Comparative Study of IIR Notch Filters for Suppressing 60-Hz Interference in Electrocardiogram Signals

Transcription

1 5() June 3, pp. 7-8 International Science Press, India I J E C Comparative Study of IIR otch Filters for Suppressing 6-Hz Interference in Electrocardiogram Signals ing Li* Abstract: This paper gives the performance analysis of three well-known infinite impulse response (IIR) notch filters in two aspects. One is the amplitude property of the frequency response functions based on the measure of rectangular coefficient. The other is the phase property of the frequency response functions with respect to linearity. The analysis results suggest that the amplitude property of IIR notch filters seems satisfactory but the phase property may be unsatisfactory. The demonstration with a real ECG signal exhibits that an ECG signal may be considerably corrupted by nonlinear phase distortions. Keywords: Electrocardiogram (ECG) signal processing, power line interference, interference suppression, IIR digital filters. ITRODUCTIO Electrocardiogram (ECG) plays a key role in quantitatively diagnosing cardiopathies. A high quality of ECG signals is obviously a basis of achieving high quality diagnostic analysis of these signals. However, since the electrical behavior of the human heart is measured by electrodes applied externally on the human body, the amplitude of an ECG signal is low, approximately ~ 5 mv (Talmaon [], Levkov et al. []). The low amplitude ECG signals are usually contaminated by various disturbances. Among them, the power line interference in 5 Hz (e.g., in China) or 6-Hz (e.g., in USA) is the most common one. This is because the strong 5 Hz or 6-Hz electromagnetic fields on the mains exist in modern hospitals and homes. Therefore, techniques for removing the power line interferences are desired. Various types of digital filters and hardware circuits have been proposed in this regard, see e.g., Talmaon [, pp. 5], Hamilton [3], artens et al. [4], Ahlstrom and Tompkins [5], Pei and Tseng [6], Dotsinsky and Stoyanov [7], Ramos et al. [8], Ider et al. [9]. However, adopting notch filters in the standard sense remains a question (Breithardt et al. []). For this reason, this paper gives the performance analysis of notch filters discussed in [3,4,5,6,7]. The present analysis is for the 6-Hz interference notch filters but the analysis procedure is also usable for 5 Hz ones. The present results suggest that the amplitude property of the notch filters in [3,4,6,7], which is measured by the rectangular coefficient, is satisfactory in the sense that their rectangular coefficients are approximately equal to but their phases are nonlinear. A case study with a real ECG signal is used to demonstrate that the phase nonlinearity of the filters may seriously cause distortions of an ECG signal. The remaining article is organized as follows. The preliminaries regarding digital filters are briefed in Section. The performance analysis of the notch filters in [3,4,5,6,7] is given in Section 3. Discussions are in Section 4, which is followed by conclusions. * School of Information Science & Technology, East China ormal University, Shanghai 4, PR. China, mli@ee.ecnu.edu.cn, ming_lihk@yahoo.com

2 8 I J E C 5() 3. PRELIIARIES OF DIGITAL FILTERS The functionality of a digital filter is to transform an input sequence x(nt) of the filter to its output sequence y(nt), where n is the index of the sequence number and T the sampling interval. There are two categories of digital filters. One is infinite impulse response (IIR) filter and the other finite impulse response filters (FIR) (Harger [], Lam [], itra and Kaiser [3]). One class of linear difference equations used to characterize a linear time-invarying digital filter is given by where a =. Rewiritng () by a y( nt kt ) br x( nt rt ), () k k r Then, the transfer function of () is expressed by r k () r k y( nt ) b x( nt rt ) a y( nt kt ). r br z Y ( z) r ( ). X ( z) k ak z k H z (3) By factoring of a polynomial (Korn and Kong [4,.7-]), one has r r r r b z b ( z zr ), (4) k k k k a z a ( z zk ). (5) Eqs. (4) ~ (5) imply that there are ( + ) zeros z r (r =,,, ) and ( + ) poles z k (k =,,, ) in H(z) expressed by (3) (Vegte [5]). Doing the inverse of Z transform yeilds the impulse function h(n) written by r br z r ( ) [ ( )], k ak z k h n Z H z Z (6) where Z represents the inverse of Z transform. ote that k a z k k k ak z. k (7)

3 Comparative Study of IIR otch Filters for Suppressing 6-Hz Interference in Electrocardiogram Signals 9 Then, one immediately sees that h(n) is a function that is non-zero over an infinite length of time. Hence, IIR. The frequency response function of a digital filter is written by H e H z (8) j f ( ) ( ) j f. ze The amplitude of H(e jf ) is denoted by A( f ) = H (e jf ) and the phase of H(e jf ) by j( f ) jf Im[ H( e )] j f tan. Re[ H ( e )] An IIR filter is stable in the sense of bounded input and bounded output if all poles of the transfer function have absolute values smaller than one. In other words, all poles must be located within a unit circle in the z-plane. In contrast to IIR filters, finite impulse response filters (FIR) have fixed-duration impulse responses. Based on FIR filters, a linear difference equation to characterize a linear time-invarying digital filter is expressed by y( n) br x( n r). (9) r The transfer function corresponding to (9) is expressed by r H ( z) b z. () r An FIR filter is always stable as its system function has no poles. Its impulse function is given by Hence, FIR. r br, r h( r). (), elsewhere Usually, IIR filters do not have linear phase property but the phases of FIR ones are always linear. IIR filters are recursive but FIR ones are not. Due to the recursive property of IIR, in general, an FIR filter is time consuming and expensive in comparison with an IIR one. Therefore, IIR filters are often preferred in the design of digital filters in engineering though their phases are nonlinear (Lam [, --3]). Three types of notch filters analyzed in this paper are all IIR. 3. PERFORACE AALYSIS The ideal amplitude property of a notch filter in the sense of low pass to remove the 6-Hz power line interference is expressed by j f A( f ) H ( e ) u( f ) u( f (6 )) u( f (6 )), () where > is an infinitesimal. Unfortunately, the above is physically unrealizable according to the Paley- Wiener criterion (Papoulis [6]). In practical terms, the above can only be approximated. Therefore, there is a side effect of a practical notch filter. ore precisely, it may suppress useful frequency components in a way while eliminating the 6-Hz interference. The side effect is expected to be minimized. That is, the functionality of a filter is to suppress the 6-Hz interference as much as possible while keep the information in the other frequency components as much as possible. In fact, medical scientists are paying efforts on

4 I J E C 5() 3 seeking for useful information from electrocardiograms, see e.g., Rijnbeek [7], Zywietz et al. [8], Farrell and Rowlandson [9]. Thus, a notch filter is expected to approximate to () as much as possible. In this paper, we use the rectangular coefficient to measure the amplitude property of a frequency response function. Without the generality losing, we use the normalized frequency response function in what follows. By normalized frequency response function, we mean that Denote B.7 the 3-dB bandwidth. Then, H e (3) ( j f ). max Denote B. the bandwidth for H e j f ( ).77. f B.7 (4) H e Then, the rectangular coefficient is defined by j f ( ).. f B..7 Rec.. (5) B (6) B According to (6), one sees that Rec for for the ideal filter expressed by (). Fig. shows the plot of the amplitude of the ideal notch filter for removing the 6-Hz interference. As a matter of fact, Rec is a parameter to measure how much similar of the amplitude of a frequency response function to an ideal notch filter in the form of low pass. For example, suppose there are two filters. Filter has the rectangular coefficient Rec while filter has Rec. Then, if Rec < Rec, we say that filter has better amplitude response than that of filter..5 A(f) Figure : The Amplitude of Ideal otch Filter for Removing the 6-Hz Interference 4. COPARATIVE STUDY OF OTCH FILTERS I ECG SYSTES 4. otch Filter H (z) in [3,4] The transfer function of the 6-Hz notch filter (IIR) discussed by Hamilton [3] and artens et al. [4] is given by cos(6 T ) z z H( z), (7) r cos( 6 T) z r z

5 Comparative Study of IIR otch Filters for Suppressing 6-Hz Interference in Electrocardiogram Signals where T is the sampling interval. The parameter r is used to adjust the bandwidth of the notch filter. The bandwidth of the filer becomes narrow and the transient response time tends to be larger when the value of r increases. It corresponds to the difference equation given by where = cos(6t). where y(t) = ry (t nt) r y (t nt) + x(t) (x) (t nt) + x )(t nt), (8) The amplitude of the frequency response is Denote the phase of H (e jf ) by j f a jb A( f ) H( e ), (9) c jd a = cos ( 6 T) cos ( ft) + cos ( ft), () b = sin ( f T) cos ( 6 T) sin ( f T), () c = r cos ( 6 T) cos ( f T) + r cos ( ft), () d = r sin ( ft) r cos ( 6 T) sin ( ft) (3) j f Im[ H( e )] j f Re[ H( e )] phi( f ) tan. (4) In the cases of r =.98 and T =.4 s, A( f ) and phil (f) are indicated in Fig. and Fig., respectively..5 A(f) phi(f) Figure : Frequency Response of (7) in the Cases of r =.98 and T =.4 s.. Amplitude. Phase It can be seen from Fig. that A( f ) has a high Rec. In fact, A(59.75) =.77, (5) and A (59.94) =.. (6) Thus, one has B B (7).7 Rec.988..

6 I J E C 5() 3 ote. This filter has satisfactory amplitude property since Rec but its phase is nonlinear, see Fig otch Filter H (z) in [5] Ahlstrom and Tompkins [5] studied an IIR notch filter whose transfer function is expressed by The difference equation for H (z) is 56 z H ( z). z (8) y(nt) = y (nt T) + x (nt) x(nt 56T) (9) The amplitude of the frequency response function is given by j f cos( f 56 T ) j sin( f 56 T) A f H e cos( ft) j sin( ft ) ( ) ( ). (3) The phase of H (e jf ) is j f Im[ H ( e )] j f Re[ H ( e )] phi( f ) tan. (3) In the case of T =.65, we have A( f ) and phi( f ) as shown in Fig A(f) phi(f) Figure 3: Frequency Response of (8) in the Case of T =.65 s.. Amplitude. Phase Since and we have A(6.6) =.77, (3) A(54.55) =., (33) B 6.6 (34) B Rec ote : This filter has the functionality to eliminate dc offset because A() =. ote 3: Due to the sudden change at f = 3 Hz as shown in Fig. 3, the phase of this filter is never linear. ote 4: In practice, T =.65 s requires over high sampling rate. Thus, that is replaced by Tm =.65, where m < is a constant. For example, if T =. s, one can set m =.65 [5].

7 Comparative Study of IIR otch Filters for Suppressing 6-Hz Interference in Electrocardiogram Signals otch Filter H 3 (z) in [6,7] In the reference papers by Pei and Tseng [6], Dotsinsky and Stoyanov [7], a notch filter was discussed. Its transfer function is written by ( a ) a z ( a ) z H ( ). 3 z az az (35) The difference equation for is expressed by y( nt ) a y( nt T ) a y( nt T ).5[( a ) x( nt) a x( nt T) ( a ) x( nt T ), (36) where y and x are filtered and nonfiltered samples, respectively, and a a cos( ) tan( / ), (37) tan( / ). (38) tan( / ) The variables and have to be computed for the interference frequency f (Hz), the signal sampling rate f s (Hz), and the desired low and high cut-off frequencies f L and f H of the notch filter: f, (39) f s fh fl. (4) f Let f = 6 Hz, f s = 5 Hz, f L = 59 Hz, and f H = 6 Hz. Then, =.8, =.4. Consequently, cos( ) tan( / ) a.44, a.975. tan( / ) tan( / ) The frequency response function of is given by s H e ( a) a cos( ft ) ( a) cos(4 ft) a cos( ft ) a cos(4 ft ) j[ a sin( ft ) a sin(4 ft )] jf 3( ), j[a sin( ft) ( a)sin(4 ft )] a cos( ft ) a cos(4 ft) j[ a sin( ft ) a sin(4 ft )] (4) The amplitude of the frequency response function is given by A f ( a) a cos( ft ) ( a) cos(4 ft ) a cos( ft ) a cos(4 ft) j[ a sin( ft ) a sin(4 ft )] j[a sin( ft ) ( a)sin(4 ft )] a cos( ft ) a cos(4 ft ) j[ a sin( ft ) a sin(4 ft )] 3( ). (4)

8 4 I J E C 5() 3 The phase of H 3 (e j f ) is j f Im[ H3( e )] jf Re[ H3( e )] phi3( f ) tan. (43) Fig. 4 and Fig. 4 show the plots of A3(f) and phi3 (f), respectively. A3(f).5.5 phi3(f) Figure 4: Frequency Response of (4).. Amplitude. Phase Since and we have A3(59.6) =.77, (44) A3(59.9) =., (45) B 59.6 (46) B Re c. ote 5. The amplitude property of the filter H 3 (z) is satisfactory since Rec but its phase is nonlinear (Fig. 4 ) Discussions The previous discussions exhibit that H (e jf ) and H 3 (e jf ) have the similar high rectangular coefficient but the rectangular coefficient of H (e jf ) is low. We summarize the analysis results for three filters discussed previously in Table. Table Summary of the Analysis Results System function H (e jf ) H (e jf ) H 3 (e jf ) Rectangular coefficient.988 (high).488 (fair).985 (high) It can be easily seen from the high rectangular coefficient that the amplitude of the frequency response function of the notch filters in [3,4,6,7] to suppress the 6-Hz interference is good enough. evertheless, their phases are nonlinear, which people in the field might be unsatisfactory, because the nonlinear phase of a notch filter may cause distortions of ECG signals.

9 Comparative Study of IIR otch Filters for Suppressing 6-Hz Interference in Electrocardiogram Signals 5 We now use a real ECG signal x(t), which is with the file name in the Records in the IT-BIH Arrhythmia Database. It is available to download freely from the web site with the link []. Fig. 5 indicates that signal. ote that where FFT stands for the fast Fourier transform. Figs. 6 and show its normalized amplitude of the spectrum its phase, respectively. x(t) Figure 5: A real ECG Signal x(t) X(f).5 phi_x(f) Figure 6: Spectrum of x(t).. Amplitude of X(f).. Phase of X( f ) We now purposely add a 6 Hz interference to x(t). Denote n(t) =.5cos (ft) for f = 6 Hz. (47) Figs. 7 and indicate the plot of n(t) and its spectrum, i.e., ( f ) = FFT[n(t)] respectively. 4 3 n(t) (f) t Denote Figure 7: Sixty Hz Interference. Time Signal. Spectrum g(t) = x(t) + n(t). (48) Then, the signal x(t) is seriously contaminated by n(t), as can be seen from Fig. 8. Therefore, if we filter g(t) by H (e j f ) all frequency components of x(t) should pass through H (e jf ) with the 6 interference suppressing. Denote y(t) and Y( f ) the time series and the spectrum of the output of H (e jf ), respectively. Then,

10 6 I J E C 5() 3 g(t) G(f) Figure 8: Signal x(t) is Contaminated by a 6 Hz Interference Time Signal Spectrum Y f H e G f (49) j f ( ) ( ) ( ), y( t) IFFT[ Y ( f )], (5) where IFFT implies the inverse of FFT. Figs. 9 and show the amplitude of Y( f ) and its phase, respectively. Y(f) phi_y(f) Figure 9: Spectrum of y(t) Amplitude Phase Fig. 9 clearly indicates that the 6 Hz interference is satisfactorily eliminated by the notch filter H (e j f ). According to y(t) = IFFT[Y ( f )], we have a reconstructed signal in time as indicated in Fig.. y(t) Figure : Time Signal y(t) By comparing Fig. 5 to Fig., we see that the R waves have been completely reconstructed but others, such as P, Q, T and U waves, are seriously contaminated in a way, see Fig.. We shall explain the possible reason below. x(t), y(t) Figure : Comparing x(t) to y(t). Solid line: y(t). Dot line: x(t)

11 Comparative Study of IIR otch Filters for Suppressing 6-Hz Interference in Electrocardiogram Signals 7 In appearance, the phase of y(t) is the same as that of x(t). However, there is the phase distortion caused by the filter H (e j f ). Fig. shows the comparison between the phase of X(f ) and that of Y(f ). From Fig., we see that the phase of X(f ) is completely overlapped with that of Y(f ) except that there is an obvious phase distortion for f being near 6 Hz. This is natural because the filter H (e j f ) has considerably nonlinear phase for f being near 6 Hz and its phase quite linear for other frequencies (see Fig. ). phi_x(f), phi_y(f) Figure : Comparing the Phase of X( f ) to that of Y( f ). Solid Line: the Phase Y( f ). Dot Line: the Phase of X( f ) ote that the X(f ) is totally overlapped with Y(f ) (see Fig. 3), implying that H (e j f ) has satisfactory amplitude response. Therefore, we conclude that the reason why there are distortions shown in Fig. may be the nonlinearity of the phase of H (e j f ) although its phase is only considerably nonlinear for f being near 6 Hz. X(f), Y(f) Figure 3: Comparing X(f ) to Y(f ). Solid Line: Y(f ). Dot Line: X(f ) As the nonlinear phase is an intrinsic property of IIR filters while FIR notch filters have linear phases, we suppose that to find an FIR notch filter that has high rectangular coefficient may be a way to overcome the shortcoming of nonlinear phase of an IIR filter though an FIR filter may be time consuming. 5. COCLUSIOS We have analyzed three well-known IIR digital notch filters for suppressing 6-Hz interference in ECG signals. The present results exhibit that the filters in [3,4] and [6,7] have satisfactory rectangular coefficient but their phases are nonlinear. We have shown a case of applying the filter H (e j f ) to remove 6 Hz interference with a real ECG signal. The analysis of this case implies that the nonlinearity of the phase of a filter may cause serious distortions of the ECG signal being analyzed. Consequently, to design FIR notch filters that have high rectangular coefficient may be a way to reduce the side effect of notch filters for removing 6-Hz interference.

12 8 I J E C 5() 3 ACKOWLEDGEETS This work was supported in part by the ational atural Science Foundation of China under the project grant numbers 65735, , and REFERECES [] J. L. Talmaon, Pattern Recognition of the ECG: A Structured Analysis, Ph.D. Thesis, (983). [] C. Levkov, G. ichov, R. Ivanov, and I. K. Daskalov, Subtraction of 5 Hz Interference from the Electrocardiogram, edical and Biological Engineering and Computing, (4), (984), [3] P. S. Hamilton, A Comparison of Adaptive and onadaptive Filters for Reduction of Power Line Interference in the ECG, IEEE Trans. Biomedical Engineering, 43() (996), 5 9. [4] S.. artens,. ischi S. G. Oei, and J. W. Bergmans, An Improved Adaptive Power Line Interference Canceller for Electrocardiography, IEEE Trans. Biomedical Engineering, 53(), (6), 3. [5]. L. Ahlstrom and W. J. Tompkins, Digital Filters for Real-time ECG Signal Processing using icroprocessors, IEEE Trans. Biomedical Engineering, 3(9), (985), [6] S. C. Pei and C. C. Tseng, Elimination of AC Interference in Electrocardiogram using IIR otch Filter with Transient Suppression, IEEE Trans. Biomedical Engineering, 4(), (995), 8 3. [7] I. Dotsinsky and T. Stoyanov, Power-line Interference Cancellation in ECG Signals, Biomedical Instrumentation & Technology, 39(), (5), [8] R. Ramos, A. ànuel-làzaro, J. D. Río, and G. Olivar, FPGA-Based Implementation of an Adaptive Canceller for 5-6Hz Interference in Electrocard, IEEE Trans. Instrumentation and easurement, 56(6) 7, [9] Y. Z. Ider,. C. Saki, and H. A. Gcer, Removal of Power Line Interference in Signal-averaged Electrocardiography Systems, IEEE Trans. Biomedical Engineering, 4(7), (995), [] G. Breithardt,. E. Cain,. El-Sherif,. C. Flowers, V. Hombach,. Janse,. B. Simson, and G. Steinbeck, Standards for Analysis of Ventricular Late Potentials using High-resolution or Signal-averaged Electrocardiography. A Statement by a Task Force Committee of the European Society of Cardiology, the American Heart Association, and the American College of Cardiology, Circulation, 83(4), (99), [] R. O. Harger, An Introduction to Digital Signal Processing with ATHCAD, PWS Publishing Company, (999). [] H. Y. F. Lam, Analog and Digital Filters: Design and Realization, Prentice-Hall, (979). [3] S. K. itra and J. F. Kaiser, Handbook for Digital Signal Processing, John Wiley & Sons, (993). [4] G. A. Korn and T.. Korn, athematical Handbook for Scientists and Engineers, cgraw-hill, (96). [5] J. Van de Vegte, Fundamentals of Digital Signal Processing, Prentice Hall, (3). [6]A. Papoulis, The Fourier Integral and Its Applications, cgraw-hill, (96). [7] P. R. Rijnbeek, J. A. Kors, and. Witsenburg, inimum Bandwidth Requirements for Recording of Pediatric Electrocardiograms, Circulation, 4 (5), (), [8] C. Zywietz, D. Celikag, and G. Joseph, Influence of ECG easurement Accuracy on ECG Diagnostic Statements, Journal of Electrocardiology, 9(Supplement) (996), [9] R.. Farrell and G. I. Rowlandson, The Effects of oise on Computerized Electrocardiogram easurements, Journal of Electrocardiology, 39(4), (6), S65 S73. []