# Denoising Directional Room Impulse Responses with Spatially Anisotropic Late Reverberation Tails
## 3 Plane Wave Decomposition and Spatial Analysis  
The result of Equation (9) is valid for incident directions distributed around a single reference point, which is precisely the representation enabled by the SHD (assuming the proper corrections for the mode strengths of the SMA have been applied). From this modal representation, a direction-dependent view can be constructed by plane-wave decomposition (PWD) for a given set of \"steering\" or \"look\" directions around the unit sphere. The practical implementation of the PWD for finite-order SHD signals can be seen from a beamforming perspective as the \"natural\" or maximum directivity index (max-DI) beamformer, with coefficients \(\alpha_{l}=1\ \forall\ l\in[0,L]\), and a main lobe peak-to-null half-width \(\Theta_{\text{PN}}\approx\pi/L\)[19]. The finite-order PWD can thus be written simply in matrix form:  
\[\mathbf{x}_{\text{PWD}}(f,t)=\mathbf{Y}\mathbf{x}_{\text{SHD}}(f,t), \tag{10}\]  
where \(\mathbf{x}_{\text{PWD}}\) is the signal column vector corresponding to the \(D\) look directions \(\Omega_{d}\), \(\mathbf{Y}\) is the \(D\times(L+1)^{2}\) decomposition matrix of elements \(y_{d,n}=\alpha_{l}Y_{l,m}(\Omega_{d})\) (with indices \(n=l^{2}+l+m\) and \(d\in[1,D]\)), and \(\mathbf{x}_{\text{SHD}}\) is the length \((L+1)^{2}\) signal column vector in the SHD. The beamforming view of the PWD highlights the problem of lobe overlap when analyzing several look directions; in any finite-order implementation, the resulting spatial-domain signals will inevitably include contributions from directions other than their target look direction \(\Omega_{d}\), which may compromise their independence and hence the coherence analysis described above. The first crucial constraint is that the decomposition matrix \(\mathbf{Y}\) must not introduce any linear dependence itself; since any look direction \(\Omega_{d}\) can be written as a linear combination of any number of others, the number of linearly independent signals is restricted by the number of spherical harmonics, i.e., \(D\leq(L+1)^{2}\) (which follows from the fact that the spherical harmonics are linearly independent basis functions). However, if the PWD is to be useful as a spatial representation for denoising anisotropic DRIRs, the resulting (denoised) signals must also be transformed back into the SHD for posterior use in HOA-based spatialization systems. Since the rank of the SHD re-encoding matrix \(\mathbf{Z}\) will be limited by \(D\leq(L+1)^{2}\) from the PWD step, complete linear independence throughout the full process can only be ensured if \(D=(L+1)^{2}\). As such, the re-encoding step can be written  
\[ \begin{split}\mathbf{Z}\mathbf{x}_{\text{PWD}}&= \mathbf{Z}\mathbf{Y}\mathbf{x}_{\text{SHD}}\\ &=\mathbf{x}_{\text{SHD}}\text{ if }\mathbf{Z}\mathbf{Y}=\mathbf{I}_{(L+1)^{2}}. \end{split}  \tag{11}\]  
The signal overlap due to the beamforming directivity lobes can be further quantified by analyzing the interactions between the directivity functions of each look direction \(w_{d}(\Omega)\), obtained by evaluating the PWD with respect to \(\Omega_{d}\):  
\[w_{d}(\Omega)=\mathbf{y}^{\text{T}}(\Omega)\mathbf{y}(\Omega_{d}), \tag{12}\]  
where \(\mathbf{y}(\Omega)\) is the length \((L+1)^{2}\) column vector of elements \(y_{n}(\Omega)=\alpha_{l}Y_{l,m}(\Omega)\) and \(\cdot^{\text{T}}\) represents the simple transpose. As the beamforming patterns \(w_{d}(\Omega)\) are axisymmetric, the dimensionality of the interaction between two look directions \(\Omega_{d}\) and \(\Omega_{d^{\prime}}\) can be reduced to a single relative angle \(\Theta\) along the great circle joining the two. Figure 1 shows the interaction between the directivity functions of two adjacent look directions at different relative distances (central angles) \(\Delta\Omega\) apart. Figure 1b in particular highlights a critical value of the relative angle \(\Theta\), the main lobe peak-to-null half-width \(\Theta_{\text{PN}}\).