# Time Domain Spherical Harmonic Processing with Open Spherical Microphones Recording
## 2 Problem Formulation
### Equivalent Spherical Harmonic Decomposition of a Sound Field in Time-Space Domain  
While the frequency domain spatial sound field capture is well established as explained in Section 2.1, in this paper, our objective is to investigate the possibility of an analogous spherical harmonic analysis in time domain. In a similar fashion to (1) and (4), we now consider the relationship between sound pressure \(s(\mathbf{x},t)\) recorded by a spherical microphone array and the time domain spherical harmonic coefficients, denoted as \(a_{nm}(t)\). It is desirable to have these time domain coefficients \(a_{nm}(t)\) independent of the measurement radius. Thus, we only need to record \(a_{nm}(t)\) to obtain the sound field over the entire region of interest \(\Omega\). A time domain method can directly extract \(a_{nm}(t)\), thus avoiding the Fourier transformation of signals. As a time domain analysis is usually with real-valued components, we rewrite (1) in the form of  
\[S(\mathbf{x},k)=\sum_{n=0}^{N}\sum_{m=-n}^{n}i^{n}\alpha_{nm}(k)\frac{j_{n}(kr)}{i ^{n}}\ Y_{nm}(\theta,\phi), \tag{5}\]  
where \(i=\sqrt{-1}\), in order to make the inverse Fourier transform of all terms to be real. Taking the inverse Fourier transformation of (5), we obtain  
\[s(\mathbf{x},t)=\sum_{n=0}^{N}\sum_{m=-n}^{n}a_{nm}(t)*p_{n}(t,r)\ Y_{nm}(\theta, \phi), \tag{6}\]  
where \(*\) denotes the convolution operation,  
\[a_{nm}(t)\overset{\mathcal{F}}{\longrightarrow}\ i^{n}\alpha_{nm}(k), \tag{7}\]  
where \(\overset{\mathcal{F}}{\longrightarrow}\) denotes the Fourier transform operator,  
\[p_{n}(t,r)\overset{\mathcal{F}}{\longrightarrow}\ \frac{j_{n}(kr)}{i^{n}}, \tag{8}\]  
which is given by  
\[p_{n}(t,r)=\left\{ \begin{array}{cc}\frac{\varepsilon}{\varepsilon}P_{n}( \frac{t}{\varepsilon})&-\frac{r}{\varepsilon}\leq t\leq\frac{r}{\varepsilon} \\ 0&|t|>\frac{r}{\varepsilon}\end{array} \right., \tag{9}\]  
where \(P_{n}(\cdot)\) is the Legendre function. The proof of (9) is given in Appendix A. We note that every component in (6) is real valued. Equation (6) shows how to reconstruct the sound pressure at \(\mathbf{x}=(r,\theta,\phi)\) with the recorded time domain spherical harmonic coefficients \(a_{nm}(t)\). We consider an alternative time domain filter to obtain \(a_{nm}(t)\) from the recorded signals rather than taking the inverse Fourier transform of (4) since \(1/j_{n}(kr)\) is unbounded when \(j_{n}(kr)=0\). Note that \(j_{n}(kr)\) as a filter has order dependent zeros when \(j_{n}(kr)=0\). As a result, \(1/j_{n}(kr)\) approaches infinity at these frequencies, making it unstable to have an inverse Fourier transform. In other words, the z-transform of \(p_{n}(t,r)\) given in (9), has zeros on the unit circle because of Bessel zeros, refers to a non-minimum phase system. In this case, the inverse system of \(p_{n}(t,r)\), with the frequency response of \(1/j_{n}(kr)\) is not stable. As a result, we first define  
\[b_{nm}(t,r)\triangleq a_{nm}(t)*p_{n}(t,r), \tag{10}\]  
which has a frequency response of  
\[b_{nm}(t,r)\smash{\mathop{\longrightarrow}\limits^{\mathcal{F}}}\ \ i^{n}\alpha_{nm}(k)\frac{j_{n}(kr)}{i^{n}}= \alpha_{nm}(k)j_{n}(kr). \tag{11}\]  
Since \(Y_{nm}(\theta,\phi)\) is independent to both frequency and time, \(b_{nm}(t,r)\) can be obtained by integrating (6) over a sphere of radius \(r\) such that  
\[b_{nm}(t,r)=\int_{0}^{2\pi}\int_{0}^{\pi}S(r,\theta,\phi,t)Y_{nm}(\theta,\phi )\sin(\theta)d\theta d\phi. \tag{12}\]  
If we regularly place \(Q\geq(N+1)^{2}\) omni-directional microphones on a sphere of radius \(R_{Q}\), we can estimate the integration in (12) with a finite summation such that  
\[b_{nm}(t,R_{Q})\approx\sum_{q=1}^{Q}S(\mathbf{x}_{q},t)Y_{nm}(\theta_{q},\phi_{q}). \tag{13}\]  
To simplify the implementation, we sample the signals with sampling time T such that \(t=\
u T=\
u/F_{s}\), where \(\
u\) is the time index and \(F_{s}\) is the sampling frequency. We rewrite (10) as  
\[