 Research
 Open Access
ECG signal performance denoising assessment based on threshold tuning of dualtree wavelet transform
 Oussama El B’charri^{1}Email authorView ORCID ID profile,
 Rachid Latif^{1},
 Khalifa Elmansouri^{2},
 Abdenbi Abenaou^{1} and
 Wissam Jenkal^{1}
 Received: 7 November 2016
 Accepted: 30 January 2017
 Published: 7 February 2017
Abstract
Background
Since the electrocardiogram (ECG) signal has a low frequency and a weak amplitude, it is sensitive to miscellaneous mixed noises, which may reduce the diagnostic accuracy and hinder the physician’s correct decision on patients.
Methods
The dual tree wavelet transform (DTWT) is one of the most recent enhanced versions of discrete wavelet transform. However, threshold tuning on this method for noise removal from ECG signal has not been investigated yet. In this work, we shall provide a comprehensive study on the impact of the choice of threshold algorithm, threshold value, and the appropriate wavelet decomposition level to evaluate the ECG signal denoising performance.
Results
A set of simulations is performed on both synthetic and real ECG signals to achieve the promised results. First, the synthetic ECG signal is used to observe the algorithm response. The evaluation results of synthetic ECG signal corrupted by various types of noise has showed that the modified unified threshold and wavelet hyperbolic threshold denoising method is better in realistic and colored noises. The tuned threshold is then used on real ECG signals from the MITBIH database. The results has shown that the proposed method achieves higher performance than the ordinary dual tree wavelet transform into all kinds of noise removal from ECG signal.
Conclusion
The simulation results indicate that the algorithm is robust for all kinds of noises with varying degrees of input noise, providing a high quality clean signal. Moreover, the algorithm is quite simple and can be used in real time ECG monitoring.
Keywords
 ECG
 Denoising
 Dual tree wavelet transform
 Threshold tuning
 Realistic noise
Background
Recently, wavelet transform (WT) that localizes features in time–frequency domain has been emerged widely in ECG signal denoising [1]. Generally, removing noises based on WT can be divided in two mainly methods. The first method is based on WT modulus maxima by holding the maximum information on the original ECG signal, which lead to a large amount of calculation [2], while the second method used by Donoho and Johnstone [3, 4] threshold the decomposed wavelet coefficients then reconstruct the signal using inverse wavelet transform. Although the efficiency of WT based thresholding method in ECG denoising, it suffers from some shortcomings like aliasing that brings artifacts in the denoised signal when the wavelet coefficients are processed [5]. In order to overcome those shortcomings, the dual tree wavelet transform (DTWT) has been introduced with new properties that can enhance the reconstructed ECG signal [6]. The DTWT was tested on ECG signal denoising applying soft thresholding on magnitude nonlinearity [7]. However, the optimal decomposition level together with threshold value and function was not taken into consideration.
A substantial amount of studies focused their work on removing commonly known noise such as white noise. Generally, a reliable denoising algorithm is able to remove noise from the acquired ECG signal, such as powerline interference, baseline wander, muscle noise and motion artifact and other noises with, which in different levels leads to misjudgment and deletion of standard ECG identification for the ECG feature extraction and decreases the degree of diagnostic accuracy. Moreover, with the modern telehealthcare systems involving transmission and storage of ECG, noise also arises due to poor channel conditions. A noisy ECG may hinder the physician’s correct evaluations on patients. Therefore, removing noise from ECG signal and preprocessing has become an exclusive requirement.
On the other hand, wavelet thresholding is a viable technique for noise reduction, the value of the threshold is usually application dependent and difficult to fix in practice. Wavelet threshold function mainly includes hard thresholding and soft thresholding. The basis of these methods is quite simple, and they are easy to use in practice. Hard thresholding can retain the abrupt information in the signal, but it may generate oscillations in the reconstructed signal known as PseudoGibbs phenomenon [8, 9]. Soft thresholding can further smoothen the signal than hard thresholding, and has a good continuity. However, the reconstructed signal may be distorted and has a blurred edge. Furthermore, the amplitude of the reconstructed signal will decrease significantly, in particular, the amplitude of the R wave in QRS complex will attenuate greatly, which is a crucial parameter for heart diagnosis. All these shortcomings are detrimental for cardiovascular diagnostic accuracy.
To overcome aforementioned limitations and to provide an efficient tool for the extraction of highresolution ECG signals from recordings contaminated with background noise, the dual tree wavelet transform, which has elegant computational structure [10, 11] is investigated in this paper. The results are obtained by performing extensive simulation studies on threshold tuning. This threshold tuning is performed by varying the threshold value and function as well as the optimal decomposition level, which affects the algorithm performance on removing the noise. This performance is assessed by using a wide range of noises that are of major concern.
This paper is organized as follows; the “Methods” section is dedicated to the theoretical background on the dual tree complex wavelet transform. In this section, the materials used and the proposed algorithm are also presented. For quantitative and qualitative assessments of the algorithm performance, a set of simulations is performed in the “Results” section. These simulations are discussed and explained in the “Discussions” section. Finally, the conclusion of this study is provided in the “Conclusions” section.
Methods
Data acquisition
Synthetic ECG signal
In order to obtain freenoise ECG signal, a synthetic ECG signal is used. The dynamical model used, which is introduced by McSharry [12], can generate a realistic artificial ECG waveform. The signal is created by coupling three ordinary differential equations. The user can settle various parameters including the ECG sampling frequency, number of beats, mean heart rate and waveform morphology. In this work, the default synthetic ECG parameters were taken.
Colored noises
Colored noise type based on β value
β value  Noise type 

0  White 
1  Pink (flicker noise) 
2  Brown (Brownian motion) 
−1  Blue 
−2  Violet (purple noise) 
ECG acquisition from ECG databases
To work with real ECG signal, two databases were used. The first database is the MITBIH Arrhythmia Database [13]. It includes 48 annotated recordings. Each record lasts about 30 min and is sampled at a frequency of 360 Hz with 11bit resolution over a 10 mV range. The signals are extracted from two channel ambulatory ECG recordings. About 29 records are collected from a mixed population of inpatients; the remaining records are collected from outpatients. The second database used is the MITBIH noise stress test database [14]. This database can be classified into two classes of records. The first class includes 3 recordings of noise typical in ambulatory ECG recordings. These real noise records are baseline wander (BW), muscle artifact (MA), and electrode motion (EM) artifact. They are created using physically active volunteers and standard ECG recorders, leads, and electrodes while the second class contains 12 records that are created from two signals (118 and 119) of the MITBIH Arrhythmia database by adding the EM noise. All the records contained in this database are about 30 min in length having sampling frequency of 360 Hz with 12bit resolution. We are interested in the first class since it gives us the ability to add calibrated amounts of real noise to any freenoise ECG signal. All the ECG data along with further information about these records can be collected from the two described database via [13, 14], respectively.
Wavelet analysis
Wavelet transform
Since its inception, the WT has become the most powerful tool for analyzing signals in many fields of research including the analysis of nonstationary signals. Unlike the traditional Fourier transform (FT), WT provides a time–frequency analysis that can detect local, transient or intermittent components in the studied signal. It is a linear transform, which can refine a signal into multiresolution representation using a scaled and shifted form of the mother wavelet. For practical applications, Mallat [15] has introduced a reliable and efficient algorithm to calculate the discrete wavelet transform (DWT).
Despite its success in several areas of research, the DWT suffers from several drawbacks like oscillations around singularities, shift variance and lack of directionality. Besides, the aliasing issue appears when wavelet coefficients are threshold, which causes distortion to the reconstructed signal.
Dual tree wavelet transform
Wavelet thresholding
Wavelet threshold functions

Hard thresholding.
$$R = \left\{ {\begin{array}{*{20}l} {s,} \quad {\left s \right \ge Th} \\ {0,} \quad {\left s \right Th} \\ \end{array} \begin{array}{*{20}l} { } \\ { } \\ \end{array} } \right.$$(5) 
Soft thresholding
$$R = \left\{ {\begin{array}{*{20}l} {sign(s)(\left s \right  Th),} & \quad {\left s \right \ge Th} \\ {0,} & \quad {\left s \right < Th} \\ \end{array} \begin{array}{*{20}l} { } \\ { } \\ \end{array} } \right.$$(6) 
Semisoft thresholding (S–S)
$$R = \left\{ {\begin{array}{*{20}l} { } \\ {\begin{array}{*{20}l} {s,} & \quad {\left s \right < Th_{2} } \\ {sign(s)\frac{{\left( {s  Th_{1} } \right)*Th_{2} }}{{Th_{2}  Th_{1} }},} & \quad {Th_{2} < \left s \right \le Th_{1} } \\ {0,} & \quad {\left s \right < Th_{1} } \\ \end{array} } \\ { } \\ \end{array} } \right.$$(7) 
Nonnegative Garrote thresholding (NNG)
$$R = \left\{ {\begin{array}{*{20}l} {sign(s)\left( {\left s \right  \frac{{Th^{2} }}{\left s \right}} \right),} & \quad {\left s \right \ge Th} \\ {0,} & \quad {\left s \right < Th} \\ \end{array} } \right.$$(8) 
Hyperbolic thresholding (HYP)
$$R = \left\{ {\begin{array}{*{20}l} {sign(s)\sqrt {s^{2}  Th^{2} } ,} & \quad {\left s \right \ge Th} \\ {0,} & \quad {\left s \right < Th} \\ \end{array} } \right.$$(9)
In these techniques, the variable \(R\) refers to the resulting signal from threshold function, \(s\) represents the wavelet coefficients and \(Th\) is the threshold value. In the case of semisoft thresholding, this function introduces two threshold values \(Th_{1}\) and \(Th_{2}\), where \(Th_{1} < Th_{2}\).
Threshold value selection
Among the critical parameters, that affect the quality of noise suppression, is the threshold value. According to the selected value, the denoised ECG signal could either retain some interferences or have some distortion and discontinuities, depending on whether the threshold value was too small or overly large value. The common threshold values used in the literature [16, 17] are defined as follows:

Universal threshold
$$Th_{1} = \sigma \sqrt {2\log N}$$(10) 
Universal threshold level dependent
$$Th_{2} = \sigma_{j} \sqrt {2\log n_{j} }$$(11) 
Universal modified threshold level dependent
$$Th_{3} = \sigma_{j} \frac{{\sqrt {2\log n_{j} } }}{{\sqrt {n_{j} } }}$$(12) 
Exponential threshold
$$Th_{4} = 2^{{\left( {\frac{j  J}{2}} \right)}} \sigma_{j} \sqrt {2\log N}$$(13) 
Exponential threshold level dependent
$$Th_{5} = 2^{{\left( {\frac{j  J}{2}} \right)}} \sigma_{j} \sqrt {2\log n_{j} }$$(14) 
Minimax threshold
$$Th_{6} = 0.3936 + 0.1829 \times \left( {\frac{{\log n_{j} }}{\log 2}} \right)$$(15) 
The modified unified threshold [17]
$$Th_{7} = \sigma_{j} \frac{{\sqrt {2\log (N)} }}{\log (j + 1)}$$(16)
The value of \(\sigma\) is calculated from each level detail coefficients except the universal threshold case, which is calculated from the first level detail coefficients.
The proposed algorithm
Freenoise ECG signal
To have a clean real ECG signal seems to be difficult. In our study, we first use the synthetic ECG signal for visual performance evaluation of noise reduction algorithm. These signals can be assumed as nearly freenoise signals. Afterward, we process the other signals of the described databases.
Noise generator
To generate noise, we create a function in Matlab that can generate various types of noises, including white noise, colored noise (flicker, Brownian noise, blue, and purple), baseline wander noise (BW), electromyogram noise (EM), and motion artifact (MA). These interferences are inserted into a clean ECG signal with a desired value of signaltonoiseratio.
DTWT ECG decomposition level
Zeroing approximation coefficients
Since the decomposition level is determined, the approximation coefficients magnitude at level \(J\) are set to zero to suppress the baseline wander noise.
Details magnitude threshold
The details coefficients representing the high frequency of the signal are quantified up to a level \(X\). This level \(X\) is determined empirically through a set of simulations. To select the optimal values, all threshold values and functions presented in this work have been tested on the algorithm.
Reconstruct ECG signal
The processed details coefficients at each level, all together with the vanished approximation coefficients at level \(J\) are inversely transformed using the inverse dual tree complex wavelet transform to get the clean ECG signal.
Evaluation parameters
To quantify the algorithm performance and compare it with other methods, we took the most significant and widespread parameters in literature. In the following expressions, \(x_{n} \left( n \right)\) means the studied original signal \(x_{0} \left( n \right)\) corrupted by noise, while \(x_{r} \left( n \right)\) represents the denoised signal that is reconstructed from the algorithm.

Mean square error

Signaltonoiseratio
From Eqs. (24) and (21), the greater the \(SNR_{imp}\) is, the better denoising performance is achieved. Conversely, as the \(MSE\) parameter is small as the distortion is low in the reconstructed signal.
Results
A set of analyses is performed to achieve the promised results. For simplicity and practical implementation reasons, the studied signal is decomposed using ‘farras’ filter [18].
Baseline wander removal
Threshold tuning
The SNR improvement for various threshold values and functions
Threshold function  SNR _{ in } (dB)  Th _{ 1 }  Th _{ 2 }  Th _{ 3 }  Th _{ 4 }  Th _{ 5 }  Th _{ 6 }  Th _{ 7 } 

Soft  −5  3.20389  8.98377  0.25384  1.62297  1.47873  8.25054  9.91075 
0  3.20123  8.53072  0.25589  1.61892  1.47919  4.49104  8.72283  
5  3.19821  7.74215  0.26424  1.63262  1.50036  −0.02684  6.82608  
S–S  −5  1.68416  3.31768  0.00031  0.02849  0.01377  8.51847  11.02541 
0  1.67689  3.33545  0.00033  0.03438  0.01846  4.55702  10.40910  
5  1.66978  3.50081  0.00042  0.05118  0.02850  −0.02684  9.22058  
HYP  −5  2.35838  5.39335  0.00599  0.23616  0.18567  8.49517  10.96391 
0  2.35281  5.34901  0.00627  0.24495  0.19283  4.56183  10.45635  
5  2.34758  5.38295  0.00713  0.27530  0.21732  −0.02684  9.39921  
NNG  −5  2.80969  7.16647  0.01163  0.42158  0.34169  8.31101  10.62271 
0  2.80490  7.04860  0.01217  0.43125  0.35085  4.50462  10.16068  
5  2.80054  6.89711  0.01375  0.47023  0.38413  −0.02684  9.03560  
Hard  −5  1.42053  2.61142  0.00031  0.01829  0.00901  8.87432  10.86859 
0  1.41609  2.60485  0.00029  0.02486  0.01048  4.67213  10.25426  
5  1.40484  2.77515  0.00042  0.03542  0.02116  −0.02684  9.32698 
Denoising results based on the optimized threshold
Denoising results using the synthetic ECG signal
ECG denoising performance for various types of noise
ECG signal corrupted by 5 dB SNR _{ in }  SNR _{ imp }  MSE 

Flicker noise  5.62309  0.00399 
Brownian noise  14.83924  0.00048 
Blue noise  13.19360  0.00070 
Purple noise  14.65982  0.00050 
BW noise  15.24564  0.00044 
EM noise  5.65466  0.00397 
MA noise  7.21953  0.00277 
Combined noise  10.85304  0.00120 
Denoising results using MITBIH signals
Discussions
The simulation results are established in three separate steps. Before we discuss each subsection result, it should be noted here that to exhibit the algorithm performance and to provide a visual noisy ECG signal on all the figures, the input \(SNR\) noise is added with values representing a strong background noise. In the first subsection, we can distinguish the result of baseline wander denoising since it is independent of threshold process of the detail coefficients. The BW noise used here is a real noise and is taken from noise stress database [14]. From Fig. 4, it is obvious that BW noise is perfectly removed without introducing distortion to the original ECG signal.
To choose the optimal threshold, all the previously described threshold functions and values are tested in the threshold tuning subsection. According to Table 2, the modified unified threshold value \(Th_{7}\) gives the best result over all other threshold values. It can effectively reduce all kinds of high frequency and low frequency noises while preserving the amplitude and characteristics of ECG signals. By against, the threshold functions exhibit low variation in SNR improvement. It may be noted that the hard, soft and semisoft functions slightly surpasses the two other functions. Once these threshold parameters are settled, we vary both the decomposition level and the input noise to observe the suitable threshold level. We can see that the amount of added noise and the decomposition level greatly influence the choice of the threshold function. By analyzing the added noise percentage in Fig. 5a and b, the semisoft function is most suitable for positive values of \(SNR_{in}\), while the hyperbolic function is most adapted when \(SNR_{in}\) is negative. For the remainder of the simulations, the hyperbolic function is opted since it provides a low distortion in the entire range of \(SNR_{in}\). On the other hand, the threshold decomposition level is set empirically by selecting the level that affords a small \(MSE\) value regardless the background power noise. From Fig. 5b, we can see that the optimal decomposition level is 3. Since the signal is decomposed to a level based on its sampling frequency, the coefficients of the details magnitude are threshold up to level \((J  4)\). This means that the threshold level will change according to the sampling frequency of the original signal.
The last subsection of results brings together all the set parameters to evaluate the algorithm performance. This evaluation is performed initially on the synthetic ECG signal. From Fig. 6, we can clearly see the efficiency of this method on colored noises. In the case of flicker noise, we can notice a minor distortion in the denoised signal. However, the useful information of the signal remains intact. The combined noise that represents the real noise is also tested on the synthetic ECG signal, as seen in Fig. 7. The input noise value, as well as the weights, were chosen to have a visual strong background noise. Although the ECG signal has strong background noise, the reconstructed signal preserves the QRS complex. The P and T waves have been slightly distorted in some parts of the signal.
To summarize noise removal from the synthetic ECG signal, we calculated the \(SNR\) improvement and \(MSE\) for all kinds of noises using 5 dB \(SNR\) input. For comparison purposes, this value of input noise is chosen as the one used in a recently published work [19]. From Table 3, we can observe that flicker, EM, and MA noises are difficult to remove from ECG signal. This inconvenience is remarkable through \(SNR\) improvement values.
The assessment is also expanded to signals from the MITBIH Database. In Figs. 8 and 9, we worked with two noisy ECG signals. We can visually observe the effectiveness of the algorithm even when changing the lead as illustrated in Fig. 9. In Fig. 10, we chose the record no. 119 from MITBIH Arrhythmia database that has some ectopic beats. We corrupted this record by a real noise. We can note that these ectopic beats are preserved in the reconstructed signal. Figure 11 shows the denoising process of the record no 111 (Fig. 11a). We corrupted this noisy ECG signal by flicker noise (Fig. 11b), which is an electronic noise that is always present in some passive components like the resistors in the ECG recorder. We can observe that the algorithm can effectively remove this mixture of noises (Fig. 11c). In Fig. 12a, we took another noisy ECG signal from the MITBIH database. We apply the conventional white noise to this record as illustrated in Fig. 12b. We can clearly see the robustness of the algorithm on the white noise denoising. We can notice a minor distortion in the denoised signal (Fig. 12c). However, the clinical parameters like R peaks can be easily detected in the denoised signal.
Comparison of the conventional WT and DTWT based methods with the proposed DTWT for MITBIH arrhythmia records
Record no.  SNR  

DWT  DTWT  Proposed  
100  42.6534  45.8309  98.5023 
101  48.4172  49.3955  100.1488 
102  45.4918  49.8727  100.8777 
103  52.2014  55.4198  106.6181 
104  54.1384  57.5161  105.5597 
105  62.8240  66.1887  97.0638 
106  47.4558  51.8933  93.6908 
107  54.5761  55.5218  105.0257 
108  43.3222  48.1617  90.3493 
109  51.2570  53.7656  113.8190 
111  33.9796  37.2333  96.6799 
Conclusions
In this study, threshold tuning of dual tree wavelet transform was applied to reduce noise in ECG signals. The initial simulations were conducted on synthetic ECG signal and were extended to MITBIH arrhythmia database. Threshold tuning was performed empirically based on the optimal threshold function, the optimal threshold value, and the suitable decomposition level. The study was extended to realistic and colored noises. The effectiveness of the proposed method was assessed through quantitative evaluation and visual inspection using a set of simulations from the standard database. The proposed technique achieves outstanding results over ordinary DTWT based ECG denoising methods in the presence of all kinds of noises. Furthermore, the proposed algorithm is simple to embed in real time application and is able to be investigated in QRS identification as well, which is the purpose of our future work
Abbreviations
BW: baseline wander; CN: combined noise; DWT: discrete wavelet transform; DSP: digital signal processing; DTWT: dual tree wavelet transform; ECG: electrocardiogram; EM: electromyogram; FT: Fourier transform; HYP: hyperbolic; Ma: motion artifact; MSE: mean square error; NNG: nonnegative garrote; SS: semisoft; WT: wavelet transform.
List of symbols
\(\beta {:}\) the slope of the power spectral density of colored noise; \(\propto{:}\) represents the direct proportionality between two variables; \(\sigma {:}\) the standard deviation of the first detail coefficients level; \(\sigma_{j} {:}\) the standard deviation of the detail coefficients at j level; \(\sigma_{v} {:}\) the variance of a given signal; \(\psi (t) {:}\) the analytic dual tree wavelet function; \(\psi_{h} (t) {:}\) the real part of the dual tree wavelet function; \(\psi_{g} (t){:}\) the imaginary part of the dual tree wavelet function; \(\left {\psi (t)} \right {:}\) the magnitude (or modulus) of the dual tree wavelet function; \(\angle \psi \left( t \right) {:}\) the argument (or phase) of the dual tree wavelet function; \(ciel {:}\) rounds a real number to the nearest integer greater than or equal to that number; \(d_{j} {:}\) the detail wavelet coefficients of the jth decomposition level; \(f {:}\) the frequency of a given signal; \(f_{BL} {:}\) the baseline wander frequency; \(F_{max} {:}\) the Nyquist frequency (sampling frequency); \(j {:}\) the jth level (or scale) of the dual tree wavelet coefficients; \(J {:}\) the last decomposition level of the dual tree wavelet transform; \(MAD {:}\) the median absolute deviation of a given signal; \(median {:}\) the median value of a given signal; \(MSE {:}\) the mean square error of an estimator; \(N {:}\) the length of the original signal; \(n_{j} {:}\) the length of the jth level of the dual tree wavelet coefficients; \(R {:}\) the reconstructed signal from a threshold function; \(s {:}\) the wavelet coefficients of a given signal; \(sign {:}\) an odd function that extracts the sign of the wavelet coefficients; \(SNR_{imp} {:}\) the signaltonoiseratio improvement; \(SNR_{in} {:}\) the input signaltonoiseratio; \(SNR_{out} {:}\) the output signaltonoiseratio; \(Th {:}\) the threshold value; \(wbw {:}\) the weight of baseline wander noise; \(wem {:}\) the weight of electromyogram noise; \(wma {:}\) the weight of motion artifact noise;; \(X{:}\) the threshold level of the algorithm (is defined empirically from Fig. 5); \(x_{n} \left( n \right) {:}\) the noisy ECG signal; \(x_{0} \left( n \right) {:}\) the original ECG signal; \(x_{r} \left( n \right) {:}\) the denoised EC signal.
Declarations
Authors’ contributions
OE, RL, KE, AA and WJ have equally contributed to the manuscript; both were also involved in the algorithm programming and the results analysis. All authors read and approved the final manuscript.
Acknowledgements
Not applicable.
Competing interests
The authors declare that they have no competing interests.
Availability of data and materials
The datasets generated during and/or analysed during the current study are available in:
[MITBIH Arrhythmia Database (mitdb)], [http://physionet.org/cgibin/atm/ATM].
[MITBIH Noise Stress Test Database (nstdb)], [http://physionet.org/cgibin/atm/ATM].
[ecgsyn], [https://www.physionet.org/physiotools/ecgsyn/Matlab/].
Ethics approval and consent to participate
ECG signal data are taken from PhysioNet database, it is freely accessible via the link given in the Availability of data and materials section and is supported by the National Institute of General Medical Sciences (NIGMS) and the National Institute of Biomedical Imaging and Bioengineering (NIBIB) under NIH Grant Number 2R01GM10498709.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
Authors’ Affiliations
References
 Prochazka A, Kingsbury NG, Rayner PJW, Uhlir J. Signal analysis and prediction. Springer Science + Business Media New York: Birkhäuser Basel; 1998.View ArticleMATHGoogle Scholar
 Yzu QR. Realization and improvement of the modulus maximum denoising method based on wavelet transformation. J Nanjing Univ Posts Telecommun. 2009;29:74–9.Google Scholar
 Donoho DL, Johnstone IM. Ideal Spatial adaption by wavelet shrinkage. Biometrika. 1994; 425–55.Google Scholar
 Donoho DL. Denoising by softthresholding. IEEE Trans Inf Theory. 1995;41:613–27.MathSciNetView ArticleMATHGoogle Scholar
 Selesnick IW, Baraniuk RG, Kingsbury NG. The dualtree complex wavelet transform. IEEE Signal Process Mag. 2005;22:123–51.View ArticleGoogle Scholar
 Kingsbury NG. The dualtree complex wavelet transform: a new technique for shift invariance and directional filters. Proc 8th IEEE DSP Work Utah 1998; paper 86.Google Scholar
 Wang F, Ji Z. Application of the dualtree complex wavelet transform in biomedical signal denoising. Biomed Mater Eng. 2014;24:109–15.Google Scholar
 Coifman RR, Donoho DL. Translationinvariant denoising. In: Antoniadis A, Oppenheim G, editors. Wavelets and statistics. New York: Springer; 1995. p. 125–50.View ArticleGoogle Scholar
 Bui TD. Translationinvariant denoising using multiwavelets. IEEE Trans Signal Process. 1998;46:3414–20.View ArticleGoogle Scholar
 Kingsbury N. Complex wavelets for shift invariant analysis and filtering of signals. Appl Comput Harmon Anal. 2001;10:234–53.MathSciNetView ArticleMATHGoogle Scholar
 Kingsbury N. A dualtree complex wavelet transform with improved orthogonality and symmetry properties. IEEE Int Conf Image Process. 2000;2:375–8.Google Scholar
 McSharry PE, Clifford GD, Tarassenko L, Smith LA. A dynamical model for generating synthetic electrocardiogram signals. IEEE Trans Biomed Eng. 2003;50:289–94.View ArticleGoogle Scholar
 Moody GB, Mark RG. The impact of the MITBIH arrhythmia database. IEEE Eng Med Biol. 2001;20:45–50. doi:10.13026/C2F305.View ArticleGoogle Scholar
 Moody GB, Muldrow WE, Mark RG. A noise stress test for arrhythmia detectors. Comput Cardiol. 1984;11:381–4. doi:10.13026/C2HS3T.Google Scholar
 Mallat SG. A theory for multiresolution signal decomposition: the wavelet representation. IEEE Trans Pattern Anal Mach Intell. 1989;11:674–93.View ArticleMATHGoogle Scholar
 Castillo E, Morales DP, Garcia A, MartinezMarti F, Parrilla L, Palma AJ. Noise suppression in ECG signals through efficient onestep wavelet processing techniques. J Appl Math. 2013. doi:10.1155/2013/763903.MathSciNetMATHGoogle Scholar
 Liu S. A novel thresholding method in removing noises of electrocardiogram based on wavelet transform. J Inf Comput Sci. 2013;10:5031–41.View ArticleGoogle Scholar
 Abdelnour F, Selesnick I. Symmetric nearly orthogonal and orthogonal nearly symmetric wavelets. Arab J Sci Eng. 2004;29:3–16.MathSciNetMATHGoogle Scholar
 Awal MA, Mostafa SS, Ahmad M, Rashid MA. An adaptive level dependent wavelet thresholding for ECG denoising. Biocybern Biomed Eng. 2014;34:238–49.View ArticleGoogle Scholar