{nm}(\tau,k)\).  
In (5), the model steering matrix is \(\mathbf{Y}^{H}(\Phi)\) and does not contain microphone locations and the mode strength components. The complex-valued \(\hat{\mathbf{x}}_{nm}(\tau,k)\) is called a spherical harmonic complex-valued feature (SHC).