Electrocardiogram Denoised Signal by Discrete Wavelet Transform and Continuous Wavelet Transform

Akram Aouinet & Cherif Adnane
Signal Processing: An International Journal (SPIJ), Volume (8) : Issue (1) : 2014 1
Electrocardiogram Denoised Signal by Discrete Wavelet
Transform and Continuous Wavelet Transform
Akram Aouinet akram_aouinet@yahoo.fr
Faculty of Sciences of Tunis/ Laboratory of
Signal Processing/ Physics Department
University of Tunis-Manar
TUNIS, 1060, TUNISIA
Cherif Adnane adnane.cher@fst.rnu.tn
Faculty of Sciences of Tunis/ Laboratory of
Signal Processing/ Physics Department
University of Tunis-Manar
TUNIS, 1060, TUNISIA
Abstract
One of commonest problems in electrocardiogram (ECG) signal processing is denoising. In this
paper a denoising technique based on discrete wavelet transform (DWT) has been developed. To
evaluate proposed technique, we compare it to continuous wavelet transform (CWT).
Performance evaluation uses parameters like mean square error (MSE) and signal to noise ratio
(SNR) computations show that the proposed technique out performs the CWT.
Keywords: Continuous Wavelet Transform, Denoising, Discrete Wavelet Transform,
Electrocardiogram Signal.
1. INTRODUCTION
Electrocardiogram gives information from generation and propagation of electric signals in the
heart. This activity is measured and recorded for more than a hundred years [1]. One cardiac
cycle in an ECG signal consists of the P, Q, R, S, and T waves, and gives important information
on diagnosing cardiac diseases. Most of the clinically useful information in the ECG is found in
the amplitudes and intervals defined by its features. ECG signal are characteristically corrupted
by noise from electric interference, baseline wandering, and electromyography [2]. Processing is
necessary to cancel these noises while conserving information. Therefore the development of
accurate and quick method for automatic ECG denoising is of major importance.
In this paper, we propose a denoising technique using discrete wavelet transform (DWT), to
evaluate proposed technique we compare it to continuous wavelet transform. The method
described is robust and simple to implement. Finally, the ECG signals used in the experiments
are obtained from MIT-BIH arrhythmia database.
2. MATERIAL
2.1 Wavelet Transform
The Fourier transform is a useful tool to analyze the frequency components of the signal, but it is
well suited only to the study of stationary signals where all frequencies have an infinite coherence
time. But the wavelet transform replaces Fourier transform’s waves by a family generated by
translations and dilations of a window called a wavelet, it take two arguments: time and scale.
The original wavelet transform is called mother wavelet and is employed for generating all basis
functions [3]. A set of functions is constructed by scaling and shifting the mother wavelet ψሺtሻ.

Those functions are expressed as follow:
ψୟ,ୠሺtሻ =
ଵ
√ୟ
ψ ቀ
୲ିୠ
ୟ
ቁ ሺ1ሻ
Where a ∈ Rା
− ሼ0ሽ, b ∈ R.
2.2 Continuous Wavelet Transform
The original signal can be reconstructed by an appropriate integration and this is performed after
projecting the given signal on a continuous family of frequency bands. A continuous wavelet
transform (CWT) is used to divide a continuous-time function into wavelets. Mathematically, the
continuous wavelet transform of a function x(t) is defined as the integral transform of f(t) with a
family of wavelet functions, ψୟ,ୠሺtሻ:
CWT(a,b)=
ଵ
√ୟ
‫׬‬ xሺtሻ . ψ∗ାஶ
ିஶ
ቀ
୲ିୠ
ୟ
ቁ dt ሺ2ሻ
Were the superscript * is the complex conjugate and ψୟ,ୠ
∗
represents a translated and scaled
complex conjugated mother wavelet.
The mother wavelet ψ is invertible when it verifies the condition of admissibility which is stated as
[4]:
න
หψ෡ሺωሻห
ω
ାஶ
ିஶ
dω < ∞ ሺ3ሻ
2.3 Discrete Wavelet Transform
The Discrete Wavelet Transform (DWT), which is based on sub-band coding is found to yield a
fast computation of wavelet transform. It is easy to implement and reduces the computation time
and resources required [5].
The DWT is defined by the following equation:
ωሺj, kሻ = ∑ ∑ xሺkሻ. 2ି
୨
ଶൗ
ψሺ2ି୨
.୩ n − kሻ୨ ሺ4ሻ
Where ψሺtሻ is a time function with finite energy and fast decay called the wavelet. The DWT
analysis can be performed using a fast, pyramidal algorithm related to multirate filter banks [6].
The goal of using DWT in an algorithm of filtering biomedical signals is the possibility of choosing
the signal’s coefficients with a significant energy and discards the others that have a very low
percentage of all energy.
Many mother wavelets are used for computing the wavelet transform and Morlet is one of them. It
is expressed as follow [7].
φሺtሻ =
ଵ
√ୟ
∙ exp ൤− ቀ
୲
୘బ
ቁ
ଶ
൨ expሺjω଴tሻ ሺ5ሻ
This analytic wavelet contains a central frequency and having an exponentially decaying time
support, while the orthonormal wavelet frequency support of discrete wavelet transform and WPT
covers a broader bandwidth. Therefore the proposed Morley’s wavelet is more “frequency
focused” along each scale [8].

2.4 Database
The preliminary tests of the two denoising techniques application were mad on the ECG signals
of MIT-BIH Arrhythmia Database. This Database contains many data sets of electrocardiogram
signals, mostly abnormal or unhealthy electrocardiograms, but it also contains normal
electrocardiograms that can be used as a reference base [9].
3. THE PROPOSED DENOISING TECHNIQUE
The objective is to apply the technique of discrete wavelet transform and the technique of
continuous wavelet transform on electrocardiograms signals and compared the obtained results
to decide that these two techniques gives a better result. In order to evaluate the performance of
these proposed techniques, we used the:
• signal to noise ratio (SNR) improvement measure:
impሾdBሿ = SNR୭୳୲୮୳୲ − SNR୧୬୮୳୲ (6)
= 10 × log ቂ
∑ |୶ౚሺ୧ሻି୶ሺ୧ሻ|మ
౟
∑ |୶ሺ୧ሻି୶౤ሺ୧ሻ|మ
౟
ቃ
Where xd and x represent respectively the denoised and the clean ECG signals and xn denotes
the noisy ECG one.
• Mean Square Error (MSE) expressed by the following equation is also used:
MSE =
ଵ
୒
∑ ൫sሺnሻ − s෤ሺnሻ൯
ଶ୒ିଵ
୬ୀ଴ (7)
Where sሺnሻ and s෤ሺnሻ are respectively the clean and the denoised signals.
Figure 1 shows the block diagram of the proposed technique and they will be detailed in the next
paragraph.
FIGURE 1: The Two Methods Chosen For Dinoising.
The ECG signals taken from MIT-BIH arrhythmia database are converted in to Matlab format
(.mat files) and are sampled at 360 Hz with a resolution of 11 bits.
Input ECG signal
Discrete Wavelet Transform Continuous Wavelet Transform
Heursure Minimaxi Rigrsure Sqtwolog
mln sln one mln sln one mln sln one mln sln one

• 'Rigrsure' uses for the soft threshold estimator, a threshold selection rule based on Stein's
Unbiased Estimate of Risk (quadratic loss function). One gets an estimate of the risk for a
particular threshold value t. Minimizing the risks in t gives a selection of the threshold value.
• 'Sqtwolog' uses a fixed-form threshold yielding minimax performance multiplied by a small
factor proportional to log(length(X)).
• 'Heursure' is a mixture of the two previous options. As a result, if the signal to noise ratio is
very small, the SURE estimate is very noisy. If such a situation is detected, the fixed form
threshold is used.
• 'Minimaxi' uses a fixed threshold chosen to yield minimax performance for mean square error
against an ideal procedure. The minimax principle is used in statistics in order to design
estimators. Since the de-noised signal can be assimilated to the estimator of the unknown
regression function, the minimax estimator is the one that realizes the minimum of the
maximum mean square error obtained for the worst function in a given set.
• 'One' for no rescaling.
• 'Sln' for rescaling using a single estimation of level noise based on first-level coefficients.
• 'Mln' for rescaling done using level-dependent estimation of level noise.
4. RESULTS AND DISCUSSION
Table 1 gives the average results of calculation of SNR applied to 30 ECG signals using the
technique DWT.
Mean Results of 30 ECG Signals Tested
Input
SNR
(dB)
Output SNR (dB)
Heursure Minimaxi Rigrsure Sqtwolog
Mln sln One Mln Sln One mln sln one mln sln One
-5 2.28 2.5 1.55 2.7 3.04 1.55 3.38 -0.11 2.01 2.3 3.11 2.01
0 5.9 7.16 3.16 6.64 7.25 3.14 7.63 3.44 3.63 4.17 6.8 3.61
5 11.46 10.34 3.82 10.41 11.41 3.82 11.98 8.91 4.32 7.91 8.35 4.32
10 15.01 14.47 4.07 13.92 15.38 4.07 15.93 13.78 4.57 11.02 14.2 4.57
TABLE 1: Comparison Improvement SNR vs SNR (DWT Technique).
Table 1 shows that this method achieves a very good performance when we use the 'rigrsure'
and 'mln' argument. Then we use these results when we will make a comparison with the
results of the second method (CWT).

Table 2 gives the average results of calculation of SNR applied to 30 ECG signals using the
technique CWT.
TABLE 2: Comparison Improvement SNR vs SNR (CWT Technique).
Figure 2 illustrate the comparison between the improvement values of SNR obtained by discrete
wavelet transform denoising technique and those obtained by the second technique based on
continuous wavelet technique.
FIGURE 2: Output SNR Improvement vs SNR.
Table 3 gives the average results of calculation of MSE applied to 30 ECG signals using the
technique DWT.
-2
0
2
4
6
8
10
12
14
16
18
-5 0 5 10
SNRf (DWT)
SNRf (CWT)
Input SNR (dB) Output SNR (dB)
-5 -0.29
0 4.53
5 10.81
10 16.9

Input
SNR
(dB)
MSE*(10-4
)
heursure Minimaxi rigrsure Sqtwolog
mln sln one mln Sln one mln Sln one mln sln One
-5 11.21 10.52 12.91 10.45 9.48 12.91 8.78 22.4 11.5 10.88 9 11.5
0 5.07 3.64 9.04 4.21 3.55 9.04 3.32 7.38 8.12 5.86 3.94 8.12
5 1.35 1.55 7.84 1.78 1.37 7.84 1.21 2.47 7.02 3.1 1.64 7.02
10 0.61 0.68 7.32 0.81 0.55 7.32 0.48 0.71 6.72 1.52 0.72 6.74
TABLE 3: Comparison Improvement SNR vs SNR (DWT Technique).
Table 3 shows that this method achieves a very good performance when we use the 'rigrsure'
and 'mln' argument. Then we use these results when we will make a comparison with the
results of the second method (CWT).
Table 4 gives the average results of calculation of MSE applied to 30 ECG signals using the
technique CWT.
TABLE 4: MSE variation vs SNR (CWT technique)
Figure 3 illustrate the comparison between the improvement values of MSE obtained by discrete
wavelet transform denoising technique and those obtained by the second technique based on
continuous wavelet technique.
Input SNR (dB) MSE
-5 0.0331
0 0.0192
5 0.0160
10 0.0154

FIGURE 3: MSE versus Input SNR of the Two Methods.
The obtained values show that the discrete wavelet transform denoising technique gives better
results than continuous wavelet technique.
Figure 4 and Figure 5 show some examples of denoised of ECG signals corrupted by white
Gaussian Noise at 5db.
FIGURE 4: (a) Original ECG signal, (b) Filtered ECG signal by DWT, (c) Mean Square Error.
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
-5 0 5 10
MSE(DWT)
MSE(CWT)

FIGURE 5: (a) Original ECG signal, (b) Filtered ECG signal by CWT, (c) Mean Square Error.
5. COMPARATIVE STUDY
In this part benchmarking research with other authors was provided to strengthen the results
found in this work.
Burhan Ergen [10] proves in their work that the rigresure method has showed a better
performance than the other methods in terms of SNR level especially when the decomposition
level is higher than 6.
Vijay S. Chourasia et al. [11] shows that the developed wavelet “fetal” along with rigresure and
soft threshold provides the best performance for denoising the fetal phonocardiography signals.
Zhang xizheng et al.[12] shows, four threshold denoising and default threshold denoising
methods (rigrsure, minimaxi, sqtwolog and heursure) are improved SNR, and have removed
some noise. From the denoising effect, the principle of rigrsure threshold and minimaxi threshold
denoising are better than the other methods, because the two denoised SNR higher than
heursure and sqtwolog denoising methods and its RMSE are smaller.
6. CONCLUSION
ECG signals are corrupted by noise from electric interference, electromyography, and baseline
wandering. In this paper we propose a denoising technique based on smoothing of discrete
wavelet transform coefficients. To evaluate the proposed technique, we compare it to smoothing
of continuous wavelet transform. The results obtained from the SNR computation show that the
SNR improvement values obtained by the proposed technique outperform the denoising
technique based on continuous wavelet transform. Results obtained from MSE computation, also
show that the MSE improvement values obtained by using the proposed technique are better
than those obtained by second technique and this is by reducing the difference between the
original and denoised signals.

7. REFERENCES
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Electrocardiogram Denoised Signal by Discrete Wavelet Transform and Continuous Wavelet Transform

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