Cardiac cycle calculation

The last step is to recognize the heart sound components. Considering the quasi-periodic nature of heart sounds, this step can be more efficiently accomplished if the cardiac cycle is calculated. In some studies, the cardiac cycle was calculated by using the partial autocorrelation function (PACF) [10, 15]. However, because of the inherent defects of PACF, the calculating results are not satisfactory. In order to overcome the shortcomings of PACF, this study proposes a cardiac cycle calculation method based on the unbiased autocorrelation function (UACF), considerably improving the applicability.

The PACF and UACF are defined as (9) and (10), respectively.

where m = 0, 1, …N - 1, N is the sampling number of E~NO. Figure 3 demonstrates the PACF and UACF.

PACF and UACF of heart sound: a is the thresholded heart sound envelope E~NO; b is the PACF of a; c is the UACF of a. The peaks that are marked by red circles are primary peaks, while the peaks unmarked are side peaks

For PACF, as m increases, the number of sampling points involved in multiplication and accumulation decreases gradually. Consequently, the primary peaks of the PACF exhibit gradual attenuation (see in Fig. 3b), causing the side peaks between the primary peaks to be interference of the cardiac cycle calculation. By contrast, the UACF only averages the terms that are involved in multiplication and accumulation, thereby overcoming the drawback of primary peak’s attenuation in the PACF.

Figure 3c shows that there is a relatively significant difference in amplitude between the primary peaks and side peaks. To further expand the amplitude difference, square energy, which is defined in (11), is utilized. After obtaining the energy signal, thresholding is used to remove the side peaks. The energy signal of the UACF and the energy signal after thresholding are denoted as E_R^′ and E~R′, respectively.

where σ = 0.4 × max(E_R^′).

The last peak of E~R′ is likely to be a side peak that escaped from thresholding because of its relatively higher amplitude (see in Fig. 4c). This kind of side peak is usually generated by the multiplication and accumulation of S1 and S2. In order to calculate the cardiac cycle accurately, the last peak of E~R′ is forcibly removed regardless of whether it is a side peak or a primary peak (see in Fig. 4d). Finally, the UACF of E~R′ with the last peak removed is calculated, and this UACF is denoted as R_Final (see in Fig. 4e). Then, thresholding is performed on R_Final with a threshold of 0.5 × max (R_Final), leaving only primary peaks in the thresholded R_Final. The intervals between adjacent peaks of thresholded R_Final are then calculated and averaged to obtain the average cardiac cycle. Figure 4 shows the main procedures of cardiac cycle calculation.

The main procedures of cardiac cycle calculation: a is R^′(m), the UACF of a heart sound with S4; b is E_R^′(m), the energy signal of R^′(m); c is E~R′m; d is c without the last peak; e is R_Final, the UACF of d