Wavelet threshold denoising technique in ECG Signal Processing
Cardiac cell depolarization and repolarization electrophysiological phenomenon is the basis of the heart operation. ECG records the heart cell depolarization and repolarization process, to a certain extent reflects the heart of the objective of the physical condition of the site and thus of great significance for clinical medicine.  as a result of the human body as the detection of ECG changes in state and time, on the one hand with the more obvious characteristics of non-stationary, on the other hand also contains a lot of interference, such as frequency interference, electrical interference, such as respiratory disturbance, additive or multiplicative mixed with the ECG, causing distortion of the ECG in order to cover up the original ECG waveform characteristics in the information, so that the whole ECG waveform vague and difficult to identify the diagnosis. Traditional methods of removal of RC interference filter, digital filter compensation, as well as the baseline to baseline drift and so on, but there are a number of shortcomings. In this paper, in recent years with a new time-frequency analysis theory - the theory of wavelet transform (WT: Wavelet Transforms) applied to the measurement of ECG using wavelet multi-resolution multi-scale features, will carry out decomposition of ECG, signals of different frequency bands became apparent in the wavelet decomposition of the different scales for signal reconstruction, the removal of high-frequency interference and baseline drift where the scale of information, so that after the reconstruction of the signal no longer contains an interference component in order to accurately assess the ECG of the characteristic parameters and the desired detection of the ECG waveform, and then extract the value of diagnostic information.
2, the principle of wavelet threshold denoising
2.1 Wavelet Transform
Fourier analysis is broken down into a series of signals of different frequencies of the superimposed sine wave, the same wavelet analysis is to decompose the signal into a series of superimposed wavelet function, wavelet function which is a mother wavelet function and scale expansion, after a shift to the . The definition of wavelet transform is to be called a basic wavelet (also called the mother wavelet) displacement b of the function to do, and then in different scales to be analyzed with a signal x (t) do with the plot: one for x (t ) is a square integrable function (recorded as ), Φ (t) is the basic wavelet or mother wavelet (MW) function, and satisfy the conditions to allow
Referred to as x (t) of the wavelet transform. Where: Wx (a, b) is x (t) of the wavelet transform, a> 0 is a scaling factor; b reflect the displacement, and its value is to be negative, superscript * on behalf of the conjugate , Is the basic wavelet and the scale of the displacement expansion. Style (1) is not only continuous variables, and a and b is also a continuous transformation, it is called continuous wavelet transform (CWT). Style (1) of the equivalent frequency domain is expressed as:
Where X (ω), Ψ * (aω) are x (t) and Ψ (t) the Fourier transform.
Slave (1) (2), we can see that if x (t) for the signal function, the wavelet transform is a signal and wavelet function of the inner product is to satisfy certain additional conditions the signal filter, this additional condition is reflected in the wavelet function and choice of wavelet factor. Wavelet transform provides a good localization properties, it can in the time domain, frequency domain can also be localized in the observation position. The use of high-frequency small-scale a value, low-frequency large-scale use of a value, analysis of high and low frequency, but the analysis of frequency band analysis is consistent quality factor , if you want in the time domain of the more detailed observations , the more we may need to reduce the scope of observation and analysis of the frequency to increase. Wavelet transform by the use of this mathematical microscope with the characteristics and frequency domain band-pass characteristics of the signal can be separated from the necessary, carry out an analytical study.
2.2 Wavelet Thresholding Denoising Algorithm Analysis
The so-called noise threshold, that is, in accordance with certain pre-set threshold value changes signal wavelet compression coefficient, and then be compressed in order to achieve noise reduction coefficient reconstruction purposes. At present, the most widely used is made hard Donoho threshold and soft threshold denoising method. Because in the wavelet domain, the signal energy concentrated in a relatively few locations, but noise is generally more widely distributed, according to the instantaneous nature of the signal reflected in the coefficient of some large and some small coefficient more by the noise and signal energy generated by the mutation, so wavelet threshold denoising is an effective use of signal and noise signal in the wavelet transform of the singularity of the performance of different characteristics to remove noise, to retain an effective signal. ECG of the main ingredients in the 100Hz frequency below the noise in the interference EMG 5 ~ 2000Hz, so the signal is compared with the ECG, EMG is a high-frequency interference. So to pass multi-resolution analysis of wavelet analysis method to show small-scale wavelet decomposition in the direct removal of the electromyographic interference, to achieve high-frequency electrical interference filter, and then through the threshold band will overlap with part of ECG interference cancellation of the EMG. And then to deal with the wavelet coefficients after the wavelet reconstruction image after the ECG waveform. Mainly divided into the following steps: (1) of the observed multi-scale decomposition of signals, by the time domain into the wavelet domain, the observed signal wavelet coefficients; (2) of the estimated noise and choice of threshold, the wavelet coefficients of the threshold operation, new wavelet coefficients; (3) from a revised reconstruction of wavelet coefficients to be the original signal. 
2.2.1 Selection of the basic wavelet
In the use of wavelet transform signal processing method of the process, the choice of wavelet function is very important function of different wavelet decomposition of signals, you can highlight the different characteristics of the signal characteristics. As signal processing is the role of wavelet band-pass filter, so the equivalent for the symmetric and antisymmetric linear phase and generalized linear phase. If a band-pass filter is not linear-phase or generalized linear phase, it will produce distortion of the signal through. In order to avoid signal distortion, the experimental selection with compactly supported, symmetric and antisymmetric nature of the spline wavelet. Repeated experiments and simulation show that the increase in the number of spline curve more and more smooth, but as a result of increased bandwidth, in addition to weakening the effects of noise. After careful consideration, and finally select the three B-spline wavelet function as the wavelet decomposition of the ECG and synthetic. Three B-spline wavelet of polynomials as follows:
2.2.2 the choice of scale
Scale wavelet transform and signal-one correspondence between the frequency of the relationship, in order to correctly identify the ECG, it is also necessary to choose the correct features of the correct scale. After several digital simulation and analysis, the experiments show that, QRS wave of energy focused on the scale 23. 23 as the center in order to measure whether large-scale or smaller, QRS wave energy will be gradually reduced. As for the low-frequency T-wave, its energy concentrated in the 24 scale. In a larger scale 2 j (j ≥ 5) on, QT wave energy attenuation become larger, while the interference energy has become a lot bigger-scale operation at the same time the greater the amount. Therefore, choose only from 21 to 24 of the four-scale decomposition of the ECG and synthesis.
2.2.3 Threshold function selection and the determination of the threshold t
Threshold threshold function is divided into hard and soft thresholds are based wavelet coefficients djk for, djk 'for thresholding wavelet coefficients after, if the threshold by the hard way:
If soft-threshold approach:
Threshold because of the hard threshold function as a result of discontinuity will cause a larger variance, and unstable, small changes in data-sensitive comparison. Therefore, using soft-threshold approach. 
Determination of the threshold wavelet shrinkage de-noising is the most critical step, the threshold is too small, the large variance of data due to smoothing; threshold is too large, the data would have been smooth, the singularity of the signal may be lost. Wavelet coefficients of the threshold operation, in two ways, one for each threshold the wavelet coefficients to operate, and the second is to block threshold operation practices. By the singularity of signal theory, noise in the ECG has a negative singularity, the magnitude and scale with the density decreased, while the opposite signal. Therefore select the threshold can not be a single, adaptive threshold selection in this article to overcome this drawback of threshold selection formula is as follows: One, N sampling points for ECG, j-class-based measure, z is a constant, this article from the experiment z = 1. 
3, the experimental steps and results of analysis
This paper used experimental data from standard ECG MIT-BIH database, as shown in Figure 1, the sampling rate of 360Hz, A / D conversion precision of 12. By adding Gaussian white noise standard ECG signal noise pollution simulation, signal to noise ratio is 10dB, as shown in Figure 2.
Figure 1 Standard ECG
Figure 2 ECG signal with noise
First of all, we make use of three B-spline wavelet on the ECG signal contains noise for binary discrete wavelet transform, scale check for 4, and calculate the scale of the signal wavelet coefficients, the transformation results as shown in Figure 3: based on soft thresholding value method, the use of adaptive threshold settings to adjust the threshold wavelet coefficients to remove the ECG of random noise, and finally, after the adjustment of the wavelet transform coefficients for inverse transform, so that was after the Noise signal data, draw simulation diagram shown in Figure 4:
Figure 3 ECG of the four-scale wavelet decomposition
Figure 4 under the soft-thresholding with adaptive threshold de-noising of ECG after
4, Summary and Outlook
In this paper, the wavelet threshold noise cancellation of ECG methods, the experiment showed that the de-noising method for noise suppression of ECG is very effective in eliminating noise maintained after the ECG waveform of the basic characteristics of adaptive selection threshold method with self-adaptive for non-stationary de-noising of ECG processing, with the traditional de-noising of ECG methods have obvious advantages. At the same time, wavelet thresholding de-noising techniques for various excellent properties, have been the concern of many researchers has been the concern of many researchers has greatly broadened the scope of wavelet denoising, these studies will greatly enrich the theory of wavelet denoising, to promote wavelet denoising technique to achieve greater development.
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