time-frequency mapping of the qrs complex in normal subjects and in postmyocardial infarction...

Journal of Electrocardiology Vol. 27 No. 1 1994

Time-frequency Mapping of the QRS Complex in Normal Subjects and in Postmyocardial Infarction Patients

Peter Novak , MD,* Zh ix ing Li, P h D , t Vera Novak , MD, PhD,* a n d

Robe r t Hata la , MD, PhD*

Abstract: The effect of myocardial infarction upon the frequency content of the QRS complex was analyzed. Three bipolar signal-averaged surface electrograms, recorded during the early (10-15 days) and late (6 months) chronic phases of myocardial infarction, were analyzed in 61 patients and 11 healthy subjects. All patients were free of ventricular arrhythmia during 6 months of follow-up examinations. Time-frequency analysis of the QRS complex was based on the modified Wigner distribution, which is well suited to examine nonstationary character of data. Standard t ime-domain analysis for the presence of late potentials was used for comparison. High-frequency (>-90 Hz) components, separable from the dominant low-frequency components (<90 Hz), were found in all groups. They were present throughout the QRS complex and were peaking in its middle portion. The high-frequency components were found significantly higher in postinfarction patients in both early ( P < .007) and late chronic stage myocardial infarction (P < .05) compared to healthy subjects. Patients who tested positive for late potentials (24%) also had elevated high-frequency components; however, a comparable increase was also observed in late potential negative patients. Furthermore, the high-frequency component increase occurred in all patients earlier in the QRS than in its terminal 40 ms, where late potentials are traditionally evaluated. It is concluded that high-frequency components are an integral part of the QRS complex under physiologic conditions and persist in variable amount throughout its duration. The high-frequency components are increased in patients after myocardial infarction not associated with ventricular arrhythmia, and their elevation is not limited to the terminal QRS complex. Key words: electrocardiography, myocardial infarction, ventricular activation, time-frequency mapping, Wigner distribution, late ventricular potentials.

Early studies have shown that an altered depolari- zation after myocardial infarction can be manifested

From the *Research Center, H~pital du Sacr~-Coeur de Montreal and Department of Medicine, Universit~ de Montreal, and "]-Institut de G~nie Biomedical, Ecole Polytechnique, Universit~ de Montreal, Montreal, Canada.

Reprint requests: Dr. Peter Novak, the Lady Davis Institute for Medical Research, The Sir M. B. Davis-Jewish General Hospital, McGill University, 3755 Chemin de la C6te-Ste-Catherine, Mon- treal, Quebec, H3T 1E2 Canada.

as an increased number of notches and slurs within the QRS complex. 1-5 In contrast, spectral analysis revealed reduction of high-frequency energy in patients after myocardial infarction. 6-s Standard time- domain analysis is based on the concept of late ventricular potentials defined as high-frequency (usually above 25 Hz) low-amplitude potentials at the terminal 40 ms of the QRS complex. 9 Late potentials are present in 21% to 44% of patients recovering from an acute myocardial infarction, and this finding is

49

50 Journal of Electrocardiology Vol. 27 No. 1 January 1994

associated with a high risk of sustained ventricular tachycardia. 10, ~ ~ However, since time- domain analysis is limited to the terminal QRS complex, late potentials may not be detected if the abnormal myocardial region is activated earlier during the QRS complex, l°,~x Furthermore, patients with bundle branch blocks are generally excluded from this analysis or modified criteria must be used. 13

Spectral analysis was also used to search for unique frequency features that characterize late ventricular potentials 14 and for the analysis of the entire cardiac cycle. 15 However, results of spectral analysis are difficult to reproduce16-1 ~ since the basic requirement for the signal stationarity is not fulfilled. More advanced techniques, such as spectrotemporal mapping 2°,21 and, in particular, turbulence analysis, 22 reduced some undesirable properties of spectral analysis. These dynamic techniques were found to be more sensitive than time-domain analysis for the detection of patients prone to the ventricular arrhythmias following myocardial infarction.

In this study we have applied time-frequency mapping based on the modified Wigner distribution 23 for analysis of the QRS complex. Time-frequency mapping is more suitable for analysis of signals with time- varying spectrum than conventional methods be- cause both time and frequency characteristics of the signal are simultaneously available and consequently, the assumption of the signal stationarity is not required. This makes it possible to trace high- frequency components with substantially higher time resolution than the standard Fourier transform or autoregressive models.

We used time-frequency mapping to determine the frequency content of the QRS complex in healthy subjects and in patients with acute and remote myocardial infarction. Thus, the aim of this study was to test the hypothesis that abnormal ventricular activation due to uncomplicated myocardial infarction (ie, free of ventricular arrhythmias in the first 6 months) differs in its frequency content from physiologic ventricular activation.

Materials and Methods

Concept of Time-frequency Mapping

The ventricular activity can be expressed in a two- dimensional plane where one axis is time and the second is frequency. The summed projection in the time axis is the QRS complex (more precisely, the square of the QRS complex), while the summed pro-

jection in the frequency axis is its spectrum. In other words, a square of the QRS complex at a chosen time instant is equal to the sum of all frequency components at that time. In mathematical language:

f P(t,f)df = Is(t)[ 2,

f P(t,f)dt = S(f),

where S(f) denotes a spectrum of the signal s(t) and P(t,f) stands for a time-frequency distribution (t = time, f = frequency).

The duality between time and frequency is a basic concept of time-frequency mapping. The advantage of this approach is that it enables more detailed analysis of high-frequency components of the QRS complex than standard Fourier methods. In addition, it eliminates the assumption of signal stationarity.

Since only the time projection (eg, the QRS complex) is known, the problem is to reconstruct the time-frequency plane from the QRS complex. In 1932 Wigner 24 described a function (Wigner distribution) that turned out to be a good candidate for reconstruction of the time-frequency plane from time functions.

However, undesirable properties of the basic definition of the Wigner distribution limited its applications to only simple monocomponent signals. If two or more frequencies are concurrently present in the signal, the Wigner distribution generates spurious peaks called cross-terms. This spurious interference occurs between the signal frequencies. 25 For example, if there will be frequency components at 5 and 15 Hz, respectively, there will be a spurious peak at 10 Hz. An increasing interest in Wigner analysis follows the report of Martin and Flandrin 26 who showed the effective way of cross-terms suppression. The cross-terms feature cosine amplitude depen- dency, thus they can be reduced by a variety of smoothing methods.

In addition to smoothing, the time-frequency estimation can be improved by transforming the electrocardiographic signal into its analytic form prior to computation of the Wigner distribution. This trans- formation does not affect the frequency content of the electrodardiographic signal, it only removes the redundant negative frequencies. Consequently, the interference between the negative and positive frequencies is eliminated. It also increases frequency resolution by a factor of two and reduces error at the DC (0 Hz) level.

Time-frequency mapping comprises the above- mentioned modification of the Wigner distribution, such as analytic transform and smoothing of the Wig-

Frequency Content of the QRS Complex • Novak et al. 51

ner distribution. 23 We experimentally found that a Gaussian function is superior for frequency smoothing and the rectangular function for time smoothing. Smoothing decreases both time and frequency resolution proportionally to the degree of smoothing. De-

gree of time smoothing is controlled by parameter M, while frequency smoothing is controlled by parameter alpha (Appendix, equation 1).

The relationship between the classical Fourier spectrum and Wigner distribution can be demon-

Fig. 1. Comparisons between the modified Wigner distribution and the Fourier spectrum. The signal with decreasing frequency (a) was reversed resulting in the signal (b). The signal (c) is the summat ion of signals a and b. The time-frequency mapping (d) of the signal c was performed with the following parameters: N = 128, M = 9, q(k) = Gaussian window with alpha = 2.5, h(k) = 1. For the Fourier analysis (e), the 128-point data window was multiplied by the Hanning window and the square of the 128-point fast Fourier transform was taken. The data w indow was then shifted one sample ahead until the spectra from all remaining segments were obtained. Spu- rious peaks at 28 and 50 Hz at a time interval of 8 0 - i 5 0 ms dominate in the Fourier analysis (e). There is also an error at the DC level (at 0 Hz). The real frequencies are suppressed and smeared out. Poor est imation of the Fourier spectrum is due to time variat ion of the signal frequency within the data window. The t ime-frequency mapping follows the signal structure well, and the two main frequencies dominate over the Wigner plane (d).

a - -

b ii

0 30 80 90 1 20 1 50 180 210 240

Time (ms)

dIl.

"~ ~ J ~ l l l ~ / ~ : - ~ l l l l l l l l t l l l l l l ~ . , ~ - ~ H I 7~F~d!J~JJJJ/J

~ l l l l l ~ ..,; .~

~p


strated on the time-frequency plane. When the frequency components do not change during the ana- lyzing time interval (eg, the signal is stationary), then the Fourier spectrum is equal to the sum of the frequency components along the frequency axis (over that time). In this case, the Wigner and Fourier ap- proaches are compatible. Noncompatibility between Wigner distribution and Fourier transform arises in the nonstationary condition. For such signals, the Fourier spectrum is corrupted with substantial portions of unpredictable spurious peaks. In Figure 1 the signal with known theoretical spectrum is presented. Two components with increasing/decreasing frequencies were mixed together. Thus, at each time instant, there are only two frequencies. Looking at Figure 1, the fast Fourier transform generates a substantial portion of peaks without physical meaning since these peaks do not have counterparts in the signal. They result from breaking the assumption on the signal stationarity. On the other hand, the modified Wigner distribution is much closer to the theoretical spectrum and real frequencies dominate on the time-frequency plane.

Standard time-domain analysis is defined on a vector summation. For direct comparability with time- frequency mapping, we defined a "vectorized" Wig- ner distribution (Appendix, equation 2) that, similar to vector summation, depends on the contribution of all electrocardiographic leads (X, Y, and Z).

Patients

The study comprises three groups of subjects: 11 healthy subjects with a normal electrocardiogram, no history of cardiovascular disease, and free of any medication; 31 patients with recent myocardial infarction ( i0 -15 days); and 30 patients with remote myocardial infarction (5-6 months). Left ventricular function was assessed with radionuclide or contrast ventriculography in postmyocardial infarction patients. The myocardial infarction was localized by the standard 12-lead electrocardiogram and/or ventriculography. All patients were free of ventricular tachycardia up to 6 months following the acute myocardial infarction and did not receive antiarrhythmic medication. Clinical characteristics of the groups are sum- marized in Table 1.

Data Analysis

Three electrocardiographic surface orthogonal bipolar X, Y, and Z leads were recorded with a PC- based acquisition system (Predictor, Corazonix, OK)

Table 1. C h a r a c t e r i s t i c s o f P a t i e n t s

Group 1 Group 2 Group 3 (Healthy) (Recent MI) (Remote MI)

Number 11 31 30 Sex (M/F) 6/5 24/9 22/8 Age (years) 39 -4- 5.3 61 -+ 2.3 54 __ 2.4 Heart rate (beats/ 75 _+ 7.6 68 + 2.2 63 _+ 2.3

min) MI specification

Anterior - - 11 7 Inferior - - 13 17 Non Q wave - - 9 6

Left ventricular - - 44.6 _+ 2.4 45.6 _+ 2.5 ejection fraction (%)

QRS durat ion 95 -+ 3.2 114.5 _+ 3.7 109.4 _+ 3.8 (ms)

Conduct ion - - 6 6 abnormal i ty

MI = myocardial infarction.

and sampled at 2,000 Hz over a bandwidth of 0.05-1,000 Hz. The X, Y, and Z leads were averaged after passage through a template recognition pro- gram generated from a 3-second display of normal sinus rhythm. All subsequent QRS complexes were tested against the template with a cross-correlating technique. 14 The beats with a correlation coefficient ->.99 were averaged until the final level of noise was below 0.3 ~V. The median number of averaged beats was 215 (range, 82-875 beats).

Numerical computing was performed with 64 bits precision, except for the trigonometric terms, which are computed with 80 bits accuracy. Entirely written in C + + language, the time-frequency mapping system runs on an IBM-PC compatible computer (pro- cessor 386 or higher).

The DC level, as well as the low-frequency trend (< 10 Hz), were removed by a fourth-order polynom- ial. The detrended signal was converted into its analytic form by the Hilbert transform. 27 The analytic signal was evaluated by the modified Wigner distribution with the following parameters: N = 256, q(k) = Gaussian, window 2s with alpha = 2.9, h(p) = 1, M = 12. Thus, the spectral estimator was computed for every 0.5-ms time instant with a frequency resolution of 3.9 Hz. The computational details can be found in our previous paper. 23 Then the vectorized Wigner distribution was computed according to equation 2 in the Appendix.

Low-frequency components were identified as local maxima of the Wigner distribution in the band <90 Hz, while high-frequency components were identified as local maxima in the band ->90 Hz. The level of spectral noise was computed by averaging the frequency content over the band 500-999 Hz. High-frequency components were quantified by the high-frequency index defined by peak power of high-

uV 40.00

20.00

0,00 0

A

58

6O 120

152

180 240 300 ms


2 0 5 0 1 O 0 1 5 0 2 0 0

B F r e q u e n c y ( H z )

O

. . . a m , - •

Fig. 2. Time-domain plots (A) and the smoothed Wigner distribution (B,C) of the signal-averaged electrocardiogram of a healthy 47-year-old subject. For the time-domain plots, the signal in the last 40 ms of the QRS complex is highlighted. Note absence of late potentials in the time-domain plots (QRS duration = 94 ms, RMS = 18.29 p,V, LAS40 = 37.5 ms). For the Wigner distribution, low-frequency components dominate in the 2 0 - 6 0 Hz band. High-frequency components are present in the 8 0 - 1 6 0 Hz band throughout the QRS complex. RMS = root-mean-square voltage of the signal in the last 40 ms; LAS40 = duration of the low-amplitude signal -<40 p,V.


uV

40.00 I

20.00

0 .00 '

60 120 180 240 300 A ~s

A

0 5 0 1 O 0 1 5 0 2 0 0 B F r e q u e n c y ( H z )

O

• ? . . ~ ~ . , c ~ ' ~ "~

Fig. 3. Time-domain plots (A) and the smoothed Wigner distribution (B,C) of the signal-averaged electrocardiogram from a 44-year-old patient 2 weeks after inferior myocardial infarction. Time-domain analysis for the presence of late potentials was negative (QRS duration -- 98.5 ms, RMS = 78.73 /~V, LAS40 -- 23.5 ms): The smoothed Wigner distribution reveals clearly discernible high-frequency components in the 125-180 Hz band. (B) The heavy line designates the peaks of high- frequency components. Abbreviations as in Figure 2.

25000

20000

~ 15000

10000

I

5OOO

0

C o n t r o l R e c e n t MI R e m o t e MI n=11 n=31 n=30

Fig. 4. Comparison of high-frequency components obtained from the three groups. High-frequency components are significantly increased in both postmyocardiat infarction groups compared to healthy subjects. MI - myocardial infarction. *P < .05; **P < .007.

frequency components/spectral noise. These parameters were obtained for each 0.5-ms time instant within the QRS complex. The time instant corresponding to the maximal value of the high-frequency index was stored for further analysis as well.

Time-domain analysis for the presence of late potentials was used for comparison. Late potentials were analyzed on the vector summat ion of the X, Y, and Z leads according to the Gomes criteria 2'~.3° with a bidirectional filter at 40 Hz. The QRS duration was measured on the filtered vector summation. Late potentials were considered present if duration of the filtered QRS complex duration was longer than 114 ms and one or both of the following criteria were fulfilled: ( 1 ) a root -mean-square voltage of the terminal 40 ms of the QRS < 20 ~V and (2) low-ampli tude (-<40 ~V) signal duration ->38 ms.

Average values of high-frequency components over the entire QRS complex from the vectorized Wigner distribution were compared by analysis of variance. The between-groups comparison was performed by the post hoc Tukey test. In addition, the interaction between high-frequency components and time (recent, remote), localization (anterior, inferior), and type (with/without Q wave) of myocardial infarction was studied by analysis of variance. The relationship between high-frequency components and both left ventricular ejection fraction and QRS duration was analyzed by the correlation analysis. For graphic purposes only, the log (a + bx) where x is the value of the Wigner distributions is presented in Figures 2 -4 .

Results

Normal Ventricular Activation

All Wigner distributions showed a dominant band below 80 Hz with the local max ima between 30 and 50 Hz (low-frequency components) . The crest of this

Frequency Content of the QRS Complex • Novak et al.' 55

dominan t activity was typically smooth in t ime direc- tion (Fig. 2). High-frequency components were seen as na r row-band activity concentrated at frequencies higher than the dominant low-frequency components clearly separated f rom the level of the spectral noise. High-frequency components continuously spread f rom the dominant low-frequency components or were seen as a separate frequency band, but they were clearly separable by their local max ima of the Wigner distributions. Typically, high-frequency components cover the band 8 0 - 2 0 0 Hz and are detectable throughout the QRS complex. There was no ponderance of high-frequency components with re- spect to the specific lead (X, Y, or Z).

Ventricular Activation After Myocardial Infarction

A typical finding in a patient after recent myocardial infarction is shown in Figure 3. The dominant low-frequency components cover the band 2 0 - 8 0 Hz, while high-frequency components are detectable in the band 125-180 Hz (designated by heavy lines that ride the crests of their respective peaks). High- frequency components start to increase about 22 ms after QRS onset, peak at 45 ms, and both their frequency and magni tude decrease toward the terminal QRS. Finally, they gradually merge with low-frequency components at the end of the QRS. Late potentials were not present in this patient.

High-frequency components were detectable before the terminal 40 ms of the QRS complex in all patients. They peaked at 48.5 _+ 1.9 ms from QRS onset in patients with anterior myocardial infarction, at 50.4 +_ 1.4 ms in patients with inferior myocardial infarction, and at 52.2 +_ 2.1 ms in patients with non Q wave infarction. The differences in timing were not significant. The dominant peak of the high-frequency components did not coincide in t ime with the peak of R or S waves.

Analysis of variance of all groups showed significant overall difference (P < .009) be tween the average values of high-frequency components (quantified by the high-frequency index). High-frequency components were significantly increased in both myocardial infarction groups ( P < .007 for the recent myocardial infarction group and P < .05 for the remote myocardial infarction group, Fig. 5) compared to the control group. High-frequency components were lower in patients with remote myocardial infarction compared to the recent infarction group, but this difference was not significant. Furthermore, sta- tistical compar ison of the high-frequency components localized in the terminal 40 ms of the QRS


uV 40.00

20.00

0.00 0

A

vl

186

-,,,-I m . ==i

A ==i

60 120 180 240 300

ms

- g

0

C

0 5 0 1 O 0 1 5 0 2 0 0 B F r e q u e n c y ( H z )

Fig. 5. Time-domain plots (A) and the smoothed Wigner distribution (B,C) of the signal-averaged electrocardiogram from a 49-year-old patient 2 weeks after anterior myocardial infarction. Time-domain plots reveal late potentials at the terminal QRS (QRS duration = 121 ms, RMS = 17.83 p~V, LAS40 = 44 ms). For the smoothed Wigner distribution, high-frequency components are increased at the second half of the QRS complex. (B) The heavy line designates the peaks of high-frequency components. (C) A time instant corresponding to the terminal 40 ms of the QRS complex is designated by a heavy line on the back-forward oblique projection. Only part of the high-frequency components manifest as late potentials. Remarkable increasing of high-frequency components, however, is seen 62 ms before the QRS offset. Abbreviations as in Figure 2.

Frequency Content of the QRS Complex • Novaket al. 57

complex did not show any significant difference between the three study groups (P < .05).

Interactions between other factors were not significant. High-frequency components were independent of both QRS duration (r = .23) and left ventricular ejection fraction ( r = - . 0 2 ) . Late potentials, according to the above-described t ime-domain criteria, were identified in 16 myocardial infarction patients (9 patients with recent infarction and 7 with remote infarction). These patients had an increase of high- frequency components at the terminal QRS complex, but the high-frequency components peaked before the terminal 40 ms. Furthermore, none of our patients had high-frequency components localized ex- clusively at the terminal 40 ms of the QRS. An example of a patient with late potentials detected by time- domain analysis and increased high-frequency components in the terminal QRS is shown in Figure 4. Here, low-frequency components are concentrated in the 10-80 Hz range. Further increase of frequency content above 80 Hz is observed in the second half of the QRS. High-frequency components spread from the dominant low-frequency components and are discernible from 68 ms of the QRS and later. These high-frequency components cover the 8 0 - 2 0 0 Hz band with local maxima at 105 Hz.

Discussion

Applications of Fourier analysis were expected to improve the detection of abnormal activation pat- terns. However, the implicit assumption of the signal stationarity is in contrast with real situations and this discrepancy could explain the controversial reports that use Fourier-based analysis. An elegant way to reduce the requirement on the signal stationarity is to compute the successive spectra over shorter segments (spectrotemporal mapping) using the fast Fourier transform algorithm 2° or autoregressive model, xl These techniques showed that high-frequency components of the terminal QRS complex could be present in both patients with and without ventricular tachycardia. Kelen et al. 22 further ex- tended the concept of spectrotemporal mapping to- wards the whole QRS complex. They also improved the spectral estimation by computing the Fourier spectrum of differences between successive samples ("turbulence analysis"), thus further reducing a requirement on the stationarity. Turbulence analysis achieved higher accuracy than late-potential analysis (94% vs 73%) for the prediction of patients with inducible sustained monomorphic ventricular tachycardia.

These methods, however, do not solve the main

limitation of Fourier-based analysis (eg, the assumption of the signal stationarity). The Fourier transform generates spurious peaks even for very short segments if the frequency content within this data seg- ment changes. Furthermore, the resulting frequency resolution for such short segments is very low.

The purpose of this study was to overcome these limitations by means of a more adequate signal processing method. It focuses on the t ime-frequency characteristics of ventricular activation in normal subjects and in postmyocardial infarction patients without ventricular arrhythmia. It uses a novel signal processing approach, already successfully applied for the analysis of heart rate, blood pressure, and respiratory rhythms 23 in delineation of the respiratory drive upon the cardiovascular system 31 and in the detection of patients prone to the neurally mediated syncope. 3x Our method is based on the modified Wigner distribution and we suggest for it, in accordance with its original description, 23 the term time-frequency mapping. This should at the same time stress the differences between our method and Fourier analysis. The main implications of our study are: (1) high- frequency components of the QRS complex can be reliably detected and traced in time by means of time- frequency mapping; (2) high-frequency components are present throughout normal ventricular activation; (3) these components are increased in patients after myocardial infarction not associated with ventricular arrhythmia, and the elevation of high-frequency components is not limited to the terminal QRS complex; and (4) the results of t ime-frequency mapping are not influenced by QRS duration, and thus patients with conduction defects do not have to be a priori excluded.

Considering the possibility that high-frequency components are due to spurious interference, it is well known that the interference occurs between signal frequencies. Therefore, this mechanism would suggest the presence of spurious peaks in the range of 60 to 80 Hz (eg, between high- and low-frequency components) , but it cannot explain the presence of high-frequency components.

High-frequency components are lead-dependent, but their spatial distribution on the body surface (X, Y, or Z leads) did not correlate with the site of the infarction (anterior vs inferior). Whether this is due to a certain overlap of regions viewed by each lead and, thus, might be eliminable by more specific surface mapping lead arrays remains to be established.

Late potentials were present in 24% of the patients with myocardial infarction; these patients did not have malignant ventricular arrhythmias and/or sudden death during a 6-month follow-up period. No late potentials were detected in the control group.


However, even in patients with late potentials, a substantial portion of the high-frequency components was detectable before the terminal 40 ms of the QRS complex (ie, the QRS complex portion where, by definition, late potentials are detected). Thus, time- frequency mapping indicates that late potentials are only "the tip of the iceberg" of high-frequency components, but substantial portions of high-frequency components could be localized before the terminal 40 ms of the QRS complex.

The source(s) of the high-frequency components detected on the body surface both in healthy and myocardial infarction patients remains to be clarified. They might originate from the intramyocardial cell- to-cell conduction during the tangential activation wave spread mainly in subepicardial myocardial lay- ers. The anisotropy of this conductive medium, further increased in infarcted muscle, might be a crucial factor in generating high-frequency signals. Bolick et al. 33 have shown that all myocardial infarctions comprise different amounts of patchy zones and that their border zones are characterized by a mixture of healthy fibers and interdigitizing connective tissue, that is, the highly anisotropic structures generally be- lieved to be the arrhythmogenic substrate. 3a If this substrate functions as a "generator" of high frequencies during its activation, then, accordingly, these frequencies should be detectable during the QRS complex in all myocardial infarction patients. This concept does not contradict our findings of detectable high-frequency components during the QRS complex in normal subjects. As pointed out in the classic papers of Spach et al., 35,36 even normal myocardium shows a certain degree of both morphologic and functional anistropy and, thus, could be the source of high-frequency components. Our results, suggesting discernible high-frequency components during QRS in normal subjects and their increase in myocardial infarction patients, could represent the frequency counterpart of the above-mentioned findings. Fur- thermore, they are fully compatible with the original morphologic studies of the QRS complex, suggesting a variable proportion of high-frequency components ("notches and slurs") increasing in postmyocardial infarction patients. ~-5 Interestingly enough, to our knowledge none of the previously published studies have demonstrated the expected elevation of high- frequency components in patients with myocardial infarction not associated with ventricular arrhythmias.

Study Limitation

Obviously, this pilot study comprises only a limited number of patients, and much larger groups have to

be analyzed to get well-defined normality criteria. The increased high-frequency components possibly reflect the extent of the myocardial damage following an infarction. Therefore, a reliable detection of abnormal high-frequency components is highly desir- able and might have important clinical impact. The proposed method seems to be a further step in the search for an ideal tool. This concept remains to be studied further in myocardial infarction patients with malignant arrhythmias, who must be compared to their arrhythmia-free counterparts.

Acknowledgments

The authors thank Ilias Iakovidis, PhD, for methodo- logic suggestions, and Mrs. Nicole Poitras for techni- cal assistance.

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Appendix

Signal Processing Background

Time- f requency m a p p i n g 23 based o n the Wigne r

d is t r ibut ion 24 gives a s i m u l t a n e o u s t ime- f requency represen ta t ion of the signal.

The cross Wigne r d i s t r ibu t ion of two analyt ic signals, f and g, modif ied by t ime a nd f requency smoo t h i ng is g iven by:


= ]q(k)2[ WDfg (n,m) N k=o

[%' (1) × h(p) f(n + p + k) g

L p = M + 1

] - - 2 i ~ k m

*(n + p - k ) | e ~ , J

where n denotes the discrete time and m the frequency. The function q(k) of the length N applies for frequency smoothing, while h(p) of the length 2M-1 applies for time smoothing. The auto Wigner distribution is computed with f = g. If the Wigner distribution is computed from real signals, then the frequency resolution reduces to a factor of two.

The Wigner distribution is compatible with the t ime-domain signal and this allows definition of the vectorized WDxyz as follows:

(2) WDxyz (m,n) =

x/WD~ (m,n) + WD 2 (m,n) + WD2z (m,n),

where WDx(m,n), WDf(m,n) , and WDz(m,n) desig- nate the Wigner distribution obtained from the X,Y, and Z leads, respectively. If V stands for the vector summation (V = ~v/(X 2 + y2 + Z 2), it can be shown that WDxyz(m,n) and WDv(m,n) are mathematically equivalent, for example, by using the relation 6 in Martin and Flandrin. 26 The Wigner distribution can also be directly computed from the vector summation V. It is preferable to compute the WDxyz according to equation 2, al though this requires even more computer power. The Wigner distribution is computed on the data with removed DC level and low-frequency trend. Practically, these operations on the vector summation are numerically less stable than those on separate leads. The WDxyz enables direct comparison to the results of t ime-domain analysis of the vector summation XYZ.

time-frequency mapping of the qrs complex in normal subjects and in postmyocardial infarction...

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