# A Deep Learning Method for DOA Estimation with Covariance Matrices in Reverberant Environments
## 2 Data Model  
A spherical microphone array with a radius \(r\) and \(L\) mutually independent and isotropic sensors in standard spherical coordinate [26] is considered. The center of the array coincides with the origin of the coordinate system and the microphones are located at the position \(r_{l}=r[\cos\varphi_{l}\sin\theta_{l},\sin\varphi_{l}\cos\theta_{l},\cos \theta_{l}]^{T}\), where \(\theta_{l}\) and \(\varphi_{l}\) represent the elevation and azimuth of the \(l\)-th sensor, respectively, \(l=1,\cdots,L\). Figure 1 indicates the spherical coordinate system used for localization. Figure 1: Spherical coordinate system. An acoustic far-field signal \(s_{d}(t)\) emitted from the location \(\mathbf{r}_{d}=(r_{d},\theta_{d},\phi_{d})\) is supposed. The distance between the source and the center of the spherical microphone array is \(r_{d}\), and \(\theta_{d}\) and \(\phi_{d}\) mean the elevation and azimuth of the \(d\)-th source, respectively. Thus, the output vector \(\mathbf{x}(t)\) of the spherical microphone array can be obtained as  
\[\mathbf{x}(t)=\mathbf{A}\mathbf{s}(t)+\mathbf{v}(t), \tag{1}\]  
where \(\mathbf{x}(t)=\left[x_{1}(t),\cdots,x_{L}(t)\right]^{T}\), \(\mathbf{A}\) is the steering matrix that represents the transfer functions from the source signals \(\mathbf{s}(t)=\left[s_{1}(t),\cdots,s_{D}(t)\right]^{T}\) to the array microphones and holds the DOA information, \(\mathbf{v}(t)=\left[v_{1}(t),\cdots,v_{L}(t)\right]^{T}\) is the noise vector, \(t\) is the time index, \(D\) represents the number of sources. When source signals are wideband ones, short-time Fourier transform (STFT) [27] must be considered to construct a signal model in the time-frequency domain from the spatial domain as follows  
\[\mathbf{x}(\tau,k_{h})=\mathbf{A}(k_{h})\mathbf{s}(\tau,k_{h})+\mathbf{v}( \tau,k_{h}), \tag{2}\]  
where \(\tau\) indicates the index of time frame, \(k_{h}=2\pi f_{h}/c\) is the wavenumber corresponding to the frequency bin \(f_{h}\), \(h\) represents the frequency bin index, \(c\) is the speed of the sound, \(\mathbf{x}(\tau,k_{h})=\left[x_{1}(\tau,k_{h}),\cdots,x_{L}(\tau,k_{h})\right] ^{T}\), \(\mathbf{v}(\tau,k_{h})=\left[v_{1}(\tau,k_{h}),\cdots,v_{L}(\tau,k_{h})\right] ^{T}\), and \(\mathbf{A}(k_{h})\), the corresponding steering matrix [28], can be decomposed as  
\[\mathbf{A}(k_{h})=\mathbf{Y}(\Omega)\mathbf{B}(k_{h}t)\mathbf{Y}^{H}(\Phi), \tag{3}\]  
where \(\mathbf{Y}(\Omega)=\left[\mathbf{y}(\theta_{1},\varphi_{1}),\cdots,\mathbf{y }(\theta_{L},\varphi_{L})\right]^{T}\) is the spherical harmonic matrix containing the information of the locations of all the microphones and \(\mathbf{Y}(\Phi)=\left[\mathbf{y}(\theta_{1},\phi_{1}),\cdots,\mathbf{y}( \theta_{D},\phi_{D})\right]^{T}\) is the spherical harmonic matrix only dependent on DOAs of D sources, \(\left[\cdot\right]^{T}\) indicates transpose, \(\left[\cdot\right]^{H}\) is the conjugate transpose, the spherical harmonics vector \(\mathbf{y}(\theta,\phi)\) is expressed as  
\[\mathbf{y}(\theta,\phi)=\left[Y_{0}^{0}(\theta,\phi),Y_{1}^{-1}(\theta,\phi), Y_{1}^{0}(\theta,\phi),Y_{1}^{1}(\theta,\phi),\cdots,Y_{N}^{N}(\theta,\phi) \right]^{T}, \tag{4}\]  
where \(Y_{n}^{m}(\theta,\phi)\) is the spherical harmonic function with order \(n\) and degree \(m\) described as  
\[Y_{n}^{m}(\theta,\phi)=\sqrt{\frac{(2n+1)(n-m)!}{4\pi(n+m)!}}P_{n}^{m}(\cos \theta)e^{im\phi}, \tag{5}\]  
where \(i^{2}=-1\), \(P_{n}^{m}(x)\) is the associated Legendre function of order \(n\) and degree \(m\). When \(m\) is positive, \(P_{n}^{m}(x)\) is defined as  
\[P_{n}^{m}(x)=(-1)^{m}(1-x^{2})^{\frac{m}{2}}\frac{d^{m}}{dx^{m}}P_{n}(x), \tag{6}\]  
for the negative parts, it can be obtained from the positive parts as  
\[P_{n}^{-m}(x)=(-1)^{m}\frac{(n-m)!}{(n+m)!}P_{n}^{m}(x), \tag{7}\]  
where \(P_{n}(x)\) is the associated Legendre polynomial given by  
\[P_{n}(x)=\frac{1}{2^{n}n!}\frac{d^{n}}{dx^{n}}{(x^{2}-1)}^{n}. \tag{8}\]  
\(N\) is the highest order of the spherical harmonic function, \(n=0,\cdots,N\) and \(m=-n,\cdots,0,\cdots,n\).