# Time Domain Spherical Harmonic Processing with Open Spherical Microphones Recording
## 1 Introduction  
Spherical harmonic analysis has been widely used for spatial acoustic signal processing for years [1]. Sound field recordings can be decomposed into a set of orthogonal spatial basis functions and respective coefficients when an appropriately designed spherical microphone array is used [2, 3]. The spherical harmonic decomposition has the advantage that a given sound field can be analyzed over a continuous spatial region rather than a set of distributed points [4]. This has embraced a wide range of algorithms in three-dimensional (3D) audio signal processing such as: sound intensity analysis [5], sound field diffusive analysis [6], beamforming [7, 8], source localization [9, 10], and spatial Active Noise Control (ANC) [11, 12]. A spatial ANC system aims to reduce the unwanted acoustic noise [13] over a space in order to create a silent zone for people. Multiple microphones are used to record the residual noise, and multiple loudspeakers are used to generate the anti-noise field. The recording's accuracy of the residual sound field can highly influence the performance of an ANC system. Furthermore, recording efficiency is also important, as ANC usually focuses on low frequency and time-variant noise. As a result, an accurate and low latency algorithm for residual sound field recording is desired [14]. The sound field recording step in a spatial ANC system focuses on obtaining the location independent spherical harmonic coefficients that represent the residual sound field inside a region of interest. This is different to real time spherical harmonic beamforming or directivity analysis which focuses on extracting source location information from the spatial recording. Moreover, spatial ANC mainly focuses on reducing the sound field inside the spherical microphone array (the region of interest). While other spatial recording applications may focus on analysing the sound field exterior to the array. Additionally, although most of the spatial audio applications utilize a rigid spherical array [15; 16; 17] for its convenience to build and use, an open spherical array is considered to be more suitable for a spatial ANC system. This is because users should be able to enter and move within the ANC region of interest that is surrounded by the spherical microphone array [12; 18]. Furthermore, there exists previous work focusing on optimising the open array for spherical harmonic recording [19; 20], and for spatial ANC systems [21]. However, we consider the optimisation of the open microphone array design to be outside of the scope of this paper, and instead focus on a time-domain recording algorithm. Real-time spatial beamforming systems illustrate that applications with strict delay requirements can highly benefit from the small latency and efficient computation of time domain processing [22; 23]. By posing the signal processing algorithm in the time domain, system performance can be optimized with real-valued lower order filters [24], and lower modeling delays [25]. Specifically, for a spatial ANC system, the system delay which includes the filter group delay (signal processing algorithm), the A/D and D/A converter, and the data processing delay, should be less than the acoustic delay from the reference microphones to the secondary loudspeakers in order to satisfy causality [26]. Furthermore, a longer signal processing delay slows down the convergence speed of the adaptive filtering and may lead to an unstable system [27; 28]. Therefore, it is worthwhile to consider a time domain spherical harmonic decomposition method to achieve sound field recording with an open spherical array for the application of spatial ANC. Frequency domain spherical harmonic recording has been well developed with various optimised filters [29; 30; 31]. One benefit of developing the method in the frequency domain is that the influence of the spherical Bessel zeros can be easily removed by avoiding the estimation of the coefficients at these erroneous frequency bins [19; 32; 33; 34].