# Denoising Directional Room Impulse Responses with Spatially Anisotropic Late Reverberation Tails
## 2 Stochastic Model for Anisotropic Late Reverberation  
We assume that at any given time in the stochastic late reverberation domain (i.e., above the Schroeder frequency and after the mixing time), the anisotropic sound field can be seen from the surface of an SMA (\(\Omega\equiv(\theta,\phi)\in S^{2}\)) as an infinite number of independent plane waves subject to an arbitrary amplitude density distribution (which we will write homogeneous to a power) over the incident directions \(\Omega\):  
\[x(f,\Omega,t)=\sqrt{P(f,\Omega,t)}\Phi(f,\Omega,t), \tag{1}\]  
where \(P(f,\Omega,t)\) is the power envelope and \(\Phi(f,\Omega,t)=e^{i\varphi(f,\Omega,t)}\) is the plane wave phase function such that \(|\Phi(f,\Omega,t)|=1\ \forall\ f,\Omega,t\) and \(\mathrm{E}\left\{\Phi(f,\Omega,t)\Phi^{*}(f,\Omega^{\prime},t)\right\}=\delta _{\Omega,\Omega^{\prime}}\), where \(\mathrm{E}\left\{.\right\}\) denotes mathematical expectation and \(\delta_{\Omega,\Omega^{\prime}}\) is the Kronecker delta. The HOA format [13] is based on the spherical harmonic (SH) decomposition of the sound field measured by an SMA. For the continuous function \(x(f,\Omega,t)\) defined on \(S^{2}\), the ideal SH transform is written as follows [14]:  
\[X_{l,m}(f,t)=\int_{\Omega\in S^{2}}x(f,\Omega,t)Y_{l,m}(\Omega)d\Omega, \tag{2}\]  
where \(Y_{l,m}(\Omega)\) are the real spherical harmonics of order \(l\in\mathbb{N}_{0}\) and degree \(m\in[-l,l]\). Using an SMA, this transform is approximated by sampling over the \(Q\) transducer positions using a weighted finite sum, with the specific weights chosen such that the sum approaches the continuous integral of Equation (2) (e.g., by least-squares minimization [15]). As a result of this discretization, the maximum encoding order \(L\) is limited such that the \((L+1)^{2}\times Q\) encoding matrix must have \(K=(L+1)^{2}\leq Q\) nonvanishing singular values [16]. Truncating the SHD to a finite maximum order has the added consequence of introducing a spatial aliasing limit \(ka\simeq L\), where \(k=2\pi f/c\) is the wavenumber (with \(c\) the speed of sound and \(a\) the SMA radius) [17]. Finally, to obtain a representation of the measured sound field that is independent of the measurement array, a correction for the mode strengths (or holographic functions) of the SMA must be applied. In HOA, the theory describing optimal correction filters is provided by [PERSON] and [PERSON] [13]. In the SHD, a spatially incoherent sound field can be defined per component as [7]  
\[\widehat{X}_{l,m}(f,t)=\sqrt{\widehat{P}_{l,m}(f,t)}\widehat{\Phi}_{l,m}(f,t), \tag{3}\]with \(\widehat{\Phi}_{l,m}(f,t)\) stochastically independent such that  
\[\mathrm{E}\left\{\widehat{\Phi}_{l,m}(f,t)\widehat{\Phi}_{l^{\prime},m^{\prime}}^ {*}(f,t)\right\}=\delta_{l,l^{\prime}}\delta_{m,m^{\prime}}. \tag{4}\]  
Since classic reverberation theory and its accompanying algorithms are based on energy decay or power envelopes, we take the power of the (ideal, infinite-order) inverse SH transform of Equation (3),  
\[ \begin{split}\widehat{P}(f,\Omega,t)&=\mathrm{E} \left\{\widehat{x}(f,\Omega,t)\widehat{x}^{*}(f,\Omega,t)\right\}\\ &=\sum_{l=0}^{\infty}\sum_{m=-l}^{l}\sum_{l^{\prime}=0}^{\infty} \sum_{m^{\prime}=-l^{\prime}}^{l^{\prime}}\mathrm{E}\left\{\widehat{X}_{l,m}( f,t)Y_{l,m}(\Omega)\widehat{X}_{l^{\prime},m^{\prime}}^{*}(f,t)Y_{l^{\prime},m^{ \prime}}^{*}(\Omega)\right\}\\ &=\sum_{l=0}^{\infty}\sum_{m=-l}^{l}\sum_{l^{\prime}=0}^{\infty} \sum_{m^{\prime}=-l^{\prime}}^{l^{\prime}}\sqrt{\widehat{P}_{l,m}(f,t) \widehat{P}_{l^{\prime},m^{\prime}}(f,t)}\mathrm{E}\left\{\widehat{\Phi}_{l,m} (f,t)\widehat{\Phi}_{l^{\prime},m^{\prime}}^{*}(f,t)\right\}Y_{l,m}(\Omega)Y_ {l^{\prime},m^{\prime}}^{*}(\Omega)\\ &=\sum_{l=0}^{\infty}\sum_{m=-l}^{l}\widehat{P}_{l,m}(f,t)|Y_{l,m }(\Omega)|^{2}\end{split}  \tag{5}\]  
and observe that since both \(\widehat{P}_{l,m}(f,t)\) and \(|Y_{l,m}(\Omega)|^{2}\) are real and non-negative, \(\widehat{P}(f,\Omega,t)\) cannot be constructed to be arbitrarily anisotropic by acting on \(\widehat{P}_{l,m}(f,t)\) alone.