# Limitations and Performance Analysis of Spherical Sector Harmonics for Sound Field Processing
## 4 Limitations on Radial Extrapolation  
Sound field radial extrapolation is an essential property of SHs. With (1), we can extrapolate the sound field from the measured sphere to another sphere with a different radius. This property enables us to analyze the sound field over the space with a measurement on a sphere and is implemented in many acoustics applications, such as ANC [14] and sound field reproduction [34]. As SSHs are not solutions of the Helmholtz equation, shown in Section 3, the radial function is not theoretically guaranteed to present the sound field changes in the radial direction. Therefore, analyzing the feasibility of conducting the radial extrapolation with SSHs is important. In our previous work [28], we proposed a sector sound field radial extrapolation method using the mapping relationship between SHs and SSHs, where we conducted radial extrapolation with SHs and then mapped SHs to SSHs to reconstruct the sound field in the extrapolated sector region. However, obtaining accurate SH coefficients from a restricted measurement region required a dual-array measurement and a large angular range of the measurement region. In this section, we investigate whether SSHs possess a similar extrapolation property to SHs and can decompose the sound field with radially independent coefficients. As shown in Figure 2, the radial extrapolation of a spherical sector sound field involves expanding or contracting the sectorial sound field from the measured region, \(\mathbb{M}~{}(\theta\in[\theta_{M},\pi],\phi\in[0,2\pi),r=R_{M})\), to a target sectorial region, \(\mathbb{T}~{}(\theta\in[\theta_{M},\pi],\phi\in[0,2\pi),r=R_{T})\), which shares the same angular range for \(\theta\) and \(\phi\) but has a different radius. This process can be divided into two cases: the exterior and interior cases. In the exterior case, illustrated in Figure 2a, the sound sources are located inside a sphere, \(\mathbb{S}~{}(\theta\in[0,\pi],\phi\in[0,2\pi),r\leq R_{S},R_{S}<R_{M})\), and we expand the sound field to an outer target region,\(\mathbb{T}\left(R_{T}\geq R_{M}\right)\). Conversely, in the interior case shown in Figure 2b, the sound sources are positioned outside a spherical region, \(\mathbb{S}\left(\theta\in[0,\pi],\phi\in[0,2\pi),r\geq R_{S},R_{S}>R_{M}\right)\), and we contract the sound field to an inner target region, \(\mathbb{T}\left(R_{T}\leq R_{M}\right)\). To test the performance of radial functions in SSH radial extrapolation, we create a similar equation, Equation (18), with reference to (1):  
\[P(r,\theta,\phi,k)=\sum_{n=0}^{\infty}\sum_{m=-n}^{n}B_{n}^{m}(k)R_{n}(kr)T_{n} ^{m}(\theta,\phi), \tag{18}\]  
where \(B_{n}^{m}(k)\) is the radial independent SSH coefficient. If (18) is valid, \(B_{n}^{m}(k)\) can be calculated by  
\[B_{n}^{m}(k)=\mathbf{M}_{q}^{\ddagger}\mathbf{p}_{q^{\prime}}, \tag{19}\]  
where \((\cdot)^{\ddagger}\) denotes the pseudo-inverse operation and \(\mathbf{M}_{q}\) denotes a \(Q\times(N_{s}+1)^{2}\) matrix that contains the multiplication of SSHs and the radial function, expressed as  
\[\mathbf{M}_{q}\!\!=\!\!\left[ \begin{array}{cccc}T_{0}^{0}(\theta_{1},\phi_{1})R _{0}(kr_{1})&\cdots&T_{N_{s}}^{N_{s}}(\theta_{1},\phi_{1})R_{N_{s}}(kr_{1})\\ \vdots&\ddots&\vdots\\ T_{0}^{0}(\theta_{Q},\phi_{Q})R_{0}(kr_{Q})&\cdots&T_{N_{s}}^{N_{s}}(\theta_{Q}, \phi_{Q})R_{N_{s}}(kr_{Q})\end{array} \right]. \tag{20}\]  
\(\mathbf{p}_{q}=[P(\mathbf{x}_{1},k),P(\mathbf{x}_{2},k),...,P(\mathbf{x}_{Q},k)]^{T}\) is a \(Q\times 1\) vector that contains the sound pressure measurements over the sector region by \(Q\) microphones (\(Q\geq(N_{s}+1)^{2}\)), with the coordinate \(\mathbf{x}_{q}=(r_{q},\theta_{q},\phi_{q})\). Simulations are conducted to test the validity of (18) for both exterior and interior cases. To calculate the extrapolation error, \(L=5214\) points are placed over \(\mathbb{T}\) according to the nearly uniform sampling.