- Research
- Open Access

# Detection of the valvular split within the second heart sound using the reassigned smoothed pseudo Wigner–Ville distribution

- Abdelghani Djebbari
^{1}Email author and - Fethi Bereksi-Reguig
^{1}

**12**:37

https://doi.org/10.1186/1475-925X-12-37

© Djebbari and Bereksi-Reguig; licensee BioMed Central Ltd. 2013

**Received:**15 September 2012**Accepted:**19 November 2012**Published:**30 April 2013

## Abstract

### Background

In this paper, we developed a novel algorithm to detect the valvular split between the aortic and pulmonary components in the second heart sound which is a valuable medical information.

### Methods

The algorithm is based on the Reassigned smoothed pseudo Wigner–Ville distribution which is a modified time–frequency distribution of the Wigner–Ville distribution. A preprocessing amplitude recovery procedure is carried out on the analysed heart sound to improve the readability of the time–frequency representation. The simulated S2 heart sounds were generated by an overlapping frequency modulated chirp–based model at different valvular split durations.

### Results

Simulated and real heart sounds are processed to highlight the performance of the proposed approach. The algorithm is also validated on real heart sounds of the LGB–IRCM (Laboratoire de Génie biomédical–Institut de recherches cliniques de Montréal) cardiac valve database. The A2–P2 valvular split is accurately detected by processing the obtained RSPWVD representations for both simulated and real data.

## Keywords

- Phonocardiography
- Heart sound
- Valvular split
- Reassigned Smoothed Wigner–Ville distribution

## Introduction

It is well known that digital phonocardiography is a powerful tool for assessing the pulmonary artery pressure than Doppler echocardiography [1]. Xu *et al.*[2] found out that the pulmonary artery pressure is correlated with the A2–P2 valvular split.

The split within the S1 and the S2 heart sounds emerged as an indicator of several valvular diseases alongside the Doppler echocardiography (DE). However, the DE is inaccurate in approximately 50% of patients of normal pulmonary artery pressure (PAP), 10–20% of patients with increased PAS, and 34–76% of patients with chronic obstructive pulmonary disease, a weak Doppler signal or a poor signal to noise ratio (SNR) [2]. Indeed, Fisher *et al.*[3] studied the accuracy of the DE in hemodynamic assessment of the pulmonary hypertension (PH). They demonstrated that DE can usually overestimate and underestimate the PAP in PH patients. This can be partly explained by inaccuracies of the right atrial pressure estimation as well as poor Doppler imaging of the transtricuspid regurgitant blood flow. Moreover, Rich *et al.*[4] compared the Doppler echocardiography (DE) with the right sided heart catheterisation (RHC) as an invasive measure of the PAP in 160 patients with PH. They found out that the DE is inaccurate in estimating the PAP in 50.6% of patients at a bias of 8.0 mmHg. Therefore, the DE–based estimation of the PAP is not reliable to diagnose the PH or to assess the efficacy of therapy.

In contrast, the split duration as well as the dominant frequency of P2 are increased in pulmonary hypertension and are considered as reliable parameters to estimate the PAP. Xu *et al.*[2] found that the duration between the onsets of the aortic (A2) and the pulmonary (P2) components within the S2 heart sound (S2) allow accurate measurement of the PAP through advanced digital signal processing techniques. However, this split duration is limited (<100 ms) and still difficult to measure since these components are often overlapping and are of frequency modulated chirp behaviour [5, 6]. The separation of the valvular components of both the S1 and the S2 heart sounds remains a problematic issue. Indeed, several studies reported the complexity of analysing such a transient signals formed by overlapping chirps [7, 8]. Xu *et al.* proposed a nonlinear transient chirp model to simulate the A2 and P2 components of the S2 heart sound [5]. They also proposed a dechirping approach using the Wigner–Ville distribution (WVD) to estimate the instantaneous frequency (IF) of the aortic (A2) and the pulmonary (P2) components. However, they reported weak energies at the beginning and the end of each chirp component to recover the frequency modulated behaviour of heart sounds in the time–frequency plane. This is due to the weak amplitude of all of the valvular heart sound chirps at their onsets and their ends [5].

The A2–P2 valvular split can be originated under physiological or pathological conditions. In normal subjects, a physiological split of the S2 heart sound can occur during inspiration as a result of the delayed pulmonary pressure to raise over the right intraventricular pressure which closes the pulmonary valve. Cardiac pathologies such as the right bundle branch block and the pulmonary stenosis may induce a wide S2 split [9, 10].

In a previous paper [11], we applied the Smoothed pseudo Wigner–Ville distribution (SPWVD) on aortic stenosis and normal heart sounds to quantify their different spectral content within the time–frequency plane. The time-frequency representations we previously obtained by the SPWVD yield global quantification of the valvular intracardiac activity and should be improved through reassignment [12, 13] to adequately represent the intracardiac valvular activity within each heart sound.

Santos *et al.*[14] proposed an A2–P2 valvular split detection algorithm based on the instantaneous frequency calculated by the Hilbert transform. However, the instantaneous frequency of a multicomponent signal cannot be estimated as the derivative of the phase of its analytic signal which is calculated through the Hilbert transform [15]. Therefore, a powerful method that takes into account the multicomponent behaviour of heart sounds should be investigated to represent the intracardiac valvular activity.

In this paper, We developed a new algorithm based on the Reassigned smoothed pseudo Wigner–Ville distribution (RSPWVD) to accurately detect the A2–P2 valvular split within simulated and real S2 heart sounds. Firstly, we recovered the onset and the end amplitude of each valvular component by an envelope recovery procedure we developed. Secondly, we reconstructed the IF of simulated heart sounds at a higher time–frequency resolution by the RSPWVD. We recovered their time–frequency content at several valvular split durations (from 30 to 60 ms at a step of 10 ms). Subsequently, we processed real S2 heart sounds of the LGB–IRCM (Laboratoire de Génie biomédical–Institut de recherches cliniques de Montréal) cardiac valve database to validate the algorithm in real conditions.

This paper is organised as follows. Section “Phonocardiographic data”, entitled ‘Phonocardiographic data’, presents simulated and real S2 heart sounds to be processed by the developed algorithm. Section “Time–frequency analysis”, recalls the theoretical background of the WVD, Smoothed pseudo WVD (SPWVD) and the Reassigned SPWVD (RSPWVD). Section “Detection algorithm of the A2–P2 valvular split” presents the detection algorithm of the A2–P2 valvular split within S2 heart sounds. Sections “Detection of the A2–P2 valvular split in simulated heart sounds” and “Detection of the A2–P2 valvular split in real S2 heart sounds of the LGB–IRCM cardiac valve database” present the detection results of the algorithm applied on simulated S2 heart sounds at various valvular split durations and real S2 heart sounds (LGB–IRCM cardiac valve database) respectively.

## Phonocardiographic data

### Valvular heart sound model

Tran *et al.* developed a heart sound simulator by combining a set of equations to model several phonocardiogram behaviours [16]. This model is formed by a linear chirp with an amplitude adjusted according to clinically recorded S1 and S2 heart sounds. Xu *et al.* extended this model to a narrow–band non–linear chirp signal with a fast decreasing IF over the time–frequency plane to model the valvular heart sound [5, 6]. This decreasing frequency behaviour is generated by the decaying aortic and pulmonary pressures after the end of systole and during the beginning of early diastole. The modulated frequency content of the valvular sound is of chirp nature rather than linear. Indeed, in previous works, we confirmed that heart sounds are narrow–band non–linear chirp signals [17, 18].

Xu *et al.* discussed the exponentially damped sinusoid model [19, 20], the matching pursuit method [21, 22], and the linear chirp model as modelling approaches of heart sounds. They found out that the transient nonlinear chirp signal they developed is the suitable model for the analysis–synthesis of the valvular heart sounds.

*et al.*[5, 6] to generate simulated valvular sounds to study the performance of the detection algorithm we developed. The valvular non–linear chirp model is defined by an amplitude and a phase functions according to (1) as follows:

where *a*(*t*) and *φ*(*t*) represent the IF and the phase of the valvular sound respectively.

where (*a*_{A 2}(*t*),*φ*_{A 2}(*t*)) and (*a*_{P 2}(*t*),*φ*_{P 2}(*t*)) denote the amplitude and the phase of the A2 and the P2 valvular sounds respectively. The split duration interval denoted by *d*_{
s
} separates the beginning of A2 and P2 components. The simulated valvular non–linear chirp component duration is set to 60 ms [5].

### The LGB–IRCM cardiac valve database

The PCG signals of the LGB–IRCM cardiac valve database were recorded in the IRCM (Institut de recherches cliniques de Montréal) and the Montreal General Jewish Hospital in Quebec (Canada). This database is set up with PCG and ECG signals recorded from 172 patients with a prosthetic heart valve in the aortic or the mitral position. All patients signed an inform consent form attesting their assent to take part in the recording of the LGB–IRCM cardiac valve database.

The patient was placed in dorsal decubitus in a recording room. The back of the bed was raised to have 45° between the bust of the patient and the horizontal plane. After that, a thorax auscultation had been carried out to localise the auscultation areas on the chest of the patient. Subsequently, the recording was carried out after 5 minutes rest and calm breathing.

A precordial multi–sites recording was carried out from the aortic, pulmonary, left ventricular, and apical auscultation areas. For each PCG recording, the ECG (derivation II) signal was simultaneously acquired.

## Methods

### Time–frequency analysis

where *x*(*t*) represents the signal to be analysed and *ϕ*_{
T
F
}(*u*,*Ω*) denotes the kernel of the time–frequency distribution.

#### Wigner–Ville Distribution (WVD)

where *x*(.) denotes the signal to be analysed.

*x*(

*t*);

where *h*^{∗}(*t*) is the sliding window.

The spectrogram is calculated by a linear then a bilinear operations. Firstly, the linear operator consists of a Fourier transform, and secondly the squared modulus as a bilinear operator is applied to the signal to be analysed. In contrast, the WVD begins with a quadratic estimation of the energy and then a Fourier transform is applied to the signal according to (4) [28]. The WVD combines the time and the frequency representations with some required properties to adequately represent a given signal *x*(*t*) in the time–frequency plane [27]. A summary of these nice properties can be found in the appendix in section* “Appendix: Properties of the Wigner–Ville distribution”.

*x*(

*t*) formed by 2 components

*x*

_{1}(

*t*) and

*x*

_{2}(

*t*), the WVD can be written;

where ${W}_{{x}_{1}}$ and ${W}_{{x}_{2}}$ are auto–terms, also know as signal–terms, are the WVDs of *x*_{1}(*t*) and *x*_{2}(*t*) which are assumed to be analytic as well as *x*(*t*). Whereas the term ${W}_{{x}_{1}\xb7{x}_{2}}$ represents cross–terms, also known as outer–artefacts, which appear midway between ${W}_{{x}_{1}}$ and ${W}_{{x}_{2}}$ within the time–frequency plane. Therefore, the WVD of *x*(*t*) is formed by the WVDs of its constituents and the cross–terms which represents their cross–Wigner–Ville distribution (XWVD). Unfortunately, when the analysed signal contains several components, the obtained time–frequency representation becomes unreadable.

#### Smoothed Pseudo Wigner–Ville Distribution (SPWVD)

*g*and

*h*through the kernel

*ϕ*

_{ T F }(

*u*,

*Ω*) =

*g*(

*u*)

*H*(

*Ω*) according to (7) [29];

where *g* and *h* are two real even windows with *h*(0) = *G*(0) = 1.

The SPWVD has a separable smoothing kernel (*g*(*t*),*H*(*f*)) which provides an independent control of the time and frequency resolutions. For a zero time–resolution, i.e., *g*(*t*) = *δ*(*t*), the calculated SPWVD has no time smoothing (where *H*(*f*) is the Fourier transform of *H*(*f*)). Thus, the resulting time–frequency distribution is known as the pseudo–WVD (PWVD).

Smoothing the time–frequency distribution affects the time–frequency localisation of the signal content. Therefore, a trade–off between interference attenuation and time–frequency localisation occurs to ensure a good time–frequency representation [30–34].

#### Reassigned Smoothed Pseudo Wigner–Ville Distribution (RSPWVD)

The reassignment method was first applied by Kodera *et al.*[35, 36] to the spectrogram to surpass its unavoidable Gabor–Heisenberg inequality [34, 37] to provide a better time–frequency representation. Auger *et al.*[12] studied the reassignment method and demonstrated its effectiveness to improve the readability of all the bilinear time–frequency representations. This method rearranges the coefficients of the time–frequency distribution around new zones to yield a high resolution TFR. Thus, this method can be used as a complement to any bilinear time–frequency distribution.

*t*,

*ω*) within the time–frequency plane. This is the sum of the terms

*ϕ*

_{ T F }(

*u*,

*Ω*)

*W*

*V*(

*x*;

*t*-

*u*,

*ω*-

*Ω*) which represents the weighted coefficients of the WVD in the points at the vicinity of (

*x*;

*t*-

*u*,

*ω*-

*Ω*). Therefore, the distribution is concentrated at the time–frequency support of the kernel

*ϕ*

_{ T F }(

*u*,

*Ω*). Unfortunately, the cross–terms are attenuated at the cost of spreading the auto–terms of the analysed signal. The modified time–frequency distribution attributes new values to each coefficient to its neighbouring centre of gravity according to (8) and (9) as follows;

where *δ*(.) denotes the Dirac impulse.

The reassigned time–frequency distribution is not bilinear. However, it is time and frequency shift–invariant, and respects the energy conservation property. Moreover, its powerful property perfectly localises chirps [12].

- 1.
Compute the Smoothed pseudo–WVD of the signal,

- 2.
Evaluate the local centres of gravity

*t*_{ a }(*t*,*ν*) and*ν*_{ a }(*t*,*ν*) of the calculated SPWVD in every point of the time–frequency representation, - 3.Assign the energetic content to the new point within the time–frequency plane according to (11);${\left|{F}_{x}(t,\nu )\right|}^{2}\to {S}_{x}\left({t}_{a}(t,\nu ),{\nu}_{a}(t,\nu )\right)$(11)

Therefore, the reassignment method improves the readability of the calculated time–frequency representation by boosting the time and frequency resolutions [12, 13, 38].

### Detection algorithm of the A2–P2 valvular split

Xu *et al.* demonstrated that heart sounds are formed by overlapping chirp components which are generated by the closures of the intracardiac valves [5, 6]. Unfortunately, the restrictive weak amplitude at the onset and the end of each chirp component confined the time–frequency chirp shape of each component at its highest amplitude domain [5].

- 1.
Envelope recovery: the signal is consecutively multiplied with its complement to 1 until the correlation between the latest consecutive signals exceeds 99.90%.

- 2.
Calculation of the Reassigned–SPWVD (RSPWVD) of the signal.

- 3.
Estimation of the IF by detecting frequencies of the RSPWVD coefficients with highest intensities over the time domain.

- 4.
Processing the IF as unidimensional curve to detect the split inflection by localising the maximum and the minimum having the maximum amplitude difference.

- 1.Calculation of the envelope of the signal by the Hilbert transform which is given by to (12) [27];$\begin{array}{ll}H\left\{x\left(t\right)\right\}& =x\left(t\right)\ast \frac{1}{\mathrm{\pi t}}\phantom{\rule{2em}{0ex}}\\ =\frac{1}{\pi}\mathrm{p.v.}\left\{\underset{-\infty}{\overset{+\infty}{\int}}\frac{x\left(\tau \right)}{t-\tau}\mathrm{d}\tau \right\}\phantom{\rule{2em}{0ex}}\end{array}$(12)

- 2.
Detection of the split by localising the midway local minimum of the envelope at the higher amplitude.

The discrete IF is calculated as the inverse of the detected periods of heart sounds through the time domain. This detection is carried out by localising time–instants of zero–crossings of the analysed heart sounds. This discrete IF sequence enables to back up the discussion of the final IF detected from the RSPWVD.

## Results and discussion

Firstly, we applied the algorithm we developed on simulated heart sounds at various A2–P2 valvular split durations to show its ability to retrieve these splits within the time–frequency plane. Secondly, we processed several real S2 heart sounds of the LGB–IRCM cardiac valve database presented in section “The LGB–IRCM cardiac valve database” to validate the algorithm in real conditions. Time–frequency representations and the A2–P2 split detection results for a sample data file selected from the LGB–IRCM cardiac valve database are presented. The Hilbert envelope detection approach (section “Detection algorithm of the A2–P2 valvular split”) and the discrete IF are used as complementary tools to check the obtained results.

### Detection of the A2–P2 valvular split in simulated heart sounds

*et al.*[5] during time–frequency analysis of heart sounds, we calculated another version of heart sounds which we call the envelope recovered heart sound as presented in section “Detection algorithm of the A2–P2 valvular split”. Figure 9 illustrates the envelope recovered sound of the S2 heart sound of Figure 5. This adjusted sound benefits of the same amplitude for its entire time support.

The Hilbert envelope detection approach retraces the instantaneous power of this S2 heart sound and shows an amplitude variation which is related to the A2–P2 split. This split is localised at 36 ms from the onset of the A2 valvular component at an error of 6 ms from the original split (30 ms).

The recovery of the heart sound is carried out until reaching a cross–correlation between consecutive steps of 99.90% as presented in section “Detection algorithm of the A2–P2 valvular split”. This procedure of the A2–P2 split detection algorithm is not CPU time consuming and still a vital step for the time–frequency analysis. It should be noticed that 8 iterations are sufficient to reach the desired cross–correlation rate for this simulated S2 heart sound.

The RSPWVD detection method is based on the variation of the IF of the S2 heart sound within the time–frequency plane rather than following variations in the amplitude of the signal as carried out by the Hilbert envelope detection approach. Therefore, the RSPWVD–based A2–P2 detection is accurate and improves the detection in comparison to the Hilbert envelope approach.

**The A2–P2 valvular split detected by the RSPWVD–based detection method**

Split (ms) | 30 | 40 | 50 | 60 |
---|---|---|---|---|

A2–P2 | 25.5 | 37 | 38.5 | 49 |

A2–P2 | 35.5 | 51 | 52.5 | 63.5 |

A2–P2 | 30.5 | 44 | 45.5 | 56.25 |

A2–P2 | +0.5 | +4 | +4.5 | -3.75 |

The A2–P2 split measurements summarised in Table 1 confirms the ability of the RSPWVD–based method to detect the A2–P2 split. The RSPWVDs of Figures 15 & 18 continues to show the inflection behaviour at the A2–P2 split zone and provides 44 and 56.25 ms as measured values of the simulated split values of 40 and 60 ms.

### Detection of the A2–P2 valvular split in real S2 heart sounds of the LGB–IRCM cardiac valve database

The PCG signal of Figure 4 corresponding to the sample data file (10001.11) of the LGB–IRCM cardiac valve database is considered for analysis to show the ability of the proposed algorithm to detect the valvular split on real data. This PCG signal is segmented in systole and diastole by detecting the peak of the R–wave and the end of the T–wave by an algorithm presented in [39] which uses both the amplitude and the curvature of the ECG waves. This algorithm provides correct detection of the overall ECG–PCG signals of the LGB–IRCM cardiac valve database.

*et al.*, the weakness of the valvular sounds at their extremities restricts the energy bursts in the high amplitude time support of the analysed sounds within the time–frequency plane [5].

## Conclusions

The A2–P2 valvular split detection algorithm we developed is mainly based on the Reassigned Smoothed Pseudo Wigner–Ville Distribution (RSPWVD). The Reassignment mixed to the smoothing achieved both in time and frequency domains by the SPWVD provides a higher readability to the obtained RSPWVD. The preprocessing envelope recovery procedure we proposed adapts the analysed heart sounds to the WVD which is optimal for analysing frequency modulated chirps. The performance of the algorithm is confirmed on simulated heart sounds at various split durations (30, 40, 50 and 60 ms). The A2–P2 valvular split is localised at the frequency inflection in the obtained RSPWVD. The developed algorithm is validated on real heart sounds of the LGB–IRCM cardiac valve database and retraces the inflection of the A2–P2 valvular split of the S2 heart sound within the time–frequency plane. The discrete IF is estimated for both simulated and real data to confirm the results obtained through the RSPWVD. Therefore, the proposed algorithm deals adequately with detecting the A2–P2 valvular split and confirms the chirp behaviour of heart sounds. Thus, we demonstrated through the algorithm we developed that the A2–P2 valvular split can be accurately detected by time–frequency analysis using the RSPWVD.

## Consent

The patients were contacted in confidentiality by their treating cardiologist and they signed an inform consent form attesting their assent to take part in the study allowing to record the LGB–IRCM cardiac valve database. The signals were recorded at the Institut de recherches cliniques de Montréal (IRCM) and at the Montreal General Jewish Hospital in Quebec (Canada).

## Appendix: Properties of the Wigner–Ville distribution

- 1.
The WVD is a member of the Cohen’s class with a weighting function

*g*(*ν*,*τ*) = 1. - 2.
Realness: The WVD is of real values over the time–frequency plane which makes it suitable for representing the energy of the analysed signal.

- 3.Time and frequency marginals: As shown in (13), the integration of the time–frequency distribution over time yields the spectral density of the signal. As stated in (14), the integration of the time–frequency distribution over frequency yields the instantaneous power of the analysed signal
*x*(*t*), as follows;$\underset{-\infty}{\overset{+\infty}{\int}}{W}_{x}(t,f)\phantom{\rule{1em}{0ex}}\mathrm{d}t=\left|X\right(f){|}^{2}$(13)$\underset{-\infty}{\overset{+\infty}{\int}}{W}_{x}(t,f)\phantom{\rule{1em}{0ex}}\mathrm{d}f=\left|x\right(t){|}^{2}$(14)

*X*(

*f*) denotes the Fourier transform of the signal

*x*(

*t*), and

*W*

_{ x }(

*t*,

*f*) represents its WVD.

- 4.Global energy: Integration of the WVD over the time–frequency plane yields the global energy
*E*_{ x }of the analysed signal as follows;$\underset{-\infty}{\overset{+\infty}{\iint}}{W}_{x}(t,f)\phantom{\rule{1em}{0ex}}\mathrm{d}t\phantom{\rule{1em}{0ex}}\mathrm{d}f={E}_{x}$(15) - 5.Instantaneous frequency (IF): the first moment of the WVD with respect to frequency of the analytic signal yields the IF as follows;$\frac{\underset{-\infty}{\overset{+\infty}{\int}}f{W}_{x}(t,f)\phantom{\rule{1em}{0ex}}\mathrm{d}f}{\underset{-\infty}{\overset{+\infty}{\int}}{W}_{x}(t,f)\phantom{\rule{1em}{0ex}}\mathrm{d}f}=\frac{1}{2\pi}\phantom{\rule{1em}{0ex}}\frac{\mathrm{d}}{\mathrm{d}t}\left[\text{arg}x\right(t\left)\right]$(16)
- 6.Time delay (TD): the first moment of the WVD with respect to time of the analytic signal of yields the TD as follows;$\frac{\underset{-\infty}{\overset{+\infty}{\int}}t{W}_{x}(t,f)\phantom{\rule{1em}{0ex}}\mathrm{d}t}{\underset{-\infty}{\overset{+\infty}{\int}}{W}_{x}(t,f)\phantom{\rule{1em}{0ex}}\mathrm{d}t}=-\frac{1}{2\pi}\phantom{\rule{1em}{0ex}}\frac{\mathrm{d}}{\mathrm{d}f}\left[\text{arg}X\right(f\left)\right]$(17)
- 7.
The WVD is limited to time–frequency support defined by the duration and the bandwidth of the analysed signal

*x*(*t*). - 8.Convolution invariance: The WVD of the time–convolution of two signals (
*x*_{1}(*t*)and*x*_{2}(*t*)), yields the time– convolution of their respective WVDs$\left({W}_{{x}_{1}}(t,f)\text{and}{W}_{{x}_{2}}(t,f)\right)$ as follows;$\begin{array}{ll}{x}_{3}\left(t\right)& ={x}_{1}\left(t\right)\underset{t}{\ast}{x}_{2}\left(t\right)\phantom{\rule{2em}{0ex}}\\ \Rightarrow {W}_{{x}_{3}}(t,f)& ={W}_{{x}_{1}}(t,f)\underset{t}{\ast}{W}_{{x}_{2}}(t,f)\phantom{\rule{2em}{0ex}}\end{array}$(18) - 9.Modulation invariance: The WVD of the frequency–convolution of two signals (
*x*_{1}(*t*) and*x*_{2}(*t*)) yields the frequency–convolution of their respective WVDs$\left({W}_{{x}_{1}}(t,f)\text{and}{W}_{{x}_{2}}(t,f)\right)$ as follows;$\begin{array}{ll}{x}_{3}\left(t\right)& ={x}_{1}\left(t\right)\xb7{x}_{2}\left(t\right)\phantom{\rule{2em}{0ex}}\\ \Rightarrow {W}_{{x}_{3}}(t,f)& ={W}_{{x}_{1}}(t,f)\underset{f}{\ast}{W}_{{x}_{2}}(t,f)\phantom{\rule{2em}{0ex}}\end{array}$(19)

The WVD is time and frequency invariant. Furthermore, a range of peaks of the IF and TD of the analysed signal are directly readable on the WVD [40, 41]. The WVD covers the spectral bandwidth of the analysed signal. Moreover, fluctuations of the maximum frequency of the analysed signal is well represented over the time domain by the WVD [42].

## Declarations

### Acknowledgements

The authors would like to thank Prof. Louis–Gilles Durand (Institut de recherches cliniques de Montréal, Quebec, Canada) for putting the LGB–IRCM cardiac valve database at our disposal as well as his valuable technical advices to achieve this work.

## Authors’ Affiliations

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