# A Measure Based on Beamforming Power for Evaluation of Sound Field Reproduction Performance
## 5 Discussion  
A time window was used in the experiment to exclude the late reflections based on the concept of mixing time. In general, as the window becomes shorter, both the spatial and beam-power error becomes smaller because the direct sound becomes more predominant. Controlling such a window length can be useful to check whether or not the sound field is close to a free-field condition. Early reflections were included in this study, which occasionally shifted the peak in the beam-power. This might be related to the fact that early reflections of 5-10 ms can shift the perceived source position in human listening [35; 49]. This relation needs to be further studied, but it is beyond the scope of the present study.  
As shown in Figure 4, the beam-power distribution shows the difference between the desired and the reproduced fields with respect to the assumed direction. The shift of the propagating direction of the direct sound is shown in the main-lobe, and the effect of excessive reflections is shown mainly in the side lobes. This means that the beam-power distribution gives more information than the spatial distribution does.  
In addition, as shown in Figure 6, if there are no reflections, it is expected that the beam-power error can detect the difference in the main lobe as effectively as the spatial error. Thus, for reproduced sound fields by simple panning methods, such as VBAP (vector-based amplitude panning) [50], the beam-power error is expected to be as large as the spatial error. On the other hand, because of the directional filter, the difference in the side lobes can be underestimated compared with the spatial error.  
Recently, another measure called planarity has been proposed for evaluation of the reproduced fields [51]. The planarity quantifies the similarity to a plane wave sound field using the ratio between the intensity component in the direction of the plane wave and the total energy flux by the beamforming technique. Although the planarity and the proposed beam-power error use the beamforming method in common, the main difference is that the planarity does not take the shape of the main lobe into consideration because the planarity is calculated as the weighted sum of the beam-power. This makes the planarity insensitive to the difference in the wavefronts. Figure 10 shows the planarity for the case considered in Figure 6. The planarity decreases as the angle difference increases as expected, but the change is not significant. For example, the spatial error and the beam-power error increases by around 15 dB for the shift of five degrees, while the planarity decreases from 0.970 to 0.967. Consequently, it is concluded that the beam-power error is more effective to detect the difference in wave fronts.  
The experimental example used a specific reproduction method (2.5D HOA) in a specific listening room, and sound fields in the controllable region on the horizontal plane, which is frequency dependent (\(kr<N\)), were used to obtain the beam-power distribution. However, this measure can be used in other  
Figure 10: Planarity with respect to the \(\Delta\)\(\phi\) in the same condition of Figure 6.  
circumstances. For example, a frequency-independent spherical region that just includes the listener's head can be chosen, because the sound field in this region is directly related to the sound pressure that the listener would have at two ears. If 3D reproduction methods are used, and the elevation angle of the virtual source is also of concern, the beamformer output can be extended to the two-dimensional case that the beamformer output is expressed as \(B(\theta_{c},\phi_{c})\).  
The present study is limited to the physical evaluation of sound fields. Subjective tests are needed to prove that the proposed measure has good agreements with human localization in future works.