# A Review of Sound Field Control
## 2 Sound Field Reproduction
### Typical Techniques for Sound Field Reproduction  
Sound field reproduction includes binaural sound field reproduction and large region sound field reproduction. Binaural sound field reproduction is a virtual sound field reproduction method based on the perception characteristics of human ears, which does not generate the original sound field in the spatial region. The best listening area of this technology is generally small. Sound field reproduction over a large-scale region aims to provide better listening experience to more listeners. There exist a number of techniques for the reproduction: pressure matching (PM), higher order Ambisonics (HOAs), and wave field synthesis (WFS). The PM method based on the least-square (LS) principle aims to minimize the sum of squared errors between the desired and reproduced sound pressure at the control points [9]. The conventional PM method usually activates all the candidate loudspeakers to minimize the reproduction error. However, in many cases, the desired sound field can be accurately reproduced using only a few loudspeakers. The least absolute shrinkage and selection operator (Lasso) can be adopted to limit the number of active loudspeakers and optimize the positions of the loudspeakers. The two-stage Lasso-LS optimization further reduces the reconstruction error [23]. The first stage using Lasso can effectively reduce the number of activated speakers, and then the reproduction error is reduced by the second stage using LS. However, these algorithms are designed in the frequency domain, which means that the loudspeaker weights are independently optimized at each control frequency. The PM method in the time domain combined with the framework of the group Lasso can optimize the number and position distribution of loudspeakers at all control frequencies simultaneously [24]. In recent years, some studies introduce the velocity matching [25; 26]. When the loudspeaker array is non-uniformly spaced, the velocity matching method can improve the accuracy of the reproduced sound pressure and intensity direction. The HOA approach expands the sound field in orthogonal modes and minimizes the modal coefficients error between the original sound field and the reproduced sound field. It was first proposed by [PERSON] in 1973 [27] based on first-order spherical harmonic decomposition, which uses four loudspeakers located at the vertices of a regular tetrahedron. This approach is only applicable to small-scale sound field reproduction. The HOA based on higher order spherical harmonic decomposition was then proposed to enlarge the reproduction area [7; 28; 29]. [PERSON] and [PERSON] [8] analyzed the reproduction error based on the HOA method and presented the quantitative relationship between the radius and frequency of the reproduction area, the order of the spherical harmonic decomposition, and the number of loudspeakers required. The performance of the HOA reproduction in terms of the size of reproduction area and the upper frequency limit improves with the increased order of the spherical harmonic decomposition. An \(N\)th-order 3D Ambisonics requires at least \((N+1)^{2}\) loudspeakers. The concept of WFS was introduced by [PERSON] in 1993 [5] based on the Kirchhoff-Helmholtz integral. The sound field in a bounded and source-free region is determined by the pressure and normal velocity on the boundary of the region. This boundary condition is generated by a continuous distribution of secondary sources. Then the transformation from continuous sources to discrete loudspeakers leads to spatial aliasing effects, which were analyzed in [30]. Early WFS research focused on the sound field reproduction in a 2D plane using linear and planar loudspeaker arrays [31; 32]. The 2.5D WFS was proposed to synthesize the field of 3D sources using 2D array configurations by deriving the 2.5D operator [33; 34]. Local Sound Field Synthesis (LSFS) features a more accurate reproduction in a small region. The region is smaller than the area surrounded by the loudspeaker distribution. [PERSON] et al.