# Limitations and Performance Analysis of Spherical Sector Harmonics for Sound Field Processing
## 5 Near Edge Distortion Problem  
The sound field decomposition and reconstruction are fundamental steps in SSH-based sound field processing. For a spherical sector region, the related SSH coefficients \(\Gamma_{n}^{m}(r,k)\) can be found by  
\[\Gamma_{n}^{m}(r,k)=\mathbf{L}_{q}^{\intercal}\mathbf{p}_{q}, \tag{22}\]  
where \(\mathbf{L}_{q}\) denotes the \(Q\times(N_{s}+1)^{2}\) matrix of SSH, expressed as  
\[\mathbf{L}_{q}= \begin{bmatrix}T_{0}^{0}(\theta_{1},\phi_{1})&\cdots&T_{N_{s}}^{N_{ s}}(\theta_{1},\phi_{1})\\ \vdots&\ddots&\vdots\\ T_{0}^{0}(\theta_{Q},\phi_{Q})&\cdots&T_{N_{s}}^{N_{s}}(\theta_{Q},\phi_{Q}) \end{bmatrix} . \tag{23}\]  
After estimating \(\Gamma_{n}^{m}(r,k)\) with (22), the sound field can be reconstructed using (3). While this process is similar to sound field decomposition and reconstruction in SH processing, it can lead to severe distortions around the edges of the spherical sector region. These distortions are caused by mapping the spherical sector to a whole sphere. As illustrated in Figure 10, the points over the two latitudinal edges of the sector region (\(\theta=\theta_{1},\phi\in[\phi_{1},\phi_{2})\) or \(\theta=\theta_{2},\phi\in[\phi_{1},\phi_{2})\)) are mapped to the two poles of the sphere (\(\theta=0,\phi=0\) or \(\theta=\pi,\phi=0\)) during the mapping process of shifted associated Legendre polynomials. Similarly, the points over the two longitudinal edges of the sector region (\(\theta\in[\theta_{1},\theta_{2}]\), \(\phi=\phi_{1}\) or \(\theta\in[\theta_{1},\theta_{2}]\), \(\phi=\phi_{2}\)) are mapped to one longitudinal line of the sector region (\(\theta\in[\theta_{1},\theta_{2}]\), \(\phi=0\)) during the mapping process of scaled exponential functions. These mappings lead to information loss over the edge of the sector region, causing distortion around the sector edge. We investigate this distortion problem using simulations. For the exterior case, we place three unit strength point sources with the Cartesian coordinates \((0.5,0,0)\) m, \((0,0.5,0)\) m, and \((0,0,0.5)\) m. For the interior case, we place three unit strength point sources with the Cartesian coordinates \((3,0,0)\) m, \((0,3,0)\) m, and \((0,0,3)\) m. We analyze the sound field reconstruction distortion over a sector region \(\mathbb{T}\) with \(\theta_{1}=1/3\pi,\theta_{2}\ =\ 2/3\pi,\phi_{1}\ =\ 0\), \(\phi_{2}=\pi\), and \(R_{T}=1\) m. The measurement region \(\mathbb{M}\) is the same as \(\mathbb{T}\). Microphones are placed over \(\mathbb{M}\) with a nearly uniform distribution, with \(Q=100\) for the 300 Hz setup and \(Q=300\) for the 700 Hz setup. The reconstruction error over elevation direction \(\epsilon_{e}(\theta,k)\) and azimuth direction \(\epsilon_{a}(\phi,k)\) are defined as  
\[ \begin{split}\epsilon_{e}(\theta,k)&=20\text{log}_ {10}(\frac{1}{\sum_{l=1}^{L_{a}}|P_{T}(R_{T},\theta,\phi_{l},k)|}\\ &\times(\sum_{l=1}^{L_{a}}|P_{T}(R_{T},\theta,\phi_{l},k)-\hat{P} _{T}(R_{T},\theta,\phi_{l},k)|)),\end{split}  \tag{24}\]  
Figure 9: Extrapolation error \(\epsilon\) for the interior case with a point source, different settings of frequencies, sizes of \(\mathbb{T}\), and distances between \(R_{T}\) and \(R_{M}\): (**a**) extrapolation error with 300 Hz; (**b**) extrapolation error with 500 Hz; (**c**) extrapolation error with 700 Hz; (**d**) extrapolation error with 900 Hz. \[\epsilon_{a}(\phi,k) =20\text{log}_{10}(\frac{1}{\sum_{l=1}^{L_{e}}|P_{T}(R_{T},\theta_{l},\phi,k)|} \tag{25}\] \[\times(\sum_{l=1}^{L_{e}}|P_{T}(R_{T},\theta_{l},\phi,k)-\hat{P_{T} }(R_{T},\theta_{l},\phi,k)|)),\]  
where \(L_{a}\) is the number of samples along the azimuth direction, and \(L_{e}\) represents the number of samples along the elevation direction. In the simulation, we set \(L_{a}=L_{e}=203\), resulting in a total of 41,209 sampling points over region \(\mathbb{T}\). The results depicted in Figures 11 and 12 reveal that the distortion appears around all four edges of \(\mathbb{T}\) in both the exterior and interior cases, and the distortion is more severe as it approaches the edges.