# Head-Related Transfer Functions for Dynamic Listeners in Virtual Reality
## 2 HRTF Sets and Analysis
### Objective Analysis  
The objective analysis used two different auditory models, both implemented in the Auditory Modeling Toolbox [31], to gain objective metrics on the deviations in azimuth and elevation localization cues between the measured, modeled, and generic HRTF sets. We used a binaural azimuth estimation model [32] to obtain interaural time and level difference (ITD; ILD) estimates for each participant (\(N=12\)) and the KU100 generic binaural head. The equivalent rectangular bandwidth (ERB) center frequency was set to 1000 Hz for ITD modeling and 4300 Hz for ILD modeling. Figure 1 displays the auditory-model-based estimates for the ITD (upper panel) and ILD (lower panel). Focusing on the maximum ITDs, the \(\phi=90^{\circ}\) azimuth horizontal plane mean ITD for the set of measured HRTFs is \(723\)\(\upmu\)s (SD = \(27\)\(\upmu\)s) and for the modeled HRTFs \(669\)\(\upmu\)s (SD = \(26\)\(\upmu\)s). For the KU100 generic head, the modeled ITD at 1 kHz is \(786\)\(\upmu\)s. The difference in mean ITDs between the individual sets and compared to the generic set are above the just noticeable difference value of \(20\)\(\upmu\)s found in previous literature [33]. Additionally, a paired sample Wilcoxon signed rank test showed that there is a statistically significant difference between the ITD populations (\(p=0.002\)) and one-sample Wilcoxon signed rank tests showed significant median differences compared to the generic head ITD (measured \(p<0.001\), modeled \(p<0.001\)). The deviations in ILD are more difficult to summarize. Still, a consistent ILD discrepancy can be observed in the azimuth range \(\phi=\{30,...,70\}^{\circ}\), where the measured and generic HRTFs yield larger ILD values compared to the modeled HRTFs. Further to the side the ILD predictions become more varied between the HRTF sets, before converging to similar values beyond \(\phi=160^{\circ}\) behind the head. For localization estimates in elevation, we used a sagittal-plane sound localization model [34]. This model assumes the human brain holds an internal _template_ set of learned HRTFs, which can be compared with an incoming sound filtered with a _target_ HRTF that can either be the person's own HRTF or a generic HRTF with some degree of deviation from the template. The model predicts the percentage of quadrant errors (\(\theta>90^{\circ}\)) and the root-mean-square (RMS) local elevation errors in degrees (\(\theta\leq 90^{\circ}\)) based on a comparison  
Figure 1: ITD in the ERB band centered at 1.0 kHz (bandwidth 133 Hz) and the ILD in the ERB band centered at 4.3 kHz (bandwidth 489 Hz) based on an auditory model by [PERSON] et al. [32]. The boxplots show the sample median and the first and third quartiles, any data within 1.5 times the interquartile range, and any outliers outside the range. Azimuth \(\phi=0^{\circ}\) corresponds to the direction in front of the listener and positive angles are clockwise towards behind the head. process between the target representation and template HRTFs. Here, the prediction was conducted for the elevation angles from \(-70^{\circ}\) to \(+250^{\circ}\) in \(2.5^{\circ}\) increments in the \(0^{\circ}\) azimuth sagittal plane. Table 1 displays the prediction results averaged over all elevation angles for the cases: (I) template and target are the modeled HRTFs; (II) template and target are the measured HRTFs; (III) template are the measured HRTFs, target the modeled HRTFs; (IV) template are the modeled HRTFs, target generic HRTFs; and (V) template are the measured HRTFs, target generic HRTFs. The results are further depicted graphically in Figure 2. The RMS local elevation errors for matching template and target HRTFs are significantly lower than the RMS elevation error averages when combining measured templates with modeled target HRTFs. Similar large elevation error values are observed when combining generic target HRTFs with the modeled or measured HRTF templates.