# Sparse Plane Wave Approximation of Acoustic Modes to Address Basis Mismatch
## 1 Introduction  
Recently, the reconstruction of a low-frequency sound field in an enclosed space has generated interest among researchers. It is significant for many applications, including sound field auralization [1], room equalization [2], sound field control [3; 4; 5], etc. The reconstruction problem usually involves interpolating the impulse responses at unmeasured positions from the impulse responses measured by a discrete set of microphones. The interpolation methodology can avoid the extensive experimental effort caused by dense point-by-point measurements. However, according to the study on the _plenacoustic function_[6], the Nyquist sampling theorem must be applied, requiring multiple microphone-based measurements for higher cut-off frequency. This contrasts with the goal of employing the minimum number of microphones to recover the sampled sound field faithfully. The framework of _compressive sensing_ (CS) [7; 8] compensates for the lack of observation information by exploiting prior knowledge, so the inherent sparsity of a low-frequency enclosed sound field can help reduce the number of microphones. Under this framework, most methods formulate a physical propagation model and expand the limited set of measurements into finite-order _elementary waves_[9]. Commonly used elementary wave (or propagation model) includes the spherical harmonic function [10; 11], plane wave (PW) [12; 13; 14; 15; 16; 17], and free-field Green function [13]. The PW approximation is widely used because [PERSON]'s theory [18] shows that any homogeneous sound field in an enclosed convex region can be approximated by a finite number of PWs, independent of boundary conditions [19]. The current PW approximation methods can be classified into two types. The frequency-domain method [11; 13; 14; 15; 16] approximates the steady-state sound field by a finite sum of PWs for each frequency and uses the \(\ell_{1}\)-norm relaxation to solve the PW complex amplitudes (PWAs). In previous studies working with the frequency domain, different sparsity promoting regularizations were compared [13], and a two-step \(\ell_{1}\)-norm minimization method was developed by expressing PWs in the spherical harmonic domain to address the high mutual coherence of the sensing matrix when PWs are densely discretized [14]. The modal structure was incorporated into the PW model, and the proposed method could predict frequency response functions between the unmeasured source-receiver pairs [16]. The time-domain method [12; 17] decomposes the homogeneous wave equation into the superposition of damped harmonic eigen-PWs. It first uses the matching pursuit or the rational fraction polynomials global curve fitting method to identify the acoustic modes, and then conducts the sparse PW approximation for each mode. Compared with the sound field in the frequency domain, the eigen-PWs of a single mode have substantially better sparsity, especially in rectangular enclosures. Therefore, this study focuses on the time-domain method's PW approximation of acoustic modes. However, the PW approximation has an inherent drawback in that the spatially continuous PW directions must be discretized, and regardless of how finely the directions are discretized, there is always a mismatch between the pre-discretized direction and the actual direction, which is referred to as the _basis mismatch_[20]. This will inevitably reduce the approximation accuracy. A dictionary or sensing matrix comprising dense PW directions is required to address this issue. This produces a high mutual coherence for the sensing matrix and has the disadvantage of increased computational time. A standard method used to tackle the basis mismatch in acoustics is to adaptively optimize pre-discretized grids based on parametric dictionary learning. For inhomogeneous sound field recording, a grid-pruning-based Newton's method was utilized to optimize monopole positions [21]. Joint sparse recovery and a parametric dictionary learning problem were formulated based on the Bayesian framework for source localization [22].