# Limitations and Performance Analysis of Spherical Sector Harmonics for Sound Field Processing
## 2 Spherical Sector Harmonics  
This section introduces the background theory of SSHs and SSHs and highlights the differences between them. As illustrated in Figure 1, spherical coordinates \((r,\theta,\phi)\) are used to specify the position of a point, where \(r\) represents the radial distance from the origin, \(\theta\in[0,\pi]\) is the polar angle between the radial line and the positive z-axis, and \(\phi\in[0,2\pi)\)is the azimuthal angle, measured as the counterclockwise rotation of the radial line around the positive x-axis on the xy-plane [29]. A sound field over a spherical region can be decomposed by SHs to  
\[P(r,\theta,\phi,k)\approx\sum_{n=0}^{N_{h}}\sum_{m=-n}^{n}A_{n}^{m}(k)R_{n}(kr)Y_{n }^{m}(\theta,\phi), \tag{1}\]  
where \(k=2\pi f/c\) is the wave number with \(f\) as the frequency and \(c\) as the speed of sound propagation; \(A_{n}^{m}(k)\) is the radial independent SH coefficient; \(R_{n}(kr)\) is the radial function, which is the solution of the Helmholtz equation in the radial direction for the spherical coordinate; \(N_{h}\geq\lceil kr\rceil(\lceil\cdot\rceil\) is the ceiling operation) [30] is the truncation order of SH decomposition; and \(Y_{n}^{m}(\cdot)\) is the spherical harmonics of \(n\)-th order and \(m\)-th degree, defined by  
\[Y_{n}^{m}(\theta,\phi)=\sqrt{\frac{(2n+1)(n-m)!}{4\pi(n+m)!}}\mathcal{P}_{n}^{ m}(\text{cos}\theta)e^{im\phi}, \tag{2}\]  
where \(\mathcal{P}_{n}^{m}(\cdot)\) is the associated Legendre polynomial of \(n\)-th order and \(m\)-th degree, and \(e^{im\phi}\) is the exponential function of \(m\)-th degree. For the interior case, where the source is located outside the sphere, \(R_{n}(kr)=j_{n}(kr)\) is the first kind of spherical Bessel function of the \(n\)-th order; for the exterior case, where the source is inside the sphere, \(R_{n}(kr)=h_{n}^{(2)}(kr)\) is the spherical Hankel function of the second kind of \(n\)-th order. Generally, the sound field can be a combination of these two cases with distinct radial-independent SH coefficients for the interior and exterior parts [31]. Similarly, a sound field over a spherical sector region (\(\theta\in[\theta_{1},\theta_{2}]\), \(\phi\in[\phi_{1},\phi_{2}]\)) can be decomposed by SSHs to [16]  
\[P(r,\theta,\phi,k)\approx\sum_{n=0}^{N_{h}}\sum_{m=-n}^{n}\Gamma_{n}^{m}(r,k) T_{n}^{m}(\theta,\phi), \tag{3}\]  
where \(T_{n}^{m}(\theta,\phi)\) is the spherical sector harmonics of \(n\)-th order and \(m\)-th degree, \(\Gamma_{n}^{m}(r,k)\) are the SSH coefficients, and \(N_{s}\geq\lceil kr\rceil\) is the truncation order of SSH decomposition [16]. \(T_{n}^{m}(\theta,\phi)\) can be expressed as [16]  
Figure 1: Illustration of the coordinate system. \[T_{n}^{m}(\theta,\phi)=\left\{ \begin{array}{c}K_{n}^{m}\mathcal{P}_{n}^{m}(q_{1} \text{cos}\theta+q_{2})e^{imu\phi}\\ \quad\quad\forall\ 0\leq n\leq\infty,0\leq m\leq n,\\ (-1)^{|m|}T_{n}^{|m|*}(\theta,\phi)\\ \quad\quad\forall\ -n\leq m<0,\end{array} \right. \tag{4}\]  
where \(K_{n}^{m}\) is the normalization constant, which can be calculated by  
\[K_{n}^{m}=\sqrt{\frac{(2n+1)(n-m)!q_{1}u}{4\pi(n+m)!}}, \tag{5}\]  
where \(q_{1}\) and \(q_{2}\) are the shift coefficients that transform \(\mathcal{P}_{n}^{m}(\cdot)\) to the shifted associated Legendre polynomials \(\mathcal{P}_{n}^{m}(q_{1}\text{cos}\theta+q_{2})\), and \(u\) is the scale coefficient that changes the exponential function \(e^{imu\phi}\) to the scaled exponential functions \(e^{imu\phi}\). These coefficients depend on the spherical sector region, represented by  
\[ \begin{array}{l}q_{1}=2/(\text{cos}\theta_{1}-\text{cos}\theta_{2}),\\ q_{2}=-(\text{cos}\theta_{1}+\text{cos}\theta_{2})/(\text{cos}\theta_{1}-\text {cos}\theta_{2}),\\ u=2\pi/(\phi_{2}-\phi_{1}).\end{array}  \tag{6}\]  
As \(\theta_{2}>\theta_{1}\) and \(\phi_{2}>\phi_{1}\), based on (6), we have \(q_{1}\in[1,\infty),q_{2}\in(-\infty,\infty)\) and \(u\in[1,\infty)\). When \(\theta_{1}=0,\theta_{2}=\pi\), \(\phi_{1}=0\), and \(\phi_{2}=2\pi\), SSHs reduce to SH.