# Binaural Rendering with Measured Room Responses: First-Order Ambisonic Microphone vs. Dummy Head
## 3 Experiment II: Dummy-Head MOBRIR vs. ARIR
### Rendering with Measured Ambsonic RIR and the Ambsonic Spatial Decomposition Method (ASDM)  
As depicted in Figure 4, dynamic binaural rendering from room responses measured in Ambisonics (ARIRs) is modular and consists of three blocks: (i) a multi-channel convolution of the source signal with an order \(N\) upscaled ARIR, (ii) an efficient rotation [32; 33] corresponding to the head orientation of the listener, (iii) and multi-channel convolution with an Ambsonic binaural renderer. For efficient and low-effort measurements of the ARIR, we use a compact first-order tetrahedral spherical microphone array and denote the discrete-time B-format ARIRs \(h(t)\), \(x(t)\), \(y(t)\), \(z(t)\) as the responses of a Soundfield ST450 array. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Method** & ref & \(8^{\circ}/4\)k & \(15^{\circ}/2\)k & \(30^{\circ}/2\)k & \(60^{\circ}/1\)k & \(8^{\circ}\)lin & \(15^{\circ}\)lin & \(30^{\circ}\)lin & \(60^{\circ}\)lin \\ \hline ref & - & **2.19** & **1.69** & **0.90** & 0.02 & 0.00 & 0.00 & 0.00 & 0.00 \\ \hline \(8^{\circ}/4\)k & **1.00** & - & **1.78** & **1.69** & **0.15** & 0.0 & 0.00 & 0.00 & 0.00 \\ \hline \(15^{\circ}/2\)k & **1.06** & **1.70** & - & **0.44** & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 \\ \hline \(30^{\circ}/2\)k & **0.86** & **0.82** & **1.33** & - & 0.02 & 0.00 & 0.00 & 0.00 & 0.00 \\ \hline \(60^{\circ}/1\)k & **0.09** & **0.15** & **0.90** & **0.86** & - & 0.00 & 0.00 & 0.00 & 0.00 \\ \hline \(8^{\circ}\)lin & 0.01 & 0.03 & 0.02 & **0.09** & **0.83** & - & **0.16** & **1.38** & **0.39** \\ \hline \(15^{\circ}\)lin & 0.01 & 0.00 & 0.02 & 0.03 & **0.09** & **0.90** & **-** & **2.19** & 0.00 \\ \hline \(30^{\circ}\)lin & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.02 & **0.88** & - & 0.00 \\ \hline \(60^{\circ}\)lin & 0.00 & 0.00 & 0.01 & 0.00 & 0.00 & **1.06** & **1.70** & **0.33** & - \\ \hline \hline \end{tabular}
\end{table} 
Table 2: \(p\)-values (Wilcoxon signed-rank test with Bonferroni–Holm correction) for ratings of continuity in Experiment I. The upper triangle corresponds to the pink noise, the lower triangle to music as source signal. Insignificant differences (\(p\)-values \(\geq\) 0.05) are indicated by bold numbers. Similar to the spatial decomposition method SDM [34], the Ambsonic SDM (ASDM) assigns a direction of arrival (DOA) \(\mathbf{\theta}(t)\) to each discrete-time sample \(t\) of the omni-directional RIR component \(h(t)\), cf. [22]. For the DOA estimation, we suggest using the pseudo-intensity vector (PIV) [35] for the frequencies between \(200\,\mathrm{Hz}\) and \(3\,\mathrm{kHz}\). Here, the upper frequency limit is chosen below the spatial aliasing frequency \(f_{a}=\frac{c}{2\pi r_{ST450}}\approx 3.6\,\mathrm{kHz}\) for \(r_{ST450}=1.5\,\mathrm{cm}\) and the low cut minimizes low-frequency disturbance in the DOA estimation. We perform a zero-phase band limitation (e.g., by MATLAB's filtfilt) denoted by \(F_{200-3\mathrm{k}}\) and a zero-phase temporal smoothing \(F_{L}\) of the resulting PIV using a moving-average Hann window in the interval \([-L/2;L/2]\) for \(L=16\) to get the DOA estimate  
\[\mathbf{\theta}(t) =\frac{\mathbf{\tilde{\theta}}(t)}{\|\mathbf{\tilde{\theta}}(t)\|},\quad \text{with} \tag{5}\] \[\mathbf{\tilde{\theta}}(t) =F_{L}\left\{F_{200-3\mathrm{k}}\{h(t)\}\ F_{200-3\mathrm{k}} \left\{ \begin{bmatrix}x(t)\\ y(t)\\ z(t)\end{bmatrix} \right\}\right\}\]  
as Cartesian unit vector \(\mathbf{\theta}(t)\). In a first step, the ASDM-upscaled ARIR re-encodes every time sample at the detected DOA  
\[\tilde{h}_{nm}(t)=Y_{n}^{m}[\mathbf{\theta}(t)]\,h(t), \tag{6}\]  
where \(Y_{n}^{m}(\mathbf{\theta})\) are the N3D-normalized, real-valued spherical harmonics of order \(n\) and degree \(m\), cf. [23], evaluated at the direction \(\mathbf{\theta}\), and the maximum order \(n\leq N\) can be chosen freely. In the late diffuse part of the response, the implicit assumption of there being only a single DOA per time sample does not hold.