## 2 Theoretical Basis
### The Sound Prediction Based on ARM  
From Equation (7), it could be seen that, when calculating the equivalent source strength in the ESM, the measurement points should not be less than the equivalent sources. Considering the engineering cost, it is important to reduce the quantity of measurement points for the sound prediction. As the ARM has been proven to have good convergence with the increase of ARM order, it offers possibility to reduce the scale of the measurement system [31]. Based on the theory of ARM, two methods could be designed to apply for the sound prediction. One is combined with the theory of compressed sensing (CS), which is called the CMESM [28]. The other is combined with the reconstruction of structural vibration, which is called the VR-ESM for convenience. Both methods are modified with the SFS to construct the ARMs more accurately when the influences of reflection sounds on the structural vibration could not be neglected. Firstly, the theory of ARM should be introduced briefly. ARM explain the sound radiation of a structure from the perspective of sound power \(\mathbf{W}\). In the work [38], [PERSON] gave out the relationship between the sound power and the equivalent source strengths:  
\[\mathbf{W}=\mathbf{Q}^{H}\mathbf{R}\mathbf{Q} \tag{14}\]  
where \(\mathbf{R}\) is the power resistance matrix and the expression of \(\mathbf{R}\) in free field is as follows:  
\[\mathbf{R}=\left[ \begin{array}{cccc}1&\text{sinc}(kr_{12})&\dots&\text{sinc}(kr_ {1N})\\ \text{sinc}(kr_{21})&1&\dots&\text{sinc}(kr_{2N})\\ \dots&\dots&\dots&\dots\\ \text{sinc}(kr_{N1})&\text{sinc}(kr_{N2})&\dots&1\end{array} \right] \tag{15}\]  
where \(\text{sinc}(. )\) is the sinc function and \(\text{sinc}(\alpha)=\sin\alpha/\alpha\), \(r_{ij}\) is the distance between the ith and the jth equivalent sources. Obviously, \(\mathbf{R}\) is a real symmetric matrix. Using the eigenvalue decomposition, \(\mathbf{R}_{0}\) could be expressed as follows:  
\[\mathbf{R}_{0}=\mathbf{T}^{H}\mathbf{\Lambda}\mathbf{T} \tag{16}\]  
where \(\mathbf{\Lambda}\) is a real diagonal matrix of the eigenvalues that are listed in descending order, and \(\mathbf{T}\) is an orthogonal matrix composed of eigenvectors that contains an orthogonal basis of \(N\)-dimensional space. The column vectors of \(\mathbf{T}\) are regarded as the equivalent source ARMs. [PERSON] proposed the CMESM to reduce the scale of the measurement system on the holographic surface [28]. The usage of ARM in the theory of CMESM enhances the sparsity of equivalent source strengths and makes the sparsity of measurement points possible. Still, it could break through the limit of Nyquist sampling law. CMESM is able to reconstruct the sound field generated by a sound source with arbitrary shape and has less limitation to the sparsity of true source distribution. The power resistance \(\mathbf{R}\) could be different for the half-space from the free field when the effect of reflection sounds could not be neglected. As \(\mathbf{R}\) derives from the interactions among the equivalent sources [25], it could fall to three categories: the power resistance related to the equivalent sources inside the real structure \(\mathbf{R}_{0}\), the power resistance related to the image equivalent sources inside the virtual structure \(\mathbf{R}_{i}\), and the mutual power resistance related to the interactions among the equivalent sources and their mirrors \(\mathbf{R}_{m}\). Thus, the total radiated sound power \(\mathbf{W}\) consists of four components:  
\[\left\{ \begin{array}{l}\mathbf{W}_{0}=(\mathbf{q}_{0}+\mathbf{q}_{s})^{H}\mathbf{R}_{0}(\bm {q}_{0}+\mathbf{q}_{s})\\ \mathbf{W}_{i}=\mathbf{q}_{R}^{H}\mathbf{R}_{i}\mathbf{q}_{R}\\ \mathbf{W}_{m1}=(\mathbf{q}_{0}+\mathbf{q}_{s})^{H}\mathbf{R}_{m}\mathbf{q}_{R}\\ \mathbf{W}_{m2}=\mathbf{q}_{R}^{H}\mathbf{R}_{m}(\mathbf{q}_{0}+\mathbf{q}_{s})\end{array} \right.