## 2 Theory Background and Methodology
### Compressed ESM Based on the Equivalent Redundant Dictionary
#### 2.2.2 Construction of the Equivalent Redundant Dictionary  
When the actual sound sources are not sparse, it is necessary to construct an appropriate basis to sparsely decompose the sound field. In this paper, the SVD method is applied to determine this basis. By applying the SVD to the transfer matrix \(\mathbf{G}\), Equation (5) can be rewritten as Equation (9) [27; 28]:  
\[\mathbf{P}_{R}=\mathbf{G}\mathbf{q}=\mathbf{U}\mathbf{S}\mathbf{V}^{H}\mathbf{q}=\sum_{i=1}^{N}\mathbf{u}_{i} \lambda_{i}\mathbf{v}_{i}^{H}\mathbf{v}_{i}\mathbf{q}=\sum_{i=1}^{N}\mathbf{u}_{i}\mathbf{\alpha} _{i}=\mathbf{U}\mathbf{\beta} \tag{9}\]  
where \(\mathbf{U}\) and \(\mathbf{V}\) are unitary matrices, and \(\mathbf{u}_{i}\) and \(\mathbf{v}_{i}\) represent column vectors of \(\mathbf{U}\) and \(\mathbf{V}\), respectively. \(H\) denotes the conjugate transpose, \(\mathbf{S}\) is a real diagonal matrix, and \(\lambda_{i}\) denotes singular values. \(\mathbf{\beta}\) is the coefficient vector of the basis. In Equation (9), \(\mathbf{U}\) is an orthogonal matrix of singular vectors and contains a set of basis vectors of \(N\)-dimensional space, which has been well-studied in literature [11; 27]. Similar to the method presented by [PERSON] et al. [23], the basis can also be regarded as acoustic modes [29]. Since the modes are spatially continuous, \(\mathbf{U}\) is especially suitable for the spatially extended sound source. The acoustic signal can be reconstructed well only by a few column vectors of \(\mathbf{U}\), which correspond to the large singular value. This enables the sparsity of the solution to be enhanced when the SVD is utilized, and therefore the ESM in the CS framework can be solved. However, in most practical applications, the characteristics of the source are not clear [30; 31]. It is difficult to choose a suitable basis or dictionary for the sparse representation of source distribution. To solve this problem, a new equivalent redundant dictionary, \(\mathbf{D}=[\mathbf{\phi},\mathbf{U}]\), is constructed by combining Equation (8) and (9). Therefore, the sound pressure on the reconstruction plane can be expressed as Equation (10):  
\[\mathbf{P}_{R}=\mathbf{D}\mathbf{\gamma} \tag{10}\]  
In this equivalent redundant dictionary \(\mathbf{D}\), \(\mathbf{\phi}\) is the equivalent dictionary used in CESM, which is especially suitable for the sources distributed spatially sparse. \(\mathbf{U}\) can be viewed as \"acoustic mode\", which is applicable for the spatially extended source. Therefore, by taking advantage of \(\mathbf{\phi}\) and \(\mathbf{U}\) comprehensively, \(\mathbf{D}\) is expected to be appropriate whether the sources are spatially sparse or spatially extended. According to Equations (3) and (5), the pressure \(p\) can be reformulated as Equation (11):  
\[\mathbf{p}=\mathbf{A}\mathbf{q}=\mathbf{A}( \mathbf{G})^{+}\mathbf{P}_{R} \tag{11}\]  
and substituting Equation (10) to Equation (11) yields Equation (12):  
\[\mathbf{p}=\mathbf{A}(\mathbf{G})^{+}\mathbf{P}_{R}=\mathbf{A}(\mathbf{G})^{+}\mbox{\boldmath $D$}\mathbf{\Upsilon}=\mathbf{D}\mathbf{e}\mathbf{\Upsilon} \tag{12}\]  
where \(\mathbf{D}\mathbf{e}=\mathbf{A}(\mathbf{G$ })^{+}\mbox{\boldmath$D}\), and the superscript \"+\" denotes the Moore-Penrose inverse. Using the \(l_{1}\) norm regularization, Equation (12) can be solved by minimizing the cost function as Equation (13):  
\[\mathbf{\Gamma}_{\lambda}(\mathbf{\Upsilon})=||\mathbf{p}-\mathbf{D}\mathbf{e}\mathbf{\Upsilon }||_{2}^{2}+\lambda||\mathbf{\Upsilon}||_{1} \tag{13}\]  
Equation (13) is equivalent to the \(l_{1}\) norm minimization problem, see Equation (14):  
\[\underset{\mathbf{\Upsilon}}{\mbox{\rm argmin}}||\mathbf{ \Upsilon}||_{1}\ \ \mbox{subject to}\ ||\mathbf{p}-\mathbf{D}\mathbf{e}\mathbf{\Upsilon}||_{2}\leq\varepsilon \tag{14}\]  
where \(\varepsilon\) represents the tolerance.