# Time Domain Spherical Harmonic Processing with Open Spherical Microphones Recording
## 3 Filter Design for Obtaining Time Domain Spherical Coefficients
### Stability of Ideal Inverse Filter  
Due to the challenges mentioned in Section 2, rather than directly using (14), we pre-design a series of filters \(\rho_{n}(\
u,r)\) such that  
\[b_{nm}(\
u,R_{Q})*\rho_{n}(\
u,R_{Q}) \tag{16}\] \[=a_{nm}(\
u)*p_{n}(\
u,R_{Q})*\rho_{n}(\
u,R_{Q})=a_{nm}(\
u),\]  
where  
\[p_{n}(\
u,R_{Q})*\rho_{n}(\
u,R_{Q})\approx\delta(\
u). \tag{17}\]  
We note here that \(\rho_{n}(\
u,r)\) should be order \(n\) dependent but mode \(m\) independent, as is the same property with \(p_{n}(\
u,r)\).  
However, we can never achieve a precise \(\delta(\
u)\) in (17), as the energy of measured sound pressure at the frequency bins of Bessel zeros has been filtered to zero by \(p_{n}(\
u,R_{Q})\). Therefore, we refrain from designing the inverse filter at these zero positions. Instead, we modify \(\delta(\
u)\) to \(\hat{z}_{n}(\
u)\) such that its frequency response \(\hat{Z}_{n}(f)\) is given by  
\[\hat{Z}_{n}(f)=\left\{\begin{array}{ll}1&\quad|j_{n}(kr)|\geq\epsilon\\ 0&\quad|j_{n}(kr)|<\epsilon\end{array}\right., \tag{18}\]  
where \(\epsilon\) is a small positive constant threshold which satisfies \(j_{n}(kr)\approx 0\) when \(|j_{n}(kr)|<\epsilon\). For a fixed \(R_{Q}\), both \(j_{n}(2\pi fR_{Q}/c)\) and \(\hat{Z}_{n}(f)\) can be seen as a function of \(f\). Figure 1 shows \(j_{n}(2\pi fR_{Q}/c)\) and \(\hat{Z}_{n}(f)\) with \(\epsilon=1/40\) of the first four orders of \(n\).  
From Figure 1, we can see that \(\hat{Z}_{n}(f)\) is a superposition of a series of rectangular windows, meaning its inverse Fourier transformation, \(\hat{z}_{n}(\
u)\),of sinc functions. In practice, due to inherent properties of \(j_{n}(2\pi fR_{Q}/c)\), for a given maximum frequency \(f_{\max}\), the number of active spherical harmonic orders is up to \(N\approx\lceil kR_{Q}\rceil\)[1]. We use the same truncation limit when designing \(\hat{Z}_{n}(f)\), resulting in \(\hat{z}_{n}(\
u)\) to be a superposition with a finite number of sinc functions. The necessity and the influence of this truncation on frequency \(f_{\max}\) will be further discussed in Section 4.1.  
Let us define \(w^{(n)}\) in radian (rad), such that \(j_{n}(w^{(n)}F_{s}R_{Q}/c)=\epsilon\), where \(\epsilon\) is the positive threshold we explained in the last paragraph. Therefore, \(w^{(n)}\) can be considered as the edges of window in \(\hat{Z}_{n}(f)\) (see Figure 1). Given the vector of \([w^{(n)}_{1},w^{(n)}_{2},w^{(n)}_{3},\cdots]\), we can write \(\hat{z}_{n}\) as  
\[\begin{split}\hat{z}_{n}(\
u)&=\sum_{s=1}^{S}(w^{(n )}_{2s}-w^{(n)}_{2s-1})\text{sinc}\big{(}\frac{w^{(n)}_{2s}-w^{(n)}_{2s-1}}{2} \
u\big{)}\\ &\quad\times\cos(\frac{w^{(n)}_{2s}+w^{(n)}_{2s-1}}{2}\
u),\end{split} \tag{19}\]  
where \(S\) is the number of rectangular windows in \(\hat{Z}_{n}(f)\) for \(-f_{\max}\leq f\leq f_{\max}\). Furthermore, \(w^{(n)}\)s are dependent on the radius of the microphone array \(R_{Q}\), sampling frequency \(F_{s}\) and the speed of sound \(c\), but the value of \(w^{(n)}F_{s}R_{Q}/c\) remains constant for each order \(n\) such that \(|j_{n}(w^{(n)}F_{s}R_{Q}/c)|=\epsilon\). The first four order of \(w^{(n)}\) is given in Table 1 with the highest frequency limit of \(f_{\max}=2047\) Hz and sampling frequency \(F_{s}=48,000\) Hz. Note that for the zero-th order, we set \(w_{1}=8.9\times 10^{-4}\) to block DC component in practice.  
If we have a series of concentric spherical microphone arrays with the radii of \(r_{1},r_{2},\cdots\), the value of \(w^{(n)}F_{s}r_{q}/c\) would be different from a single sphere model, which can be calculated by \(|j_{n}(w^{(n)}r_{1}/F_{s}c)+j_{n}(w^{(n)}r_{2}/F_{s}c)+\cdots|=\epsilon\).