White Gaussian Noise Energy Estimation and Wavelet Multi-threshold De-noising for Heart Sound Signals

White Gaussian noise (WGN) commonly exists in acquisition and transmission of heart sound (HS) signals. The energy distributions of WGN and HS in wavelet decomposition levels (WDLs) are explored. The statistical analysis indicates that for WGN, energy proportions of detail portions of WDLs and energy proportion of the 2nd WDL are fixed. This finding is verified by using Monte Carlo test. Moreover, for a HS signal recorded under sampling frequency of 4 kHz, energies of the 1st and 2nd WDL are almost the same, which are validated by theoretical analysis and practical observation on three HS benchmark datasets. Based on these findings, equations estimating WGN energy and signal to noise ratio (SNR) for a noisy HS signal are created. In addition, a novel energy distribution-based wavelet multi-threshold de-noising approach (ED-WMTD) is proposed to reduce WGN. In which, firstly based on the energy distribution in WDLs and estimated energy of WGN, WGN energy in detail portion of each WDL is figured out. Then, soft-threshold method is adopted. The best threshold in a WDL is defined as the one by which the energy loss of noisy HS signal in this WDL is most similar to the energy of WGN in detail portion of this WDL. The accuracy of such a WGN energy estimation method is evaluated by average error of SNR estimation. ED-WMTD is assessed using mean square error and compared with four generally used WMTD methods. Experimental results show that this novel HS de-noising approach not only filters out HS noise effectively but also well retains its pathological information.