# Binaural Rendering with Measured Room Responses: First-Order Ambisonic Microphone vs. Dummy Head
## 2 Experiment I: Dynamic Rendering of Multiple-Orientation Binaural Signals
### Linear Interpolation with Switched High-Frequency Phase (LISHPh)  
For both the left and right ear, the interpolated ear signal in a horizontal set of orientations is obtained by a combination of the corresponding signals \(x_{q}(t)\) and \(x_{q+1}(t)\) belonging to the head orientation closest to the current orientation of the listener \(\varphi(t)\), where \(t\) is the discrete-time index. With MOBRIRs measured for \(Q\) equi-angular orientations on the horizon (around the Cartesian \(z\)-axis), and \(\Delta\varphi\) as azimuthal resolution, the indices of the two closest BRIRs (or recorded ear signals) are \(q=\left\lfloor\frac{\varphi(t)}{\Delta\varphi}\right\rfloor\), and \(q+1=\left\lceil\frac{\varphi(t)}{\Delta\varphi}\right\rceil\), where \(\left\lfloor\cdot\right\rfloor\) and \(\left\lceil\cdot\right\rceil\) are the floor and ceil functions, respectively, and the ear signals \(x_{q}(t)\), and \(x_{q+1}(t)\) are obtained by convolution; see Figure 1(a).  
In a broadband linear interpolation, the resulting ear signal is obtained as  
\[x(t)=(1-\alpha)x_{q}(t)+\alpha x_{q+1}(t), \tag{1}\]  
where the interpolation weight is obtained by \(\alpha=\left\lceil\frac{\varphi(t)}{\Delta\varphi}\right\rceil-\frac{\varphi( t)}{\Delta\varphi}\). However, the linear combination of delayed signals produces comb-filtering introducing severe colorations in the resulting signal. In particular, the maximum delay \(\tau=\frac{\tau}{c}\sin(\Delta\varphi)\) between adjacent HRIRs is estimated by a simplistic head model, where \(r=8.5\,\mathrm{cm}\) is the head radius and \(c=343\,\mathrm{m/s}\) is the speed of sound. The maximum delay is observed between ear signals of the head orientation \(0^{\circ}\) and those of the orientations \(\pm\Delta\varphi\)  
Figure 1: Binaural room impulse response measurement setup.  
for a frontal source. To avoid destructive interference, artifact-free linear interpolation can only be achieved below  
\[f_{max}=\frac{1}{2\tau}=\frac{c}{2r}\frac{1}{\sin(\Delta\phi)}. \tag{2}\]  
Spectral artifacts of linearly interpolated BRIRs are comparable with those of HRIRs when the direct sound dominates. For interpolation of BRIRs from a diffuse field, the same (worst-case) frequency limit for destructive interference holds. In particular, if the contribution of frontal sounds is pronounced, the interpolated result partly contains the destructive interference at \(f_{max}\).  
The LISHPh method is employed [16; 18; 20] to avoid noticeable spectral artifacts around and above \(f_{max}\), regardless of the acoustic scenario. As depicted in Figure 2b, the signal in the lower band is processed in the time domain by applying the linear weights as in Equation (1), and the signal in the high-frequency band is obtained by magnitude interpolation  
\[X(k)=\{(1-\alpha)|X_{q}(k)|+\alpha|X_{q+1}(k)|\}\;\mathrm{e}^{i\angle(k)}, \tag{3}\]  
where \(k\) is the frequency index of a short-time Fourier transform (STFT) frame, \(i^{2}=-1\), and \(\angle(k)\) is the phase argument which is switched between \(\angle(k)=\measuredangle_{q+1}(k)\) for \(\alpha\geq 0.5\) and \(\angle(k)=\measuredangle_{q}(k)\) else. Whenever the phase argument of a narrow-band signals has to make a transition by \(\pi\), switching can theoretically become audible and can only be avoided by spectrogram inversion [17; 18]. We avoided the additional effort, as the negative influence of the switching noise turned out to be inaudible for speech, music recordings, noise, etc. with suitable block size settings [20].