## 2 Reflection Power Estimation Using Eigenbeam Spatial Correlation Model
### Problem Formulation  
Consider a convex room with a single sound source and a spherical microphone array of radius \(R\) with \(Q\) omnidirectional microphones centered at a location \(O\), as shown in Figure 1. Let the spherical coordinate \(\mathbf{y}_{o}=(r_{o},\theta_{o},\phi_{o})\) denote the sound source location with respect to \(O\). Similarly, the \(q\)th microphone element is located at \(\mathbf{x}_{q}=(R,\theta_{q},\phi_{q})\) for \(q\in\{1,2,\cdots,Q\}\). In this paper, all the elevation angles are \(\in[0,\pi]\) downwards from the Z-axis and the azimuth angles are \(\in[0,2\pi)\) counterclockwise from the X-axis. We treat the room as a linear time-invariant (LTI) acoustic transmission system whose dynamic behavior is represented by the RIRs derived from the spherical microphone array measurements. Let \(H(\mathbf{x}_{q},\mathbf{y}_{o},t,k)\) be the room transfer function (RTF), between the source at \(\mathbf{y}_{o}\) and the microphone element at \(\mathbf{x}_{q}\), obtained from the short-time Fourier transform (STFT) of the RIR. Here, \(t\) is the STFT temporal frame index and \(k=2\pi f/c\) is the wavenumber with \(f\) and \(c\) representing the frequency and speed of sound, respectively. Since the incident sound field at the receiver contains the direct sound and the reflections, we can decompose the RTF \(H(\mathbf{x}_{q},\mathbf{y}_{o},t,k)\) as  
\[H(\mathbf{x}_{q},\mathbf{y}_{o},t,k)=H_{\text{d}}(\mathbf{x}_{q},\mathbf{y}_{o},t,k)+H_{\text{ r}}(\mathbf{x}_{q},\mathbf{y}_{o},t,k) \tag{1}\]  
where \(H_{\text{d}}(\mathbf{x}_{q},\mathbf{y}_{o},t,k)\) and \(H_{\text{r}}(\mathbf{x}_{q},\mathbf{y}_{o},t,k)\) are the direct path and reflected path components, respectively. Assuming that the distance between \(\mathbf{y}_{o}\) and \(\mathbf{x}_{q}\) is significantly larger than the aperture size of the microphone array, we can represent \(H_{\text{d}}(\mathbf{x}_{q},\mathbf{y}_{o},t,k)\) and \(H_{\text{r}}(\mathbf{x}_{q},\mathbf{y}_{o},t,k)\) as a composition of plane waves in the spatial domain as  
\[H_{\text{d}}(\mathbf{x}_{q},\mathbf{y}_{o},t,k)=G_{\text{D}}(t,k|\mathbf{y}_{o})e^{ik\phi _{q}.\mathbf{x}_{q}} \tag{2}\]  
\[H_{\text{r}}(\mathbf{x}_{q},\mathbf{y}_{o},t,k)=\int_{\hat{\mathbf{y}}}G_{\text{R}}(t,k, \hat{\mathbf{y}}|\mathbf{y}_{o})e^{ik\hat{\mathbf{y}}.\mathbf{x}_{q}}d\hat{\mathbf{y}} \tag{3}\]  
where \(G_{\text{D}}(t,k|\mathbf{y}_{o})\) is the direct path gain with respect to \(O\), \(\hat{\mathbf{y}}_{o}\) is the unit vector along the source direction, \(i=\sqrt{-1}\), \(G_{\text{R}}(t,k,\hat{\mathbf{y}}|\mathbf{y}_{o})\) is the gain of the reflected plane wave arriving from the direction \(\hat{\mathbf{y}}=(1,\theta,\phi)\), and \(\int_{\hat{\mathbf{y}}}d\hat{\mathbf{y}}=\int_{0}^{2\pi}\int_{0}^{\pi}\sin\theta d\theta d\phi\). Here, we have modeled  
Figure 1: Geometric illustration of the spherical microphone array centered at the coordinate origin and the single sound source located at \(\mathbf{y}_{o}=(r_{o},\theta_{o},\phi_{0})\). the reflection gain \(G_{R}\) as a non-isotropic directional distribution function that varies with frequency and time to comprehend a real room with inhomogeneous surfaces that have frequency-dependent wall impedance and damping coefficients. By examining \(\mathbb{E}\left\{H_{\mathrm{d}}H_{\mathrm{d}}^{*}\right\}\) based on (2), where \(\mathbb{E}\{\cdot\}\) represents the statistical expectation operator, we can express the direct path power as  
\[P_{D}(t,k|\mathbf{y}_{o})=\mathbb{E}\left\{|G_{D}(t,k|\mathbf{y}_{o})|^{2}\right\} \tag{4}\]  
where \(|\cdot|\) denotes the absolute value.