Fundamentals of Time- and Frequency-Domain Analysis of Signal-Averaged Electrocardiograms R. Martin Arthur, PhD

Transcription

1 CORONARY ARTERY DISEASE, 2(1):13-17, Fundamentals of Time- and Frequency-Domain Analysis of Signal-Averaged Electrocardiograms R. Martin Arthur, PhD Keywords digital filters, Fourier transform, linear systems, spectral analysis The mathematical framework established for the computer-aided analysis of linear systems has been used both to generate and to study signal-averaged ECGs. In this context a system is a mathematical model for any physical operation or transformation that takes an input and produces an output. A system is linear if the output due to two inputs delivered simultaneously is the sum of the outputs due to the two inputs delivered individually, i.e., superposition and proportionality apply. There is a powerful and extensive methodology in place for analyzing and designing digital systems which are both linear and time invariant [1-2]. One of the reasons for the large body of techniques developed for such a system is that a linear, time-invariant system is fully characterized by either its impulse response or by its frequency response. The impulse response of a system is the time-domain waveform produced when an impulse is its input. A system s frequency response is its gain and phase shift at each possible frequency of a sinusoid used as system input. In addition to characterizing the system, either of these two descriptions can be used to determine the output of the system. In this selective review three fundamental aspects of discrete-time, linear-system analysis applied to signalaveraged ECGs are examined. First, to illustrate the concepts of impulse and frequency response, the signalaveraging process itself is cast within the linear-system framework to determine the effect of misalignment in the averaging process on the frequency content of the resultant signal-averaged ECG. Second, in order to design filters to extract features of interest in a given cardiac condition, it is imperative first to determine the frequencies which distinguish that condition. Basic methods of spectral analysis are reviewed. Third, the design and implementation of filters to uncover the time-domain behavior of frequency components of interest are discussed. I. SIGNAL AVERAGING Features such as late potentials or high-frequency components of the ECG may be masked by typical noise levels at the body-surface. Signal averaging, which may be done either spatially or temporally, improves the signal-to-noise ratio. Spatial averaging methods combine ECGs recordedr simultaneously from several closely spaced sites. Temporal signal averaging, which has been used most often to enhance detection of late potentials, assumes that the same From the Department of Electrical Engineering, Washington University, St. Louis, MO 6313 ECG occurs repetitively at a single site. If the noise which contaminates the ECGs is uncorrelated, then the noise level is reduced by the square root of the number of ECGs in either the spatial or temporal average. For 1 ECGs in the average, uncorrelated noise is reduced by a factor of 1. The effect on the signal of temporal alignment or misalignment can be combined into a single linear-system operation. If all of the ECGs are aligned, only the noise is reduced, the signal is unchanged. The impulse response of a linear system that represents this process is an impulse at time zero. If, however, the ECGs are misaligned because of jitter, the signal, as well as the noise, will be reduced. Misalignment will not affect reduction of uncorrelated noise. The following example demonstrates the properties of the linear-system framework, which can be used to describe as a single linear system the misalignment of any number of ECGs by any amount which may occur during temporal averaging. The effect on the ECG of signal misalignment due to jitter which may occur during the averaging process is shown in Figure 1. In this example of a 1-beat average, 7 beats were correctly aligned, 1 were misaligned one sample interval early, 1 one sample interval late, 5 two samples early, and 5 two samples late. Because the impulse responses of linear systems in parallel add and because the impulse response of a delay system is just a delayed impulse, this misalignment scheme is characterized by the sequence just described. That combined impulse response for this example along with its Fourier transform, i.e., its frequency response are shown in Figure 1. Loss of high frequency components in the ECG due to this particular misalignment scenario is seen in the frequency response in the lower panel of Figure 1. The result for this example, that high frequencies are reduced more than low frequencies, however, is true for any form of misalignment. II. FREQUENCY-DOMAIN ANALYSIS Although time- and frequency-domain representations of a signal contain the same information, features which distinguish cardiac conditions may be easier to separate in the frequency domain. Once distinguishing frequencies are known, filters can be designed to extract their behavior in the time domain. The spectrum of a signal, like the frequency response of a system, is given by its Fourier transform. If a signal starts and ends with a value of zero, adding leading or trailing zeros does not influence its Fourier transform. Thus the frequency content throughout the cardiac cycle between isoelectric regions (zeros) is given by the fast Fourier transform (FFT) of samples of the original ECG

2 2 CORONARY ARTERY DISEASE, 2(1):13-17, Impulse Response 6 Frank Z Lead Sample Position Frequency Response Time, msec Magnitude Response Fraction of Sample Frequency Fig. 1. Effect on the ECG of misalignment or signal jitter in the averaging process. Upper panel) The impulse response of a system in which 7 ECGs are correctly aligned, 1 are aligned one sample interval too early, 1 are one sample too late, 5 are two samples too early, and 5 are two samples too late. Separation between bars is the sample interval used in acquiring the ECG, which is typically 1 msec. Lower panel) The frequency response of this system, which was found by padding the impulse response with zeros for a total of 124 samples, then taking the FFT of the padded sequence. The sample frequency is the reciprocal of the sample interval. Thus for a sample interval of 1 msec, the sample frequency is 1 khz. Because the highest frequency in the frequency response is half of the sample frequency, the highest frequency for a 1 khz sample rate is 5 Hz. If all ECGs were correctly aligned, the frequency response would be for all frequencies. Because the impulse response is symmetrical about the origin, its frequency response has zero phase. padded with zeros to make the total number of samples a power of 2 [21,22]. The FFT describes the spectrum of the original ECG, as long as it was sampled at a rate at least twice the highest frequency present in the original ECG. Padding the cardiac-cycle ECG with zeros will change where the FFT samples the continuous spectrum of the original signal, not the nature of the spectrum. By padding with enough zeros, any frequency in the original signal can be recovered. Adding a constant to each sample in the padded ECG, i.e., shifting the baseline affects only the DC component or zero-frequency line in the spectrum. Figure 2 shows the spectrum of a signal-averaged Frank Z lead over the cardiac cycle. The cardiac cycle was 717 samples long. These samples, taken at a 1kHz rate with 12 bits of precision (dynamic range of 72 ), began and ended with zeros and were padded with zeros for a total of 124 samples, then the FFT was calculated. The magni- Radians Phase Response Fig. 2. Spectrum of an ECG over an entire cardiac cycle. Upper panel) A signal-averaged Frank Z lead sampled at 1 khz with 12 bits of precision. Middle and lower panels) The magnitude and phase of the spectrum, respectively. Because the ECG began and ended with zero values, its spectrum was found by padding the ECG with zeros for a total of 124 points, then calculating the FFT of the padded sequence. Estimates of the spectrum of the interval indicated by the vertical bars in the upper panel are presented in Fig. 3. tude spectrum, is like that of most ECGs, in that it falls quickly with increasing frequency. Most of the energy in the ECG, but not necessarily all of the distinguishing features, is in the first 1 Hz. The phase spectrum changes very rapidly; its principal part is limited to a range between ±π. Phase information is required to properly place the frequency components in the time domain. Clearly the magnitude spectrum or its square, the power spectral density, is of more interest than the phase spectrum in identifying distinguishing features. Several methods for estimating the spectra of bioelectric events are available. Each method has advantages, as

3 ARTHUR: FUNDAMENTALS OF SIGNAL-AVERAGED ECG ANALYSIS 3 well as limitations, depending on the characteristics of the signal of interest. Most analyses of signal-averaged ECGs have concentrated on portions of the ECG rather than on the whole cardiac cycle. Spectral estimation from an interval of a signal, such as the terminal QRS complex or the ST segment of the ECG, is usually done using the sample spectrum, which is calculated from the FFT of the interval of interest after multiplying it by a window function to isolate it [23-25]. Multiplication by a window function limits frequency resolution and causes the DC value of the segment to affect the whole spectrum, not just the DC component. A further limitation of methods using the sample spectrum is that the spectral estimates have a large standard deviation. To reduce the standard deviation of FFT spectral estimates, the data interval may be divided into overlapping segments, which are windowed, transformed, then averaged to produce a periodogram. Figure 3 shows Welch-method periodogram estimates of the spectral density over both the cardiac cycle and the 13 msec ST interval indicated in the upper panel of Figure 2. Segmenting the waveform provides spectral estimates that are more statistically stable. The increased stability with segmenting the data interval, however, comes at the expense of reduced frequency resolution. Parametric techniques for spectral estimation provide higher resolution than the periodogram. They do not use window functions, nor do they assume that the signal is zero outside the interval analyzed. These methods are based on assuming a time-series model for the random process which produced the signal. The data are used to determine parameters of the model. Power spectral density is then calculated from the model rather than directly from the data. Figure 3 compares the Welch periodogram to two autoregressive models, which have rational system functions. All three methods are similar over the cardiac cycle for which no window function is needed. The Welch periodogram estimate differs markedly from the autoregressive methods for the short interval, primarily because of the window which affects the periodogram, but not the autoregressive estimates. The three different estimates of power spectral density in Figure 3 represent the same data. Any one of the estimates may be valid depending on the nature of the signal. Clearly the choice of window function and the length and overlap of segments for averaging affects the periodogram result. Likewise, the choice of time series model and the number of parameters in the model affects the autoregressive estimates. Differences in these and other choices in spectral estimation methods obviously may make comparison of results from different investigators difficult. III. TIME-DOMAIN ANALYSIS Once spectral features of interest have been identified their effect in the time domain may be brought out by appropriate digital filtering of the samples of the ECG [26,27]. A linear, time-invariant digital filter may be implemented in the time domain as a difference equation or as a convo Cardiac Cycle Power Spectral Density Welsh: 8 sections Burg: 16th Order Yule Walker: 64th ST Segment Power Spectral Density Welsh: 8 sections Burg: 16th Order Yule Walker: 64th Fig. 3. Estimates of the power spectral density on the ECG shown in Figure 2. Upper panel) Estimates over the cardiac cycle. Welch periodogram estimates are based on the FFT. Compare these estimates to the FFT magnitude shown in Figure 2. Lower panel) Estimates over the 13-msec interval indicated in Figure 2 from late in QRS to early in the T wave. This short interval was extracted by a Blackman-Harris window for periodogram estimates. The periodogram estimate is quite different from those using the Burg and Yule-Walker autoregressive models. lution sum. If the filter is based on a rational polynomial, such as a Butterworth, Chebyshev, or Bessel polynomial, then a difference equation is likely to require fewer arithmetic operations to produce an output than using the impulse response of the filter, which has been windowed, in the convolution sum. Although these polynomial filters are easy to design, they have nonlinear phase characteristics, which means that some of the frequency components that pass through the filter will be shifted compared to their position in the original ECG. Bidirectional processing forces the phase of the filter to zero, so that no phase distortion occurs. This result is obtained if the signal is passed through a the filter, the samples in that output are reversed along the time axis, passed through the filter again, and finally the second output is reversed along the time axis. An alternative to bidirectional filtering is to design a

4 4 CORONARY ARTERY DISEASE, 2(1):13-17, BandPass Filter Response Hz Hz Frank X Lead Hz Hz Time, msec Fig. 4. Filtering an ECG. Upper panel) The magnitude of the frequency response of two bandpass filters. They have passbands of 13 to 56 Hz and 7 to 128 Hz, respectively. Both filters have zero-phase responses and therefore introduce no phase distortion and no phase shift. Lower panel) The result of filtering a signalaveraged Frank X lead ECG is shown over one cardiac cycle with each of the filters. For clarity the baseline of the output of the Hz filter was displaced to -1 and that of the Hz filter output was displaced to -3. filter that has no phase distortion. Such a filter can be implemented with the convolution sum in a single step. For example, filters that minimize the maximum error between the actual and desired filter characteristics for a given number of samples in the impulse response of the filter can be designed using readily available tools [28]. The upper panel of Figure 4 shows two optimal, bandpass filters designed using the Remez exchange algorithm. Both have 255-point impulse responses. This length is needed to provide sharp cutoff at the band edges. This cutoff is much sharper than that provided by a low-order (4 pole) Butterworth filter. Use of these zero-phase filters is demonstrated in the lower panel of Figure 4. The frequency response of each filter was multiplied by the FFT over the cardiac-cycle of the ECG shown. The filter output was obtained by computing the inverse FFT of the frequency-domain products. The effect of zero phase is two-fold. There is no phase distortion, so that the frequencies that are passed by the filter have the same relative timing as they had in the original ECG. Further the zero-phase filter insures they appear at exactly the same time, i.e., they are precisely aligned with the original waveform as shown in Figure 4. IV. CONCLUSIONS The strength of the linear-systems framework for ECG analysis is its ability to easily characterize, then extract either time- or frequency-domain features of interest. The vast body of methods and the range of applications to whichthosemethodshavebeenandremaintobeappliedto the study of signal-averaged ECGs can only be suggested in a brief review. Although most studies have examined late-qrs-to-st intervals, these methods will certainly be exploited to determine distinguishing spectral, temporal, and spatial features of other portions of the cardiac cycle, as well as of the entire cardiac cycle. V. Appendix 1. Papoulis, A., Signal Analysis, McGraw-Hill, New York, Gabel, R. A. and R. A. Roberts, Signals and Linear Systems, Wiley and Sons, New York, Oppenheim, A. V. and A. S. Willsky, Signals and Systems, Prentice-Hall, Englewood Cliffs, N. J., Ziemer,R.E.,W.H.Tranter,andD.R.Fannin,Signals and Systems: Continuous and Discrete, 2nd edition, MacMillan Publ., New York, References 1-4 are examples of texts that present a balanced treatment of both continuous and discrete-time linear systems. 5. Gold, B. and C. M. Rader, Digital Processing of Signals, McGraw-Hill, New York, Oppenheim, A. V. and R. W. Schafer, Digital Signal Processing, Prentice-Hall, Englewood Cliffs, N. J., Peled, A. and B. Liu, DigitalSignalProcessingTheory, Design, and Implementation, Wiley and Sons, New York, Tretter, S. A., Introduction to Discrete-Time Signal Processing, Wiley and Sons, New York, Jong, M. T., Methods of Discrete-Time Signal Processing and Systems Analysis, McGraw-Hill, New York, Ahmed, N. and T. Natarajan, Discrete-Time Signals and Systems, Reston Publ., Reston, VA, Stanley, W. D., G. R. Dougherty, and R. Dougherty, DigitalSignalProcessing, 2nd edition, Reston Publ., Reston, VA, Cadzow, J. A., Foundations of Digital Signal Processing and Data Analysis, MacMillan Publ., New York, DeFatta,D.J.,J.G.Lucas,andW.S.Hodgkiss,Digital Signal Processing: A System Design Approach, Wiley and Sons, New York, Kuc, R., Introduction to Digital Signal Processing, McGraw-Hill, New York, Proakis, J. G. and D. G. Manolakis, Introduction to Digital Signal Processing, MacMillan Publ., New York, Bellanger, M., Digital Processing of Signals Theory and

5 ARTHUR: FUNDAMENTALS OF SIGNAL-AVERAGED ECG ANALYSIS 5 Practice, 2nd edition, Wiley and Sons, New York, Johnson, J. R., Introduction to Digital Signal Processing, Prentice-Hall, Englewood Cliffs, N. J., Lynn, P. A. and W. Fuerst, Introduction to Digital Signal Processing with Computer Applications, Wiley and Sons, New York, A. V. Oppenheim and R. W. Shaefer, Discrete-Time Signal Processing, Prentice Hall, Englewood Cliffs, N. J., Stearns, S. D. and D. R. Hush, Digital Signal Analysis, 2nd edition, Prentice-Hall, Englewood Cliffs, N. J., 199. References 5-2 are examples of texts that concentrate on the characteristics of discrete-time or digital linear systems including the FFT and filter design. VI. Annotated references and recommended reading o Of interest oo Of outstanding interest 1. Cooley, J. W. and J. W. Tukey, An algorithm for the machine computation of complex Fourier series, Mathematics of Computation, vol. 19, pp , (o) First modern description of the method for calculating the discrete Fourier transform known as the FFT 2. Bergland, G. D., A guided tour of the fast Fourier transform, IEEE Spectrum, vol. 6, pp , (oo) A basic and comprehensive introduction to the FFT 3. S. M. Kay and S. L. Marple, Jr., Spectrum analysis a modern perspective, Proc. IEEE, vol. 69, pp , (o) A thorough review of spectral analysis mathematics 4. S. L. Marple, Jr., Digital Spectral Analysis with Applications, Prentice-Hall, Englewood Cliffs, N. J., (oo) The spectral estimates in Figure 3 were generated from programs included with this text 5. Candy,J.V.,Signal Processing The Modern Approach, McGraw-Hill, New York, (o) A text which covers discrete-time systems with an emphasis on spectral analysis methods 6. Introduction to Digital Filtering, eds. R. E. Bogner and A. G. Constantinides, Wiley and Sons, New York, (o) Example of a text that focuses on techniques for the design and implementation of digital filters 7. Hamming, R. W., Digital Filters, 3rd edition, Prentice- Hall, Englewood Cliffs, N. J., (o) Example of a text that focuses on techniques for the design and implementation of digital filters 8. Programs for Digital Signal Processing, eds. Digital Signal Processing Committee, IEEE Acoustics, Speech, and Signal Processing Society, IEEE Press, New York, (o) The bandpass filters in Figure 4 of this article were designed using one of the programs in this collection.