# Time-Domain Sound Field Reproduction with Pressure and Particle Velocity Jointly Controlled
## 1 Introduction  
Spatial sound field reproduction aims at faithfully reproducing the sound field within an extended region of space so that listeners inside the region could experience the replication of the original sound field as realistically as possible. Such a system normally uses multiple loudspeakers as secondary sources to control sound radiation and nowadays has been widely used in various audio applications such as in theaters, cinemas, concerts, home entertainment systems, etc. The exploration of spatial sound field reproduction has never stopped, up to now, many approaches have been developed for this technology, including wave field synthesis (WFS), Ambisonics, and least-squares (LS)-based multi-point control [1]. The WFS approach was first proposed by [PERSON], which is based on the Huygens-Fresnel integral to represent a sound field within a bounded region of the space by a continuous distribution of monopole and dipole secondary sources arranged on the boundary of that region [2,3]. For practical implementation, an array of equally spaced loudspeakers are used to approximate the continuous distribution of secondary sources. Reproduction artifacts due to the spatial discretization of the continuously-distributed secondary sources and the finite size of the array were investigated [4]. In WFS, a large number of closely spaced loudspeakers is necessary for accurate sound field reproduction. Another well-known sound field reproduction technique, Ambisonics, was designed based on Huygens principle [5,6]. The system adopts the zero and first order spherical harmonic decomposition of the original sound field into four channels, and from a linear combination of these four channels to derive the loudspeaker driving signals. This low-order system is optimum only at low frequencies and when the listener stays at the central sweet spot. Later, Higher-Order Ambisonics (HOA) based on the higher-order spherical harmonic decomposition of a sound field was developed for high reproduction frequencies and large reproduction regions [7, 8, 9]. A typical Ambisonic or HOA system uses a circular or spherical loudspeaker array geometry. In addition, the spherical harmonic expansionorder increases with the reproduction frequency and radius of the reproduction region, thus in most time HOA also requires densely distributed loudspeakers to match all the spherical harmonics to the given order to avoid spatial aliasing [10]. For arbitrarily placed loudspeakers, a simple approach in sound reproduction is the least-squares (LS) based multi-point control, which uses multiple microphones as matching points to derive the least-square solution as the loudspeaker weights [11]. This approach is a typical inverse problem, and Tikhonov regularization is commonly adopted to obtain the loudspeaker weights with limited energy and to improve the system robustness. With this method, the placement of both loudspeakers and matching microphones greatly affects control accuracy and filter stability [12]. In addition, the acoustic characteristics of loudspeakers affect the reproduction results, and its modeling though measurements has been discussed in previous work [13]. While most existing work focuses on controlling sound pressure only, some recent work started to investigate controlling the particle velocity in sound field reproduction [14]. A joint control of sound pressure and particle velocity has also been proposed for single-zone [15] and multi-zone sound field reproduction [16]. A general finding is that particle velocity assisted sound field reproduction is especially suitable to a non-uniformly spaced loudspeaker array with reduced number of loudspeakers and control points required. The extension to intensity based sound field reproduction has also been investigated [17; 18]. So far, the particle velocity assisted sound field reproduction system has only been developed in the frequency domain. In this work, we propose a time-domain sound field reproduction algorithm with both sound pressure and particle velocity jointly controlled.