# Limitations and Performance Analysis of Spherical Sector Harmonics for Sound Field Processing
## 4 Limitations on Radial Extrapolation
### Interior Case  
#### 4.2.1 Plane Wave  
For the interior case, we first validate the extrapolation with a unit-amplitude plane wave source arriving from the direction \(\theta_{A}=180^{\circ}\), \(\phi_{A}=0\). The setups for \(\mathbb{M}\) and the microphone placement are the same as in the exterior case. We estimate the sound field for different inner concentric sector regions with \(R_{T}=R_{M}\) and \(0.6R_{M}\). The results are shown in Figure 5 below. We observe that (i) when \(R_{T}=R_{M}\), we can accurately estimate the plane wave shape over \(\mathbb{T}\); (ii) compared to the exterior case, when \(R_{T}\
eq R_{M}\), the estimation error is significantly larger. Then, we analyze the extrapolation performance with different settings of frequencies (\(f=300\), \(500\), \(700\), and \(900\) Hz), sizes of \(\mathbb{T}\) (\(\theta_{M}=45^{\circ},75^{\circ},105^{\circ}\), and \(135^{\circ}\)), and distances between \(R_{T}\) and \(R_{M}\). We consistently set \(R_{M}=1\) m and \(Q=\left[1.3(N_{\mathrm{s}}+1)^{2}\right]\). With \(0<R_{T}\leq R_{M}\), we only analyze the distances between \(R_{T}\) and \(R_{M}\) from \(0\) to \(0.7\lambda\). The results are shown in Figure 6, which demonstrates that (i) \(\epsilon\) increases sharply within \(0.1\lambda\) distance; (ii) \(\epsilon\) reaches the largest value when the distance is in the range of \(0.3\lambda\) to \(0.5\lambda\) and decreases or fluctuates around the peak value when the distance increases further; (iii) compared to Figure 4, we have a larger \(\epsilon\) in Figure 6, and the dependency of the error on the frequency and the size of \(\mathbb{T}\) is less regular in Figure 6.  
We additionally explore the impact of the arrival direction by varying \(\theta_{A}\) from \(0^{\circ}\) to \(180^{\circ}\) in increments of \(10^{\circ}\), while keeping \(\phi_{A}=0^{\circ}\) and \(f=300\) Hz. We conduct simulations with varying \(\theta_{M}\) settings and distances between \(\mathbb{T}\) and \(\mathbb{M}\). The results are illustrated in Figure 7, leading us to the following observations: (i) the extrapolation error \(\epsilon\) exhibits  
Figure 4: Extrapolation error \(\epsilon\) for the exterior case with 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. Figure 5: Spherical sector sound field radial extrapolation for the interior case with a plane wave, \(\theta_{M}=120^{\circ}\), \(R_{M}=1\) m, \(f=300\) Hz, \(Q=100\): (**a**) true sound field with \(R_{T}=1\) m; (**b**) estimated sound field with \(R_{T}=1\) m; (**c**) true sound field with \(R_{T}=0.6\) m; (**d**) estimated sound field with \(R_{T}=0.6\) m.  
symmetry around \(\theta_{A}=90^{\circ}\); (ii) for \(R_{T}=R_{M}\), \(\epsilon\) is small and particularly responsive to changes in \(\theta_{A}\), especially with larger \(\theta_{M}\); (iii) when \(R_{T}=R_{M}\), \(\epsilon\) rises from its minimum value to a peak at \(\theta_{A}=90^{\circ}\), and then descends back to its minimum again; (iv) if \(R_{T}\
eq R_{M}\), \(\epsilon\) is larger and less sensitive to directions in \(\theta_{A}\); (v) with \(R_{T}=R_{M}\), \(\epsilon\) diminishes with the increment of \(\theta_{M}\), whereas with \(R_{T}\
eq R_{M}\), \(\epsilon\) rises with the increment of \(\theta_{M}\). Figure 6: Extrapolation error \(\epsilon\) for the interior case with a plane wave, 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.