Signal de-noising method based on wavelet decomposition

FENG Hao,SHI Xiao-dan,HUANG Xiao-min,ZHANG Zhi-jie

(Key Laboratory of Instrumentation Science & Dynamic Measurement (North University of China),Ministry of Education,Taiyuan 030051,China)

Signal de-noising is a classic issue and the de-noising method can be divided into classical filter method and modern filter method.The classical filter method is mainly based on sampling theorem and Fourier transform,besides,the separation of signal and noise is achieved by a digital filter.It has advantages of high accuracy,stability and flexibility due to the fact that it realizes filtering through numerical calculations[1-2].However,when the real signal and noise coincide in the frequency domain,the filtering effect of the digital filter will be deteriorated.Digital filter can remove most of the noise; nevertheless,it also reduces the signal peak and creates a certain delay for transient signal[3].In thermocouple dynamic calibration experiments,a reduction of signal amplitude will increase dynamic error and affect modeling accuracy.Signal delay will affect the output signals’ synchronization.The digital filter method is ineffective because the measured signal is not stable,more vulnerable to interference and the signal-to-noise ratio is low in transient process.In this case,the use of modern filtering method can achieve more satisfactory results.

1 Theoretical basis of wavelet analysis

The greatest advantages of wavelet analysis are “localization” natures and “mathematical microscope” natures; therefore,it is suitable for analysis of nonlinear and non-stationary signals.It overcomes the defects of traditional Fourier transform and has a good time-frequency localization performance,therefore,wavelet theory is widely used in signal analysis,image processing,military electronic countermeasure,weapons intellectualization,computer classification and identification,synthetic language,medical imaging and diagnosis,fault diagnosis,numeric analysis,seismic analysis,fractal theory,fluid turbulence,equation solving ,celestial mechanics and signal de-noising[4].For example,f(t) is a square integrable signal,namelyf(t)∈L2(R),then the continuous wavelet transformf(t) is defined as

(1)

The inner product form off(t) is defined as

WTf(a,b)=〈f,Ψa,b〉,

(2)

Ψj,k(t)=2j/2Ψ(2jt-k).

(3)

The transformation form is given by

(4)

Then the wavelet transform coefficient after discretization is

(5)

Discrete wavelet transform is mainly used in signal processing,which is achieved by Mallat algorithm.First,wavelet transform is conducted on larger scale signal,and then lower-frequency portion is selected to carry on wavelet transform on half the original scale.Fig.1 depicts the single-step process of fast wavelet transform (FWT) algorithm[5-6].

Fig.1 Decomposition algorithm for single-step process

Reconstruction operation is the inverse transformation of wavelet transform,which is a process of superposition approximate coefficients and detail coefficients to get original signal.The filter graph is shown in Fig.2.

Fig.2 Reconstruction algorithm for single-step process

Reconstruction process starts from the lowest approximate coefficientcAjand detail coefficientscDjthrough low and high frequency reconstruction filter(Lo_RandHi_R).This process continues until the original signal is obtained.Generally,various processes are conducted on decomposition coefficients.For example,wavelet threshold de-noising is a way of reaching the purpose of de-noising by adding a threshold on decomposition wavelet coefficients to reconstruct the signal containing a small amount of noise.

2 Signal de-noising method based on wavelet

There are many kinds of wavelet de-noising methods,such as modulus maxima method,threshold de-noising method and block de-noising method,etc.Among them,the most commonly used is the threshold de-noising method,and at present many de-noising methods are based on threshold de-noising[7].Wavelet threshold de-noising method is a kind of nonlinear de-noising method which can obtain a better visual effect and achieve an approximate optimum under the minimum mean square error condition; thus,it has been further studied and widely used.Hard threshold and soft threshold are two kinds of methods in the wavelet threshold method.As for de-noising of signal with wavelet threshold de-noising,first,a certain series of decompositions are conducted on signal.Signal decomposition series are related to sampling rate of signal and frequency distribution of the noise in the signal,and then the threshold is used at all levels of wavelet coefficient[8].Determining a threshold value is the core of threshold de-noising algorithm,and the selection of threshold value can influence the quality of noise reduction.At present,there are four kinds of commonly used threshold selection rules,namely,the sqtwolog rule,rigrsure rule,heursure rule and minimax rule[9].However,the thresholds determined by these methods are not always the most optimal in practice,which need to be adjusted according to wavelet decomposition coefficient.With the wavelet analysis toolbox in the Matlab platform,not only the threshold can be flexibly settled,but also the wavelet can be intuitively decomposed.In this paper,the output signal of infrared radiation thermometer in temperature calibration is analyzed in Fig.3 and its spectrum diagram is shown in Fig.4.Dynamic test signals are generally non-stationary signals which are characterized by a short delay,fast mutation and other properties[10].

Fig.4 Spectrum of signal

3 Application of wavelet technology in signal de-noising

In wavelet analysis,the most important theoretical problem is how to select a wavelets basis.Based on the features of the test signal and the natures of wavelet basis,the following analyses are obtained[11].

1) Useful signals of dynamic test signals are mainly reflected through mutation portion of the signals.Therefore,the selected wavelets have the properties of compact support in the time domain and fast decay in the frequency domain.

2) The more similar the shapes of the selected wavelet basis and the signal are,the more features of signal are extracted by wavelet transform and the more accurately the characteristics of signal are analyzed by wavelet basis.

3) When wavelet noise reduction is conducted on the dynamic test signal,signal distortion should be minimized.Therefore,a symmetry wavelet basis should be selected.

4) Higher order vanishing moments characteristics of ideal wavelet show that the transient part of the dynamic test signals should be highlighted to the maximum extent during noise reduction and that when selecting wavelet basis,it should have a certain order vanishing moment.As many times de-noising experiments have been conducted combined with specific models,in this paper,symlets wavelet family are selected to be wavelet basis function in analysis of dynamic testing signal[12-13].As an example of output signal of infrared radiation thermometer in the dynamic calibration process for thermocouple,the signal’s sampling frequency is 1 kHz.The signal is interfered by a fixed frequency,and the minimum interference is 100 Hz with very large strength.According to the properties of the wavelet decomposition,the 100 Hz interference and the low-frequency component of signal can be separated by at least 3-level decomposition.The level of signal wavelet decomposition is 4 in this experiment,during which threshold of wavelet coefficient is dealt with in detail to complete wavelet threshold de-noising.From Fig.5,it can be seen that most of the noise is removed through the digital filter,however,the peak of signal is reduced and a delay of signal occurs with delay time 0.1 s .

Fig.5 Filtering results of digital filter

The results can be analyzed by Fig.6 and Fig.7,where most of the signal energy is concentrated in the low-frequency part and high-frequency noise energy is small,and the noise is not uniform which has a power frequency interference.Noise and useful signal in the frequency band are not completely separated in the transient signal.The digital filter method filters out the high frequency component and loses the useful information.The experimental result is shown in Fig.8.As depicted,this method is effective to remove fixed frequency and high frequency noise without delay.Therefore,effect of the wavelet threshold de-noising is superior to that of the digital filter de-noising in transient process.

Fig.6 Signal spectrum after filtering

Fig.7 Signal spectrum after filtering

Fig.8 De-noising effect of wavelet threshold

4 Conclusion

In this paper,digital filter method and wavelet de-noising method are elaborated during the transient signal de-noising process.And the disadvantages of classical filter method are analyzed and the application of wavelet analysis is introduced in the signal de-noising during dynamic testing.Wavelet is conveniently and intuitively decomposed and all levels of thresholds are flexibly set with wavelet analysis toolbox on the Matlab platform.The wavelet de-noising method is effective to remove the fixed frequency and high-frequency noise in the transient signal.Furthermore,the synchronization of the useful components in original signal and the signal after de-noising is also satisfactory.

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