## 2 The Parallel Dual-Q-Factors Bases Sparse Decomposition  
The resonance property of a signal is defined by the quality factor Q, the high Q-factor base represents the continuous oscillation component, and the low Q-factor base represents the transient oscillation component. Figure 1 shows the concept of resonance properties of signals and the time-domain waveform of the signal. In Figure 1, when Q = 1, it shows the single-period pulse signal, and they are defined as the low resonance signals because of the small quality factor Q. When Q = 3 it shows the multi-period pulse signals, and they are defined as the high resonance signals because of the large quality factor Q.  
The RSSD method is actually a sparse representation using the high and the low tunable Q-factor wavelet transform (TQWT) [35; 38]. The TQWT is fully discrete, has the perfect reconstruction property, is modestly overcomplete, and is based on the two-channel filter bank illustrated in Figure 2[39].  
Figure 1: The time-domain diagram of low Q-factor and high Q-factor.  
Figure 2: Analysis and synthesis filter banks for the dual-Q wavelet transform.  
In Figure 2, \(x(n)\) is the original signal, \(H_{0}(w)\), \(H_{1}(w)\) and \(H^{*}_{0}(w)\), \(H^{*}_{1}(w)\) are the frequency response functions of the analysis and synthesis filters respectively; \(v_{0}(n)\), \(v_{1}(n)\) are the sub-band signals obtained after decomposition. \(y_{0}(n)\), \(y_{1}(n)\) are the synthetic signals. The scaling parameters are given by:  
\[\beta=\frac{2}{Q+1}\alpha=1-\frac{\beta}{r} \tag{1}\]  
where, \(\beta\) is the high-pass factor, \(\alpha\) is the low-pass factor, and \(r\) represents the redundancy.  
Suppose that the observed signal x can be represented as:  
\[x=x_{1}+x_{2}x,x_{1},x_{2}\in R^{N} \tag{2}\]  
where \(x_{1}\) and \(x_{2}\) are components with different oscillation behavior.  
The signal sparse decomposition method based on the compound Q-factor basis uses the morphological component analysis (MCA) method to separate the components nonlinearly in the signal according to the oscillation characteristics [40; 41] and establishes the optimal sparse representation of the high resonance component and the low resonance component. The goal of MCA is to estimate \(x_{1}\) and \(x_{2}\) individually. Assuming that the signals \(x_{1}\) and \(x_{2}\) can be sparsely represented in base functions (or frames) \(S_{1}\) and \(S_{2}\) respectively, (obtained by TQWT), they can be estimated by minimizing the objective function [42]:  
\[J(w_{1},w_{2})=\|x-S_{1}W_{1}-S_{2}W_{2}\|_{2}^{2}+\lambda_{1}\|W_{1}\|_{1}+ \lambda_{2}\|W_{2}\|_{1} \tag{3}\]  
where  
\(W_{1}\)--the transform coefficient of signal \(x_{1}\) under frame \(S_{1}\);  
\(W_{2}\)--the transform coefficient of signal \(x_{2}\) under frame \(S_{2}\);  
\(S_{1}\)--the filter banks of tunable-Q wavelet with the high quality factor;  
\(S_{2}\)--the filter banks of tunable-Q wavelet with the low quality factor;  
\(\lambda_{1}\)--weight parameter;  
\(\lambda_{2}\)--weight parameter.  
In the sparse signal decomposition method based on the compound Q-factor bases, the objective function \(J\) is minimized through iterating and updating the coefficient \(W_{1}\) and \(W_{2}\) by using the split augmented Lagrangian searching algorithm (SALSA).  
Eventually, it effectively separates the high resonance component and low resonance component, and extracts the transient impulse component. After \(W_{1}\) and \(W_{2}\) are obtained, the estimated high and low resonance components (solved by matching) are shown as follows:  
\[\dot{x_{1}}=S_{1}W_{1}^{*},\ \dot{x_{2}}=S_{2}W_{2}^{*}. \tag{4}\]