# Time Domain Spherical Harmonic Processing with Open Spherical Microphones Recording
## 1 Introduction
However, when we consider a time domain method, we can not simply avoid the Bessel zeros because we do not apply a Fourier Transformation to separate the Bessel zero frequency components from the others. Meanwhile, there are also several works relate to time domain spatial audio signal processing. In [35], [PERSON] and [PERSON] give a time domain description of the free-space Green's function in the spherical harmonic domain. This provides a solution to decompose the free-space channel between a loudspeaker and microphone into the time-space domain. This work only targets the free-space Green's function, and as a result, the method is highly limited to the application of free space sound field reproduction. In [36], a time domain wave field synthesis method is presented. Although an IFFT is applied to derive the time domain solution, the work still demonstrates that time-domain wave field synthesis can be beneficial to time-varying spatial acoustic applications. In [37], [PERSON] and [PERSON] offer a time domain representation of the spherical harmonic equation. They relate the time domain spherical harmonic coefficients to the sound pressure, but do not include the method of obtaining the time domain coefficients from a given recording. Time domain beamformers are designed in [38; 39] with the IFFT of spherical harmonics. These papers show certain advantages for finite impulse response (FIR) filtering based signal processing systems. Overall, these time domain approaches illustrate the advantages of time domain signal processing, however, they remain unable to obtain location-independent spherical harmonic coefficients. This makes them ill-suited for spatial ANC systems, as these location-independent coefficients provide necessary information about the continuous residual sound field inside the region of interest. In this paper, we propose a FIR filter based time domain spherical harmonic analysis method to accurately record spatial sound fields with an open spherical microphone array for the purpose of spatial ANC. We note that this work focuses solely on the problem of sound field recording, and that the spatial ANC application acts purely as motivation to our problem. Therefore, with spatial ANC in mind, the recording method prioritizes a minimum processing delay, a bandwidth of interest (low frequencies for typical noise scenarios), and a practical array geometry (open sphere surrounding a quite zone). Employing the recording method in an actual ANC system, and its evaluation, is out of the scope of this paper. The novelty of the presented work is the investigation of time domain spherical harmonic coefficients. These time domain coefficients match the properties of conventional frequency domain spherical harmonic coefficients. That is, the coefficients are location independent within the region of interest, and they represent the continuous sound field over the space. Additionally, these coefficients are obtained in time domain, which relieves the block processing constraint (and can do sample-by-sample processing) and results in lower system delay. Hence, the proposed method is considered to be highly beneficial to spatial ANC systems. We organise the main body of this paper as follows: In Section 2 we detail the background of the frequency domain spherical harmonic algorithm for spatial sound field recording. Additionally, we introduce the time domain equation of spherical harmonic decomposition, while addressing the challenges of recording time domain spherical harmonic coefficients. The filter's design and implementation to obtain time domain spherical harmonic coefficients is presented in Section 3, along with error analysis. Effects of truncation and filter length are shown in Section 4 via initial simulations of filter performance. Section 5 presents simulation results for the proposed method's estimation of spherical harmonic coefficients, as well as sound field reconstruction performance at a point and over space, verifying the effectiveness of the proposed theory and design.