## 4 Perceived Distance Control in Spatial Audio with Sound-Field-Based Methods  
Physically, a sound field in air can be entirely specified by the sound pressure \(p(\mathbf{r},t)\) as a function of the time \(t\) and vector \(\mathbf{r}\) of the receiver position. Sound-field-based methods aim to reconstruct the original (target) sound field or pressure function \(p(\mathbf{r},t)\) as exactly as possible in an appropriate receiver region. The target sound field cannot be accurate for several listeners since the position is crucial for spatial cues. Ambisonics is a typical example of sound-field-based methods, which is based on the principle of the spatial harmonics decomposition and reconstruction of a sound field [31; 35; 36]. In ambisonics, sound fields are decomposed into a linear combination of directional harmonics with different orders (azimuthal Fourier decomposition for horizontal sound fields and directional spherical harmonics functions decomposition for spatial sound fields), and the loudspeaker signals are derived by matching the reconstructed sound field with each order approximation of the target field in the directional harmonics domain. A sufficiently high-order ambisonics (HOA) is theoretically able to be used to reconstruct the free-field plane wave caused by a far-field source within a local receiver region. This is a significant fact. According to the spatial Fourier theorem, an arbitrary sound field within a source-free region can be linearly decomposed into plane waves incident from various directions. Therefore, a sufficiently high-order Ambisonics is theoretically able to be used to reconstruct an arbitrary sound field within a local receiver region, including the early reflections and the late diffused reverberation sound field of a target room or environment, so as to exactly recreate the spatial perception information related to reflections, including source distance perception information. Moreover, by matching the reconstructed sound field with each order approximation of a target spherical wave, a near-field compensated higher-order Ambisonics (NFC-HOA) is theoretically able reconstruct the curve wavefront caused by a virtual point source within a local receiver region and thereby recreate the distance perception cues of a proximal source in a free field [37]. In practice, the Ambisonics signals, which include directional localization and distance perception information, can be obtained by simulation or recording with circular or spherical microphone arrays. Wave field synthesis (WFS) is another example of sound-field-based methods [38; 39]. According to the Huygens-Fresnel principle and Kirchhoff-Helmholtz boundary integral equation, the frequency-dependent pressure \(P(\mathbf{r},f)\) in an arbitrary, closed, and source-free space is determined by the pressure and its normal derivative on the surface. Accordingly, the sound field within the boundary surface can be theoretically reconstructed by an array of dipole and monopole loudspeakers arranged on the boundary surface. The signals of dipole and monopole loudspeakers are directly proportional to the target pressure and its normal derivative on the boundary surface, respectively. In certain situations, the target sound field can be reconstructed by a simplified array including only one type of loudspeaker (such as a monopole loudspeaker). WFS is theoretically able to reconstruct an arbitrary target sound field within an extended region enclosed by the loudspeaker array. By using a time-reversal technique, WFS is able to create the sound field of a focused virtual source that is located at a proximal distance within the loudspeaker array [40]. Therefore, similar to Ambisonics, WFS is theoretically able to be use to accurately create various directional localization and distance perception cues, including distance perception cues related to direct and environmental-reflected sound.