A QRS detection method using analog wavelet transform in ECG analysis

Transcription

1 A QRS detection method using analog wavelet transform in ECG analysis 20th June 2005 Abstract Low power implementable devices like the pacemaker need good sensing circuits to correctly analyze the cardiac signal and take appropriate actions. Existing methods are reaching their limits on sensing abilities. An approximation of the wavelet transform (WT), that can be implemented in an analog dynamic translinear circuit, can be used to further the advances of cardiac signal sensing and lead to better pacemakers. A method is presented for automatic QRS complex detection in an ECG signal for use with an analog implementation of the wavelet transform. By using the modulus maxima in the WT and the time differences between them, QRS complex detection rates are up to 98% on the MIT/BIH database. 1 Introduction For medical implantable devices such as pacemakers, it is important that the power consumption is as low as possible. In normal conditions, the power consumption in the digital domain is significantly higher than in the analog domain. Therefore it is important to perform as many computations as possible in the analog domain. In this study we focus on pacemakers. The efficiency of pacemakers greatly depends on its ability to correctly analyze heart signals and to take appropriate actions. The first version of a pacemaker was introduced as early as This device however was only able to pace the heart asynchronously, at a steady rate, and did not have the ability to sense the hearts current activity. Later devices did have the ability to sense the cardiac activity and thus could avoid competition between paced and intrinsic rhythms. Clinical, surgical and technological developments have since then proceeded at a remarkable rate [3]. Much research has been done in the field of ECG (electro cardiogram) analysis. Some of the most successful methods make use of the Wavelet Transform (WT) [7, 5]. These methods however, are not well suited for use in the analog domain. The wavelet transform is a computational complex method and requires a lot of processing power. Trying to implement this digitally in an implantable device would be futile as power consumption would be far to high for any power source to last long. Using the WT for sensing and detection purposes is still favorable though, as the method offers many advantages over other methods. One solution would be to do the WT in the analog part of the circuit, as this can be done in low-power and the computational delay would be minimized. It is however generally not possible to compute the WT exactly in analog electronic circuits. Recent developments have made it possible to compute an approximation of the WT in the analog domain [6, 2]. This method can now be used to further develop a method that is well suited for implementation in a low-power electronic device, that could possibly further improve the quality of todays intra cardiac devices. It should be noted here that the method is only given on a conceptual level; issues concerning integrated circuit (IC) design are left out and no circuit designs are presented. For testing this method MATLAB s Simulink was used. Simulink is a software package for modeling, simulating and analyzing dynamic systems. It supports linear and nonlinear systems, modeled in continuous time, sampled time or a hybrid of the two. In section two the method for the analog WT approximation is presented, then in section three some issues that have to be dealt with when working in the analog domain are discussed. In section four the method for QRS complex detection is explained and after that the results and conclusions are presented. 2 Theory The fundamentals of system theory and of the wavelet transformation and the relation between them are briefly discussed here. Wavelet transformation is a succesful method in signal analysis, due to of its good estimation of both time and frequency localizations. These features make it a well suited for local analysis of fast time varying and nonregular signals. The continuous wavelet transformation of a signal f(t) is given by the convolution integral of f(t) with the time inverted and scaled wavelet function

2 Bachelor project ψ(t): W (t, ) = 1 f(τ)ψ( τ t )dτ. (1) When the scale factor is small the wavelet function is contracted in the time domain and the frequency response of the wavelet is shifted to higher frequencies, thus fine details or fast oscillations of the signal are extracted. In the same way, more global signal properties and low frequency oscillations of the signal are extracted when is large. It is generally not possible to implement the wavelet transform exactly in analog electronic circuits [6]. But when we look at linear filters, which are well understood and relatively easy to implement in analog circuits, a possible solution presents itself. The output of a linear filter of finite order with an input signal f(t) is the convolution of that signal with the impulse response h(t) of the linear system [9]: y(t) = f(τ)h(τ t)dτ. (2) So when the impulse response h(t) of the system satisfies h(t) = 1 ψ( τ t ) the analog wavelet transform of f(t) is realized. A hardware implementation of this system has to be strictly causal: there can be no response of the system before any non-zero input has been presented. This means that for t < 0 both the signal f(t) and the impuslse response h(t) have to be equal to zero. Most wavelet function however do not share this property. The wavelet function used has to be time shifted so that most of its energy is preserved at t > 0, to obtain a good approximation of the WT. The choice of the time-shift t 0 is an important issue in the quality of the WT and otherwise for the order of the linear filter: the quality of the WT improves as t 0 is increased and more energy of the wavelet is at t > 0, but the order of the linear system also increases greatly because there is a lot of dead time from t > 0 as the wavelet is very flat near t = 0. When the wavelet does not have compact support, this means that it cannot be time shifted so that all of the energy of the wavelet is at t > 0. Due to this a small truncation error will be present in the WT. For the impulse response to exactly mimic the desired wavelet function, the wavelet must have a rational Laplace transform so that H(s) = Ψ(s). For most wavelets this is not the case. So, the impulse response is an approximation of the wavelet function, this causes a small error in the WT. It is also important to note that time-shifting the wavelet function also introduces a time-shift in the WT. This time shift also depends on the dilation factor (see Figure 1. A method to approximate ψ(t) in a linear system is described in [6]). Figure 1: The Gaussian wavelet and the impulse response from the linear system For the purpose of IC design it is useful to have a state-space system to represent the linear filter for the WT. The state-space representation here is of the form ẋ(t) = Ax(t) + Bu(t), (3) y(t) = Cx(t) + Du(t). (4) Where x(t) is the state vector, u(t) is the input signal and y(t) is the output of the filter, the direct feedthrough matrix D is set to zero to achieve strict causality. The impulse response function h(t) and its Laplace transform H(s) are given by: h(t) = Ce At B, (5) H(s) = C(sI A) 1 B. (6) As stated before, if h(t) is used to approximate a time-shifted and time-reversed wavelet function ψ(t) = ψ(t 0 t), the output of the linear filter is the approximate wavelet transform W h (t, ) of the input signal u(t). To use the multiscale feature of the WT the approximate wavelet function has to be dilated and scaled. This can be obtained by scaling the coefficient matrices A and C of the state-space system, for scale ẋ(t) = 1 Ax(t) + Bu(t), (7) y(t) = 1 Cx(t). (8) The wavelet transform for signal f(t) at scale and time t is then given by: W h (t, ) = 1 f(τ)h(t 0 + t τ )dτ. (9) (v. 20th June 2005, p.2)

3 Bachelor project 3 QRS detection In this section some background information on ECG analysis and other QRS detection methods are highlighted. In ECG analysis the single most important feature is the QRS complex. Because all other features, like the P and T waves and the on- and offset of the QRS complex are defined relative to the QRS complex, see Figure 2. The P and the T wave occur respectivly before and after the QRS complex, without knowledge of the QRS location P and T waves are hard to distinguish from eachother. know from its form that a rising edge of a uniform wave (like the QRS complex) corresponds to a negative minimum and the dropping edge corresponds to a positive maximum, see Figure 3. The modulus of the local WT maxima and minima (MM) across different scales that relate to the same edge in the signal are named the modulus maxima line, see Figure 4. From now on a local maximum or minimum in the WT of the ECG will be referred to as a MM. When the scale becomes small the zero crossing of the modulus maxima pair correspond to the peak in the signal. In [8] it was also shown that a relation exists between the decay of the wavelet modulus maxima over different scales and the regularity of the signal, expressed as the Lipschitz exponent. Figure 2: A standard ECG with characteristic P, R and T waves. Most QRS detectors can be divided in to two stages: a filtering stage and a decision stage. The filtering stage is used to emphasize the QRS complex and to reduce noise and the influence of the other waves in the ECG signal (P and T waves). Typically first a bandpass filter is applied to the signal to reduce noise and to suppress P and T waves and then put through a non-linear stage to enhance the QRS complex. Then the QRS enhanced signal is thresholded and some decision logic is used for the final stage of detection. Wavelet transformation has proven to be a very efficient tool in the analysis of ECG signals [3]. Its ability to automatically remove noise and to cancel out undesired fenomena such as baseline drift are a benefit over other techniques. Furthermore, the time localization ability of the WT overcomes the need for fixed duration windowing techniques to detect time-varying transients. The multi-scale feature of the WT overcomes the need for fixed bandpass filters, which do not adopt well to the time-varying morphology of the QRS complex. When a smoothing function like the Gaussian function is used as a wavelet, one can prove that sharp variations in the signal relate to local maxima and minima in the wavelet transform across different scales [8]. When the first derivative of the Gaussian function is used, we Figure 3: The relation between the convolution integral of the wavelet function multiplied by a uniform wave and the appearance of local maxima in the WT.(a) The wavelet and the signal. (b) The wavelet and the signal multiplied. (c) The convolution integral Successful methods have been developed using these principals. One such method will be discussed here in more detail to give an idea of how the WT of an ECG can be used for automatic QRS detection. In [7] all modulus maxima, at a characteristic scale greater then a certain threshold, are searched for and located, giving the set of MM at the largest scale. Subsequently other MMs at lower scales are searched in the neighborhood of these MMs at the largest scale. After obtaining the MM sets of the used scales, a pruning technique is used to delete redundant or isolated MM. In this step the Lipschitz exponent determined by the modulus maxima line is used (v. 20th June 2005, p.3)

4 Bachelor project Figure 4: 3D plot of the WT of a typical ECG signal, the fat line is the modulus maxima line. to detect noise or other artifacts that have to be eliminated. At last, the zero crossing points of the remaining modulus maxima pairs at the smallest scale are used to determine the location of the R peak. 4 An analog QRS detection method In this section the developed method is presented and explained in more detail. Some of the problems encountered are dealt with and the general motivations for the design decisions are given. A method as described above requires huge quantities of memory and processing power, so it is not very useful for a low-power IC implementation. The principals of the method however, can be adopted to construct a method that is more suited for such an implementation. When developing a method while keeping in mind that it has to be applicable in a low power IC implementation, there are certain points that are important: As many operations as possible should be done in the analog domain as these can be performed very power efficient. To convert analog sensor information to the digital domain, an A/D (analog to digital) converter is required. Depending on the number of bits, this is a very power consuming operation. The number of bit conversions should be kept at a minimum. The use of complex mathematical operations, which are not implementable in the analog domain, require advanced digital circuits which are not disirable. The use of digital memory should be kept at a minimum. With the use of the wavelet transformation, the most important features of the signal are characterized by local maxima and minima of the wavelet transform. When analyzing a standard ECG signal with sharp peaks without noise, detection is easy, as only one pair of modulus maxima will occur at important events and the modulus maxima of the QRS complex will be significantly larger than others (Figure 4). However, when different morphologies of the QRS complex and noise are present in the signal detection becomes more dificult. Duplicate R waves or sharp noise peaks in the neighborhood of the QRS complex can result in more than two modulus maxima. The presence of high and sharp P or T waves can also be difficult to handle, as their WT is hard to distinguish from that of a QRS complex. Using pruning heuristics, to eliminate false or redundant peaks, can deal with this effect and greatly increase detection rates. It is now important to note what features of the WT signal can be measured and used efficiently in a low-power analog circuit for QRS detection. What can be measured and processed efficiently, are the height of the modulus maxima and the time events at which the last modulus maxima occurred. The method described in this paper uses these two features to do the QRS detection. The method developed can be separated in three different stages, being: The analog wavelet transformation. A modulus maxima detection and logging circuit. A logic circuit for QRS detection The analog wavelet transform By using n state space systems the WT can be computed in parallel at n scales. The coefficient matrices for the state space system (A,B,C) are scaled according to the scale to obtain W h (t, ), see 7, 8. In this method 3 scales of the WT are used for detection. The first derivative of the Gaussian function is used as the mother wavelet. This wavelet has very good time and frequency localization properties [7]. In this kind of setting it is also preferable to use the first derivative of the Gaussian function as a wavelet because its form is so straigthforward: it has only one vanishing point and two peaks. As a result of this, a wavelet tranformation using such a wavelet is easier to interpret (see Figure 3). This makes detection in an analog system more straightforward. Dyadic 1 scales are used to minimize redundancy in the WT [7]. Here scales 4, 8 and 16 are used. The signal is thus separated in 3 different channels. In other work often smaller scales are also used, to determine more precisely the location of the R peak and the onand offset of the QRS complex. In this case however, we are not interested in the exact position of the QRS complex. Because the delay time of the WT s at different scales is not the same, the responses to one event 1 An number x is dyadic when there exists an a Z such that x = 2 a (v. 20th June 2005, p.4)

5 Bachelor project in the ECG signal occur at a different time in the WT. For example, the modulus maxima pair at the smallest scale will already have occurred before the first MM of the highest scale (see Figure 5). Using the zero-crossing of the WT to determine the exact location of the R peak thus becomes very hard as it is not easy to determine which events at a small scale relate to events a larger scales. Especially when there is a lot of noise in the ECG signal. The WT at the smallest scale will have a lot of interference of noise and more modulus maxima will be present than at larger scales, where the high frequency of the noise in not picked up by the WT. The energy of the QRS complex is best preserved at scales 8 and 16 [7]. But for QRS complexes with more high frequency components the energy is higher at scale 4 then at scale 8 and for QRS complexes with more low frequency components the energy at scale 16 is higher that at scale 8. This effectively means that the output signal is equal to the input signal when both the input signal is greater then the threshold and the derivative of the signal is greater than zero, which means that the input signal is rising. The same operation is done for the negative channel and so there are 2 signals now for each scale. The output signal peak is then run through a triggered system which works as follows. When the peak signal suddenly decreases (just after the local maximum of the current modulus maxima is reached) the system holds the current value of peak and the timing circuit logs the current time, see Figure 6. Figure 6: The outputs of the peak detector circuit showing the values of pw pos,16 and tw pos,16 Figure 5: The response of the analog wavelet transform at 3 scales Modulus maxima detection Here the WT signal at a specific scale is separated in a positive and negative channel W hpos and W hneg, where W hpos = W h (t, ) and W hneg = 1 W h (t, ). At each channel the signal is compared to an adaptive threshold ɛ pos, or ɛ neg,. And the first derivative of the WT is used to determine the exact location of the local maximum as follows. For the positive channel: fw hpos = d dt W hpos Now the output signal of this stage, called peak, is defined as: { Whpos (fw peak = hpos > 0 W hpos > ɛ ) 0 (fw hpos 0) The timing circuit does not have to make use of an actual clock with a hour-minute-second time format. As only the time difference between to detected peaks is important, the timing circuit can make use of any linear increasing signal to measure the time difference between two events. The triggered system thus has two outputs. 1) The value of the top of the last detected peak, or when the value of the input signal is still increasing, the current value of the input signal. 2) The time instant at which the last peak was detected. This is done at each of the three scales, giving a total of 12 channels. pw pos, : the positive peak value of the last detected peak at scale. pw neg, : the negative peak value of the last detected peak at scale. tw pos, : the time event of the last detected positive peak at scale tw pos, : the time event of the last detected negative peak at scale (v. 20th June 2005, p.5)

6 Bachelor project Figure 9 shows a diagram of this. QRS detection Now these signals are used to determine the threshold and to determine whether an event should be classified as a QRS complex or not. For the QRS detection the time differences between a positive and a negative peak at each scale is measured as t = tw pos, tw neg,. (10) The value of t has to be in a certain interval with upper limit Ipos and lower limit Ineg to qualify as the valid time interval between two modulus maxima of a QRS complex. Whether the positive or the negative modulus maxima comes first, depends on the morphology of the QRS complex, so t can be positive or negative. So Ineg < t < Ipos in order to indicate a valid QRS complex. At large scales the wavelet function, or in this case the impulse response of the linear filter will be dilated. This implies that the time-shift induces in the original approximation will also be dilated. So not only will t be larger for large scales, the time delay between the QRS peak in the ECG and the occurance of the peaks in the WT will be larger than at smaller scales, see Figure 7. t 1, 2 could also be measured as the delay between two modulus maxima with negative sign, but as most QRS complexes are more or less uniform of shape that would make little difference. t 1, 2 also has to be in a certain interval I 1, 1 and 1 has to be the larger scale. A a result I 1, 1 cannot be negative, as the time delay at large scales is greater that the delay at smaller scales. Therefore 0 < t 1, 2 < I 1, 1 to indicate a valid QRS complex. Final detection The final decision whether a QRS complex should be detected or not is made by combining all measured values t 1, 2 and t. When on all scales the values are in their respective intervals, a QRS complex is detected. This can be done by simply obtaining boolean values for each interval test and combining them with logical AND operators, see Figure 8. Figure 8: The measured time differences between consecutive peaks across scales. The fat line indicates when all time differences are in the desired intervals and a QRS is detected At last when a QRS complex is detected the value for the threshold ɛ sgn, is updated as being Figure 7: The time delay between the actual R peak and the occurrence of the peak in the WT at different scales. N stands for normal beat and V for Premature Ventricular Contraction. The Premature ventricular contraction beats have a higher bandwidth and the time delay in the WT is greater as can be seen in the top picture. As the time delay between the occurrences of the modulus maxima in the WT between different scales depends on the bandwidth of the QRS complex. This is also measured as t 1, 2 = tw pos,1 tw pos,2. (11) { α pwsgn, pw ɛ sgn, = sgn, < tbth ɛ sgn, pw pos,16 pw neg,16 tbth (12) Where α is a scaling factor which determines the actual value of the threshold and sgn is either pos or neg. If α pw sgn, becomes too high as a result of an unusual QRS complex, which results in an MM pair with extreme high amplitude, detection is affected and MM pairs with lower amplitude are not detected as QRS complexes anymore. To counter this, α pw sgn, is not updated when pw pos,16 or pw neg,16 is too high, This value is called tbth, as in to b ig t hreshold. Only the largest scale is used with this threshold because the WT of the largest scale usually has the highest amplitude, so checking all scales is (v. 20th June 2005, p.6)

7 Bachelor project redundant. Note that a detection can only occur as late as the last peak on the highest scale in the WT arrives. This inevitably means the a delay will exist between the actual QRS complex and its detection. The exact delay depends on the scales used and on the form of the QRS complex. A wide QRS complex will have a greater delay than a narrow one. The choice of α has a significant impact on detection rate: when α is too small other characteristics like P or T waves or low-frequency noise are being detected. On the other hand, when α is not small enough QRS complexes with small bandwidth will not always be detected. It is often a difficult trade-off between the false positive (FP) rate (when a detection is made when it should not) and the false negative (FN) rate (when a detection should be made but it is not). In this case, when too many false positives occur the heart may be put under unneccecary stress from pacemaker impulses. When more false negatives occur, the pacemaker might not respond when it should. 5 Implementation For the implementation of the developed method MAT- LAB s Simulink has been used. Simulink is a software package for modeling, simulating, and analyzing dynamic systems. It supports linear and nonlinear systems, modeled in continuous time, discrete time, or a hybrid of the two. Although Simulink is mainly used to simulate real-world phenomena or dynamic systems, it can be used to model a detection system like the one presented here. Some of the design decisions for the developed method are based on available features and the limitations of Simulink. The time logging circuit for example uses a so called triggered subsystem which hold its last output when not triggered and calculates a new output when it is triggered (the time of the last detected MM is updated when peak suddenly decreases, see section 4). This acts as a sort of memory, although the output of the subsystem is still a continuous signal. 6 Results Here we made use of the MIT/BIH [1] arrhythmia database to evaluate the algorithm. The MIT/BIH database is a huge database of annotated ECG signals of various patients. Only lead 1 out of 2 was used for evaluation. The following values are used for the parameters I 4,16 = 30 0 < t 4,16 < 30, Ineg 4 = 40, Ipos 4 = < t 4 < 40, : Ineg 8 = 40, Ipos 8 = < t 8 < 40, Ineg 16 = 40, Ipos 16 = < t 16 < 40, thtb = The value of forty and minus forty for the intervals match the bandwith of the broadest QRS complex in the database, which is approximatly 120ms or 40 data samples at 360Hz (The sample rate of the MIT/BIH database). In Figure 10 the results at different values of α are plotted. Sensitivity is often used as a performance indicator in the medical field and is measured as sens = tp tp + fn. and the for further comparison an error indication is measured as error = fp + fn detectedbeats. Figure 10: Results at different values of α Figure 9: A part of the Simulink model showing the peak detection and logging circuit It is clear that at higher values of α the sensitivity decreases and the error rate increases, mainly due to the increasing numbers of false positives. The best results are obtained at α=0.3. There the avarage between the sensitivity and the error rate is highest, it depends on what is more important for the specific application of the detection method. See also Table 2, results for other (v. 20th June 2005, p.7)

8 Bachelor project Record number Total beats Detected beats tp (beats) fp (beats) fn (beats) sens (%) err (%) mitdb ,96 0,04 mitdb ,89 0,16 mitdb ,16 12,15 mitdb ,00 0,00 mitdb ,69 7,04 mitdb ,74 7,80 mitdb ,09 3,54 mitdb ,64 16,43 mitdb ,00 1,00 mitdb ,00 0,00 mitdb ,27 1,16 mitdb ,59 7,59 mitdb ,67 0,66 mitdb ,40 15,20 mitdb ,93 0,08 mitdb ,05 19,85 mitdb ,78 1,97 mitdb ,93 0,37 mitdb ,64 1,57 mitdb ,00 0,29 mitdb ,82 0,46 mitdb ,78 0,88 mitdb ,61 0,77 mitdb ,73 35,34 mitdb ,00 0,00 Total: ,58 5,37 Table 1: Results for the tested records of the MIT/BIH database at α=0.3 records look similar. Record 203 has a great number of QRS complexes with unusual form and the record is noisy. Records 201 and 217 have high numbers of Premature ventricular contractions (PVC). PVC s are often followed by a high T peak, which is detected as a QRS complex. α p dp tp fp fn sens 0, ,99 0, ,99 0, ,99 0, ,99 0, ,98 0, ,97 Table 2: Results for record 202, p=peaks, dp=detected peaks 7 Discussion and Conclusions By approximating a wavelet function, the impulse response of a state space system can be used to compute the WT in the analog domain. Then, using this, a QRS complex detection method for use in a low-power IC implementation has been developed. The method is based on the WT and the multi scale information of the WT are used. The method utilizes the distinct features that exist in the WT as a result of characteristic points in the ECG signal for detection. A QRS complex corresponds to a modulus maxima pair in the WT. The height of the modulus maxima and the time interval between them is a good indication of the kind of event in the ECG signal. With the multiscale information of the WT it is possible to distinguish between QRS complexes and high P and T waves or low frequency noise. The method sometimes has problems with certain morphologies of the QRS complex. Depending on the chosen values for the thresholds a trade-off can be made between the sensitivity and the error-rate of the system. These can be overcome by adding extra rules to the detection logic. Although the method has not been tested in an IC implementation, the simulation shows that very high detection rates can be achieved. Although the sensitivity performance is very good, (v. 20th June 2005, p.8)

9 Bachelor project the error rate of the method can be improved. The method has problems on some QRS morphologies, most notably the Premature Ventricular Contraction. This morphology is common in many of the records and most false positives can be explained by this fenomona. Increasing the value of α can deal with this problem, but has a negative effect on the performance on the whole dataset. Including a WT on a lower scale may also help to deal with this problem, as the WT on a small enough scale hardly responds to the T peaks followed by a PVC. Another solution to improve the method is by introducing blanking after a detected QRS complex. This means that in a certain time period after a detection no other detections can be made [4]. [7] Li, Cuiwei, Zheng, Chongxun, and Tai, Changfeng (1995). Detection of ecg characteristic points using wavelet transforms. IEEE Transactions on biomedical Engineering, Vol. 42(1), pp [8] Mallat, S. and Hwang, W.L. (1992). singularity detection and processing with wavelets. IEEE Transactions on Information Theory, Vol. 38(2), pp [9] Vaccaro, Richard J. (1995). Digital Control A State-Space Approach. McGraw-Hill, PSingapore. References [1] Goldberger, A. L., Amaral, L. A. N., Glass, L., Hausdorff, J. M., Ivanov, P. Ch., Mark, R. G., Mietus, J. E., Moody, G. B., Peng, C.-K., and Stanley, H. E. (2000 (June 13)). PhysioBank, PhysioToolkit, and PhysioNet: Components of a new research resource for complex physiologic signals. Circulation, Vol. 101, No. 23, pp. e215 e220. Circulation Electronic Pages: [2] Haddad, Sandro A.P., Houben, Richard, and Serdijn, Wouter A. (2004a). Analog wavelet transform employing dynamic translinear circuits for cardiac signal characterization. Proceedings of the 16th IFAC World Congress. Accepted. [3] Haddad, S.A.P., Houben, R., and Serdijn, W.A. (2004b). The history and development of pacemakers: an electronics perspective. Klinische Fysica, pp [4] Hamilton, P. S. and Tompkins, W. J. (1986). Quantitative investigation of qrs detection rules using the mit/bih arrhythmia database. IEEE Trans. Biomed. Eng., Vol. 33(12), pp [5] Kadambe, Shubha, Murray, Robin, and Boudreaux-Bartels, G. Faye (1999). Wavelet transform-based qrs complex detector. IEEE Transactions on biomedical Engineering, Vol. 46(7), pp [6] Karel, J.M.H., Peeters, R.L.M., Westra, R.L., Haddad, S.A.P., and Serdijn, W.A. (2005). Wavelet approximation for implementation in dynamic translinear circuits. Proceedings of the 16th IFAC World Congress. Accepted. (v. 20th June 2005, p.9)