7/27/2019 correlation techniques

1/4

1-4244-1557-8/07/$25.00 2007 IEEE

A Signal De-noising Algorithm Based on Correlation

Techniques

Li Xingye and Ma Linlin

Business School

University of Shanghai for Science and TechnologyShanghai, P. R. China

lixingye@tsinghua.org.cn, fuzzy-2001-2001@163.com

Ma Yi

Medical SchoolXian Jiaotong UniversityXian, Shanxi, P. R. China

jigujia@gmail.com

AbstractA new periodic signal de-noising method is proposed

based on correlation techniques. This new method can be

regarded as selecting the frequency components which mainly

attributable to the original periodic signal from the noisy signal

by using the correlation techniques and DFT (discrete Fourier

transform). Experimental results indicate that our proposed

method is not only effective when the noise is white noise, but also

can extract a periodic signal from the colored noise. It is alsoshown in the experiment that our method is better than waveletthresholding de-noising methods.

Keywords- correlation; de-noising; DFT

I. INTRODUCTIONPeriodic signal is a kind of common signal, and it is easy to

be corrupted by the noise when transformed in the noisycommunication channel, so it needs to be de-noised. At first,Fourier transform was used to de-noise the noisy signal, whichcan completely eliminate the high-frequency components of it.But at the same time, some important detailed information of

the signal may be also lost. After the appearance of thewavelet, the methods based on DWT (Discrete WaveletTransform) began to be used to de-noise the noisy signals.Many de-noising methods have been proposed during the pasttwenty years. In 1983, Witkin proposed the scale space filteringmethod [1]. Based on this method, Xu presented an algorithmcalled spatially selective noise filtration (SSNF) in 1994 [2].The effect of the method has a close relationship with thedefinition of the correlation coefficients, therefore this methodworks well only under some special conditions. By means ofthe idea of low pass filter, in 1994, Donoho proposed a verysimple wavelet thresholding algorithm, and then he developedit into the soft-thresholding method in 1995 [3] [4], which hasbeen shown to be effective for the signals corrupted by the

additive white Gaussian noise. Many new soft-thresholdingmethods have been proposed these years via Donohos work[5] [6] [7]. All of these methods can be used to extract theperiodic signals from the noise. However, they are verycomplicated. For example, they all need to find a properwavelet basis in the space to decompose the noisy signal,decide the decomposing level, choose the threshold functions,and compute the threshold values. Besides, these methods havebad effect on de-noising the periodic signals corrupted by thecolored noise. According to the properties of the periodicsignals, some researchers have also proposed the de-noising

methods aiming at the periodic signals. Most of these methodsare suitable to be applied to the signals corrupted by the whitenoise. Reference [8] successfully detected the periodic signalsin the noise by using the correlation techniques, but it didntpropose a method to extract the periodic signals. Reference [9]proposed a method based on [8] to extract the periodic signalsfrom the white noise. Since this method adopts the time-

domain method to de-noise the signals, it needs to estimate theperiod of the original signal first. The algorithm proposed inthis paper combines DFT and the method of [8]. It is simplerand more convenient and can extract the periodic signals fromthe noise without estimating the period of the original periodicsignal. Even the noise is colored noise, the algorithm is alsoeffective. Specially, when extracting the periodic signals, it ismuch more effective than DWT thresholding methods.

The article is organized as follows. Section II relates theproperty of auto-correlation functions and the basic principle ofour method. Section III presents the detailed procedure of thealgorithm. In Section IV, we give the results of the experimentscomparing our algorithm with wavelet thresholding methods

and the method in [9]. At last, section V gives someconclusions.

II. THEPRINCIPLEA. The Properties of the Auto-Correlation Function

Theorem: Let )t(s , Zt,t = , )t( and )t( + are uncorrelated. So

)(R)(RsV , if the value of is large enough. That is to say

)(Rs and )(R

N will have similar frequency components.

Since the periodic signal )t(s and its auto-correlation

)(Rs have the same period, )(R

N and )t(s should have the

same frequency components when is large enough. So we

can find the main frequency components coming from )t(s in

)(tV by the magnitude spectrums of )(RV .

III. THEALGORITHMSuppose )}t(v{ , 110 = N,...,,t is the finite observational

sequence of )t(V , and )t(n)t(s)t(v += , 110 = N,t ,

where )t(s is the truncation of a periodic signal with the

period T , )t(n is a zero-mean noise uncorrelated with )t(s .

We can process it as follows to extract )t(s from )t(v :

Step1 Transform the noisy signal to the frequencydomain using the N-point DFT, and get the Fourier transformcoefficients }V{ k , 110 = N,k .

Step2 Compute the auto-correlation )}(R{ v ,

110 = N, of )}t(v{ , 110 = N,t and eliminate the

beginning part of )}(R{ v , namely )(Rv 0 , )(Rv 1 ,, )m(Rv

( 0m ). In order to reserve enough information of the original

signal, m should not be too large ( m is usually defined by thetype and the power of noise).

Step3 Use the N-point DFT to transform

)}1({ ++ mRv ( )}1({ ++ mRv is padded with trailing zeros to

the sizeN, so the magnitude with the same frequency as )}t(v{ is obtained), then get the frequency coefficients }R{ k ,

110 = N,k .

Step4 According to the principle of this algorithm,when |R|

kis relatively smaller and the frequency component

of the original signal on the point k isnt seriously corrupted,

kV should also be relatively smaller. On the contrary, it means

the frequency component on the point k is seriously corrupted.

So we can use the following method to modifykV :

>

=elseV

ki|V||V|and|R||R|V

k

ikik

k

0

Step5 Transform }V{k

to time domain using theN-

point IFFT, and then get )}t(s{ , which is the signal extracted

from )}t(v{ .

Figure 1 Original signal

Figure 2 Noisy signal (SNR=-0.5520)

Figure 3 De-noised signal

Figure 4 De-noised signal With DWT method (the threshold selection

method is Heuristic SURE soft threshold)SNR=9.9551

222

Authorized licensed use limited to: University Duisburg Essen. Downloaded on April 8, 2009 at 17:38 from IEEE Xplore. Restrictions apply.

7/27/2019 correlation techniques

3/4

Figure 5 De-noised signal with our proposed

method

Figure 8 De-noised signal with our proposed method

(SNR= 8.4984)

Figure 7 De-noised signal With DWT method (the

threshold selection method is Penalize high hard

threshold. SNR=3.4494

Figure 6 Noisy signal (SNR= -6.0888)

IV. SIMULATIONEXPERIMENTBoth of the wavelet thresolding methods used in the

following two experiments adopted db3 wavelet to decomposethe noisy signal at depth 5.

Experiment 1: Uniform white noise on the interval]..[ 5050 is added to a period signal

)tsin()N/tcos()t(s += 4832 , 1000=N (Figure 1).

Then the noisy signal with the signal-to-noise ratio (SNR) of

dB.55200 is obtained (Figure 2). In this experiment, ouralgorithm and the method in Reference [9] were utilizedrespectively to de-noise this noisy signal. The auto-correlation

)}(R{ v of the white noise computed in practice cant

attenuate to zero immediately when 0> , so the effect is not

good enough when 0=m . In our experiment, the effect was

better when 6=m . The de-noised signal using these methodsare shown in Figure 3, Figure 4and Figure 5, where Figure 4

used the Heuristic SUREsoft threshold method to select thethreshold (Table 1 shows the SNR of the de-noised signal usingthe wavelet thresolding method, from which we can see that,for the signal used in our experiment, the effect usingHeuristic SURE soft threshold method is better than otherwavelet thresolding methods). From Figure 3 we know that,because the signal is seriously corrupted, it was not easy toestimate the period of the original signal by using the methodin Reference [9]. The method didnt eliminate the highfrequency noise completely. From Figure 4 we know that thesmoothness of the wave after de-noised using the waveletthresolding method is worse. Figure 5 shows that the wave

after denoised using our algorithm almost had no differencefrom the original signal. The algorithm reserved not only theperiodicity, but also eliminated all the high frequency noise,and the SNR increased about 12dB comparing with the waveletthresolding method. The conclusion obtained from theexperiment 1 is that: the de-noising method of the waveletthresolding has a very good effect on time-varying signal, butfor the stationary periodic signal, this method combined with

DFThas a better effect.

Experiment 2: AR(3) noise is added to the same periodicsignal and the noisy signal with the SNR of dB.08886 is

obtained (Figure 6). The wavelet thresolding method and ouralgorithm are adopted separately to de-noise the signal. FromFigure 6 we can see that the power of the noise is very large, sohere, the value ofm needs to be larger ( 159=m ) to eliminatethe affection of the noise. In fact, when the power of the noiseis too large, the auto-correlation of the noise will also be verylarge, so the speed of attenuating to zero will be slower, and itneeds lagging much order to get close to zero. However, theorder lagged should not be too large, otherwise, a large amountof information of the original signal will be lost. Figure 7shows the wave obtained after de-noised using Penalize highthreshold to select the threshold (Table 2 shows the SNR of the

Table 1 SNR (dB) OF De-noised Signal Using wavelet Methods

Unscaled

white noise

Scaledwhite

noise

Unscaled

white noise

Scaledwhite

noise

Soft Hard

Fixed form

threshold5.1932 7.3378 6.9210 4.4927

Rigorous

SURE9.4523 1.7560 5.8860 -0.4984

Heuristic

SURE9.9551 2.2044 9.3580 -0.2065

Minimax 7.1562 7.2179 5.5471 1.2327

Penalize

high4.6326 5.2516

Penalize

medium5.4573 7.0183

Penalize

low5.9060 7.1344

223

Authorized licensed use limited to: University Duisburg Essen. Downloaded on April 8, 2009 at 17:38 from IEEE Xplore. Restrictions apply.

7/27/2019 correlation techniques

4/4

TABLE 2 SNR (dB) of De-noised Signal Using wavelet Methods

Non- white noiseThe method to select

thresholdSoft Hard

Fixed form threshold 3.4472 3.4342

Rigorous SURE 2.6913 -2.1307

Heuristic SURE 3.4470 3.4342

Minimax 2.9381 -1.7823

Penalize high 3.4461 3.4494

Penalize medium 3.4461 3.4494

Penalize low 3.4461 3.4494

de-noised signal using the wavelet thresolding method, fromwhich we can see that the effect using Penalize high thresholdmethod is better than other wavelet thresolding methods). FromFigure 7 we know that, the middle part of the wave after de-noised using the wavelet thresolding method is similar with theoriginal signal, both ends of the signal have bigger distortions.Figure 8 shows the results adopting our algorithm to de-noise,from which we can see that the wave after de-noised almost

has no difference from the original signal, and the SNRincreased 5dB comparing with the wavelet methods. All ofthese showed the advantages of our algorithm.

V. CONCLUSIONIn this paper, a new periodic signal de-noising algorithm

based on the correlation techniques is proposed. It is simpleand convenient among the methods extracting the periodicsignal corrupted by the noise. The experiments showed thatthe periodic signal corrupted by the white noise or the colorednoise could be extracted and the wave extracted was almost

the same as the original signal. The effect of de-noising signalusing our algorithm was better than the one using thealgorithm in Reference [9] and our algorithm has moreadvantages than the method of the wavelet threshold. A largeamount of control experiments also show that, when the powerof the white noise is smaller, the SNR of the signal de-noisedusing our algorithm could increase about 20dB than the oneusing the wavelet threshold method. The advantages of our

algorithm would be more obvious when the power of thecolored noise is smaller.

ACKNOWLEDGMENT

This work was jointly supported by the Science Foundationof Shanghai (06ZR14144) and Shanghai Leading AcademicDiscipline Project (T0502).

REFERENCES

[1] A. Witkin, Scale space filtering. Proc. 8th Int. Joint Conf. ArtificialIntell., 1983.

[2] S. Mallat and W. L. Hwang, Singularity detection and processing withwavelets. IEEE Trans. Inform. Theory, 1992, Vol. 38, No. 2, pp. 617

643.

[3] D. L. Donoho and I. M. Johnstone, Ideal Spatial Adaptation by WaveletShrinkage. Biometrika, 1994, Vol. 81, pp. 425455.

[4] D. L. Donoho and I. M. Johnstone, Adapting to unknown smoothnessvia wavelet shrinkage. Journal of the American Statistical Association,1995, Vol. 90, pp. 12001224.

[5] M. Lang and H. Guo, Noise Reduction Using an Undecimated DiscreteWavelet Transform. IEEE Signal processing letlers, 1996, Vol. 3, No. 1,pp. 1012.

[6] S. Paul, Wavelet Thresholding of Multivalued Images. IEEETransactions on image processing, 2004, Vol. 13, No. 4, pp. 475483.

[7] T. R. Downie and B. W. Silverman, The Discrete Multiple WaveletTransform and Thresholding Methods. IEEE transactions on signalprocessing, 1998, Vol. 46, No. 9, pp. 25582561.

[8] Y. W. Lee, J. R. Cheatham, and J. B. Wieshner, Application ofCorrelation Analysis to the Detection of Periodic Signals in Noise.Proceedings of the I. R. E., 1950, Vol. 38, No. 10, pp. 11651171.

[9] R. J. Schilling and S. L. Harris, Fundamentals of Digital SignalProcessingUsing MATLAB. Xian Jiaotong University Press, 2005,9. pp. 300306.

224

Authorized licensed use limited to: University Duisburg Essen Downloaded on April 8 2009 at 17:38 from IEEE Xplore Restrictions apply