# Auralization of High-Order Directional Sources from First-Order RIR Measurements
## 2 Auralization of Arbitrary Source Directivity from Measurements
### Measuring the SRD ARIR: Proposed Method  
This section presents the proposed efficient SRD ARIR measurement and postprocessing in detail. _Measuring the MIMO RIRs:_ Here the multiple input multiple output (MIMO) RIRs are measured between a 6-channel compact spherical loudspeaker array (Cubelet) with a radius of 7.5 cm and the 4-channel B-format microphone array (ST450), see Figure 2. The loudspeaker array is equipped with Fountek FR58 EX drivers (2 inch coil diameter with \(\pm 3\) mm maximum linear excursion). A more detailed description on the used arrays and high-resolution directivity measurements can be found online ([[https://phaidra.kug.ac.at/o:104374](https://phaidra.kug.ac.at/o:104374)]([https://phaidra.kug.ac.at/o:104374](https://phaidra.kug.ac.at/o:104374))). _Ommi to omni RIR:_ Depending on the array geometries, an approximation of the point-to-point omnidirectional RIR \(h_{0}(t)\) can be obtained by transforming both sides of the MIMO RIRs in the SH-domain and extracting the response between the 0 th order components. If the array elements are arranged according to a spherical t-design, an approximate of \(h_{0}(t)\) is obtained summing over all channels in the array domain. Please note that the direct path in \(h_{0}(t)\) is ideally a single impulse. However, due to the non-ideal responses of the loudspeakers and microphones as well as the array geometries, even the direct path will be spread in time. A possible approach for improving the omnidirectional response is outlined in [32], but it is not employed here. A denoising of \(h_{0}(t)\) is optional but recommended when experiencing unrealistic long reverberation times. We suggest a denoising strategy that is similar to [33] and it is derived in the Appendix A. _DOA and DOD estimation:_ Due to the assumption of a temporally and spatially sparse RIR, we address a direction of arrival (DOA) \(\theta_{\text{R}}(t)\) and direction of departure (DOD) \(\theta_{\text{S}}(t)\) to each discrete time instance \(t\) of \(h_{0}(t)\). While due to reciprocity any DOA estimation method, e.g., as summarized in [34], can be employed for both DOA and DOD estimation, we use the pseudo intensity vector approach (PIV) as presented by [PERSON] et al. [35] for the DOA and an \(r_{\text{E}}\)-vector measure [36] related to the magnitude sensor response (MSR) by [PERSON] et al. [37] for determining the DOD. The DOAs are calculated for the frequencies between \(100\,\text{Hz}\) and \(2.5\,\text{kHz}\). Here, the upper frequency limit is chosen below the spatial aliasing frequencies \(f_{a}=\frac{c}{2\pi r_{5}\tau 450}\approx 3.6\,\text{kHz}\) for \(r_{5}\tau 450=1.5\,\text{cm}\) (defined by \(kr_{5}\tau 450=1\)). For the estimation of the DODs, a less restrictive rule is assumed, as it is less affected by linear interference. Here, the upper frequency limit is \(f_{a}=\frac{c}{\pi r_{c}\omega_{c}}\approx 1.4\,\text{kHz}\) for \(r_{cubl. }=7.5\,\text{cm}\) (inter-transducer arc length roughly below half a wavelength \(\frac{\pi}{2}r_{cubl.}\leq\frac{c}{2f}\)). The low cut at \(100\,\text{Hz}\) minimizes low-frequency disturbance in both the DOA and DOD estimation, respectively.