

SHORT COMMUNICATION 

Year : 2021  Volume
: 11
 Issue : 1  Page : 5261 

Realtime fast fourier transformbased notch filter for singlefrequency noise cancellation: Application to electrocardiogram signal denoising
Anis Ben Slimane^{1}, Azza Ouled Zaid^{2}
^{1} Laboratory of Advanced Technology and Intelligent Systems, Sousse University, Sousse; Department of Industrial Electronics, National Engineering School of Sousse, Sousse University, Tunisia ^{2} SysCom Laboratory, National Engineering School of Tunis, University of Tunis EI Manar, Tunis, Tunisia
Date of Submission  05Jan2020 
Date of Decision  16Feb2020 
Date of Acceptance  26Jul2020 
Date of Web Publication  30Jan2021 
Correspondence Address: Anis Ben Slimane Laboratory of Advanced Technology and Intelligent Systems, Sousse University, Tunisia, National Engineering School of Sousse, Sousse University, Sousse Tunisia
Source of Support: None, Conflict of Interest: None
DOI: 10.4103/jmss.JMSS_3_20
Despite the considerable improvement of the commonmode rejection ratio of digital filtering techniques, the electrocardiogram (ECG) traces recorded by commercialized devices are still contaminated by residual power line interference (PLI). In this study, we address this issue by proposing a novel realtime filter adapted to singlefrequency noise cancellation and automatic power line frequency detection. The filtering process is principally based on a pointbypoint fast Fourier transform and a judicious choice of the analysis window length. Intensive experiments conducted on real and synthetic signals have shown that our filtering method offers very clean ECGs, due to the suppression of spikes corresponding to the PLI and the preservation of spikes outside the filter band. In addition, this method is characterized by its low computational complexity which makes it suitable for realtime cleaning of ECG signals and thus can serve for more accurate diagnosis in computerbased automated cardiac system.
Keywords: Frequency detection, notch filter, physiological signals, power line interference removal, realtime fast Fourier transform, spectral filtering
How to cite this article: Slimane AB, Zaid AO. Realtime fast fourier transformbased notch filter for singlefrequency noise cancellation: Application to electrocardiogram signal denoising. J Med Signals Sens 2021;11:5261 
How to cite this URL: Slimane AB, Zaid AO. Realtime fast fourier transformbased notch filter for singlefrequency noise cancellation: Application to electrocardiogram signal denoising. J Med Signals Sens [serial online] 2021 [cited 2021 Feb 27];11:5261. Available from: https://www.jmssjournal.net/text.asp?2021/11/1/52/308421 
Introduction   
Electrocardiogram (ECG) is the oldest and most widely available physiological test. It allows identifying cardiovascular diseases (CVDs) while being entirely painless.^{[1]}
A clean ECG is often required for the proper treatment of cardiac ailments. However, in a real scenario, the ECG signal is typically contaminated by different kinds of noises during its acquisition and transmission, such as the highfrequency noise (additive white Gaussian noise and power line interference [PLI]) and the lowfrequency noise (baseline wandering).^{[2],[3]} Particularly, PLI noise causes a precision issue when interpreting lowamplitude waveforms like the ECG. In order to avoid the wrong identification of the characteristics of ECG signals and their impact on diagnostic accuracy, several PLI removal techniques have been developed.^{[4],[5],[6],[7]}
In this work, we propose a novel filtering method which has the particularity of eliminating noise with a single quasiinvariant frequency as it is the case for PLI. First, a pointbypoint windowed fast Fourier transform (FFT) with an appropriate choice of the analysis window size is applied on the original ECG signal to accurately estimate the instantaneous amplitude value of PLI (50 Hz or 60 Hz). Finally, the instantaneous value of the filtered signal is determined by simply subtracting the computed noise value from the instantaneous value of the noisy ECG signal.
It is worth noting that the suggested method allows both the automatic detection and instantaneous suppression of PLI even in the presence of a variable noise amplitude. The simulation results have shown that the spectral response is stable and has a very narrow band rejection, ensuring a minimum of 33 dB attenuation in the band (49.8 Hz, 50.2 Hz). Moreover, the theoretical basis of the proposed denoising technique has been studied in addition to the relationship between the length of the analysis window and the filter parameter setting.
Always as part of experimental analysis, the simulation tests have been firstly conducted on synthetic ECG waveforms with a constant PLI amplitude. Thereafter, the same tests have been carried out with variable PLI amplitudes to determine the limitations of the proposed method. Finally, the experiments have been conducted on real ECG signals to compare their spectrums before and after filtering. According to the obtained results, it has been demonstrated that our filtering method delivers an impressive ECG quality, which is confirmed by the suppression of PLI spikes while preserving those outside the filter band in the frequency domain.
Background
The ECG consists of an electrical signal generated by the heart's muscular activity. Its graphical representation is commonly used to identify CVDs. However, in reallife scenarios, the recorded ECG signal is often corrupted with different types of noise: those caused by breathing, which generates a deflection of the isoelectric line;^{[8]} those caused by the electrical activity of the muscle, which is also called electromyogram;^{[9]} and those caused by PLI,^{[10]} which is our focus. Previous studies showed the PLI noise with a frequency of 50 Hz or 60 Hz. This frequency has been subject to variations that do not exceed 0.5 Hz.^{[11],[12]}
Even though modern instrumentation amplifiers/differential amplifiers have a high commonmode rejection ratio, which attains 120 dB, the ECG recordings are still contaminated by PLI. In order to prevent the wrong identification of the ECG characteristics and its impact on diagnostic interpretation, advanced ECG denoising methods have been investigated. The most common solution is based on wavelet decomposition. The wavelet family is vital for the success of signal denoising applications, but waveletbased notch filters cannot completely cancel PLI. Indeed, the residual traces may interfere with the ECG waves (P, Q, R, S, and T), make the detection of their onsets and offsets difficult,^{[13],[14],[15],[16]} and corrupt the proper function of automatic ECG analysis. Interference can also disturb the correct measurement of the RR interval (heart rate).^{[17]}
Traditional analog and digital filters are not very selective and have the drawback of reducing or suppressing the ECG components near the power line frequency. Despite their limitations, different types of digital notch filters are still used.^{[17],[18],[19]} To avoid affecting the signal components near the power line frequency, the cutoff band must be narrow, which leads to inefficient filtering with significant power line frequency deviation. Moreover, the resulting transient time is often unacceptably long.
PLI may be suppressed by adaptive filtering approaches.^{[20],[21],[22],[23],[24]} However, this may be paid by a large transient response time.^{[17]} Moreover, adaptive and nonadaptive notch filters introduce significant distortions in the QRS and STsegment portions.^{[25]} SooChang and ChienCheng^{[18]} tried to reduce the convergence time using a vector projection to find better initial values for infinite impulse response bandstop filters. Yoo et al.^{[19]} put forward an adaptive central frequency hardware notch filter to track the frequency changes in the power line, thus reducing the bandwidth size. According to the results presented in Yoo et al.'s study,^{[19]} the signal distortion cannot be adequately evaluated because of the small size of the used examples.
The Fourier decomposition method proposed in Singh et al.'s study^{[26]} was an adaptive signal decomposition method that used the Fourier theory and aimed to remove baseline wander (BLW) and PLI. It could be considered as an adaptive and spectral denoising method. The experiments were done on reallife ECG data with synthetic BLW and PLI noise. In the best cases, the signaltonoise ratio (SNR) did not exceed 15 dB.
Some methods have presented a lack of completeness, such that the authors have not shown the effectiveness of their denoising algorithms clearly enough in order to use them for PLI removal. In other methods, sometimes, the original signal has not been provided.^{[17],[27]} This is to say that no differences between original and processed signals are shown^{[28]} and that the performance is only measured as a function of the mean square error instead of amplitude differences.
Methodology   
Data collection
To evaluate the performance of the proposed technique, we have built a new database^{[29]} using a wireless PCbased ECG device. By doing so, we have fixed the frequency sampling to 300 samples per second. The analog to digital conversion has been realized with the ADS1298 IC from Texas Instruments company, which is an eightchannel 24bit ΔΣ simultaneoussampling ADC.
For each subject, we have only recorded two limb leads (lead I and lead II) since lead III (aVR, aVL, and aVF) can be deducted automatically. The chest leads have not been collected. This database comprises one hundred subjects which have been collected by disabling the notch filter. This is to say that the recordings are contaminated with PLI at different levels.
Fast Fourier transformbased estimation of instantaneous power line amplitude
Fast Fourier transform basics
The Fourier transform Eq. (1) consists in representing a signal by a sum of sinusoids of frequencies and amplitudes as follows:
Where n is the frequency index, F_{s} is the sampling frequency, N is the number of samples, and A_{n} is the spike amplitude at frequency f_{n}.
The value f_{1} = F_{s}/N is called the frequency resolution or the frequency step; it defines the fineness of the decomposition. The lower the value, the better the resolution.
Let us consider, for example, a sinusoidal signal of a frequency equal to 50 Hz, as for the power line, with an amplitude equal to 1. If this signal is sampled at a frequency F_{e} fixed to 300 Hz and the sample number N is equal to 100, then the frequency step will be equal to 3 Hz.
This means that we will get the amplitudes of the sinusoids of frequencies belonging to the set F = {0, 3,…, 297, 300}. In this case, we find that 50 Hz is not a part of this set. Accordingly, the signal will be distributed over a plurality of sinusoidal signals whose frequencies are the closest to 50 Hz. As depicted in [Figure 1], the amplitude of these signals will tend to become weakened with an increasing distance from the signal frequency.  Figure 1: Graphical illustration of signal spectrum (f = 50 Hz, F_{S} = 300 Hz, and n = 100)
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Finding the correct representation of this sinusoidal signal requires the 50 Hz frequency to be present on the Xaxis; i.e., the 50 Hz frequency must be a multiple of the frequency step. By taking a window of 60 samples, the frequency step will be equal to 5 Hz, which is a divisor of 50 Hz. Accordingly, the signal will be represented by a single spike centered on the 50 Hz frequency whose amplitude is equal to 0.5 (half of the real amplitude), as shown in [Figure 2].  Figure 2: Graphical illustration of signal spectrum (f = 50 Hz, F_{S} = 300 Hz, and n = 60)
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We can deduce from this short analysis that it is possible to determine the amplitude of a sinusoidal signal embedded in another one through the Fourier transform.
Factors influencing filter parameters
In this section, we will determine the relationship between the frequency resolution r = F_{s}/N and the filter cutoff frequencies. Therefore, we will vary the filter size N while maintaining the sampling frequency F_{S} = 300 Hz. The used window is the Hanning one. Using the rectangle window gives a response with an increase in the oscillation amplitude when approaching the cutoff frequency. The amplitude of oscillations decreases as the frequency resolution becomes finer. This concept is illustrated in [Figure 3], whereas the corresponding zoomed version is depicted in [Figure 4].  Figure 3: Effects of frequency resolution and window shape on filter response
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 Figure 4: Effects of frequency resolution and window shape on filter response (zoomed plots)
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By varying the analysis window size N and preserving the sampling frequency F_{s}, the frequency resolution r = F_{s}/N varies. When r varies, the high and low cutoff frequencies vary [[Figure 4] for n = 120 and n = 300].
[Figure 5] shows the relationship between the frequency resolution and the cutoff frequencies. By examining the two curves, we can clearly notice that the filter response is linear with the two parameters, which makes it easier to determine the cutoff frequencies if the frequency resolution is known, and vice versa.  Figure 5: Graphical relationship between frequency resolution and cutoff frequencies
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Performance evaluation with respect to the power line interference amplitude
To adapt our method for realtime processing, an Nlength shift register is used to store the last samples. A pointbypoint FFT is then applied to this set of samples with instant processing.^{[30]} To determine the interference frequency, the spike amplitude corresponding to the 50 Hz is compared to that corresponding to the 60 Hz. It is worth noting that the power line frequency corresponds to the spike that has the highest amplitude.
Once the main frequency is set, the instantaneous value of the PLI amplitude can be subtracted from that of the noisy signal. This is expressed by Eq. (2):
Where is the estimated noise at the moment t = t_{s} (N –1)/2, s, is the estimate of the denoised signal at the same moment s, is the noisy signal, S is the Fourier transform of s, and I is the spike index corresponding to the power line frequency which can be determined according to Eq. (3):
Where f_{0} is the PLI frequency (50 or 60 Hz) and r is the frequency resolution.
To obtain a fair value of the PLI amplitude, we have to multiply the amplitude value of the spike that corresponds to the noise frequency by 2. Then, we divide it by the area of the analysis window that corresponds to the average of its coefficients. It is worth pointing out that since the window is centered with respect to the sample index , this technique will result in a time delay of about . Moreover, the choice of the window size should be fixed according to the frequency variation in the power line amplitude. The higher the frequency variation is, the more reduced the window size must be. However, when filtering physiological signals such as ECGs, we are compelled to reduce the width of the stopband to remove only the interference and preserve the useful data. This reduction in the size of the stopband requires increasing the size of the window. A compromise between the bandwidth cutoff and the temporal localization should be considered. Notably, the automated choice of the analysis window size as a function of the frequency variation in the interference amplitude will be subject of future research work. [Figure 6] represents the block diagram of the proposed filtering method.
To evaluate the performance of the proposed filtering technique, simulation tests have been conducted on two synthetic signals. The first one, “s_{1}” is expressed by Eq. (5). It consists of a sine wave “s” given by Eq. (4), which has been contaminated with the PLI of a constant amplitude. The second signal “s_{2}” is expressed by Eq. (6). It consists of the same sine wave “s” which, in turn, has been contaminated with the PLI of a variable amplitude. The graphical aspects of the two noisy signals are depicted in [Figure 7].  Figure 7: Graphical illustration of two simulated noisy signals: (a) “s_{1}” (b) “s_{2}”
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We have tested the proposed technique using a 300sample window, which corresponds to a 1 Hz frequency resolution and a stopband in the range of 48.75, 51.25 for F_{S}= 300 Hz [Figure 4].
[Figure 8] depicts the filtering result when using the Hanning window on a noisy signal altered by PLI with a constant amplitude. From this figure, we can see a 0.5 s delay corresponding to half of the analysis window length. To calculate the value of the filtered signal at t = 0.5 s, samples belonging to the interval (0, 1) are required.  Figure 8: Graphical illustration of PLI removal in case of constant amplitude
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The SNR is expressed by:
Where o is the original signal, f is the filtered one, and M is the number of signal components. This SNR has been used to quantitatively compare the original signal to its filtered version. According to our experiments, the obtained SNR is equal to 217.0 dB.
[Figure 9] shows the filtering result when using the Hanning window on two noisy signals, x_{1} and x_{2}, with varied PLI. In both signals, the noise amplitude is the same. For the first signal, x_{1}, the frequency variation in the noise amplitude is equal to 0.5 Hz, whereas in the second signal, x_{2} is equal to 0.25 Hz. After performing our filtering process on the two noisy signals, x_{1} and x_{2}, we obtain an SNR value of about 39.8 dB and 58.7 dB, respectively.  Figure 9: Graphical illustration of PLI removal in case of variable noise frequency: (a) f = 0.5, (b) f = 0.25
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It is clear that when the frequency of the noise amplitude increases, the determination of the instantaneous value of the noise becomes a difficult task. This is principally due to the fact that the analysis window has a temporal size of 1s. Thus, the interference amplitude varies several times along the window, which makes the determination of the instantaneous amplitude more difficult. This problem may be alleviated by reducing the size of the analysis window. However, this will be at the expense of the stopband width, which will be wider.
Experiments conducted on synthetic and real electrocardiogram signals
At this stage, we have conducted the first set of experiments on synthetic ECG signals to evaluate the performance of the proposed technique. Our choice has been justified by the fact that synthetic ECG signals allow a direct quantification of the difference between original and filtered signals.
The second set of experiments has been conducted on real ECG signals from our database.^{[29]} For this kind of data, the performance evaluation is based on two criteria. The first one is a visual assessment of the quality of the filtered signal. The second criterion is quantitative. More specifically, we have proposed to carry out a quantitative assessment of PLI attenuation in the stopbands and a quantitative assessment of the conservation of ECG components outside the rejection bands.
Comparisons have been made with Butterworth and wavelet notch filters. The Butterworth filter^{[8]} is set to a 4^{th} order with a rejection band (49 Hz, 51 Hz), whereas for the waveletbased filter, the algorithm suggested in Aqil and Jbari study^{[14]} is used.
Since the power line frequency varies by ± 0.5 Hz, the size of the Hanning window is selected in a way that the minimum attenuation in the range of 49.5, 50.5 or 59.5, 60.5 is not less than 20 dB. The value of the window length that provides a step, which is a divisor of almost 50 and 60, ensures that minimum attenuation of 20 dB in the two intervals is 150. This value corresponds to a frequency resolution of 2 Hz and a stopband width equal to 2 × 2.512 Hz [Figure 5]. The filter responses are presented in [Figure 10] for the two PLI frequencies: 50 Hz and 60 Hz.
The plots of the synthetic signal, the modeled noise n (expressed in Eq. 8), and the filtered signal are shown in [Figure 11].  Figure 11: Graphical illustration of (a) synthetic noisy signal, (b) modeled noise, and (c) filtered electrocardiogram signal
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The synthetic is obtained with the NI LabVIEW Biomedical Toolkit. The sampling frequency is 300 Hz and the heart rate frequency is 60. The computed SNR for this simulation test is equal to 200.9 dB.
[Figure 12] shows an example of the application of the proposed filtering technique on a real ECG signal extracted from our database.^{[29]} Particularly, this trace corresponds to lead II. It is obtained by bringing an electric cord closer to the right arm electrode. This cord is connected to the 220 V 50 Hz mains. This permits obtaining a very noisy signal and consequently testing the limit of the proposed technique under extreme conditions.  Figure 12: Graphical illustration of real electrocardiogram signal filtered with Hanning window (n = 300)
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From [Figure 12] and [Figure 13] (zoomed version), we can see a complete separation between the ECG waveform (blue) and PLI (green) whose peaktopeak amplitude is almost twice that of the ECG waveform. This visual comparison, although appears very satisfactory, cannot, in any case, prove the selectivity of the filter.  Figure 13: Graphical illustration of zoomed real electrocardiogram filtered with Hanning window (n = 300)
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For real signal filtering, the transfer function of the filter cannot be obtained (as it is the case of adaptive and waveletbased filters). For this reason, we put forward the spectra comparison criteria, as illustrated in [Figure 14]. For greater accuracy, [Figure 15] shows the zoomed version of the spectra of the noisy ECG signal and that of its filtered version, which is depicted in [Figure 12].  Figure 14: Spectrum of (a) original signal and (b) that of filtered signal using suggested method
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 Figure 15: Zoomed spectrum of original signal and filtered one using proposed method
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From these spectra, we observe that the spikes corresponding to the PLI (located near the frequencies 50 Hz, 100 Hz, and 150 Hz) have been removed without affecting the other frequencies, which demonstrates the selectivity of the suggested filtering method.
To evaluate quantitatively the selectivity capability, we propose to assess the spikes amplitude attenuation on the rejection band and their conservation outside of the rejection band using the SNR metric.
As stated before, this metric is of great use if the transfer function of the filter cannot be determined.
In our experiments, we use an analysis window of length 300 (step = N/F_{S}), which corresponds to a frequency step equal to 1. According to the graphical relationship illustrated in [Figure 5], the rejection band around 50 Hz is 48.74Hz, 51.26 Hz. Since we need to filter the 50 Hz multiple frequencies, the rejection bands (98.74 Hz, 101.26 Hz and 148.74 Hz, 150 Hz) should be considered as well.
Finally, to evaluate the noise cancellation, the SNR is computed only on the positive intervals (48.74 Hz, 51.26 Hz; 98.74 Hz, 101.26 Hz; and 148.74 Hz, 150 Hz). On the other side, to evaluate the signal conservation, the SNR is computed on the complementary intervals (0 Hz, 48.74 Hz; 51.26 Hz, 98.74 Hz; and 101.26 Hz, 150 Hz).
It is worth noting that outside the stopband, a high SNR value implies the good preservation of the signal components. Conversely, inside the stopband, the SNR value must be as small as possible. In fact, a small value indicates good noise cancellation.
According to our experiments, the computed value of the SNR outside the stopband is equal to 70.9 dB, whereas that calculated inside the stopband is equal to − 51.6 dB. This value demonstrates the accurate selectivity of the proposed filter.
Finally, the suggested method is compared to the Butterworth and wavelet notch filters. [Figure 16] shows the filtered signals for each method, and [Table 1] provides the obtained SNRs.
From [Table 1], we can see that all of the three methods can appropriately and identically remove PLI. Indeed, in the three cases, the SNR in the rejection band is approximately equal to − 51 dB. However, the proposed method has a better SNR outside the rejection band. This proves the filtering selectivity of the suggested method. Furthermore, and according to [Figure 17], we can see a less perfect filtering quality obtained by the Butterworth and wavelet filters (presence of oscillations near the QRS waves) despite a quite similar SNR for the Butterworth filter compared to our method.  Figure 17: Denoising results for Butterworth, wavelet, and our method (zoomed version)
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Conclusion   
In this study, we have presented a realtime filter for singlefrequency noise cancellation based on a pointbypoint FFT. This method has the advantage of being effective in the presence of interference with variable amplitudes. Nevertheless, to effectively suppress interference, the period of the amplitude variation must be lower or equal to the extent of the analysis window. This hypothesis has been proved according to the tests conducted on the signals extracted from the created ECG database.
The proposed filter has a stable response and a very narrow cutoff band ensuring the minimal attenuation of 33dB in the band (49.8 Hz, 50.2 Hz).
Unlike the conventional filtering techniques which allow attenuating the noise amplitude in a welldefined bandstop, the suggested method can estimate the instantaneous power line amplitude with excellent accuracy and consequently allows suppressing it completely. Besides, we have determined the relationship between the frequency resolution and the cutoff frequencies of the notch filter to adapt the filter size according to the denoising requirements.
Experiments have been conducted on synthetic ECG signals affected with PLI with constant and variable amplitudes. Based on the experimental results, it has been shown that the proposed filtering method can effectively remove PLI without introducing significant distortions in the ECG waves (P, Q, R, S, and T). By considering real ECG signals, comparisons between noisy and filtered ECG spectra outside the rejection bands have given satisfactory results.
Comparisons between our method and two others showed a similarity on PLI rejection and a better signal preservation outside the rejection band for our method.
For the purpose of quantitative assessments, the use of the SNR metric is hampered by some serious limitations in the case of signals with fine sensitive structures. In our future work, we plan to alleviate this problem using an objective quality metric that takes into consideration the structural similarity.
Financial support and sponsorship
None.
Conflicts of interest
There are no conflicts of interest.
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[Figure 1], [Figure 2], [Figure 3], [Figure 4], [Figure 5], [Figure 6], [Figure 7], [Figure 8], [Figure 9], [Figure 10], [Figure 11], [Figure 12], [Figure 13], [Figure 14], [Figure 15], [Figure 16], [Figure 17]
[Table 1]
