# Denoising Directional Room Impulse Responses with Spatially Anisotropic Late Reverberation Tails
## 3 Plane Wave Decomposition and Spatial Analysis
### Covariance Analysis in the Spatial Domain  
The decomposition matrix rank, angular separation, and flat coverage criteria described above provide a good starting point for selecting a set of \(D\) look directions for the PWD, but the spatial coherence properties of the resulting sound field will directly determine the ability of a given set to decompose an incoherent SHD field into independent directional signals. The sound field's spatial coherence can be examined using the covariance matrix of the PWD signals; following the results of Section 2.1, this should approach the identity matrix for any incoherent field of independent plane waves if the covariance calculation is made independent of the signal magnitudes by dividing out their expected values:  
\[C_{d,d^{\prime}}=\frac{\operatorname{E}\left\{s(\Omega_{d},t)s(\Omega_{d^{ \prime}},t)\right\}}{\sqrt{\operatorname{E}\left\{s^{2}(\Omega_{d},t)\right\} \operatorname{E}\left\{s^{2}(\Omega_{d^{\prime}},t)\right\}}}, \tag{14}\]  
where \(s(\Omega_{d},t)\) is the time-domain signal obtained by PWD for the look direction \(\Omega_{d}\). If we assume that the chosen set of \(D\) look directions leads to signals that sufficiently represent \(s(\Omega_{d},t)=\operatorname{Re}\left\{\mathcal{F}^{-1}\left[x(f,\Omega_{d},t)\right]\right\}\) (where \(\mathcal{F}^{-1}\left[\cdot\right]\) is the inverse Fourier transform and \(\operatorname{Re}\left\{.\right\}\) is the real part of a complex number), Equation (14) should indeed approach a spatially discretized version of Equation (9). As a result, a global sound field incoherence measure, analogous to the COMEDIE diffuseness measure proposed by [PERSON] and [PERSON] [23] in the SHD, can be defined over the set of \(D\) look directions. Denoted \(\Gamma\) in this paper, this measure is calculated using an eigendecomposition of the covariance matrix [23]:  
\[\Gamma=\frac{\sum_{i=1}^{D}|e_{i}-\bar{e}|}{2(D-1)\bar{e}}, \tag{15}\]  
where \(e_{i}\) are the eigenvalues of the covariance matrix and \(\bar{e}\) is their mean. To investigate the incoherence preservation of various sets of look directions, a plane-wave field is synthesized as 8192-sample Gaussian noise signals over 625 independent incident directions, distributed using a Sloan-Womersley hyperinterpolation grid [21] (in order to sample the space of incident directions as evenly as possible). Both a fully isotropic and a highly anisotropic version are created, the latter by applying a cardioid directivity pattern with a dynamic range of 60 dB to the synthesized signals. The signals are encoded to the 4 th-order truncated SHD and subsequently decomposed onto 5 different arrangements of 25 look directions by PWD. The mean and minimum angular separation between nearest neighbors, the flatness of the combined directivity \(W(\Omega)\), and the global resulting incoherence \(\Gamma\) are reported for each in Table 1. The incoherence measures in the anisotropic case being highly similar among the 5 sets, the 25-point Fliege t-design [22] is finally chosen due to its flatness, its relatively high angular separation, and its high isotropic incoherence. Figure 2 then shows the covariance matrices for the Fliege arrangement for the two types of synthesized signals (isotropic, Figure 2a,c; anisotropic, Figure 2b,d), in both the SHD (Figure 2a,b) and PWD look direction domain (Figure 2c,d). The incoherence measures \(\Gamma\) are included in each case to confirm that the PWD domain maintains its incoherence under highly anisotropic conditions.