## 2 Materials and Methods
### FID Decomposition  
The first step to generate basis sets of data-driven modeling is decomposing the FID signal while using truncated Hankel singular value decomposition (HSVD), which decomposes the FID signal into time domain components [8; 28]. The decomposition step is performed after the removal of the water peak with a time-domain FIR filter. The FID signal is described by four parameters, as depicted in Equation (2), below.  
\[z_{n}=x_{n}+jy_{n}=\sum\
olimits_{k=1}^{K}a_{k}e^{(-d_{k}+j2\pi f_{k})t_{n}+j \phi_{k}}, \tag{2}\]  
where parameters of the FID signal components are: \(a_{k}\)--amplitude of a single component, \(d_{k}\)--damping factor of a single component, \(f_{k}\)--frequency of a single component, \(t_{n}\)--sampling time, \(\phi_{k}\)--phase of single component, and \(j\)--is imaginary unit.  
HSVD starts with arranging data in form of an \(L\times M\) Hankel matrix.  
\[S_{H}=\left[\begin{array}{ccc}Z_{0}&\cdots&Z_{M-1}\\ \vdots&\ddots&\vdots\\ Z_{L-1}&\cdots&Z_{N-1}\end{array}\right]. \tag{3}\]  
\(L\) and \(M\) must be chosen greater than the number of expected damped sinusoids, \(K\). The \(L\) and \(M\) summation must be equal to \(N+1\), where \(N\) is number of datapoints. It is recommended to pick \(L\) and \(M\) in a way that makes \(S_{H}\) matrix as square as possible [8].  
Subsequently, the Hankel matrix \(S_{H}\) is decomposed into a product of three matrices by the application of SVD.  
\[S_{H}=U_{L\times L}\Sigma_{L\times M}V_{M\times M}, \tag{4}\]  
where \(U\) and \(V\) are unitary matrixes and \(\Sigma\) is a diagonal matrix, whose main diagonal elements are the singular values of \(S_{H}\). The next step is truncating \(S_{H}\) into \(S_{K}\), where \(K\) is the number of sinusoids, which is assumed to be necessary to model the FID signal, and it corresponds to the number of rows of matrix \(U_{K}\) and the number of columns of matrix \(V_{K}\).  
\[S_{K}=U_{K}\Sigma_{K}V_{K}. \tag{5}\]  
To estimate eigenvalues of the state matrix \(E^{H}\) corresponding to the model of (1), Equation (6), below, is formed. In (6), \(V_{K}^{(t)}\), and \(V_{K}^{(t)}\), matrices from \(V_{K}\), by omitting the first and the last row, respectively.  
\[V_{K}^{(t)}E^{H}=V_{K}^{(b)}. \tag{6}\]When the Equation (6) is solved in the least squares sense, the \(K\) eigenvalues of \(E^{H}\) lead to estimates of the damping coefficients \(d_{k}\) and frequencies \(f_{k}\)[28].  
\[\hat{z}_{n}=e^{(-\hat{d}_{k}+j/2\pi\hat{f}_{k})t_{n}} \tag{7}\]  
Some of the eigenvalues are positive (driving coefficients) and others are negative (damping coefficients). Therefore, the eigenvectors that are associated with negative eigenvalues (damping coefficients) must be kept. In next step, estimates \(z_{k}\) can be filled in model equation and by the least squares fit of the model (2) to the measure NMR signal. The amplitude \(a_{k}\) and phase \(\phi_{k}\) can be estimated:  
\[\hat{c_{k}}= \hat{a_{k}}\ e^{i\hat{\phi_{k}}} \tag{8}\]  
By substituting (8) in (2), we have:  
\[z_{n}=\sum_{k=1}^{K}c_{k}z_{k}^{n} \tag{9}\]  
In Figure 2, we show MR spectrum of an in vivo data as an example of decomposing a FID signal while using truncated HSVD.