# Mode-Matching-Based Sound Field Recording and Synthesis with Circular Double-Layer Arrays
## 1 Introduction  
Generating acoustically bright and dark zones is an important and attractive problem for acoustic communication technology, and many approaches are being investigated [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15]. Although these methods only control the acoustic contrast or energy between the bright and dark zones, multizone sound field synthesis methods also simultaneously control not only the sound pressures but also multiple sound fields in multiple regions [16; 17; 18; 19; 20; 21; 22; 23; 24; 25]. Most sound field synthesis approaches assume that the synthesis environment is a free-field and the synthesis target area is inside the loudspeaker array [26; 27; 28; 29; 30]. Figure 1a shows an example of a two-dimensional interior sound field synthesis based on higher-order Ambisonics [29] with a circular loudspeaker array in a free-field. Although a plane wave is synthesized inside the array, undesired sound pressures propagate outside the array. In addition, the actual synthesis environment is not a free-field, and reverberation occurs by undesired sound pressures propagating outside the array. Consequently, reverberation degrades the synthesis accuracy in the target area (Figure 1b). Therefore, considering the sound pressures propagating outside the array is important both to reduce the undesired sound propagation and to maintain the synthesis accuracy in the target area in the actual reverberant conditions. Considering the reverberant conditions, exterior propagation-free sound field synthesis approaches have been proposed [31; 32; 33; 34]. These approaches can synthesize a sound field inside an array without propagating undesired sound pressures outside the array (Figure 1c). As a result, reverberations do not occur, and the synthesis accuracy inside the array remains high in actualreverberant conditions. These methods are categorized as multizone sound field control approaches where the synthesized and acoustically dark zones are simultaneously set inside and outside the array. In these methods, the synthesized sound field is assumed to be a simple plane wave. To extend exterior propagation-free sound field synthesis for reproducing a primary sound field recorded by a microphone array [35], this paper focuses on sound field recording and synthesis methods without exterior propagation using arrays of microphones and loudspeakers. To achieve such a sound field recording and synthesis approach, pressure matching-based least-squares (LS) [36; 37] and generalized singular value decomposition (GSVD) [38] methods [39] are directly introduced. The LS and GSVD methods are based on the Kirchhoff-Helmholtz (KH) integral [40] and introduce double-layer arrays of microphones and loudspeakers. The LS method is, however, based on numerical inversion and requires repeated calculations to select the optimal regularization parameters. In the GSVD approach, iterative calculations are also required for finding the optimal threshold to automatically decide the singular value truncation parameters [40]. In previous works [14; 34; 41; 42], the spatial Fourier transform-based analytical approaches to sound field control without repeated calculations outperformed the conventional LS methods in terms of control accuracy. For more accurate interior sound field synthesis without exterior sound propagation than the conventional LS and GSVD approaches without iterative computations, this paper proposes an analytical approach with continuous circular double-layer source and receiver. The continuous circular double-layer source and receiver are finally discretized to circular double-layer arrays of microphones and loudspeakers. The proposed method is based on cylindrical harmonic expansion, which is a spatial Fourier transformation in cylindrical coordinates that is categorized as a mode-matching scheme.