Smoothed pseudo Wigner–Ville distribution as an alternative to Fourier transform in rats

Autonomic Neuroscience: Basic and Clinical 87 (2001) 258–267www.elsevier.com/ locate /autneu

Smoothed pseudo Wigner–Ville distribution as an alternative to Fouriertransform in rats

a,b , a a*Edmundo Pereira de Souza Neto MD , Marc-Antoine Custaud MD, , Jean Frutoso PhD, ,a a aLaurence Somody PhD, , Claude Gharib PhD MD, , Jacques-Olivier Fortrat PhD, MD

a ´ ˆ ´ ´Laboratoire de Physiologie de l’Environnement (Groupement d’Interet Public de l’Exercice), Faculte de Medecine Lyon Grange-Blanche, EA 645,8 Avenue Rockefeller, 69373 Lyon Cedex 08, France

b ˆ ´Hopital Cardio-vasculaire et Pneumologique Louis Pradel, 28 Avenue Doyen Lepine, 69394 Lyon Cedex 03, France

Abstract

Techniques for examining signals in the time and frequency domains are well-established tools. These tools have their limitations; theytell us in a broad sense where the signal component exists in the frequency domain, but they do not tell us how its frequencycharacteristics change over time. The time-frequency has become a powerful alternative for the analysis of signals. Among varioustime-frequency distribution methods, one of the most studied is the Wigner–Ville distribution. The aim of this study was to evaluate inconscious rats smoothed pseudo Wigner–Ville distribution (SPWVD) as an alternative to the fast Fourier transform (FFT) in RR intervalsand in systolic blood pressure (SBP), before and after adrenergic and cholinergic receptor blockade. Fourteen Wistar rats equipped with

21 21telemetry probe were evaluated: (1) under control conditions; (2) after injection of saline (100 ml kg i.v.); (3) after atenolol (1 mg kg21 21i.v.); (4) after atropine methyl nitrate (0.5 mg kg i.v.); and (5) after phentolamine (5 mg kg i.v.). FFT and SPWVD were applied to

RR intervals and SBP time series. Six-minute time series of RR intervals, systolic and diastolic pressures were analysed. The bias and2distribution of differences between FFT and SPWVD methods in RR intervals under base conditions were 1.460.4% (r 50.94; P,0.01)

2 2in LF/LF1HF; 1.560.5% (r 50.92; P,0.01) in HF/LF1HF and 4.861.9% (r 50.92; P,0.01) in LF/HF. In SBP the bias and2 2distribution were 1.560.8% (r 50.90; P,0.05) in LF/LF1HF and 1.760.6% (r 50.92; P,0.01) in HF/LF1HF. In the frequency

domain analysis of RR intervals and SBP there was no difference between FFT and SPWVD. The agreement between the methodsdemonstrates that in stationary signals both methods can be used interchangeably. SPWVD may be an interesting tool to analysebiomedical signals; it provides a good resolution at high frequency and a good frequency resolution at low frequencies independently ifsignals remain stationary. 2001 Elsevier Science B.V. All rights reserved.

Keywords: Fourier transform; Wigner–Ville distribution; Rats

1. Introduction completely achieved due to the Heisenberg uncertaintyprinciple. So Fourier transform methods are not generally

Frequency-domain techniques to measure heart rate or an appropriate approach in the analysis of signals withblood pressure variability are a non-invasive approach that transient components (Akay, 1995).allows an indirect evaluation of cardiovascular autonomic To capture time and frequency information, a differentcontrol. Fourier transforms are widely used tools in signal kind of signal representation is needed, which creates aprocessing. Although there are applications where Fourier function of two variables, time and frequency, that cap-transform could be used, there are others where it can tures transient effects in the signal. A more completeimpose severe limitations on spectral estimates calculated, representation of the signal can be obtained by usingthat of giving global frequency information and underlying time-frequency representations. The time-frequency meth-assumption of the periodic nature of the signals. The od has become a powerful alternative for the analysis ofsimultaneous mapping of time and frequency can never be transient signals. The time-frequency has variable time-

frequency resolution over the time-frequency plane byproviding good time resolution at high frequency and good

*Corresponding author. Tel.: 133-4-7877-7078; fax: 133-4-7801-frequency resolution at low frequencies (Cohen, 1989;7467.Flandrin, 1998). Among various time-frequency distribu-E-mail address: [email protected] (E. Pereira de

Souza Neto). tion methods one of the most studied is the Wigner–Ville

1566-0702/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved.PI I : S1566-0702( 00 )00211-3

E. Pereira de Souza Neto et al. / Autonomic Neuroscience: Basic and Clinical 87 (2001) 258 –267 259

distribution. The concept was first introduced by Wigner Wistar rats, 14 days before the study. The surgical(1932) and was re-introduced by Ville (1948). Like Fourier procedure has been described elsewhere (Brockway et al.,transform, it requires that the time series under analysis be 1991). Briefly, rats were anaesthetised with halothane (2%known at all times. The Wigner–Ville is the Fourier in oxygen) and under aseptic conditions, a midline abdomi-transform of the signal’s autocorrelation function with nal incision was made. The descending aorta was exposedrespect to the delay variable. It can be thought of as a and a fluid-filled sensor catheter was inserted in theshort-time Fourier transform where the windowing func- abdominal aorta above the iliac bifurcation. The tip of thetion is a time-scaled, time-reversed copy of the original telemetric catheter was placed upstream below the renalsignal. Mathematically, when the frequency components arteries. The abdomen was then closed and the transmitterdo not change during the analysing time interval these two anchored to the abdominal walls by suturing through a tabmethods are compatible (Novak and Novak, 1993; Flan- located along the length of the transmitter body. Transmit-drin, 1998). To our knowledge there is no study done to ter leads were sutured subcutaneously at the right shouldercompare the two methods in animals at rest and during (2 electrode) and at the left thigh (1 electrode), lead IIinduced changes of autonomic regulation in supposed registration.stationary situations. Body temperature was maintained using an electric

The aim of this study was to evaluate in conscious rats, heating pad during and after surgery until rats awakened.smoothed pseudo Wigner–Ville distribution (SPWVD) as During the postsurgical period the animals were ad-an alternative to the fast Fourier transform (FFT) in RR ministered 15 000 units penicillin s.c. for 3 days and wereintervals and in systolic blood pressure (SBP), before and monitored for general health and satisfactory blood pres-after adrenergic and cholinergic receptor blockade. sure pulse and heart rate.

Subsequently, 3 days before the study rats were re-anaesthetised with halothane (2% in oxygen) and a poly-

2. Materials and methods ethylene catheter was inserted into the lower inferior venacava via the left femoral vein. The catheter (no. 2, i.d. 0.38

2.1. Animals mm, e.d. 1.09 mm, Biotrol, Paris) was led under the skin toexit between the scapulae. The catheter was filled with

The investigation conforms to the Guide for the Care heparinized saline (50 U heparin /ml) and plugged with aand use of Laboratory Animals published by the US short piece of stainless steel wire.National Institutes of Health (NIH Publication No. 85-23,revised 1996). 2.4. Experimental protocol

During the quarantine period lasting 10 days, maleWistar rats (270620 g, IFFA-CREDO, Les Oncins, The rats were allowed to recover for at least 14 daysFrance) were housed in Plexiglass cages. All rats were after transmitter implantation and 2 days after cathetermaintained on a 12-h light /dark cycle (07:00–19:00 h insertion. Data recording was done only if the rats showed(light) and 19:00–07:00 h (dark)), controlled temperature no sign of infection and had normal weight gain. The rats(22618C) and humidity (60610%). They were fed with were kept individually in a Plexiglass cage placed abovestandard rat chow containing 0.3% sodium (Elevage UAR, the receiver.Epinay sur Orge, France) and tap water ad libitum. The study of RR intervals and blood pressure variability

was performed in each animal in a randomised order2.2. Telemetry system during 5 days (days 1–5). The conditions were: (1) under

control conditions (n514); (2) after injection of saline21The telemetry system was purchased from Data Sciences (100 ml kg i.v.; n514); (3) after b -adrenergic blockade1

21 21International (St Paul, MN). It is composed of four parts: by atenolol (1 mg kg i.v., 10 mg ml , Sigma, St Louis,an implantable transmitter (Tl 11-M2-C50-PXT), a re- MO; n514); (4) after atropine methyl nitrate injection (0.5

21 21ceiver (RLA 1020), a pressure reference module (C11 PR), mg kg i.v., 10 mg ml , Sigma; n514); (5) after21a data acquisition system with a consolidation matrix administration of phentolamine (5 mg kg i.v., 10 mg

21(BCM 100, for converting the radio-signal to electrical ml , Sigma; n514). b -Adrenergic blockade was demon-1

impulses) and a computer (with a UA10 PC processor card strated by the suppression of tachycardia to isoproterenol21for analogue signal processing). The monitoring of car- (0.2 mg kg i.v., Sigma). The efficacy of a -blockade by1

diovascular parameters by telemetry allows continuous phentolamine was assessed by the measurement of pressorrecordings in unrestrained and unstressed conscious ani- responses to an intravenous injection of an a-adrenoceptor

21mals (Rubini et al., 1993). agonist (phenylephrine 3 mg kg i.v., Sigma). The bloodpressure and heart rate responses to metacholine (0.1 mg

212.3. Surgical procedures kg i.v., Sigma) were used to ensure the efficacy ofcholinergic receptor blockade. Antagonists were repeated

Radio-telemetry transmitters were implanted in 14 male at the end of the procedure to confirm adequate blockade.

260 E. Pereira de Souza Neto et al. / Autonomic Neuroscience: Basic and Clinical 87 (2001) 258 –267

In most cases, the animals were quiet following in- at least 17 stationary sequences in each experimentaljection. condition to attenuate variable noise contributions and to

sharpen reproducible spectral peaks (Cerutti et al., 1991).All spectra of RR intervals were divided into three

2.5. Data analysisfrequency bands: very low frequency (VLF) from 0.0195to 0.25 Hz, low frequency (LF) from 0.26 to 0.75 Hz and

Recording sessions of 36 min were performed betweenhigh frequency (HF) from 0.76 to 4 Hz. VLF components

09:00 and 12:00 h. Continuous data acquisition at awere not considered in the further analysis. Absolute

sampling rate of 1000 Hz, was done on a PC Pentium 133power of each frequency band is expressed as cumulative

MHz with a 12-bit analogue-to-digital converter (AT-MIO-power across the frequency band. Normalised power was

16E-10; National Instruments, Austin, TX) equipped withobtained by dividing the cumulative power within each

software developed with LabVIEW 4.0.1 software (Nation-frequency band (LF or HF) by the sum of LF and HF; the

al Instruments).ratio of LF/HF was also calculated.

For each cardiac cycle, RR interval (from the ECGsignal), systolic, diastolic and mean blood pressures were

2.6.2. Smoothed pseudo-Wigner–Ville distributioncalculated. Premature and postextrasystolic beats were

(SPWVD)manually identified and replaced by linear interpolation.

The SPWVD is a two-dimensional function of time andThen, values with a larger deflection than three standard

frequency; a maximal time and frequency resolution isdeviations from the mean value were eliminated. These

achieved by using independent time and frequencyvalues were replaced by the mean value of the five

¨smoothing (Claasen and Mecklenbrauker, 1980; Novakpreceding and the five following beats. Such beats

and Novak, 1993; Novak et al., 1993). The SPWVD canrepresented,2% of all analysed beats.

provide a spectral profile for nearly every beat, eachThe assumption that underlies this correction is that the

depending on the preceding and subsequent events (NovakRR interval and blood pressure do not oscillate abruptly

and Novak, 1993; Novak et al., 1993, 1994).from the mean value and that the use of linear interpolation

The RR and SBP series were resampled at 18 Hz, i.e.,will introduce less high frequency noise (Aubert et al.,

twice the highest possible frequency of the signal, which is1999).

a prerequisite to study the entire spectrum (Claasen andThe selected RR interval and systolic blood pressure

¨Mecklenbrauker, 1980; Boashash, 1988; Cohen, 1989;data were transformed by linear interpolation to obtain

Novak and Novak, 1993). A non-linear band-pass filterequally spaced samples (De Boer et al., 1984) and both

was used to remove the baseline trend from all the signals.series were synchronised.

So, the very-low frequency oscillations (,0.1 Hz) wereFrom 36 min recording period for each condition (base,

removed. The detrended signal was converted into itssaline, atenolol, phentolamine and atropine) approximately

analytic form by the Hilbert transform (Ville, 1948). The6-min time series during which the rat was quiet and

use of an analytical signal with no negative frequenciesdisplayed minimal activity, were selected for analysis.

instead of the real signal is necessary to reduce cross-terms(Novak and Novak, 1993; Novak et al., 1993). The

2.6. Spectral analysis analytic signal was evaluated by the SPWVD with thefollowing parameters: the length of the temporal moving

2.6.1. Fast fourier Transform (FFT) window was 16 samples (0.88 s smoothing) and theData used for spectral analysis were generated from the frequency moving window was 128 samples. The spectrum

raw beat-by-beat time series of RR intervals and SBP by was obtained every 1.5 s, and each spectrum is an array ofcomputing values every 130 ms (Dt). The linear trend was 128 power values corresponding to 128 frequency valuesremoved to avoid its contribution to low frequencies equally spaced. The power spectrum based on SPWVD forpower. Data segments with standard deviations outside the a specific time is calculated by the data within a time95 confidence interval for the recording session were interval before and after that specific time. The length ofdiscarded. A Hanning window in the time domain was the interval is termed window length. The correlation isused to attenuate side effects (Oppenheim and Schafer, calculated at each time and summed by applying different1975). Power spectral analysis was applied to RR intervals weights to each time point. The weight of each pointand SBP by subjecting 512 (N) data point time series decreases as the distance from the specific time increases.overlapping by quarter to the Cooley–Tukey’s fast Fourier Spectra of RR intervals and SAP were divided into threetransform using the Time Series Statistical Analysis Sys- frequency bands and after averaging the 6-min values of

tem (software TSAS ). The spectrum computation allowed each frequency they were normalised as cited above. Weevaluation of a spectral modulus for each of the 256 did not study the VLF oscillations (,0.26 Hz). Normalisedfrequencies. The frequency resolution was Df51/NDt, i.e., power was obtained by dividing the cumulative power0.015 Hz, and the highest frequency was 4 Hz, where f is within each frequency band (LF or HF) by the sum of LFfrequency. and HF; the ratio of LF/HF was also calculated.

For each animal, the results represent the mean values of Normalised units allowed comparison between animals


them, we used the statistical method proposed by Blandand Altman (1986). In control conditions the mean differ-ence or bias was calculated as the difference between FFTand SPWVD and precision was the standard deviation(S.D.) of the mean bias. The limits of agreement weredefined as bias62 S.D. To facilitate comparisons betweendifferent conditions, a normalised measure, i.e., the coeffi-cient of variation (S.D. of the parameter /mean parameter3100), was also derived for each parameter.

A probability value less than 0.05 was consideredstatistically significant. All data are expressed asmean6S.D.

The statistical analyses were performed using theAbacus Concepts, StatViewE for windows (1996, version4.57, Abacus Concepts, Berkeley, CA).

3. Results

The injections did not induce any sign of restlessness orirritation. Before atenolol, isoproterenol caused an immedi-ate reduction in RR intervals amounting to 5969 ms(P,0.001 versus control); after atenolol, isoproterenolinduced no significant change in RR intervals (667 ms).

Before atropine, metacholine induced an increase in RRintervals (12067 ms; P,0.001 versus control) and areduction in SBP (27068 mmHg; P,0.001 versus con-trol). After atropine, metacholine induced no significantchange in RR intervals (865 ms) nor in SBP (2462mmHg).

The efficacy of acute a -blockade was assessed by the1

disappearance of pressor response to phenylephrine (4565mmHg before a -blockade, P,0.001 versus control; and1

2.060.5 mmHg after a -blockade).1

In the time domain analysis, saline had no effect onvariables (Table 1).

Phentolamine reduced significantly RR intervals by 25%Fig. 1. Step-by-step schema of the methodology for the estimation of the and the coefficient of variation by 10%. SBP was sig-spectral analysis in fast Fourier transform (FFT) and smoothed pseudo nificantly reduced by 22% and the coefficient of variationWigner–Ville distribution (SPWVD). See text for details.

by 14%.Atropine reduced significantly RR intervals by 22% and

or between spectral analysis methods characterised by the coefficient of variation by 42%. SBP was significantlylarge variations in total power. increased by 9% and the coefficient of variation by 14%.

A schema of the methodology is given in Fig. 1. Atenolol increased significantly RR intervals by 12%and the coefficient of variation by 31%. SBP was sig-

2.7. Statistics nificantly reduced by 9%.Frequency domain analysis (FFT and SPWVD) in

Non-parametric Friedman test associated with the Wil- control conditions can be seen in Figs. 2–4. Phentolaminecoxon matched-pairs test corrected for multiple compari- reduced significantly LF/LF1HF and increased HF/LF1

sons were performed for comparison between base, saline, HF in SBP. Atropine reduced significantly HF/LF1HF ina -, b - and cholinergic-blockades. Wilcoxon’s test for RR intervals and SBP and increased LF/LF1HF in RR1 1

paired data was used to study antagonist effects before and intervals and SBP, and LF/HF. Atenolol reduced sig-2after blockade. The coefficient of correlation (r ) between nificantly LF/LF1HF and LF/HF in RR intervals. There

FFT and SPWVD was calculated. The null hypothesis was was no difference between FFT and SPWVD (Tablesthat the measurements by the two methods were not 2–4).

2linearly related. As r measures the strength of a relation The bias and distribution of differences between FFTbetween two variables and not the agreement between and SPWVD methods in RR intervals under base con-


Table 1aTime domain measures of RR intervals, systolic and diastolic blood pressure in different experimental conditions in 14 conscious rats

Control Saline Atenolol Atropine Phentolamine

RR intervals (ms) 17069 17469 190613* 13264* 12766*CV (%) 5.2 5.1 6.8* 3.0* 4.7*SBP (mmHg) 121612 122613 110611* 132615* 9468*CV (%) 9.9 10.6 10.0 11.3* 8.5*DBP (mmHg) 9366 9266 8565* 10266* 7464*CV (%) 6.4 6.5 5.8* 5.8* 5.4*

a CV, coefficient of variation; SBP, systolic blood pressure; DBP, diastolic blood pressure. Values are expressed as means6S.D.Friedman test: *P,0.05 versus control.

Fig. 2. Spectral profiles (fast Fourier transform) of RR intervals and systolic blood pressure (SBP) obtained in one rat under control conditions. PSD,Power spectral density.


Fig. 3. Smoothed pseudo Wigner–Ville distribution (SPWVD) of RR intervals in one rat under control conditions.

2ditions were 1.460.4% (r 50.94; P,0.01) in LF/LF1 4. Discussion2HF; 1.560.5% (r 50.92; P,0.01) in HF/LF1HF and

24.861.9% (r 50.92; P,0.01) in LF/HF. In SBP the bias Frequency domain analyses have been applied exten-2and distribution were 1.560.8% (r 50.90; P,0.05) in sively over the last years to the study of the autonomic

2LF/LF1HF and 1.760.6% (r 50.92; P,0.01) in HF/ nervous system (Japundzic et al., 1990; Cerutti et al., 1991;LF1HF (Figs. 5–9). Malliani et al., 1991; Parati et al., 1995). Physiological

Fig. 4. Smoothed pseudo Wigner–Ville distribution (SPWVD) of systolic blood pressure in one rat under control conditions.


Table 2aFrequency domain measures of LF/LF1HF of RR intervals and systolic blood pressure in different experimental conditions in 14 conscious rats

RR intervals SBP

FFT SPWVD FFT SPWVD

Control 0.45360.089 0.43960.081 0.62460.094 0.63960.028Saline 0.45760.063 0.44660.070 0.61960.085 0.62760.090Atenolol 0.34860.059* 0.33960.066† 0.62960.076 0.62860.078Atropine 0.52060.053* 0.51760.037† 0.68060.061* 0.67360.050†Phentolamine 0.44760.073 0.44160.077 0.55360.098* 0.56060.093†

a LF, low frequency; HF, high frequency; SBP, systolic blood pressure; FFT, fast Fourier transform; SPWVD, smoothed pseudo Wigner–Villedistribution. Values are expressed as means6S.D.

Friedman test: *P,0.05 versus control FFT; †P,0.05 versus control SPWVD. There is no difference between FFT and SPWVD.

Table 3aFrequency domain measures of HF/LF1HF of RR intervals and systolic blood pressure in different experimental conditions in 14 conscious rats

RR intervals SBP

FFT SPWVD FFT SPWVD

Control 0.54560.079 0.56060.084 0.37560.080 0.35860.094Saline 0.54260.055 0.55360.073 0.38060.095 0.37260.089Atenolol 0.65060.061 0.65960.069 0.37060.067 0.37160.072Atropine 0.47960.049* 0.48260.025† 0.31960.043* 0.32360.046†Phentolamine 0.55260.065 0.55760.058 0.44660.072* 0.43960.084†

a LF, low frequency; HF, high frequency; SBP, systolic blood pressure; FFT, fast Fourier transform; SPWVD, smoothed pseudo Wigner–Villedistribution. Values are expressed as means6S.D.

Friedman test: *P,0.05 versus control FFT; †P,0.05 versus control SPWVD. There is no difference between FFT and SPWVD.

signals have been easily converted between the time interval the signal is stationary and calculate the spectrumdomain and the frequency domain with Fourier technique for each interval (Campbell, 1996). Fourier method alsoand autoregressive techniques, which are broadly applied suffers from the balance between temporal resolution andin the physiological sciences (Akselrod et al., 1985; Pagani frequency resolution.et al., 1986; Malliani et al., 1991). The preliminary In contrast, the time-frequency analysis method implicit-assumption of Fourier transform is data stationarity, which ly takes different data length for different frequency rangesimplies that the data must be stable during the observation in the signal and maintains both time and frequencyperiod (Kay and Marple, 1981). This is limited particu- resolution and can be readily applied to many differentlarly, for biological signals. Biological signals contain types of data. The Wigner–Ville distribution is the Fourierrhythms that may vary in the time, in the frequency and inthe amplitude of cycles (Hyndman et al., 1971; Campbell,1996). One common solution is to assume that over a short

Table 4Frequency domain measures of LF/HF of RR intervals in different

aexperimental conditions in 14 conscious rats

RR intervals

FFT SPWVD

Control 0.83160.209 0.78360.212Saline 0.84360.128 0.80660.104Atenolol 0.53560.215* 0.51460.200†Atropine 1.08560.109* 1.07260.120†Phentolamine 0.80960.187 0.79760.096

a LF, low frequency; HF, high frequency; FFT, fast Fourier transform; Fig. 5. Bias and variance of RR intervals between fast Fourier transformSPWVD, smoothed pseudo Wigner–Ville distribution. Values are ex- (FFT) and smoothed pseudo Wigner–Ville distribution (SPWVD) inpressed as means6S.D. LF/LF1HF under control conditions (base). This analysis includes all

Friedman test: *P,0.05 versus control FFT; †P,0.05 versus control data pairs. Overall, a bias (mean) of 0.014 and a precision (S.D.) ofSPWVD. There is no difference between FFT and SPWVD. 60.004 were determined.


Fig. 9. Bias and variance of systolic blood pressure (SBP) between fastFourier transform (FFT) and smoothed pseudo Wigner–Ville distributionFig. 6. Bias and variance of RR intervals between fast Fourier transform(SPWVD) in HF/LF1HF under control conditions (base). This analysis(FFT) and smoothed pseudo Wigner–Ville distribution (SPWVD) inincludes all data pairs. Overall, a bias (mean) of 0.017 and a precisionHF/LF1HF under control conditions (base). This analysis includes all(S.D.) of 60.006 were determined.data pairs. Overall, a bias (mean) of 20.015 and a precision (S.D.) of

60.005 were determined.

tion is a time-scaled, time-reversed copy of the originalsignal. It is a popular tool for time-frequency analysisbecause generally it has better resolution than the short-time Fourier transform. Calculation of power spectra isbased on correlation-like terms, which implicitly considersthe temporal relationship of the data. This makes theassumption of data stationarity unnecessary (Mansier et al.,1996; Huang et al., 1997). The main advantage of theWigner–Ville over global spectral analysis is the dynamicrepresentation of the signal structure, both time andfrequency characteristics are simultaneously available andthe temporal and frequency windows can be adapted toreduce to a minimum the interferences between the differ-

Fig. 7. Bias and variance of RR intervals between fast Fourier transform ent spectral components and to obtain maximal time and(FFT) and smoothed pseudo Wigner–Ville distribution (SPWVD) in

frequency resolutions (Martin and Flandrin, 1985). TheLF/HF under control conditions (base). This analysis includes all dataWigner–Ville distribution raises two problems: cross-pairs. Overall, a bias (mean) of 0.048 and a precision (S.D.) of 60.019

were determined. terms, and aliasing. These problems have been largelydealt with, although people are still on the lookout forbetter smoothing functions and alias-free distributiontransform of the signal’s autocorrelation function with(Novak and Novak, 1993; Flandrin, 1998). One of therespect to the delay variable. It can also be thought of as amost important properties of the Wigner–Ville distributionshort-time Fourier transform where the windowing func-is that it does not have imaginary components-like theshort-time Fourier transform.

The SPWVD can quantify RR intervals and bloodpressure variability over very short periods and allows adynamic study of the instantaneous spectral power (Jassonet al., 1997).

Novak and Novak (1993) compared the discreteWigner–Ville distribution with the Fourier transform; theblood pressure signal was obtained during slowing of therespiratory rate. The discrete Wigner distribution followscontinuously frequency changes of the signals, whereas theFFT spectral estimator is fragmented and suffers from bothtime and frequency distortion and lack of resolution.Jasson et al. (1997) using head-up tilt (inclination of the

Fig. 8. Bias and variance of systolic blood pressure (SBP) between fast table at 15, 30, 45, 60 and 908) compared the results of theFourier transform (FFT) and smoothed pseudo Wigner–Ville distribution

Wigner–Ville method with those of the classic AR algo-(SPWVD) in LF/LF1HF under control conditions (base). This analysisrithms and did not find any great differences between theseincludes all data pairs. Overall, a bias (mean) of 20.015 and a precision

(S.D.) of 60.008 were determined. two methods. However, the global variability of the RR


Bland, J.M., Altman, D.G., 1986. Statistical methods for assessingintervals measured by the smoothed pseudo Wigner–Villeagreement between two methods of clinical measurement. Lancet i,distribution was not modified by the tilt and did not307–310.

correlate with the angles of inclination conversely to other Boashash, B., 1988. Note on the use of the Wigner distribution forstudies using autoregressive algorithms (Pagani et al., time-frequency signal analysis. IEEE Trans. Acoust. Speech Signal1986; Montano et al., 1994). Jasson et al. (1997) demon- Processing 36, 1518–1521.

Brockway, B.P., Mills, P.A., Azar, S.H., 1991. A new method forstrated, as in our study that SPWVD of heart rate presentedcontinuous chronic measurement and recording of blood pressure,an extreme instability of the spectral power when plottedheart rate and activity in rat via radio-telemetry. Clin. Exp. Hypertens.against time (Figs. 3 and 4). This instability is not 13, 885–895.

detectable by FFT because it calculates an average spec- Campbell, M.J., 1996. Spectral analysis of clinical signals: an interfacetrum (Fig. 2). Mathematically, when the frequency com- between medical statisticians and medical engineers. Stat. Methods

Med. Res. 5, 51–66.ponents do not change during the time interval analysisCerutti, C., Gustin, M.P., Paultre, C.Z., Lo, M., Julien, C., Vincent, M.,these two methods are compatible but in strict terms,

Sassard, J., 1991. Autonomic nervous system and cardiovascularrigorous stationary conditions are unknown to biologicalvariability in rats: a spectral analysis approach. Am. J. Physiol. 261,

signals. In our study acceptable levels of accuracy and H1292–H1299.precision of FFT versus SPWVD were found between 1 ¨Claasen, T., Mecklenbrauker, W., 1980. The Wigner distribution: a tool

for time-frequency signal analysis. Philips J. Res. 35, 217–250.and 2%. Only in RR intervals LF/HF presented a bias atChallis, R.E., Kitney, R.I., 1991. Biomedical signal processing (in four4.8%. We think that this difference could be explained by

parts). Part 3. The power spectrum and coherence function. Med. Biol.the fact that this ratio does not maintain strictly inEng. Comput. 29, 225–241.

stationary conditions and the SPWVD, unlike FFT, was Cohen, L., 1989. Time-frequency distributions: a review. Proc. IEEE 77,able to describe the instantaneous variability of the time 941–981.

De Boer, R.W., Karemaker, J.M., Strackee, J., 1984. Comparing spectra ofseries independently of stationary situations or there wasa series of point events, particularly for heart rate variability data.an appreciation of the fractional distribution of the energyIEEE Trans. Biomed. Eng. 31, 384–387.increased by the ratio LF/HF.

´Flandrin, P., 1998. Vers le temps-frequence: des approches multiples. In:Another point is the time resolution. The more compact ´ ´Temps-frequence: Traitement du signal. Hermes, Paris, pp. 42–48.

or peaked we make the window in the time domain, the Huang, H.H., Chan, H.L., Lin, P.L., Wu, C.P., Huang, C.H., 1997.Time-frequency spectral analysis of heart rate variability duringmore time resolution is achieved (Cohen, 1989; Challisinduction of general anaesthesia. Br. J. Anaesth. 79, 754–758.and Kitney, 1991). However, in FFT a compromise has to

Hyndman, B.W., Kitney, R.I., Sayers, B.McA., 1971. Spontaneousbe reached between a temporal window of analysis largerhythms in physiological control systems. Nature 233, 339–341.

enough to have a reliable spectral estimation, but not too Japundzic, N., Grichois, M.L., Zitoun, P., Laude, D., Elghozi, J.L., 1990.wide to impact in stationarities. In our study a Gaussian Spectral analysis of blood pressure and heart rate in conscious rats:

effects of autonomic blockers. J. Auton. Nerv. Syst. 30, 91–100.window was chosen because it is more effective in cross´Jasson, S., Medigue, C., Maison-Blanche, P., Montano, N., Meyer, L.,terms reduction and the time resolution reductions (Novak

Vermeiren, C., Mansier, P., Coumel, P., Malliani, A., Swynghedauw,and Novak, 1993).B., 1997. Instant power spectrum analysis of heart rate variability

To our knowledge there is no study available comparing during orthostatic tilt using a time/ frequency-domain method. Circu-the two methods in animals at rest and during induced lation 96, 3521–3526.

Kay, S.M., Marple, Jr. S.L., 1981. Spectrum analysis: a modern perspec-changes of autonomic regulation stationary situations. Thetive. Proc. IEEE 69, 1380–1419.measures of the strength of the relation between FFT and

Malliani, A., Pagani, M., Lombardi, F., Cerutti, S., 1991. CardiovascularSPWVD and the limits of agreement showed that thereneural regulation explored in the frequency domain. Circulation 84,

were not a lot of discrepancies between the two methods in 482–492.normotensive rats. As SPWVD allows a dynamic repre- ´Mansier, P., Medigue, C., Charlotte, N., Vermeiren, C., Coraboeuf, E.,

´Deroubai, E., Ratner, E., Chevalier, B., Clairambault, J., Carre, F.,sentation of the signal without stationarity consideration,Dahkli, T., Bertin, B., Briand, P., Strosberg, D., Swynghedauw, B.,we may suppose that it is a more suitable method to study1996. Decreased heart rate variability in transgenic mice overexpres-biological signals in all physiological and pharmacologicalsing atrial b1-adrenoceptors. Am. J. Physiol. 271, H1465–H1472.

conditions. Martin, W., Flandrin, P., 1985. Wigner-Ville spectral analysis of non-stationary processes. IEEE Trans. Acoust. Speech Signal Process. 33,1461–1470.

Montano, N., Ruscone, T.G., Porta, A., Lombardi, F., Pagani, M.,References Malliani, A., 1994. Power spectrum analysis of heart rate variability

to assess the changes in sympathovagal balance during gradedAkay, M., 1995. Introduction: Wavelets in biomedical engineering. Ann. orthostatic tilt. Circulation 90, 1826–1831.

Biom. Eng. 23, 531–542. Novak, P., Novak, V., 1993. Time/ frequency mapping of the heart rate,Akselrod, S., Gordon, D., Madwed, J.B., Snidman, N.C., Shannon, D.C., blood pressure and respiratory signals. Med. Biol. Eng. Comput. 31,

Cohen, R.J., 1985. Hemodynamic regulation: investigation by spectral 103–110.analysis. Am. J. Physiol. 249, H867–H875. Novak, V., Novak, P., de Champlain, J., Le Blanc, A.R., Martin, R.,

Aubert, A.E., Ramaekers, D., Beckers, F., Breem, R., Denef, C., Van de Nadeau, R., 1993. Influence of respiration on heart rate and bloodWerf, F., Ector, H., 1999. The analysis of heart rate variability in pressure fluctuations. J. Appl. Physiol. 74, 617–626.unrestrained rats. Validation of method and results. Comput. Methods Novak, V., Novak, P., de Champlain, J., Nadeau, R., 1994. AlteredPrograms Biomed. 60, 197–213. cardiorespiratory transfer in hypertension. Hypertension 23, 104–113.


Oppenheim, A.V., Schafer, R.W., 1975. In: Digital Signal Processing. Rubini, R., Porta, A., Baselli, G., Cerutti, S., Paro, M., 1993. PowerPrentice-Hall, Englewood Cliffs, NJ. spectrum analysis of cardiovascular variability monitored by telemetry

Pagani, M., Lombardi, F., Guzzetti, S., Rimoldi, O., Furlan, R., Pizzinelli, in conscious unrestrained rats. J. Auton. Nerv. Syst. 45, 181–190.P., Sandrone, G., Malfatto, G., Dell’Orto, S., Piccaluga, E., Turiel, M., ´Ville, J., 1948. Theorie et applications de la notion de signal analytique.Baselli, G., Cerutti, S., Malliani, A., 1986. Power spectral analysis of Cable Transmission 2A, 61–74.heart rate and arterial pressure variabilities as a marker of sympatho- Wigner, E., 1932. On the quantum correction for thermodynamic equilib-vagal interaction in man and conscious dog. Circ. Res. 59, 178–193. rium. Phys. Rev. 40, 749–759.

Parati, G., Saul, J.P., Di Rienzo, M., Mancia, G., 1995. Spectral analysisof blood pressure and heart rate variability in evaluating cardiovascu-lar regulation: a critical appraisal. Hypertension 25, 1276–1286.

Smoothed pseudo Wigner–Ville distribution as an alternative to Fourier transform in rats

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