This is a preprint of an article published in International Journal on Communications Antenna and Propagation (I.Re.C.A.P.), Vol.1, N.1 February 2011, Copyright © 2011 Praise Worthy Prize
A new chirp–based wavelet for heart sounds time–frequency analysis Abdelghani Djebbari1, Fethi Bereksi-Reguig2 Abstract – The present paper investigates the ability of a chirplet model to be considered as a pattern generating a mother wavelet for time–scale analysis of heart sounds. The choice of the mother wavelet is a sensitive task since it affects considerably the obtained time–scale representation. Heart sounds are formed by frequency modulated components; it is therefore convenient to analyze them by means of chirplet rather than ordinary wavelet. Indeed, we study a heart sound valvular component chirplet model to generate an adapted mother wavelet pattern, which is then used within the Continuous Wavelet Transform (CWT). Using the Morlet wavelet and the Discrete Meyer wavelet as comparative approaches, we found that the valvular chirplet model is suitable as mother wavelet for heart sounds time–frequency representation. Indeed, we can reach 98.59% of cross–correlation between the adapted wavelet and the original chirplet for 15 as fitting polynomial order. The VCCM represents adequately more than 80% of the frequency modulated components of both S1 and S2 heart sounds. Therefore, the proposed VCCM wavelet provide better results than the Morlet wavelet and the discrete Meyer wavelet in representing heart sounds energy. Keywords: Heart sound, Phonocardiogram, Continuous Wavelet Transform, Morlet Wavelet, Discrete Meyer Wavelet, Chirplet
(the two atria and the two ventricles). Commonly, the normal PCG signal is mainly formed by four heart sounds, denoted S1, S2, S3 and S4. Additional murmurs can be observed for pathological cases. As illustrated in Fig. 1, the normal heart beat regularly generate two audible sounds; S1 and S2, denoting the beginning of the systole and the diastole phases respectively. The S3 and S4 heart sounds are usually unlikely recorded but still appearing under some physiological conditions [1]. The S1 is mainly generated by the closure of the mitral and the tricuspid valves, denoted as M1 and T1 respectively. The opening of the aortic and the pulmonary valves slightly contribute to the formation of this systolic sound. Indeed, the S1 sound is the onset of the ventricular systole which is the duration of the ventricular blood ejection. Similarly, the S2 heart sound is generated by the closure of the aortic and the pulmonary valves, denoted respectively as A2 and P2, and a slight contribution of the opening of the mitral and the tricuspid valves. It is the vibration rather than the closing of each valve which contributes considerably to form a heart sound component [1]. This valvular vibration emanating through the blood flow within heart cavities and vessels is perceived at the patient’s chest as frequency modulated components [2];[3].
Nomenclature PCG S1 S2 S3 S4 M1 T1 A2 P2 PAP VCCM TSR TFR CWT PSD FIR
Phonocardiogram signal First heart sound Second heart sound Third heart sound Fourth heart sound Mitral component of the first heart sound Tricuspid component of the first heart sound Aortic component of the second heart sound Pulmonic component of the second heart sound Pulmonary Artery Pressure Valvular Component Chirplet Model Time-scale representation Time-frequency representation Continuous Wavelet Transform Power spectral density Finite impulse response
I.
Introduction
The phonocardiogram (PCG) signal is a high quality sound recording of intracardiac activity. It collects valuable medical information relating to heart valves and cavities status. This tracing records audible and subaudible heart sounds and murmurs collected from auscultation areas well localized by experienced physicians from the chest of the patient. Heart sounds are complex signals because they are formed by the closures and openings of the four cardiac valves as well as the blood circulation inside the four myocardium chambers 92
A.Djebbari, F. Bereksi–Reguig
readability of the TFR estimated from the TSR of heart sounds. We show that browsing heart sounds within the time–scale plan by a chirplet as mother wavelet adapted to valvular component pattern can yield better results than using classical mother wavelets. The impact of this chirplet on the obtained TFR is investigated with comparison to the Morlet wavelet and the discrete Meyer wavelet. This paper is organized as follows; Section 2 entitled ‘Methods’ includes five subsections. Subsection 2.1 presents the heart sound model. Subsection 2.2 presents the theoretical background of the Continuous wavelet transform (CWT). Subsection 2.3 is focused on the chirplet model as a mother wavelet candidate within the CWT. Subsections 2.4 and 2.5 describes the Morlet wavelet and the discrete Meyer wavelet respectively. In section 3 entitled ‘Results’, normal S1 and S2 heart sounds are analyzed by both classical mother wavelets and the heart sound–based chirplet model as mother wavelet.
1
Amplitude, %
S1
S2
0.5 0 -0.5
Sy stole
Diastole
-1 0
200
400 600 Time, ms
800
Fig. 1 – One cardiac cycle heart sounds
Heart sounds are still useful to predict, diagnose and quantify the severity of cardiac pathologies. Indeed, several scientific research [4]-[10] deals with the innovation that can be inserted within digital phonocardiography through advanced signal processing methods. The timing of intracardiac events is a valuable decision element with respect to the detection of cardiac pathologies [11]-[15]. However, the temporal analysis of heart sounds remains insufficient without making use of their frequency content. Moreover, spectral methods require temporal information in term of duration and appearing instant relating to each significant spectral component within the obtained spectra. Therefore, scientific research is now more focused on the time– frequency analysis of heart sounds and murmurs rather than their spectral or temporal information separately [13];[16]-[19].
II.
Methods
Durand et al. [31] studied the transmission of heart sounds and murmurs from the myocardium to the chest in dogs. By recording the intracardiac and thoracic phonocardiograms, the transfer and coherence functions of the chest were quantified. The chest was therefore considered as a filter which affects the acoustic transmission of the valvular activity as well as the blood flow within the myocardium cavities. They named this filtering effect as heart/thorax acoustic system. This model was focused on studying the attenuation of the valvular component rather than their frequency content during the cardiac cycle.
The wavelet transform can effectively analyze non– stationary time series over the frequency domain [20]. Wavelet analysis is then one of the useful tools allowing two-dimensional representation of signals. Indeed, by introducing a scale parameter of a dilated and compressed wavelet used as analyzing function, the wavelet transform generates a Time-Scale Representation (TSR) of signals. Moreover, this TSR can be extended to the time–frequency plane by estimating the central frequency of the mother wavelet [21]. Wavelet analysis occupies a considered position among digital signal processing methods when dealing with such non–stationary signal as phonocardiogram [15];[17];[18];[22]-[25]. Thus, time–scale methods can reliably elicit information relating to the intracardiac valvular events. These vibrations are perceived as frequency variation within heart sounds [26]-[29]. The Time–Frequency Representation (TFR) of heart sounds reflects their correlation with the mother wavelet through the scale domain. Tran et al. [30] proposed a heart sound model assuming that heart sounds are formed by linear chirps. This model did not take into account the complexity of heart sounds and abridged them into linear frequency components. Xu et al. [2] delved deeply into heart sounds study and linked their content to a chirplet model they proposed. They showed that this model is convenient for representing the valvular activity which forms the main energy within heart sounds. The objective of this paper is to improve the
II.1.
The Valvular Component Chirplet Model (VCCM)
Several models have been proposed for simulating heart sounds. Tran et al. designed a heart sound simulator [30] for learning auscultation skills. The proposed heart sound simulator was based on the work of Franks [32]. It considers S1 and S2 heart sounds as linear chirps, which consists of a sinusoidal wave of instantaneous frequency varying linearly with time. This is the main defect of this model since the S1 and S2 are generated from more complex activities related to valvular and intracardiac blood flow circulation. Based on the Laplace's law, Aggio et al. [33] and Longhini et al. [34] shown that the instantaneous frequency of P2 (second component of the S2 heart sound) is proportional to the Pulmonary Artery Pressure (PAP). Bartels and Harder [35] demonstrates that the time–frequency analysis of S2 can yield an estimation of the arterial pressure. Therefore, A2 and P2 components are considered as decreasing frequency modulated chirps with an instantaneous frequency proportional to the decreasing arterial and pulmonary pressures respectively.
93
A.Djebbari, F. Bereksi–Reguig
Therefore, it can be assumed that valvular components forming S1 and S2, namely M1 and T1 for S1, and A2 and P2 for S2, are of hyperbolic modulated instantaneous frequency. Such a morphological behavior of the heart sound component has been modeled by Xu et al. [2] using a nonlinear transient chirp signal. This model, which we name the Valvular Component Chirplet Model (VCCM), is given by;
Where
A (t )
and
ϕ ( t ) denotes
1
Amplitude, %
VCCM ( t ) = A ( t ) sin (ϕ ( t ) )
The VCCM we bring out in the present paper consists of a hyperbolic frequency modulated chirp of limited duration. The chirplet amplitude is zero at t = 0 s and 250 Hz frequency which decreases to 53 Hz as illustrated in Fig. 4.
(1)
respectively
the
amplitude and the phase of the VCCM. The simulated valvular component duration is limited to 60 ms, since the duration of aortic and pulmonary valvular component lasts generally less than 50 ms [2]. The amplitude A ( t ) of the VCCM is then defined by; t t ⎛ − ⎞ − ⎛ πt ⎞ 8 16 ⎜ ⎟ A(t ) = 1 − e e .sin ⎜ ⎟ ⎜ ⎟ ⎝ 60 ⎠ ⎝ ⎠
0 -0.5 -1 0
40
60
Fig. 4 – Normalized VCCM (from 250 to 53 Hz)
Wavelet analysis relies on stretching and compressing the mother wavelet when sweeping the scale domain. Thus, it is suitable to track down the chirplet content by means of the VCCM which is therefore acceptedly typical for heart sounds time–scale analysis.
(2)
II.2.
The Continuous Wavelet Transform (CWT)
The CWT was introduced for the first time in reflection seismology by Goupillaud, Grossmann and Morlet under the name of cycle–octave transform [36]. The wavelet transform calculates the inner products between the signal and a wavelet family. The Continuous Wavelet Transform of a given signal x ( t ) is defined by
0.3
0.2
[37];
CWTa ,b { x ( t )} = x ( t )ψ a*,b ( t ) dt
∫
0.1
0
10
20
30 time, ms
40
50
60
ψ a*,b ( t ) =
Fig. 2 – Instantaneous amplitude of the VCCM
The phase ϕ ( t ) of the VCCM is given by;
(
ϕ ( t ) = 2π 24.3 t + 451.4 t
)
(3)
a
200
−1 2
100 50 0 30 time, ms
40
50
(5)
The VCCM adapted wavelet
Wavelet analysis can be viewed as correlation between the signal and the mother wavelet which is shifted at position b and dilated by scale a. Indeed, the coefficients of the wavelet transform reflect the similarity between the wavelet and the swept signal portions through the scale domain. To lift up the heart sounds energy within the time– frequency plane, it is obvious to look for a mother wavelet that can substantiate the valvular component within the TFR. The heart sound chirplet model oscillates
150
20
a
⎛ t −b ⎞ ⎟ ⎝ a ⎠
ψ⎜
ensures energy normalization.
II.3.
10
1
where parameters a and b denotes respectively the scale and the temporal position of the wavelet. The factor
250
0
(4)
(the asterisk denotes the complex conjugate). The wavelets family is defined by dilating and shifting the mother wavelet ψ a*,b ( t ) which is given by;
0
frequency, Hz
20 time, ms
The attack and decay slope are defined by the first and the second terms of equation (2) respectively. The last term limits the duration to 60 ms (Fig. 2).
amplitude
0.5
60
Fig. 3 – Instantaneous frequency of the VCCM (from 250 to 53 Hz)
94
A.Djebbari, F. Bereksi–Reguig
like a wavelet and should be a good candidate for time– scale analysis of heart sounds. The unique condition apropos of the chirplet model, to be a mother wavelet, is that its average should be zero. This is the admissibility condition for the chirplet to be viewed as a wavelet, as follows; +∞
∫−∞ ψ ( t ) dt = 0
highlights the cross–correlation between the adapted wavelet and the VCCM. TABLE I show that reaching 15 for as polynomial order lead to 98.59% as cross–correlation between the adapted wavelet and the VCCM. For 35 as polynomial order, we get 99.88% as correlation. The adapted wavelet is then so similar to the VCCM whensoever we increase the polynomial order. The constraints to be respected consist of ensuring almost zero integral, continuity at the beginning and the end of the adapted wavelet and a square norm equal to 1.
(6)
This admissibility condition ensures to the signal to be well recovered after inversion of the wavelet transform. Equation (6) can be viewed as zero–order moment condition. Another suitable property for the mother wavelet is regularity which consists of localization within the frequency domain [38]. Indeed, Fig. 5 shows the spectrum of the chirplet model which is well localized within the frequency domain. This spectrum is the Power Spectral Density (PSD) estimated by means of periodogram method [39];[40].
TABLE I CROSS CORRELATION BETWEEN THE ADAPTED WAVELET AND THE VCCM
Adapted wavelet (dashed line) & VCCM (solid line)
Polynomial order
Cross correlation, %
-3
x 10
3
02.84
5
04.32
6
38.84
7
54.43
Magnitude
3 2 1
8
66.20
Fig. 5 - Periodogram PSD estimate of the Normalized VCCM
9
91.86
The mother wavelet can have a number of vanishing moments as follows;
10
92.21
15
98.59
20
99.27
25
99.78
0 0
Mp =
50 100 Frequency , Hz
+∞ p
∫−∞ t ψ ( t ) dt = 0
150
for p = 0,1, 2,..., n
(7)
That means that the (n+1) derivatives of the mother wavelet spectrum are equal to zero at the frequency zero. The analysis is carried out within MATLAB environment by means of its Wavelet toolbox which allows us to build an admissible wavelet from a given pattern [41]. The pattern we use herein is the VCCM which proves its valuableness through the several tests we accomplished. The generated adapted wavelet is of norm equal to 1. TABLE I show that reaching 15 for as polynomial order lead to 98.59% as cross–correlation between the adapted wavelet and the VCCM. For 35 as polynomial order, we get 99.88% as correlation. The adapted wavelet is then so similar to the VCCM whensoever we increase the polynomial order. The constraints to be respected consist of ensuring almost zero integral, continuity at the beginning and the end of the adapted wavelet and a square norm equal to 1.
30
99.15
35
99.88
Cross-correlation, %
These results are also graphically illustrated in Fig. 6.
TABLE I gathers in least squares polynomial approximation results. Using this fitting approach, admissible wavelet is constructed to fit the VCCM for later detection by using the CWT. According to several values of the fitting polynomial order, TABLE I
100
50
0 10
20 30 Poly nomial order
40
Fig. 6 – Cross–correlation between the adapted wavelet and the VCCM
95
A.Djebbari, F. Bereksi–Reguig
Therefore, we can now use the VCCM within wavelet analysis since it can be modeled as an admissible mother wavelet. This model oscillates like a wavelet and ensure to heart sounds to be well analyzed. In this study, we use the Morlet wavelet are a crude wavelet and the Discrete Meyer wavelet, which is a discrete improvement of the Meyer wavelet, to get a discussion area about the qualities of the proposed VCCM adapted wavelet.
II.4.
Amplitude
1 0.5 0 -0.5 -1 40
The Morlet wavelet
45 50 55 Ef f ectiv e support
60
Fig. 8 – Discrete Meyer wavelet
The Morlet wavelet is given by; t2 2
cos ( 5t )
1
(8) Amplitude
ψ (t ) = e
−
This is a crude wavelet with [–4, 4] as theoretical effective support. However, a wider effective support can give more accurate results.
0.5
0
-0.5 40
45 50 55 Ef f ectiv e support
1
60
Amplitude
Fig. 9 – Discrete Meyer Scaling function 0.5 0
III. Results and discussion
-0.5 -1 -4
-2 0 2 Ef f ectiv e support
In a previous study [43], we have developed a data acquisition system which allows to collect PCG signals. We selected typical S1 and S2 heart sounds to be analyzed within the present study as shown in Fig. 1. We consider each heart sound separately for analysis as illustrated in Fig. 10 and Fig. 11.
4
Fig. 7 – The Morlet wavelet
As cited in equation (8) and illustrated in Fig. 7, the frequency content of the Morlet wavelet is constant. The VCCM, which is of frequency modulated nature, should give better result than the Morlet wavelet for heart sounds analysis.
Amplitude, %
II.5.
1
The Discrete Meyer wavelet
The Meyer wavelet [42] is symmetrical, ensures orthogonal and biorthogonal analysis and is infinitely derivable. However, it is not compactly supported, but decreases quickly towards 0 when t tends towards infinity. The FIR based approximation of the Meyer wavelet generates a pseudo–wavelet named the Discrete Meyer wavelet [21]. This approach allows fast wavelet coefficients calculation using the Discrete Wavelet Transform (DWT) which is not possible with the original Meyer wavelet. The discrete Meyer wavelet and its scaling function are respectively illustrated in Fig. 8 and Fig. 9. The Discrete Meyer wavelet is then considered for heart sounds analysis.
T1
M1
0.5 0 -0.5 -1 0
50
100 150 Time, ms
200
Fig. 10 – First heart sound (S1) of a normal subject
The S1 heart sound illustrated in Fig. 10 shows clearly the frequency modulated behavior of its M1 and T1 valvular components. The M1 component is more prominent within this S1 heart sound than T1. We can notice that oscillations within this S1 heart sound are of decreasing frequency and thus can obviously be considered as a chirp.
96
A.Djebbari, F. Bereksi–Reguig
1
1
P2
0.5
Amplitude, %
Amplitude, %
A2
0 -0.5 -1
A
C
B
0.5
0 0
50
100 Time, ms
150
0
50
100 150 Time, ms
200
Fig. 11 – Second heart sound (S2) of a normal subject
Fig. 12 – S1 square energy envelogram
After the systole period appears the S2 heart sound (Fig. 11) which is formed by the A2 and P2 valvular components. This heart sound is recorded after the S1 heart sound of Fig. 10 within the same phonocardiogram signal of Fig. 1. The axis limits of both S1 and S2 heart sounds are conserved through the overall TFRs to have the ability to study the interconnection between the time– frequency energy bursts and their original time appearing instants. The wavelet transform is a TSR ; however, a time– frequency representation can be generated from this TSR by estimating the central frequency of the mother wavelet throughout the scale dimension [21]. Thus, we can get TFRs by rearranging the TSR coefficients according to the frequency vector estimate. To evaluate the effectiveness of the VCCM as a mother wavelet candidate, we will use two mother wavelets, namely; the Morlet wavelet, which is a crude one, and the Discrete Meyer Wavelet, which is a discrete improvement of the original Meyer wavelet. These mother wavelets, which are used as opponent approaches, allow highlighting the ability of the VCCM to be an analyzing wavelet for heart sounds study. To allow rational comparison between the studied wavelets, common scales domain from 1 to 32 has been used for all wavelets. The TFRs are scaled by using a “jet” colormap going from blue to red colors. We begin by analyzing the first (S1) and second (S2) heart sounds using the Morlet wavelet and the discrete Meyer wavelet as basic methods. We accomplish than a time– scale analysis using the VCCM which is presented in section 2.1. In a view to better understand the TFRs we calculated, we introduce the square energy envelogram of each heart sound. Fig. 12 and Fig. 13 shows the energy distribution in accordance to time domain of S1 and S2 heart sounds respectively.
Each heart sound envelogram include two bursts. By detecting local maxima within the obtained envelograms, we can segment each heart sound temporal energy as annotated by; A, B and C for the S1 heart sound, and by D, E and F for the S2 heart sound which allow us to delimit these energy bursts.
Amplitude, %
1
F
D
0.5
E 0 0
50
100 Time, ms
150
Fig. 13 – S2 square energy envelogram
We can draw out a preliminary analysis of S1 and S2 heart sounds from the square energy envelograms of Fig. 12 and Fig. 13. This is a temporal analysis which allows appreciating the energy evolution though the time domain. We denote by; AB the duration between the lines A and B , BC the duration between the lines B and C in Fig. 12, and so on for durations DE and EF in Fig. 13 for the S2 heart sound. Both S1 and S2 contain two temporal energy bursts of different morphology. These segments are in relation with valvular activities; M1 and T1 for S1 and A2 and P2 for S2. TABLE II shows a summary of S1 and S2 heart sounds and their segments durations. We denote segments AB and DE as “First bursts” in TABLE III since they appear first within square energy envelograms of Fig. 12 and Fig. 13. Subsequently, segments BC and EF are denoted as “Second bursts” in TABLE III.
Heart sound S1 S2
TABLE II HEART SOUNDS SEGMENTS DURATIONS 2nd burst (ms) Duration (ms) 1st burst (ms) 111.51 36.38 75.13 107.48 26.76 80.72
Eyster [44] shown that the average duration of the S1 heart sound is about 131 ms and that of the S2 heart sound is around 128 ms. However, Amit et al. [45] shown that heart sound can be modulated by the respiration activity and may often present variability in
97
A.Djebbari, F. Bereksi–Reguig
-3
terms of duration. Indeed, as mentioned in TABLE II, the S1 heart sound lasts slightly longer than the S2 heart sound. This is due to the split phenomenon that can affect both S1 and S2 heart sounds.
x 10
PSD
8
TABLE III shows the contribution of each energy segment (AB and BC for S1, DE and EF for S2) as percentage of the overall temporal energy.
H
6 4 2 0 0
TABLE III TEMPORAL ENERGY PERCENTAGES WITHIN SQUARE ENERGY ENVELOGRAMS 2nd burst Heart sound 1st burst S1 61.28 38.72 S2 18.39 81.61
We notice that the main energy of the S1 heart sound is concentrated for frequencies greater than the “G” line frequency (48.25 Hz). The S2 heart sound spectral energy is localized for frequencies less than the “H” line frequency (63.25 Hz). By recalling results shown in TABLE III, we can deduce that there is an energy twist between the time and the frequency domains. Indeed, the first temporal burst of the S1 heart sound shows 61.28% and its second burst within the frequency domain is of 64.82% of energy. Moreover, the second temporal burst is of 38.72 % which is close to the first frequency burst of 35.18 %. We can understand this preliminary finding by the chirp nature of heart sounds. The energy of a heart sound over the time–frequency plane follows a hyperbolic frequency modulated behavior. Two bursts appears for both S1 and S2 heart sounds because they are both formed by two valvular components, namely M1 and T1 for S1, and A2 and P2 for S2. This energy twist is also verified for the S2 heart sound. Indeed, we observe that the temporal energy percentages are around 18 and 81% for the first and second bursts ( TABLE III). These values are inversed for the frequency domain (TABLE IV). Taking into consideration this time and frequency partition carried out for both the S1 and S2 heart sounds which paves the way to the time–frequency analysis, we can then discuss more accurately the obtained TFRs. Fig. 16, Fig. 17 and Fig. 18 shows TFRs of the S1 heart sound calculated from 1 to 32 as scales domain by means of the Morlet wavelet, the discrete Meyer wavelet and the VCCM adapted wavelet respectively. The S2 heart sound analysis is presented in Fig. 19, Fig. 20 and Fig. 21.
G
PSD
1 0.5 0 100 150 Frequency , Hz
200
250
TABLE IV SPECTRAL ENERGY PERCENTAGES WITHIN S1 AND S2 PSDS 2nd burst Heart sound 1st burst S1 35.18 64.82 S2 81.04 18.96
x 10
50
200
We quantify the contribution of each spectral energy burst as a percentage calculated over the total energy through the frequency domain. TABLE IV shows the distribution of the spectral energy in accordance to each segment. We denote by “First burst” and “Second burst” the regions localized below and above lines “G” and “H” for S1 and S2 heart sounds power spectra respectively.
-3
0
100 150 Frequency , Hz
Fig. 15 – Periodogram PSD estimate of the S2 heart sound, H: 63.25 Hz.
The first reading we can draw out from TABLE III is that the main energy of S1 heart sound is concentrated within the first burst at 61.28 % whereas the S2 heart sound energy is concentrated within the second burst at 81.61 %. Spectral analysis is another helpful discussion tool we used in the present paper which allows elucidating the calculated TFRs. The periodogram PSD estimates of each heart sound are shown in Fig. 14 and Fig. 15. We localize the local minima of each PSD estimates and we symbolize them by “v” letter shaped triangles.
1.5
50
250
Fig. 14 – Periodogram PSD estimate of the S1 heart sound, G: 48.25 Hz
As for the temporal domain, the limit between fall and rise of the power spectrum is localized. We denote by “G” and “H” lines for PSDs estimates of S1 and S2 heart sounds respectively. In Fig. 14, two energy regions are well localized and are graphically separated by the line denoted “G” at 48.45 Hz. We observe the same shape in Fig. 15. The line denoted “H” at 63.25 Hz delimitates the two energy bursts of the S2 heart sound.
98
A.Djebbari, F. Bereksi–Reguig
The same analysis approach was adopted for the S2 heart sound through Fig. 19, Fig. 20 and Fig. 21.
250
frequency, Hz
250
A
frequency, Hz
200
B
A
200
C
150
B
C
150 100 50
G
0
100
0
50
50
G 50
100 150 time, ms
200
Fig. 18 – TFR of the S1 heart sound by means of the VCCM adapted wavelet
0 0
100 150 time, ms
200
The chirp shape within the overall TFRs calculated for the S1 and the S2 heart sound reflects the frequency modulated nature of heart sounds. The TFRs calculated by means of the VCCM as mother wavelet blow up energized regions. Indeed, the region AB(f>G) reaches values around 225 Hz. The center of bursts of higher magnitude denoted by red color indicates the instantaneous frequency that can express the time– frequency behavior of the analyzed heart sound. It is obvious by comparing the TFR calculated by the VCCM (Fig. 18) to those of the Morlet wavelet (Fig. 16) and the discrete Meyer wavelet (Fig. 17) that the VCCM is suitable for heart sound analysis since the obtained TFR evidently represents the frequency modulated components of the S1 heart sound. Similarly, the TFRs of Fig. 19, Fig. 20 and Fig. 21 illustrate the analysis results of the S2 heart sound using the Morlet wavelet, the discrete Meyer wavelet and the VCCM wavelet respectively.
Fig. 16 – TFR of the S1 heart sound by means of the Morlet wavelet
We use a specific notation to lighten the text; for example, the AB region within the TFR denotes the energy region delimited by the temporal lines A and B. We denote by the AB(f
G) for frequencies greater than the G frequency, and so on. Taking into account the square energy envelogram of Fig. 12, we can recognize the high energy burst appearing in the AB region as red contours on the TFR of Fig. 16. This is confirmed by the power spectrum of Fig. 14 since the energy in the frequency domain is distributed as a narrow burst before the G line and as a spread burst after this same delimiting line. Within the TFR of Fig. 16, the AB energy in concentrated above the G frequency and the BC region is localized below this same frequency. 250
A
B
250
C
D
200 150
frequency, Hz
frequency, Hz
200
100 50
G
E
F
150 100
H
50
0 0
50
100 150 time, ms
200
0 0
Fig. 17 – TFR of the S1 heart sound by means of the discrete Meyer wavelet
50
100 time, ms
150
Fig. 19 – TFR of the S2 heart sound by means of the Morlet wavelet, Scales (1:32)
The main contribution of the TFR is the interrelationship made between the time and the frequency domains to better understand the energy distribution of the analyzed signal. Indeed, we can observe in Fig. 16, Fig. 17 and Fig. 18 that the AB narrow temporal burst appearing in Fig. 12 corresponds to a spread TFR energy region in the frequency domain as shown in Fig. 16. Furthermore, the BC energy which is of relatively reduced magnitude in comparison with the AB temporal burst.
The TFR of Fig. 21 shows the energy content of TFRs of Fig. 19 and Fig. 20, which are almost similar, on top of that is revealing energy bursts which are not detectable by the Morlet and the discrete Meyer wavelets. Indeed, the DE time–frequency region within Fig. 21 is more representative of the S2 heart sound energy. We can obviously remark that the TFR of Fig. 21 calculated by the VCCM wavelet gives the best result. Indeed, the resulting TFR is well representative of the square energy envelogram of Fig. 13 and the power spectrum of Fig. 15.
99
A.Djebbari, F. Bereksi–Reguig
decreasing in frequency to 40.08%, the chirplet based TFR boosts the AB(f
250
D
frequency, Hz
200
E
F
150 100
TABLE VII ENERGY DISTRIBUTION OF THE S1 HEART SOUND (VCCM ADAPTED WAVELET) frequency AB BC f>G 15.93 07.36 f
H
50 0 0
50
100 time, ms
150
Fig. 20 – TFR of the S2 heart sound by means of the discrete Meyer wavelet
This analysis can be extended to the second heart sound. TABLE VIII and TABLE IX show the chirp behavior through the bold text values percentages of the TFR total energy.
We introduce an energy percentage calculation within the time frequency plane to clarify the impact of the proposed VCCM adapted wavelet.
TABLE VIII ENERGY DISTRIBUTION OF THE S2 HEART SOUND (MORLET WAVELET) frequency DE EF f>H 05.33 05.52 f
250
D
frequency, Hz
200
E
F
150 100
The Morlet wavelet and the discrete Meyer wavelet can therefore regenerate the chirp shape within the time– frequency plane.
H
50 0 0
50
100 time, ms
150
TABLE IX ENERGY DISTRIBUTION OF THE S2 HEART SOUND (DISCRETE MEYER WAVELET) frequency DE EF f>H 03.99 04.12 f
Fig. 21 – TFR of S2 heart sound by means of the VCCM adapted wavelet
Tables V–X shows energy percentages calculated over the time–frequency plane for both S1 and S2 heart sounds. This approach is applied for the Morlet wavelet, the discrete Meyer wavelet and the VCCM adapted wavelet.
Comparing results of TABLE VIII and TABLE IX with those of TABLE X allow us to recognize the aptitude of the VCCM adapted wavelet to well represent heart sounds within the time–frequency plane. Indeed, the DE(f < H) energy region, which was of poor percentage over TABLE VIII and TABLE IX, is more representative of the heart sound content at 23.71% of the TFR total energy.
TABLE X ENERGY DISTRIBUTION OF THE S1 HEART SOUND (MORLET WAVELET) Frequency AB BC f>G 15.28 08.72 f
TABLE X ENERGY DISTRIBUTION OF THE S2 HEART SOUND (VCCM ADAPTED WAVELET) frequency DE EF f>H 4.83 05.43 f
We mark the energy values for each line in relation with a detected energy within the time–frequency plane as bold text. TABLE X and TABLE VI show extremities of the chirp behavior of heart sounds as 15.28% and 49.29% of the overall energy. Indeed, this can be confirmed by TFRs of Fig. 16 and Fig. 17 which presents and AB(f
In order to yield convenient representations of both S1 and S2 heart sounds, we limit contours plot at –6 dB within the VCCM TFRs of Fig. 18 and Fig. 21. The value –6 dB has been selected after several tests to get a clarified TFR. Therefore, the obtained representations are more informative about the chirplet nature of heart sounds. The center of each contour within Fig. 22 and Fig. 23 traces the frequency modulation nature of both S1 and S2 heart sounds.
TABLE VI ENERGY DISTRIBUTION OF THE S1 HEART SOUND (DISCRETE MEYER WAVELET) frequency AB BC f>G 13.50 07.78 f
By analyzing TABLE VII, we can obviously notice the chirp behavior going from high frequency domain, passing by the AB(f
100
A.Djebbari, F. Bereksi–Reguig
chirplet–based mother wavelet and found that it is a reliable tool for heart sound analysis since it yields a better TFR in comparison to that obtained by means of Morlet wavelet as well as the discrete Meyer wavelet as opponent approaches. By reaching 98.59% of cross–correlation between the adapted wavelet and the original chirplet for 15 as fitting polynomial order, we can assume that the original chirplet model can be used as analyzing wavelet. The Time–Frequency Representations (TFRs) are generated from the Time–Scale Representations (TSRs) by estimating the central frequency of the mother wavelet. The VCCM related TFRs achieve a better analysis towards heart sounds. Indeed, the time– frequency plane is well swept by the VCCM than the Morlet wavelet or the discrete Meyer wavelet. Furthermore, the VCCM represents adequately more than 80% of the analyzed S1 and S2 heart sounds energy which is in direct relation with their frequency modulated components. The Morlet wavelet and the discrete Meyer wavelet yields reduced energy percentages in comparison to those obtained by the VCCM wavelet. Therefore, the VCCM wavelet is more suitable for heart sounds characterization within the time–frequency plane.
frequency, Hz
150
100
50
0 0
50
100 150 time, ms
200
Fig. 22 – Contour plot at –6dB of the VCCM TFR of the S1 heart sound
We can easily notice the great improvement in terms of readability of the obtained TFR in Fig. 22 and Fig. 23. Therefore, the VCCM proves its ability to analyze adequately the frequency modulated content. The chirplet model can be considered as an ad hoc wavelet for heart sounds analysis since the obtained TFRs are better.
frequency, Hz
150
100
50
References 0 0
50
100 time, ms
150
[1]
Fig. 23 – Contour plot at –6dB of the VCCM TFR of the S2 heart sound [2]
TABLE XI gather in energy percentages in relation with the chirp shape within heart sounds. The sum of the following energies: AB(f > G), AB(f < G) and BC(f < G) is calculated for each mother wavelet. This “L” letter shape within Tables V–X is correlated with the chirp content of heart sounds.
[3]
[4]
TABLE XI CHIRPLET ENERGY PERCENTAGE (%) WITHIN TFR CALCULATED BY MORLET WAVELET, DISCRETE MEYER WAVELET AND THE VCCM
[5]
ADAPTED WAVELET
Heart sound Morlet S1 75.81 S2 81.13 dMeyer: discrete Meyer wavelet
dMeyer 77.06 82.91
VCCM 80.19 84.09 [6]
Therefore, we can ascertain hypotheses about the convenience of the VCCM adapted wavelet for heart sounds analysis. Indeed, TABLE XI shows that the VCCM adapted wavelet allow detecting chirplet behavior within both the S1 and the S2 heart sounds at higher energy percentages in comparison to the Morlet wavelet and the discrete Meyer wavelet.
[7]
[8]
IV.
Conclusion
The heart sounds are of modulated frequency content. Time–scale analysis as multiresolution approach allows investigating their complex content. We studied the
[9]
101
L. G. Durand and P. Pibarot, "Digital signal processing of the phonocardiogram: review of the most recent advancements," Crit Rev. Biomed. Eng, vol. 23, no. 3-4, pp. 163-219, 1995. J. Xu, L. G. Durand, and P. Pibarot, "Nonlinear transient chirp signal modeling of the aortic and pulmonary components of the second heart sound," IEEE Trans. Biomed. Eng, vol. 47, no. 10, pp. 1328-1335, Oct.2000. J. Xu, L. G. Durand, and P. Pibarot, "Extraction of the aortic and pulmonary components of the second heart sound using a nonlinear transient chirp signal model," IEEE Trans. Biomed. Eng, vol. 48, no. 3, pp. 277-283, Mar.2001. S. Choi and Z. Jiang, "Cardiac sound murmurs classification with autoregressive spectral analysis and multi-support vector machine technique," Comput. Biol. Med., vol. 40, no. 1, pp. 8-20, Jan.2010. A. F. Quiceno-Manrique, J. I. Godino-Llorente, M. BlancoVelasco, and G. Castellanos-Dominguez, "Selection of dynamic features based on time-frequency representations for heart murmur detection from phonocardiographic signals," Ann. Biomed. Eng, vol. 38, no. 1, pp. 118-137, Jan.2010. H. Tang, T. Li, and T. Qiu, "Noise and disturbance reduction for heart sounds in cycle-frequency domain based on nonlinear time scaling," IEEE Trans. Biomed. Eng, vol. 57, no. 2, pp. 325-333, Feb.2010. D. Avendano-Valencia, F. Martinez-Tabares, D. AcostaMedina, I. Godino-Llorente, and G. CastellanosDominguez, "TFR-based feature extraction using PCA approaches for discrimination of heart murmurs," Conf. Proc. IEEE Eng Med. Biol. Soc., vol. 2009, pp. 5665-5668, 2009. H. Guo and H. Huang, "[Blind source separation of heart signal and lung signal based on high-order statistics]," Sheng Wu Yi. Xue. Gong. Cheng Xue. Za Zhi., vol. 26, no. 2, pp. 413-416, Apr.2009. A. Jimenez-Gonzalez and C. J. James, "Extracting sources from noisy abdominal phonograms: a single-channel blind
A.Djebbari, F. Bereksi–Reguig
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
source separation method," Med. Biol. Eng Comput., vol. 47, no. 6, pp. 655-664, June2009. D. Kumar, P. Carvalho, M. Antunes, and J. Henriques, "Noise detection during heart sound recording," Conf. Proc. IEEE Eng Med. Biol. Soc., vol. 2009, pp. 3119-3123, 2009. A. Tassin, A. Kobeissi, L. Vitali, F. Rouleau, P. Ritter, G. Gaggini, and J. M. Dupuis, "Relationship between amplitude and timing of heart sounds and endocardial acceleration," Pacing Clin. Electrophysiol., vol. 32 Suppl 1, p. S101-S104, Mar.2009. T. Bombardini, V. Gemignani, E. Bianchini, E. Pasanisi, L. Pratali, M. Pianelli, F. Faita, M. Giannoni, G. Arpesella, R. Sicari, and E. Picano, "Post-exercise contractility, diastolic function, and pressure: operator-independent sensor-based intelligent monitoring for heart failure telemedicine," Cardiovasc. Ultrasound, vol. 7, p. 21, 2009. T. Bombardini, V. Gemignani, E. Bianchini, L. Venneri, C. Petersen, E. Pasanisi, L. Pratali, D. Alonso-Rodriguez, M. Pianelli, F. Faita, M. Giannoni, G. Arpesella, and E. Picano, "Diastolic time - frequency relation in the stress echo lab: filling timing and flow at different heart rates," Cardiovasc. Ultrasound, vol. 6, p. 15, 2008. X. Y. Zhang, E. MacPherson, and Y. T. Zhang, "Relations between the timing of the second heart sound and aortic blood pressure," IEEE Trans. Biomed. Eng, vol. 55, no. 4, pp. 1291-1297, Apr.2008. H. Nazeran, "Wavelet-based segmentation and feature extraction of heart sounds for intelligent PDA-based phonocardiography," Methods Inf. Med., vol. 46, no. 2, pp. 135-141, 2007. B. Al-Naami, J. Al-Nabulsi, H. Amasha, and J. Torry, "Utilizing wavelet transform and support vector machine for detection of the paradoxical splitting in the second heart sound," Med. Biol. Eng Comput., vol. 48, no. 2, pp. 177-184, Feb.2010. S. Babaei and A. Geranmayeh, "Heart sound reproduction based on neural network classification of cardiac valve disorders using wavelet transforms of PCG signals1," Comput. Biol. Med., vol. 39, no. 1, pp. 8-15, Jan.2009. A. Bagno, F. Anzil, V. Tarzia, V. Pengo, A. Ruggeri, and G. Gerosa, "Application of wavelet analysis to the phonocardiographic signal of mechanical heart valve closing sounds," Int. J. Artif. Organs, vol. 32, no. 3, pp. 166-172, Mar.2009. A. K. Mittra and N. K. Choudhari, "Time-frequency analysis of foetal heart sound signal for the prediction of prenatal anomalies," J. Med. Eng Technol., vol. 33, no. 4, pp. 296302, 2009. I. Daubechies, "The wavelet transform, time-frequency localization and signal analysis," Information Theory, IEEE Transactions on, vol. 36, no. 5, pp. 961-1005, 1990. P.Abry, Ondelettes et turbulence. Multirésolutions, algorithmes de décomposition, invariance d'échelles, Diderot ed. Paris: 1997. C. S. Lima and D. Barbosa, "Automatic segmentation of the second cardiac sound by using wavelets and hidden Markov models," Conf. Proc. IEEE Eng Med. Biol. Soc., vol. 2008, pp. 334-337, 2008. H. Sugiki, T. Murashita, N. Shiiya, Y. Matsui, and K. Sugiki, "Wavelet analysis of valve closing sound detects malfunction of bileaflet mechanical valve," J. Artif. Organs, vol. 11, no. 1, pp. 29-37, 2008. D. Kumar, P. Carvalho, M. Antunes, J. Henriques, Sa e Melo, R. Schmidt, and J. Habetha, "Third heart sound detection using wavelet transform-simplicity filter," Conf. Proc. IEEE Eng Med. Biol. Soc., vol. 2007, pp. 1277-1281, 2007. H. Sugiki, N. Shiiya, T. Murashita, T. Kunihara, K. Matsuzaki, T. Kubota, Y. Matsui, and K. Sugiki, "Wavelet analysis of bileaflet mechanical valve sounds," J. Artif. Organs, vol. 10, no. 1, pp. 16-21, 2007. A. A. Luisada and F. Portaluppi, "Considerations on the mechanism of the systolic click of mitral valve prolapse," Acta Cardiol., vol. 40, no. 3, pp. 301-305, 1985.
[27]
[28]
[29]
[30]
[31]
[32] [33]
[34]
[35]
[36]
[37]
[38]
[39] [40] [41] [42] [43]
[44]
[45]
J. Mazumdar and D. Woodard-Knight, "A mathematical study of semilunar valve vibration," J. Biomech., vol. 17, no. 8, pp. 639-641, 1984. M. Vukas, I. Wallentin, and A. C. Hjalmarson, "Preliminary observations on interventricular septum vibrations: an echocardiographic sign in aortic valvular stenosis," Acta Med. Scand., vol. 202, no. 5, pp. 363-366, 1977. T. N. HARRIS, L. LISKER, H. L. NEEDLEMAN, and H. A. SALTZMAN, "Spectrographic comparison of ranges of vibration frequency among some innocent cardiac murmurs in childhood and some murmurs of valvular insufficiency," Pediatrics, vol. 19, no. 1, pp. 57-67, Jan.1957. T.Tran, N.B.Jones, and J.C.Fothergill, "Heart sound simulator," Med. & Biol. Eng. & Comput., vol. 33, pp. 357359, 1995. L. G. Durand, Y. E. Langlois, T. Lanthier, R. Chiarella, P. Coppens, S. Carioto, and S. Bertrand-Bradley, "Spectral analysis and acoustic transmission of mitral and aortic valve closure sounds in dogs. Part 1. Modelling the heart/thorax acoustic system," Med. Biol. Eng Comput., vol. 28, no. 4, pp. 269-277, July1990. L. E. Franks, Signal theory. Englewood Cliffs, N.J: Prentice-Hall, 1969. S. Aggio, E. Baracca, C. Longhini, C. Brunazzi, L. Longhini, G. Musacci, and C. Fersini, "Noninvasive estimation of the pulmonary systolic pressure from the spectral analysis of the second heart sound," Acta Cardiol., vol. 45, no. 3, pp. 199-202, 1990. C. Longhini, E. Baracca, C. Brunazzi, M. Vaccari, L. Longhini, and F. Barbaresi, "A new noninvasive method for estimation of pulmonary arterial pressure in mitral stenosis," Am. J. Cardiol., vol. 68, no. 4, pp. 398-401, Aug.1991. A. Bartels and D. Harder, "Non-invasive determination of systolic blood pressure by heart sound pattern analysis," Clin. Phys. Physiol Meas., vol. 13, no. 3, pp. 249-256, Aug.1992. P.Goupillaud, A.Grossmann, and J.Morlet, "Cycle-octave and related transforms in seismic signal analysis," Geoexploration, vol. 23, pp. 85-102, 1984. O.Rioul and P.Duhamel, "Fast algorithms for discrete and continuous wavelet transforms," IEEE Trans. Inform. Theory, vol. 38, no. 2, pp. 569-586, 1992. M.Farge, "Wavelet Transforms and their Applications to Turbulence," Annual Review of Fluid Mechanics, vol. 24, pp. 395-458, 1992. P.Stoica and R.L.Moses, Introduction to Spectral Analysis Prentice-Hall, 1997, pp. 24-26. A.V.Oppenheim and R.W.Schafer, Discrete-Time Signal Processing Prentice-Hall, 1989, pp. 730-742. M.Misiti, Y.Misiti, G.Oppenheim, and J.M.Poggi, Les ondelettes et leurs applications, Hermes ed 2003. Ingrid Daubechies, "Ten lectures on wavelets," in CBMSNSF Conference series in applied mathematics 1992. A. Djebbari and F. Bereksi Reguig, "Smoothed-Pseudo Wigner-Ville Distribution of Normal and Aortic Stenosis Heart Sounds," Journal of Mechanics in Medicine and Biology, 2005. J. A. Eyster, "Studies on the venous pulse : II. the time relations of thevenous pulse and the heart sounds," J. Exp. Med., vol. 14, no. 6, pp. 594-605, Dec.1911. G. Amit, K. Shukha, N. Gavriely, and N. Intrator, "Respiratory modulation of heart sound morphology," Am. J. Physiol Heart Circ. Physiol, vol. 296, no. 3, p. H796H805, Mar.2009.
Authors’ information 1
Laboratory of Biomedical Engineering, Faculty of technology, University Abou Bekr Belkaid, Tlemcen, Algeria E-mail: [email protected] Tel. + 213 551 679 634 2 Laboratory of Biomedical Engineering, Faculty of technology, University Abou Bekr Belkaid, Tlemcen, Algeria E–mail: [email protected]
102
