## 4 Perceived Distance Control in Spatial Audio with Sound-Field-Based Methods
For practical horizontal WFS (the 2.5-dimensional WFS), the mismatched radiation characteristics of loudspeakers cause the distance dependence of the reconstructed sound field to somewhat deviate from that of the target sound field [39]. The sound-field-based methods function on the basis of sound field spatial sampling and reconstruction. Limited by the Shannon-Nyquist spatial sampling theorem, practical sound-field-based methods are only able to reconstruct the target sound field up to a certain frequency. According to this theorem, an \(L\)-order spatial Ambisonics requires \((L+1)^{2}\) loudspeakers at least, and is able to reconstruct a target sound field within a region ofradius \(r_{0}\) and up to a frequency limit of \(f_{max}\)[41]. The upper frequency is approximately related to the radius and order of ambisonics according to the following equation:  
\[f_{max}=\frac{cL}{2\pi r_{0}}, \tag{5}\]  
where \(c\) is the sound speed of 340 m/s. Therefore, within a region with a size of the mean head radius \(r_{0}\) of 0.0875 m, the preceding five-order ambisonics are able to accurately reconstruct the target sound field up to the frequencies of 0.62, 1.25, 1.87, 2.47, and 3.1 kHz, respectively. Accordingly, 4, 9, 16, 25, and 36 loudspeakers at least are required for reproduction, respectively. If an accurate reconstruction of a target sound field is desired within the same region up to a frequency limit \(f_{max}\) of 20 kHz, at least \(L=32\)-order spatial Ambisonics with \((L+1)^{2}=1089\) loudspeakers are required. Such high-order Ambisonics is infeasible in practice. Similarly, for WFS, according to spatial sampling theorem, the interval between two adjacent loudspeakers should not exceed half of the wavelength (the worst case scenario). Considering the size of a practical loudspeaker, the interval between two adjacent loudspeakers is between 0.1 and 0.3 m, at least. Accordingly, the upper frequency limit for accurate sound field reconstruction is between 1.7 and 0.57 kHz. Reconstructing a target sound field up to an upper frequency limit of 20 kHz in WFS is also infeasible. Therefore, practical sound field-based methods are usually only able to reconstruct the target sound field up to an upper frequency limit of 2-3 kHz, or up to 4 kHz, at most, thus being only able to recreate parts of directional localization and distance perception information. This is a major problem with the sound-field-based methods. In this situation, various psychoacoustic principles and means must be incorporated into the sound-field-based methods to create the desired spatial auditory events. Taking advantage of the redundancy of various directional localization information and the dominant role of low-frequency ITD in lateral localization, sound-field-based methods can create perceived virtual sources in different directions, provided that it is able to accurately reconstruct the target sound field up to 1.5-2.0 kHz. For example, theoretical analysis indicated that third-order Ambisonics are able to accurately recreate ITD and its dynamic variation with head turning up to at least 1.5 kHz for a central listening position and, thus, provide appropriate information for directional localization [28, 42]. Virtual source localization experiments also proved that spatial Ambisonics can recreate a virtual source in three-dimensional directions (including vertically). The localization performance improves as the order increases. Third- or forth-order Ambisonics usually exhibits satisfactory directional localization performance [28, 43, 44, 45]. Based on a similar psychoacoustic principle, in practice, WFS is able to recreate a perceived virtual source in different directions [46]. Similarly, the psychoacoustics and redundancy of various distance perception cues may be beneficial to perceived distance control in the sound-field-based methods.