# 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
DODs and DOAs become  
\[\theta_{\text{S}}(t) =\frac{\tilde{\theta}_{\text{S}}(t)}{\left\lVert\tilde{\theta}_ {\text{S}}(t)\right\rVert}, \text{with }\tilde{\theta}_{\text{S}}(t) =F_{\text{L}}\bigg{\{}\sum_{p=1}^{P}F_{100-1.4\text{k}}\big{\{} h_{p,0}(t)\big{\}}^{2}\Theta_{p}\bigg{\}}, \tag{6}\] \[\theta_{\text{R}}(t) =\frac{\tilde{\theta}_{\text{R}}(t)}{\left\lVert\tilde{\theta}_ {\text{R}}(t)\right\rVert}, \text{with }\tilde{\theta}_{\text{R}}(t) =F_{\text{L}}\bigg{\{}F_{100-2.5\text{k}}\big{\{}\sum_{p=1}^{P}h_{ p,0}(t)\big{\}}F_{100-2.5\text{k}}\big{\{}\sum_{p=1}^{P}h_{p,XYZ}(t)\big{\}} \bigg{\}}, \tag{7}\]  
where \(\Theta_{p}\) indicates the direction of the \(p\)-th loudspeaker, \(P=6\) is the number of array loudspeakers, \(h_{p,0}\) the RIR between the \(p\)-th loudspeaker and the W channel of the ST450 array, \(\left\lVert\cdot\right\rVert\) is the norm operator, and \(h_{p,XYZ}\) are the first-order channels of the ST450 microphone array. Both the DOA and DOD are computed using a zero-phase band limitation (e.g., by MATLAB's filtfilt with a 4 th-order Butterworth band pass) denoted by \(F_{\text{f}_{\text{i}}-\text{f}_{\text{u}}}\) and a zero-phase temporal smoothing \(F_{\text{L}}\) of the resulting estimates using a moving-average Hann window in the interval \([-L/2;L/2]\) for \(L=32\). Figure 2: Compact arrays that are used to measure the multiple input multiple output (MIMO) room impulse responses (RIRs). (**a**) Cubelet: Spherical (\(r=7.5\,\text{cm}\)) 6-channel loudspeaker array prototype with loudspeakers arranged on surfaces of a cube. (**b**) TSL ST450: 4-channel Ambisonic B-format microphone array with \(r=2\,\text{cm}\). SRD ARIR: From Equation (3), and assuming a single propagation path at a time (i.e., assuming temporal disjointness), a first version of the upscaled SRD ARIR becomes  
\[\tilde{h}_{nm}^{n^{\prime}m^{\prime}}(t)=\gamma_{n^{\prime}}^{m^{\prime}}[\mathbf{ \theta}_{\text{R}}(t)]\;h_{0}(t)\;Y_{n}^{m}[\mathbf{\theta}_{\text{S}}(t)], \tag{8}\]  
where the maximum orders \(n\leq N\) and \(n^{\prime}\leq N^{\prime}\) can be chosen freely. The multiplication of the omnidirectional RIR \(h_{0}(t)\) with the SH representations of \(\delta[\mathbf{\theta}_{\text{R}}-\mathbf{\theta}_{\text{R}}(t)]\) and \(\delta[\mathbf{\theta}_{\text{S}}-\mathbf{\theta}_{\text{S}}(t)]\) directionally sharpens the measured SRD ARIR, accordingly. However, the implicit assumption of disjointness (there being only a single DOA and DOD per time sample) is not necessarily true in the late diffuse part of the response. As a result, the temporal fluctuations of \(\mathbf{\theta}_{\text{R}}(t)\) and \(\mathbf{\theta}_{\text{S}}(t)\) cause amplitude modulation that potentially corrupt narrow-band spectral properties in \(\tilde{h}_{nm}^{n^{\prime}m^{\prime}}(t)\). A typical result thereof is a mixing of the longer low-frequency reverberation tails towards higher frequencies, causing unnaturally long reverberation there [21, 38], especially as the orders \(n,n^{\prime}\) increase. We propose a scheme for spectral correction which is similar to the one in [38] but adopted for SRD ARIR processing. In theory, the expected temporal energy decay in an ideal (isotropic) diffuse field should be identical for any source and receiver of random-energy-efficiency-normalized directivity such as the spherical harmonics; this must hold also after decomposition into frequency bands. However, less restrictively, even in non-isotropic diffuse fields, the expected energy decay is identical for subsets of source and receiver directivities that are (pseudo-)omnidirectional: Formal derivation in [38] showed that quadratic summation across same-order spherical harmonics is omnidirectional.