## 2 Materials and Methods
### Discrimination of Aortic and Pulmonary Component  
**Model Assumption:** It is assumed that the S2 is the sum of aortic and pulmonary components with the following model  
\[S_{2}(t,\theta)=a(t)+p(t-u(\theta))+v(t,\theta) \tag{1}\]  
where \(a(t)\) is the aortic component, and \(p(t-u(\theta))\) is the pulmonary component at the respiratory phase \(\theta\). \(u(\theta)\) is the time delay modulated by respiration. \(v(t,\theta)\) is random noise at the respiratory phase \(\theta\) to contaminate the second heart sound. The purpose of this article is to measure the respiratory split, \(u(\theta)\). In detail, we have further assumptions based on the mentioned observation above.  
1. \(a(t)\) is assumed to have a fixed waveform over the respiratory phase, as can been seen in Figure 6.
2. \(p(t-u(\theta))\) is the time shifted version of \(p(t)\). It is a delayed pulmonary component caused by respiration.
3. \(v(t,\theta)\) is assumed to be zero mean both over time and over respiratory phase, i.e., \(\int v(t,\theta)dt=0\) and \(\int v(t,\theta)d\theta=0\). It may be colored, non-Gaussian and non-stationary. An average operation over respiratory phase is implemented on Equation (1) \[\frac{\sum\limits_{k=1}^{K}S_{2}(t,\theta_{k})}{K}=\frac{\sum\limits_{k=1}^{K} \left[a(t)+p(t-u(\theta_{k}))+v(t,\theta_{k})\right]}{K}\] (2) where \(K\) is the number of S2s involved in operation. Based on the assumption that \(v(t,\theta)\) is zero mean, the average below produces a zero vector. \[\frac{\sum\limits_{k=1}^{K}\left[v(t,\theta_{k})\right]}{K}=\textbf{0}(t)\] (3)  
Figure 6: The second heart sounds from twelve subjects are demonstrated in joint time and respiratory phase domain. (**a–I**) subjects from No. 1 to No. 12.  
Therefore, Equation (2) becomes  
\[\frac{\sum\limits_{k=1}^{K}S_{2}(t,\theta_{k})}{K}=a(t)+\frac{\sum\limits_{k=1}^{ K}\left[p(t-u(\theta_{k}))\right]}{K} \tag{4}\]  
The average operation to the time shifted pulmonary components smooths the peaks and valleys. So, the average yields a small signal. To show the average operation clearly, an example is given in Figure 7. The S2 signals (blue lines) are over plotted. The solid bold red line is the averaged signal. It can be seen that the average of the time shifted pulmonary components produces so small a signal that it can be neglected considering the objective of this paper. Therefore, the second part in the right side of Equation (4) becomes  
\[\frac{\sum\limits_{k=1}^{K}\left[p(t-u(\theta_{k}))\right]}{K}\approx\textbf{0 }(t) \tag{5}\]  
The estimated aortic component could be obtained by  
\[\hat{a}(t)=\frac{\sum\limits_{k=1}^{K}S_{2}(t,\theta_{k})}{K} \tag{6}\]  
It means that the aortic component is discriminated by the ensemble average of S2s over respiratory phase. Furthermore, considering Equation (1), the pulmonary component could be discriminated by subtracting the aortic component from the S2s,  
\[\hat{p}(t-u(\theta))=S_{2}(t,\theta)-\hat{a}(t) \tag{7}\]  
For example, the separated components for the S2s of subject No. 4 are shown in Figure 8. It is clearly observed that the aortic components have no change, see Figure 8b. However, the estimated pulmonary components in Figure 8c have varying delay over respiratory phase. The varying delay will be estimated in the next subsection.  
Figure 7: Average operation on S2s over respiratory phase. Blue lines are the S2s and the solid red line is the average. It can be seen that the aortic component remains. Meanwhile, the average of pulmonary components produces a very small signal.