# 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 a result, a fluctuation of the DOA \(\mathbf{\theta}(t)\) causes amplitude modulation and destroys narrow-band spectral content in \(\tilde{h}_{nm}(t)\); typically, the longer low-frequency reverberation tails are hereby mixed towards higher frequencies, causing unnaturally long reverberation there [12,36] at high orders, cf. solid lines in Figure 5. However, theoretically, the expected temporal energy decay in an ideal (isotropic) diffuse field must be identical for any receiver of random-energy-efficiency-normalized directivity, such as the spherical harmonics, also after decomposition into frequency bands, and hence requires correction. Despite the mismatch, formal derivation in [22] showed that quadratic summation across same-order spherical harmonics is omnidirectional \(\sum_{m}|Y_{n}^{m}(\mathbf{\theta})|^{2}=\frac{2n+1}{4\pi}\). Hereby, ASDM-upscaled ARIRs at least displays consistent broadband energies \(\sum_{m}\|\tilde{h}_{nm}(t)|^{2}=\frac{2n+1}{4\pi}\|h(t)|^{2}\) across all spherical harmonic orders \(n\), for any sound field. To enforce consistency with spectral squares of \(h(t)\), third-octave filtering is useful, where the \(b\)th sub-band signal \(F_{b}\{h(t)\}\) with center frequency \(f_{b}\) is obtained from a bank of zero-phase filters \(F_{b}\) that is perfectly reconstructing \(h(t)=\sum_{b}F_{b}\{h(t)\}\). Figure 4: Block diagram of binaural rendering with measured Ambsonic RIRs (ARIRs). Here, _MagLS_ refers to a state-of-the art Ambisonic binaural render. For every sub band \(b\) and order \(n\), an energy decay of the ASDM-upscaled ARIR \(F_{b}\{\tilde{h}_{nm}(t)\}\) matching with the original one of \(F_{b}\{h(t)\}\) is enforced by envelope correction  
\[F_{b}\{h_{nm}(t)\}=F_{b}\{\tilde{h}_{nm}(t)\}\ w_{n}^{b}(t), \tag{7}\] \[\text{with}\ w_{n}^{b}(t)=\sqrt{\frac{2n+1}{4\pi}}\sqrt{\frac{F_{ T}\{F_{b}\{h(t)\}^{2}\}}{\sum_{m}F_{T}\{F_{b}\{h_{nm}(t)\}^{2}\}}},\]  
where \(F_{T}\{\cdot\}\) denotes temporal averaging with a time constant \(T\) (e.g., 100 ms). The energy decay reliefs for the initial and corrected result of ASDM are exemplary shown for a third octave \(f_{b}=2\) kHz and within the orders \(n=\{1,3,5,7\}\) in Figure 5.  
Finally, the ear signals are obtained by a convolution of the rotated Ambsonic signals with any state-of-the-art FIR binaural Ambsonic renderer, cf. Figure 4. Our study employs the time-invariant filters of the MagLS (magnitude-least-squares) renderer (The MagLS renderer is part of the IEM plugin suite which can be found here [[https://plugins.iem.at/](https://plugins.iem.at/)]([https://plugins.iem.at/](https://plugins.iem.at/))) defined in [24; 37] to get high-quality Ambsonic rendering already with an order \(N=3\). The filters were designed using a magnitude-least-squares optimization that disregards phase match in favor of an improved HRTF magnitude at high frequencies, and hereby avoids spectral artifacts. MagLS also includes an interaural covariance correction that offers an optimal compromise to render diffuse fields consistently [23].