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Wavelet decomposition analysis of heart rate variability in aerobic athletes
Autonomic Neuroscience: Basic and Clinical 90 Ž2001. 138–141 www.elsevier.comrlocaterautneu
Short communication
Wavelet decomposition analysis of heart rate variability in aerobic athletes Dieter Verlinde a , Frank Beckers a , Dirk Ramaekers b, Andre´ E. Aubert a,) a
Laboratory of Experimental Cardiology, UniÕersity Hospital Gasthuisberg, K.U. LeuÕen, LeuÕen, Belgium b General Internal Medicine, UniÕersity Hospital Gasthuisberg, K.U. LeuÕen, LeuÕen, Belgium
Abstract Heart rate variability ŽHRV. can be quantified, among others, in the frequency domain using digital signal processing ŽDSP. techniques. The wavelet transform is an alternative tool for the analysis of non-stationary signals. The implementation of perfect reconstruction digital filter banks leads to multi resolution wavelet analysis. Software was developed in LabVIEW. In this study, the average power was compared at each decomposition level of a tachogram, containing the consecutive RR-intervals of two groups of subjects: aerobic athletes and a control group. Compared to the controls, the aerobic athletes showed an increased power in all frequency bands. These results are in accordance with values obtained by spectral analysis using the Fourier transform, suggesting that wavelet analysis could be an appropriate tool to evaluate oscillating components in HRV, but in addition to classic methods, it also gives a time resolution. q 2001 Elsevier Science B.V. All rights reserved. Keywords: Heart rate variability; Wavelet transform; Non-stationary signals; Digital signal processing; Endurance training
1. Introduction Wavelet transform, a relative recent development, provides a general signal-processing technique that can be used in numerous biomedical applications. Its development was originally motivated by the desire to overcome the drawbacks of traditional Fourier analysis Že.g., fast Fourier transform: FFT., simultaneously providing time and frequency information of the signal. This multi resolution joint time–frequency analysis is therefore suited for the examination of non-stationary signals. Real signals, like an electrocardiogram ŽECG. or a tachogram, are mostly nonstationary. The information obtained by the wavelet decomposition can, among others, be used to compare differences in power or standard deviations at each of the wavelet levels analyzed. Power spectral determination of HRV are now routine tools for the assessment of autonomic function ŽPagani et al., 1986; Aubert et al., 1999.. Wavelet transform, on the other hand, being a very new analysis method ŽDaubechies, 1992., has only been applied in a few studies for the
)
Corresponding author. Department of Cardiology, UZ Gasthuisberg O-N, Herestr 49, 3000 Leuven, Belgium. Tel.: q32-16-345840; fax: q32-16-345844. E-mail address: [email protected] ŽA.E. Aubert..
analysis of HRV ŽGamero et al., 1996; Akay and Fischer, 1999; Wiklund et al., 1997; Petretta et al., 1999; Pichot et al., 1999; Hilton et al., 1999.. The purpose of this study was twofold: first, to develop and validate a wavelet transform algorithm for the analysis of the non-stationary characteristics of heart rate variability, and second, to apply this method to two groups of subjects: endurance-Žaerobic. trained athletes and a control group.
2. Methodology 2.1. Subjects and data recording After informed consent, two groups Žall male. were selected and compared: 10 endurance-trained athletes Žaerobic. and 10 subjects with a sedentary life style Žcontrols.. The athletes were of national competition level and trained between 6 and 9 h a week. Age ranged between 18 and 34 years, with no significant base line characteristic differences between the two groups Žaerobic: 24 " 6 years of age and control: 21.5 " 1.5 years of age.. Short time series of 10 min were obtained using ECG recordings in supine position. After RR-peak detection and
1566-0702r01r$ - see front matter q 2001 Elsevier Science B.V. All rights reserved. PII: S 1 5 6 6 - 0 7 0 2 Ž 0 1 . 0 0 2 8 4 - 3
D. Verlinde et al.r Autonomic Neuroscience: Basic and Clinical 90 (2001) 138–141
139
Fig. 1. DWT coefficients of the various decomposition levels of the tachogram in the upper left corner.
visual inspection by the operator, a file containing the consecutive RR-intervals was exported. Premature supraventricular and ventricular beats, missed beats and pauses were filtered and replaced by an interpolated value ŽAubert et al., 1999.. 2.2. Continuous waÕelet transform Just like the Fourier transform, wavelet analysis Ždiscrete wavelet transform: DWT. is calculated by taking the inner product of the original signal x Ž t . and some basis functions. Transforms measure the ‘similarity’ between the signal and the basis functions, resulting in coefficients, which indicate how close the signal is to this particular basis function. Wavelet analysis, however, has one very important simplification: all basis functions are scaled Žstretched or compressed. versions of the same prototype, called the mother wavelet c Ž t .. Details about mathematical backgrounds of wavelet transform can be found in recent textbooks such as: Young Ž1993., Strang and Nguyen Ž1997. and Mallat Ž1998..
Instrument Engineering Workbench, National Instruments, Austin, TX, USA.. LabVIEW is a programming language for the collection, analysis and display of measured data, which offers an easy, graphical way of programming. The basic idea is that the visual display of a program is more appropriate than textual, because the data flow is easier to follow. Each program created in LabVIEW is called a virtual instrument ŽVI., which can be used in another program as a sub-VI to create larger programs. The main program for wavelet decomposition has been built with several home-made sub-VIs.
2.3. Software The software used to implement the algorithm is written in a graphical language LabVIEW 5.1 ŽLaboratory Virtual
Fig. 2. Power of the HRV signal in different frequency bands.
140
D. Verlinde et al.r Autonomic Neuroscience: Basic and Clinical 90 (2001) 138–141
2.4. Statistical analysis The statistical analysis of all the results is performed with SPSS ŽScientific Packages for Social Sciences, Chicago, IL, USA.. Due to the small number of subjects, differences between the groups were analysed using a non-parametric test ŽMann–Whitney U-test.. Differences between WT and FFT were studied using a paired t-test. Statistical difference was accepted when p - 0.05. Values are presented as mean and standard deviation.
3. Results The proper functioning of the main program was first tested using mathematical test signals. More specifically, both stationary and non-stationary sine waves were used and power spectrum results were compared to values obtained with FFT method. The results were as expected and therefore, the program was applied to physiological data. The major difference between the control group and the aerobic athletes was in heart rate: 73 " 14 vs. 50 " 4.6 beatsrmin. A typical example of a 10-min tachogram of a normal subject, resampled at 2 Hz and truncated at 1024 samples is depicted in Fig. 1, left upper part. The DC component was removed from the HRV signal so no masking influence of the DC component on lower frequencies would occur. The temporal contribution of the frequency bands is presented in Fig. 1. The frequency bands are: L1: 0.5–1 Hz; L2: 0.25–0.5 Hz; L3: 0.125–0.25 Hz;
Fig. 3. Ža. Power distribution in different frequency bands calculated with FFT; Žb. power distribution for same population obtained from wavelet analysis.
L4: 0.065–0.125 Hz; L5: 0.031125–0.0625 Hz; L6: 0.015625–0.03125 Hz. Although the subject was in supine position throughout the recording, some temporal variations are seen, especially at level 1 Ž0.5–1 Hz.. Fig. 2 shows the power of all WT coefficients, corresponding to the different frequency bands from Fig. 1. The average power at each decomposition level of tachograms from the total population of controls and aerobic athletes is shown in Fig. 3. Fig. 3a shows the spectral power in the different frequency bands, calculated with FFT method, and Fig. 3b calculated with the WT. Compared to the controls, the aerobic athletes showed an increased power in all frequency bands Ž p - 0.05. except for the L1 frequency band. No statistical difference was found between results obtained from FFT and WT except for the L1 frequency band.
4. Discussion WT offers superior time resolution and time localisation compared to FFT or STFT. WT analysis is also not restricted to stationary signals. It offers rapid frequency decomposition with time resolution, useful when one is interested in a particular power spectral band over time and a potential use to assess fractal characteristics and pattern recognition ŽKadambe et al., 1999.. Another advantage of WT is the ability to detect short-lasting events of low amplitude superimposed on large signal deflections ŽGamero et al., 1996.. The advantage of the WT over FFT and autoregressive models is that it can be used on non-stationary time series and that no assumptions have to be made about model parameters, which is the case in the autoregressive models. Analysis of HRV is thought to allow insight in the modulation of heart rate by the autonomous nervous system ŽAkselrod et al., 1981., especially if the activity in the HF spectrum Žabove 0.15 Hz. has been correlated with parasympathetic modulation ŽAubert et al., 1996; Aubert and Ramaekers, 1999.. The results of this study show that aerobic athletes with a low-resting heart rate have indications of increased parasympathetic modulation in the timeas well as in the frequency-domain. Results of this study show that aerobic athletes, with a low-resting heart rate, have indications of increased power in all frequency bands compared to the control Žsedentary. group. This would imply an increased modulation of the ANS, especially of the parasympathetic component. Long-term adaptation to physical training is a complex phenomenon since it results from a mixture of several factors: biochemical, structural, metabolic, hormonal and neural. Additional modulating influences derive from genetic components, the age of the subject, the intensity of the exercise training routine and the simultaneous presence
D. Verlinde et al.r Autonomic Neuroscience: Basic and Clinical 90 (2001) 138–141
of short-time after effects from previous day’s activity. It should be stressed, however, that this study concerns only a young Ž18–34 years of age. and all male population. The importance of these findings lies in the application of HRV for risk stratification in patients. The impact of age and gender on HRV is well known. As HRV indices, especially from aerobic athletes, are different from those of sedentary subjects, it can be concluded that training level and type have to be taken into account as well, besides age and gender for prognostic stratification from HRV. Moreover, physical training has been suggested to decrease cardiovascular mortality and sudden cardiac death. It can be hypothesized further that physical activity has beneficial effects on the cardiovascular risk profile.
5. Conclusion A wavelet transform method has been developed and implemented in LabVIEW. The frequency and power information obtained by this method was in accordance with values obtained by spectral analysis using standard Fourier transform methods. The main advantage of the wavelet decomposition analysis is the additional information about the timing when the various spectral components appear in the signal. In this way, sudden amplitude andror frequency jumps can be detected. Results of this analysis showed that HRV is affected by chronic exercise, especially in endurance-trained athletes. This group has a lower heart rate and increased vagal modulation. Finally, it can be suggested that aerobic exercising helps in maintaining and developing cardiovascular fitness in a general population.
Acknowledgements The authors wish to thank Bruno Collier for collecting the data. This research has been supported with a grant from the Flemish Institute for the advancement of scientific and technological research in the industry ŽIWT..
141
References Akay, M., Fischer, R., 1999. Fractal analysis of HRV signals: a comparative study. Methods Inf. Med. 36, 271–273. Akselrod, S., Gordon, D., Ubel, F.A., Shannon, D., Barger, A.C., Cohen, R., 1981. Power spectrum analysis of heart rate fluctuations: a quantitative probe of beat-to-beat cardiovascular control. Science 213, 220–222. Aubert, A.E., Ramaekers, D., 1999. Neurocardiology: the benefits of irregularity. The basics of methodology, physiology and current clinical applications. Acta Cardiol. 54, 107–120. Aubert, A.E., Ramaekers, D., Cuche, Y., Lysens, R., Ector, H., Van de Werf, F., 1996. Effect of long term physical training on heart rate variability. IEEE Comp. Cardiol. 22, 17–20. Aubert, A.E., Ramaekers, D., Beckers, F., Breem, R., Denef, C., Van de Werf, F., Ector, H., 1999. The analysis of heart rate variability in unrestrained rats. Validation of method and results. Comp. Methods Programs Biomed. 60, 197–213. Daubechies, I., 1992. Ten Lectures on Wavelets. SIAM, Philadelphia, 397 pp. Gamero, L.G., Risk, M., Sobh, J.F., Ramirez, A.J., Saul, J.P., 1996. Heart rate variability analysis using wavelet transform. IEEE Comp. Cardiol. 177–180. Hilton, M.F., Bates, R.A., Godfrey, K.R., Chappell, M.J., Cayton, R.M., 1999. Evaluation of frequency and time–frequency spectral analysis of heart rate variability as a diagnostic marker of the sleep apnoea syndrome. Med. Biol. Eng. Comput. 37, 760–769. Kadambe, S., Murray, R., Boudreaux-Bartels, G.F., 1999. Wavelet transform-based QRS detector. IEEE Trans. Biomed. Eng. 46, 838–848. Mallat, S., 1998. A Wavelet Tour of Signal Processing. Academic Press, San Diego, 577 pp. Pagani, M., Montano, M., Porta, A., Malliani, A., Abboud, F.M., Birkett, C., Somers, C.K., 1986. Relationship between spectral components of cardiovascular variabilities and direct measures of muscle sympathetic nerve activity in humans. Circulation 95, 1141–1448. Petretta, M., Spinelli, L., Marciano, F., Vicario, M.L., Testa, G., Signorini, A., Bonaduce, D., 1999. Wavelet transform analysis of heart rate variability during dipyramidole-induced myocardial ischemia: relation to angiographic severity and echocardiographic dyssynergy. Clin. Cardiol. 22, 201–206. Pichot, V., Gaspoz, J.M., Molliex, S., Antoniadis, A., Busso, T., Roche, F., Costes, F., Quintin, L., Lacour, J.R., Barthelemy, J.C., 1999. Wavelet transform to quantify heart rate variability and to assess its instantaneous changes. J. Appl. Physiol. 86, 1081–1091. Strang, G., Nguyen, T., 1997. Wavelets and Filter Banks. WellesleyCambridge Press, Wellesley, MA, 420 pp. Wiklund, U., Akay, M., Niklasson, U., 1997. Short-term analysis of heart-rate variability by adapted wavelet transforms. IEEE Eng. Med. Biol., 113–138. Young, R.K., 1993. Wavelet Theory and its Applications. Kluwer Academic Publishing, Dordrecht, The Netherlands, 223 pp.
Short communication
Wavelet decomposition analysis of heart rate variability in aerobic athletes Dieter Verlinde a , Frank Beckers a , Dirk Ramaekers b, Andre´ E. Aubert a,) a
Laboratory of Experimental Cardiology, UniÕersity Hospital Gasthuisberg, K.U. LeuÕen, LeuÕen, Belgium b General Internal Medicine, UniÕersity Hospital Gasthuisberg, K.U. LeuÕen, LeuÕen, Belgium
Abstract Heart rate variability ŽHRV. can be quantified, among others, in the frequency domain using digital signal processing ŽDSP. techniques. The wavelet transform is an alternative tool for the analysis of non-stationary signals. The implementation of perfect reconstruction digital filter banks leads to multi resolution wavelet analysis. Software was developed in LabVIEW. In this study, the average power was compared at each decomposition level of a tachogram, containing the consecutive RR-intervals of two groups of subjects: aerobic athletes and a control group. Compared to the controls, the aerobic athletes showed an increased power in all frequency bands. These results are in accordance with values obtained by spectral analysis using the Fourier transform, suggesting that wavelet analysis could be an appropriate tool to evaluate oscillating components in HRV, but in addition to classic methods, it also gives a time resolution. q 2001 Elsevier Science B.V. All rights reserved. Keywords: Heart rate variability; Wavelet transform; Non-stationary signals; Digital signal processing; Endurance training
1. Introduction Wavelet transform, a relative recent development, provides a general signal-processing technique that can be used in numerous biomedical applications. Its development was originally motivated by the desire to overcome the drawbacks of traditional Fourier analysis Že.g., fast Fourier transform: FFT., simultaneously providing time and frequency information of the signal. This multi resolution joint time–frequency analysis is therefore suited for the examination of non-stationary signals. Real signals, like an electrocardiogram ŽECG. or a tachogram, are mostly nonstationary. The information obtained by the wavelet decomposition can, among others, be used to compare differences in power or standard deviations at each of the wavelet levels analyzed. Power spectral determination of HRV are now routine tools for the assessment of autonomic function ŽPagani et al., 1986; Aubert et al., 1999.. Wavelet transform, on the other hand, being a very new analysis method ŽDaubechies, 1992., has only been applied in a few studies for the
)
Corresponding author. Department of Cardiology, UZ Gasthuisberg O-N, Herestr 49, 3000 Leuven, Belgium. Tel.: q32-16-345840; fax: q32-16-345844. E-mail address: [email protected] ŽA.E. Aubert..
analysis of HRV ŽGamero et al., 1996; Akay and Fischer, 1999; Wiklund et al., 1997; Petretta et al., 1999; Pichot et al., 1999; Hilton et al., 1999.. The purpose of this study was twofold: first, to develop and validate a wavelet transform algorithm for the analysis of the non-stationary characteristics of heart rate variability, and second, to apply this method to two groups of subjects: endurance-Žaerobic. trained athletes and a control group.
2. Methodology 2.1. Subjects and data recording After informed consent, two groups Žall male. were selected and compared: 10 endurance-trained athletes Žaerobic. and 10 subjects with a sedentary life style Žcontrols.. The athletes were of national competition level and trained between 6 and 9 h a week. Age ranged between 18 and 34 years, with no significant base line characteristic differences between the two groups Žaerobic: 24 " 6 years of age and control: 21.5 " 1.5 years of age.. Short time series of 10 min were obtained using ECG recordings in supine position. After RR-peak detection and
1566-0702r01r$ - see front matter q 2001 Elsevier Science B.V. All rights reserved. PII: S 1 5 6 6 - 0 7 0 2 Ž 0 1 . 0 0 2 8 4 - 3
D. Verlinde et al.r Autonomic Neuroscience: Basic and Clinical 90 (2001) 138–141
139
Fig. 1. DWT coefficients of the various decomposition levels of the tachogram in the upper left corner.
visual inspection by the operator, a file containing the consecutive RR-intervals was exported. Premature supraventricular and ventricular beats, missed beats and pauses were filtered and replaced by an interpolated value ŽAubert et al., 1999.. 2.2. Continuous waÕelet transform Just like the Fourier transform, wavelet analysis Ždiscrete wavelet transform: DWT. is calculated by taking the inner product of the original signal x Ž t . and some basis functions. Transforms measure the ‘similarity’ between the signal and the basis functions, resulting in coefficients, which indicate how close the signal is to this particular basis function. Wavelet analysis, however, has one very important simplification: all basis functions are scaled Žstretched or compressed. versions of the same prototype, called the mother wavelet c Ž t .. Details about mathematical backgrounds of wavelet transform can be found in recent textbooks such as: Young Ž1993., Strang and Nguyen Ž1997. and Mallat Ž1998..
Instrument Engineering Workbench, National Instruments, Austin, TX, USA.. LabVIEW is a programming language for the collection, analysis and display of measured data, which offers an easy, graphical way of programming. The basic idea is that the visual display of a program is more appropriate than textual, because the data flow is easier to follow. Each program created in LabVIEW is called a virtual instrument ŽVI., which can be used in another program as a sub-VI to create larger programs. The main program for wavelet decomposition has been built with several home-made sub-VIs.
2.3. Software The software used to implement the algorithm is written in a graphical language LabVIEW 5.1 ŽLaboratory Virtual
Fig. 2. Power of the HRV signal in different frequency bands.
140
D. Verlinde et al.r Autonomic Neuroscience: Basic and Clinical 90 (2001) 138–141
2.4. Statistical analysis The statistical analysis of all the results is performed with SPSS ŽScientific Packages for Social Sciences, Chicago, IL, USA.. Due to the small number of subjects, differences between the groups were analysed using a non-parametric test ŽMann–Whitney U-test.. Differences between WT and FFT were studied using a paired t-test. Statistical difference was accepted when p - 0.05. Values are presented as mean and standard deviation.
3. Results The proper functioning of the main program was first tested using mathematical test signals. More specifically, both stationary and non-stationary sine waves were used and power spectrum results were compared to values obtained with FFT method. The results were as expected and therefore, the program was applied to physiological data. The major difference between the control group and the aerobic athletes was in heart rate: 73 " 14 vs. 50 " 4.6 beatsrmin. A typical example of a 10-min tachogram of a normal subject, resampled at 2 Hz and truncated at 1024 samples is depicted in Fig. 1, left upper part. The DC component was removed from the HRV signal so no masking influence of the DC component on lower frequencies would occur. The temporal contribution of the frequency bands is presented in Fig. 1. The frequency bands are: L1: 0.5–1 Hz; L2: 0.25–0.5 Hz; L3: 0.125–0.25 Hz;
Fig. 3. Ža. Power distribution in different frequency bands calculated with FFT; Žb. power distribution for same population obtained from wavelet analysis.
L4: 0.065–0.125 Hz; L5: 0.031125–0.0625 Hz; L6: 0.015625–0.03125 Hz. Although the subject was in supine position throughout the recording, some temporal variations are seen, especially at level 1 Ž0.5–1 Hz.. Fig. 2 shows the power of all WT coefficients, corresponding to the different frequency bands from Fig. 1. The average power at each decomposition level of tachograms from the total population of controls and aerobic athletes is shown in Fig. 3. Fig. 3a shows the spectral power in the different frequency bands, calculated with FFT method, and Fig. 3b calculated with the WT. Compared to the controls, the aerobic athletes showed an increased power in all frequency bands Ž p - 0.05. except for the L1 frequency band. No statistical difference was found between results obtained from FFT and WT except for the L1 frequency band.
4. Discussion WT offers superior time resolution and time localisation compared to FFT or STFT. WT analysis is also not restricted to stationary signals. It offers rapid frequency decomposition with time resolution, useful when one is interested in a particular power spectral band over time and a potential use to assess fractal characteristics and pattern recognition ŽKadambe et al., 1999.. Another advantage of WT is the ability to detect short-lasting events of low amplitude superimposed on large signal deflections ŽGamero et al., 1996.. The advantage of the WT over FFT and autoregressive models is that it can be used on non-stationary time series and that no assumptions have to be made about model parameters, which is the case in the autoregressive models. Analysis of HRV is thought to allow insight in the modulation of heart rate by the autonomous nervous system ŽAkselrod et al., 1981., especially if the activity in the HF spectrum Žabove 0.15 Hz. has been correlated with parasympathetic modulation ŽAubert et al., 1996; Aubert and Ramaekers, 1999.. The results of this study show that aerobic athletes with a low-resting heart rate have indications of increased parasympathetic modulation in the timeas well as in the frequency-domain. Results of this study show that aerobic athletes, with a low-resting heart rate, have indications of increased power in all frequency bands compared to the control Žsedentary. group. This would imply an increased modulation of the ANS, especially of the parasympathetic component. Long-term adaptation to physical training is a complex phenomenon since it results from a mixture of several factors: biochemical, structural, metabolic, hormonal and neural. Additional modulating influences derive from genetic components, the age of the subject, the intensity of the exercise training routine and the simultaneous presence
D. Verlinde et al.r Autonomic Neuroscience: Basic and Clinical 90 (2001) 138–141
of short-time after effects from previous day’s activity. It should be stressed, however, that this study concerns only a young Ž18–34 years of age. and all male population. The importance of these findings lies in the application of HRV for risk stratification in patients. The impact of age and gender on HRV is well known. As HRV indices, especially from aerobic athletes, are different from those of sedentary subjects, it can be concluded that training level and type have to be taken into account as well, besides age and gender for prognostic stratification from HRV. Moreover, physical training has been suggested to decrease cardiovascular mortality and sudden cardiac death. It can be hypothesized further that physical activity has beneficial effects on the cardiovascular risk profile.
5. Conclusion A wavelet transform method has been developed and implemented in LabVIEW. The frequency and power information obtained by this method was in accordance with values obtained by spectral analysis using standard Fourier transform methods. The main advantage of the wavelet decomposition analysis is the additional information about the timing when the various spectral components appear in the signal. In this way, sudden amplitude andror frequency jumps can be detected. Results of this analysis showed that HRV is affected by chronic exercise, especially in endurance-trained athletes. This group has a lower heart rate and increased vagal modulation. Finally, it can be suggested that aerobic exercising helps in maintaining and developing cardiovascular fitness in a general population.
Acknowledgements The authors wish to thank Bruno Collier for collecting the data. This research has been supported with a grant from the Flemish Institute for the advancement of scientific and technological research in the industry ŽIWT..
141
References Akay, M., Fischer, R., 1999. Fractal analysis of HRV signals: a comparative study. Methods Inf. Med. 36, 271–273. Akselrod, S., Gordon, D., Ubel, F.A., Shannon, D., Barger, A.C., Cohen, R., 1981. Power spectrum analysis of heart rate fluctuations: a quantitative probe of beat-to-beat cardiovascular control. Science 213, 220–222. Aubert, A.E., Ramaekers, D., 1999. Neurocardiology: the benefits of irregularity. The basics of methodology, physiology and current clinical applications. Acta Cardiol. 54, 107–120. Aubert, A.E., Ramaekers, D., Cuche, Y., Lysens, R., Ector, H., Van de Werf, F., 1996. Effect of long term physical training on heart rate variability. IEEE Comp. Cardiol. 22, 17–20. Aubert, A.E., Ramaekers, D., Beckers, F., Breem, R., Denef, C., Van de Werf, F., Ector, H., 1999. The analysis of heart rate variability in unrestrained rats. Validation of method and results. Comp. Methods Programs Biomed. 60, 197–213. Daubechies, I., 1992. Ten Lectures on Wavelets. SIAM, Philadelphia, 397 pp. Gamero, L.G., Risk, M., Sobh, J.F., Ramirez, A.J., Saul, J.P., 1996. Heart rate variability analysis using wavelet transform. IEEE Comp. Cardiol. 177–180. Hilton, M.F., Bates, R.A., Godfrey, K.R., Chappell, M.J., Cayton, R.M., 1999. Evaluation of frequency and time–frequency spectral analysis of heart rate variability as a diagnostic marker of the sleep apnoea syndrome. Med. Biol. Eng. Comput. 37, 760–769. Kadambe, S., Murray, R., Boudreaux-Bartels, G.F., 1999. Wavelet transform-based QRS detector. IEEE Trans. Biomed. Eng. 46, 838–848. Mallat, S., 1998. A Wavelet Tour of Signal Processing. Academic Press, San Diego, 577 pp. Pagani, M., Montano, M., Porta, A., Malliani, A., Abboud, F.M., Birkett, C., Somers, C.K., 1986. Relationship between spectral components of cardiovascular variabilities and direct measures of muscle sympathetic nerve activity in humans. Circulation 95, 1141–1448. Petretta, M., Spinelli, L., Marciano, F., Vicario, M.L., Testa, G., Signorini, A., Bonaduce, D., 1999. Wavelet transform analysis of heart rate variability during dipyramidole-induced myocardial ischemia: relation to angiographic severity and echocardiographic dyssynergy. Clin. Cardiol. 22, 201–206. Pichot, V., Gaspoz, J.M., Molliex, S., Antoniadis, A., Busso, T., Roche, F., Costes, F., Quintin, L., Lacour, J.R., Barthelemy, J.C., 1999. Wavelet transform to quantify heart rate variability and to assess its instantaneous changes. J. Appl. Physiol. 86, 1081–1091. Strang, G., Nguyen, T., 1997. Wavelets and Filter Banks. WellesleyCambridge Press, Wellesley, MA, 420 pp. Wiklund, U., Akay, M., Niklasson, U., 1997. Short-term analysis of heart-rate variability by adapted wavelet transforms. IEEE Eng. Med. Biol., 113–138. Young, R.K., 1993. Wavelet Theory and its Applications. Kluwer Academic Publishing, Dordrecht, The Netherlands, 223 pp.