The iterative MAP estimation ofyusing the MM algorithm is represented by:

¯

yi+1 = arg min

y (Q(y,y¯i))

=∆ x−D^{T}
1

λiΛ^{−1}_{i} +DD^{T}
−1

Dx

(2.27)

and

λi+1 =ρ(¯yi+1) (2.28)

It follows the same algorithm illustrated in Algorithm 1. The iteration process continues
until the denoised signal¯y_{i} resembles the original signal. If the denoised signal exactly equals
the original signal, the residual signalx−¯y_{i} will be the noise signal, and their variance will
bear the same value.

%^{2}_{noise}= 1

N kx−yk^{2}_{2} (2.29)

which implies

kx−y_{i}k^{2}_{2} ≤N %^{2}_{noise} (2.30)

Eq. (2.30) is used as a stop condition of the OGS-TVF. In [1], the OGS-TDF formulated with adaptive MAP estimation of regularization parameter was evaluated for heart sound denoising. In their experiment, the variance of the noise is calculated by the median absolute deviation (MAD) rule [91]. The remaining parametersα,` β,` θand group sizeK values were pre-set at 1, 50, 0.8 and 20, respectively. The denoising performance showed significant improvement over the conventional wavelet-based method.

heart sounds have the majority of the signal energy below 150 Hz with a peak at 50 Hz.

The features for heart sound segmentation problem are defined to closely meet the above description.

**2.3.1** **Envelop extraction**

The envelope helps to visualize the distribution of signal intensities at different time instances.

Based on the mechanism of producing a normal heart sound, the S1 and S2 sounds are only audible sounds separated by silent intervals. Therefore, a PCG envelope reproduces the temporal characteristic of the heart sound signal that can be used to interpret the intensity of FHS. Most envelope extraction method for HSS are derived from either Shannon Entropy, Shannon Energy, Hilbert Transform, or Homomorphic envelope [5, 16, 34, 35]. There are other primitive methods to calculate envelope based on energy (squaring) or absolute value [34].

In an energy-based envelope, the weaker signals that have smaller intensity ration are suppressed when compared to stronger intensity signals. On the other hand, the absolute value gives equal weight to all the intensities. The expressions for different envelogram methods are as under:

energy = x^{2}(t) (2.31)

absolute value = |x(t)| (2.32)

**Hilbert envelope (**Hilbert E**):** Envelope from analytic signal is a very common envelogram
method [5, 50, 92]. The analytic signal is obtained after Hilbert transform of a signal which is
given by:

y(t) = x(t) +jx(t)ˆ (2.33)

where,x(t)is the original signal and its Hilbert transformx(t)ˆ is defined as:

ˆ

x(t) = 1

πt ∗x(t) (2.34)

The Hilbert envelope is calculated from the analytic signal using the following equations:

Hilbert E =p

x^{2}(t) + ˆx^{2}(t) (2.35)

**Homomorphic envelope (**Homo E**):** Another popular envelope extraction method is using
homomorphic filter [35,62]. The advantage of homomorphic envelope is its scalable smoothing
process, which can be easily tuned. It extracts the slow varying componente(t)of a signal
that envelops the fast oscillating components o(t).

x(t) =e(t)·o(t) (2.36)

Natural logarithm operator is used to convert multiplication to summation of the two components. This additive nature makes the high-frequency component appear as a rapid variation in time.

lnx(t) = lne(t) + lno(t) (2.37)

Applying a simple low pass filter (LPF) of desired cut-off frequency will remove the unwanted frequency components. The signal is retrieved by exponentiating the filter output.

Homo E = exp [LPF [ln x(t)]] ≈ e(t) (2.38)

**Envelope moderation methods:** For the uniformity and easy detection of FHS, it is pre-
ferred that the envelope peaks corresponding to S1 and S2 should maintain more-or-less
identical intensity levels. To realise this, Shannon entropy and Shannon energy are calculated
before envelope extraction. Shannon entropy and Shannon energy emphasize the medium
intensity signals by using logarithmic weight. They are calculated as:

Shannon energy = −x^{2}(t) log(x^{2}(t)) (2.39)

Shannon entropy = −|x(t)|log|x(t)| (2.40)

**2.3.2** **Frequency domain features:**

The popular frequency domain features for HSS are power spectral density envelope and wavelet envelope [5, 6]. There will be no denoising process before the feature extraction process.

**Power spectral density (PSD) envelope:** From the spectral analysis of heart sound, it is
obvious that the spectral peak occurs approximately at 50 Hz [5]. This is true for a healthy
heart sound. However, in the case of a pathological condition such as Myocardial Infarction,
cardiomyopathy, and valvular defects, the spectral peak of the S1 and S2 sounds may deviate
depending on the elastic property or thickness of myocardium, and the pressure built up in the
heart chambers [28]. The exact peak frequencies associated with different pathological cases
are not fully established. Therefore, it is better to include spectral peak within the frequency
range of 20 Hz to 150 Hz, because, the major signal energy of FHS is below 150 Hz [19]. Thus,
the PSD envelope is calculated as the frequency of the spectral peak within the permissible
range. The Hamming window of size 50 ms is used for short-term-Fourier-transform of the
PCG [5].

**Wavelet envelope:** Wavelet analysis has been extensively used for the analysis of heart
sound signals. But choosing the wavelet family suitable for the purpose is still a difficult
decision. In the related work [5], wavelet families inclusive of Haar, Daubechies, symlet,
Coiflet, biorthogonal, and reverse biorthogonal wavelets were used to decompose the signal
into 1-5 levels. The detail coefficients and the corresponding levels that fit the frequency band
of the normal heart sound were selected. The envelope value is the sum of the absolute

values of detail coefficients. To check how well the wavelet discriminates the FHS, the ratio of the sum of the absolute values of selected coefficients corresponding to S1 and S2 sounds against other intervals and across all recordings is measured. The wavelet yielding the highest ratio value is considered the best choice.