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Wavelet Denoising for Multi-lead High Resolution ECG Signals
The main intend of this work was to find out the application of wavelet denoising in noise reduction of multichannel high resolution ECG signals. MatLab is used in this process of Denoising (noise reduction) by selecting the two main issues i.e. to select a wavelet function and on the other hand it is even more important to pick up the alternative of decomposition level on the efficiency of denoising. One of the biggest asset of this work is that the noise level is being reduced in ECG signals by the help of used denoising method. Some of the medical applications where we can find the reductions in noise levels by this method called denoising is in extrasystols and arrhythmia.
ECG signal contains the actual signal along with some noise included in it. In particular, reduction of this kind of noise included along with the actual ECG signal is considered to be one of the crucial problems while studying the electrical activity of the heart. The noise sources that are considered to be the most crucial ones are occurred due to the electrical activity of the muscles, called as EMG and the next one is due to the unstable electrode skin contact. To remove this kind of noise caused in the ECG signal an well-organized analytical tool , one of the effective method called averaging of cardiac cycles comes into the picture i.e. where the signal to noise ratio is increased. But, however this effective method of averaging of cardiac cycles could not be used in some of the medical applications such as extrasystols and arrhythmia. So, in these applications the only way to reduce the noise in the signal is to use wavelet denoising technique. By the use of this wavelet denoising, efficiency of denoising for different wavelet functions and different levels of signal decompositions were observed.
Subject and Methods:
A particular step by step procedure is being followed in this wavelet denoising method to achieve the desired noise free ECG signal.
Step-1: The 1st step is to select an mother wavelet - ψ m, n (t), which forms set of functions by compression or stretching or translation. The selected mother wavelet is as given below and is used in further process.
where, n a coefficient of time translation, ma coefficient of scale (compression).
Step-2: The 2nd step is to select number of decomposition levels of signal xi(t). In this particular step half the samples are eliminate by firstly obtaining the two complimentary high and low pass filters.
Fast wavelet transform- The filters cut frequency is equal to half of bandwidth of analyzed signal, which is amplification of discrete wavelet transform (DWT). By using these fast wavelet transform results the approximations (cAj), that corresponds to low frequency components and details (cDj), that corresponds to high frequency components contained in the signal are obtained. There by obtaining these high and low level components from the fast wavelet transform results the next levels of decomposition of signal x(t) i.e. detail coefficients cDj+1 and approximation cAj+1 are obtained in analogous way, when in place of original signal approximation coefficients of jth decomposition level are analysed in the Fig(a) below.
The procedure of Decreasing the noise content in high frequency components (known as details of signal) is depended on reducing the noise levels in the signal xi(t) in ith channel.
Step-4: In 4th step soft thresholding procedure  is followed , at which the modification of values of j-th level detail coefficients basis of fixed threshold and is given as shown below by the following notation cDj(t). In this notation cDj(t) = sgn(cDj(t))(│x│-THRj) condition is satisfied when cDj(t) is greater than threshold THRj or else cDj(t) equals to zero if cDj(t) is less than or equal to threshold THRj.
Step-5: The 5th step of the analysis is reconstruction of signal xi(t) based of approximation coefficients chosen i-th level of decomposition(cAi) and modified detail coefficients from i-th(cDi) as well as higher levels of decomposition:
Where, k(t) a scaling function from k-th level of decomposition, ψm,n(t) a wavelet functions for m=m0...mk levels of decomposition.
Noise reduction procedures were implemented in MatLab environment and Program user simultaneously has opportunity to test effectiveness of denoising process for 19 wavelet functions. Daubechies proposed functions  (db2, db3, db4, db5, db6, db7, db8), and their modifications so-called Symlets wavelets (sym2, sym3, sym4, sym5, sym6, sym7, sym8) as well as biorthogonal wavelets (bior3.3, bio4.4, bio6.8). It is also possible to select the decomposition level of analyzing signal. Main window of developed program with example of ECG signal denoising is shown in Fig(b). ECG signal from single lead is also shown in the fig(b) where Wavelet function sym8 and 5th level of decomposition were used. By analysis of different mother wavelets in multilead high resolution ECG signals the best wavelets were chosen with regards to signal morphology preservation i.e. db1 (for 4th and higher decomposition levels), sym3 (for 4th level) and sym8 (for 4th decomposition level). In Fig(c) results of signal denoising using these optimal wavelet functions are presented.
Fig(b):original signal i.e. noise included(grey line) and denoised i.e. noise free(black line)
Fig(c): Results of wavelet denoising.
A in the above Fig(c) indicates the original signal and their results of denoising procedure obtained by using wavelets: Daubechies db1, Symlets wavelets sym3 and also sym8 (all with use of 5th decomposition levels).
B in the above Fig(c) indicates magnified fragment i.e. the particular part selected in A, represented as dashed box in A of original signal and result of denoising procedure for wavelets: Daubechies db1, Symlets wavelets sym3 and sym8 for successive levels of decomposition. To be more precise and to understand clearly we can say that Grey line represents original signal with noise included and there by the black line correspond to signal after denoising process i.e. the decrease of the noise.
Interesting results of noise reduction gives also application of wavelet sym8 for 5th level of
decomposition. This function allows for good approximation of parts of ECG with lower frequency components P wave and T wave. However, it is done at expense of higher frequency components of ECG signal and affects morphology of the QRS complex.
Fig(d): Outcome of denoising procedure of ECG signal in a case of arrhythmia (lead V4)
A in the above Fig(d) represents the original signal, B represents the result of wavelet filtration (sym8 - level 5) and C represents the averaging of cardiac cycles with use of cross-correlation method.
Comparing averaged ECG signal of patient with arrhythmia, obtained by cross-correlation method (Fig. 4C), and signal after wavelet denoising (Fig. 4B) the advantage of time-frequency technique over averaging method is noticeable. In arrhythmia case morphology of ECG signal is changing form beat to beat, that is why the procedure of averaging may lead to distortion of ECG morphology, in particular to smooth out lower frequency present in signal (first of all connected with depolarization phase of heart ventricles).
In present study wavelet filtration was applied to ECG signals denoising and compared to averaging technique results. Different mother wavelets were examined to optimize denoising
procedure. The crucial choice in wavelet filtration is selection of right mother wavelet, which could fit optimally to examined signal. Suitable number of decomposition levels is also important. The best results from among tested functions with regard to noise reduction showed wavelet db1, with 1st to 4th and higher levels of decomposition as well as sym3 for 4th level of decomposition.
Increase of denoising efficiency of low frequency ECG components (T wave) in case of using sym8 function, at expense of worse approximation of higher frequencies (QRS complex), indicate validity of adaptable selection of wavelet function and the level of decomposition, depending on phase of electrical activity of the heart.
Advantage of wavelet method: There is a possibility to receive good quality signal for beat to beat analysis and possibility to have high quality signal while averaging technique is impossible, as causing morphology distortion of ECG signals.