Development of Computer Simulation of Wavelet Selection Technique for Time-Frequency Analysis

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Development of Computer Simulation of Wavelet Selection Technique for Time-Frequency Analysis

Preety D. Swami Department of Electronics

S. A. T. I., Vidisha,India Email:[email protected]

Mohammed Al-Fayoumi

Faculty of IT ,University for Graduate Studies Amman ,Jordan

Email: [email protected] P. Mahanti

Department of CSAS, University of New Brunswick Saint John, NB, E2L 4L5, Canada

Email: [email protected]

Abstract

Wavelets are used for the analysis of non-stationary signals. With its zooming in on singularities it has good ability in processing the non-stationary signals. The trick in using wavelets is to find a set of them that provides a description that is optimal in some sense to the problem at hand. If wavelet analysis in general, or the particular set chosen, is not well-suited to the problem at hand, they can be no help or, worse, lead to deeper confusion. In this work features based on duration and bandwidth of Paul, Morlet, and Gaussian derivative wavelets are described which helps in selection of either of them.

Application of these wavelets for the time frequency analysis of ECG during ventricular fibrillation is discussed .

.

Key words:CWT, Non-stationary signal, time frequency analysis, time and frequency resolution

1.Introduction

The limitations of Fourier transform in analyzing non stationary signals led to the introduction of two dimensional signal processing tools such as time-frequency and time-scale representations.

Time-frequency representations (Hlawatsch and Boudeaux- Bartels, 1992),(Cohen,1989), for example, the Short Time Fourier Transform (STFT), the Wigner Distribution (WD), the Ambiguity Function (AF), aim at describing time dependent spectral properties of signals, where as time–scale representations such as the Wavelet Transform (WT) (Rioul and Vetterli,1991), aim at extracting the localized contributions of signals which are labeled by a scale parameter. Applications of WT include classification of underwater mammals (Huynh et. al., 1998), defect detection of wheel flats of railway wheels (Jianhai et.al, 2002) acoustic emission signal processing (Serrano and Fabio,1996), studies in geophysics (Georgiou and Kumar,1995), analysis of brain evoked potential signals (Bertrand et.al.,1994), genomic signal processing (Wang and Johnson, 2002). This work

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describes the time-frequency analysis of signals using Continuous Wavelet Transform (Rao &

Bopardikar, 2002).Three wavelets namely, Paul, Morlet and Gaussian derivative are discussed.

Also their comparative performance analysis for time-frequency analysis is done.

2.Literature Review

2.1 Continuous Wavelet Transform (CWT)

The function ψ(t) is a mother wavelet or wavelet if it satisfies the following properties (Rao

&.Bopardikar,02)

The function integrates to zero:

∫

( )

∞

−

=0 dt

ψ t (1)

It is square integrable or has finite energy:

( )

∫

∞

−

〈 ∞ dt

t ²

ψ (2)

Continuous Wavelet Transform (CWT) maps a one dimensional analog signal x(t) to a set of

wavelet coefficient which vary continuously over time b and scale a, using wavelet as a basis:

( )t ((t b)/a)dt

x ψ^* − .The wavelet function ψ((t−b)/a) is used to band pass filter the signal. This can be seen as a kind of time varying spectral analysis in which scale a plays the role of a local frequency.

As a increases, wavelets are stretched and analyze low frequencies, while for small a, contracted wavelets analyze high frequencies. Therefore the time frequency extents of the wavelets vary according to a “constant Q” scheme (Rioul and Vetterli,1991), with various resolutions in time or frequency (multi resolution analysis), where as techniques based on STFT analyze the signal at constant resolution (Rioul and Vetterli,1991). Scale a is related to the local frequency f by the relation a = f0 / f, where f0 is the center frequency of the mother wavelet.

2.2 Wavelets

2.2.1 Morlet Wavelet

The most commonly used CWT wavelet is the Morlet . It is defined in the time domain as where m is the wave number.

( )

¹^/⁴ ^imt ^t ^/²

0

2

e

e t

ψ =π⁻ ⁻ ⁽³⁾

2.2.2 Paul wavelet

Paul wavelet can be described as

) 1 m ( m

m

0 (1 it)

)!

m 2 (

! m i ) 2 t

( = − ⁻ ⁺

ψ π (4)

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2.2.3 Gaussian Derivative Wavelet

The third mother wavelet used in this work is Gaussian derivative wavelet. The real component is defined as follows:

dt e

d

2 m 1

) 1 ) ( t

( _m ^-^t ^/²

m ) 1 m ( 0

2



 



 +

= −

+

Γ ψ

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The lowest derivative, 2, is known as the Marr or Mexican Hat Wavelet.

2.3 Ventricular Fibrillation

Ventricular tachyarrhythmias, in particular ventricular fibrillation (VF), are the primary arrhythmic events in the majority of patients who present with sudden cardiac death (Serrano and M.A Fabio, 1996),(Wang and Johnson, 2002). Attention has been focused upon these particular rhythms as it is recognized that prompt therapy can lead to a successful outcome. There has been considerable interest in analysis of the surface electrocardiogram (ECG) in VF centered on attempts to understand the pathophysiological processes occurring in sudden cardiac death, predicting the efficacy of therapy, and guiding the use of alternative or adjunct therapies to improve resuscitation success rates. Until recently, the ECG recorded during ventricular fibrillation (VF) was thought to represent disorganized and unstructured electrical activity of the heart. Using the new signal analysis technique based on wavelet decomposition, we have begun to reveal previously unreported structure within the ECG tracing.

The wavelet transform has become a valuable analysis tool because of its ability to elucidate simultaneously local spectral and temporal information within a signal. It overcomes some of the limitations of the more widely used Fourier Transform, which only contains globally averaged information and has the potential to lose transient or location-specific features within the signal.

The ECG of most normal heart rhythms is a broad band signal with major harmonics upto about 25Hz . During VF, the ECG becomes concentrated in the narrow band of frequencies between 5 and 7 Hz . This feature is used for VF detection algorithms.

2.4 Time Resolution vs. Frequency Resolution

It is not always possible for the CWT to resolve events in frequency. The same holds true for resolving events in time (Rao & Bopardikar, 2002) . Quantitative metrics for time and frequency resolution are based on duration and bandwidth respectively of the mother wavelet. The first moment of a mother wavelet ψ(t) is given by

∫

∞

−

∞

≡ −

2dt

| ) t (

| 2dt

| ) t (

| t 0 t

ψ ψ

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t0 , provides a measure of where ψ(t) is centered along the time axis. Similarly the centre of ψ(ω), the Fourier Transform of ψ(t),is located, along the frequency axis by taking its first moment, given by

∫

∞

−

∞

≡ −

ω ω ψ

ω ω ψ ω ω

2 d

| ) (

| 2d

| ) (

|

0 (7)

A measure of duration of the wavelet or spread in time is given by

∫

∞

−

∞

− −

≡

2dt

| ) t (

|

2 dt

| ) t ( 2| 0) t t ( t

ψ ψ

∆ (8)

This measure is known as root mean square (RMS) duration. The RMS bandwidth of the wavelet is given similarly by

∫

∞

−

∞

− −

≡

ω ω ψ

ω ω ψ ω ω ω

∆

2 d

| ) (

|

2d

| ) ( 2| 0) (

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Fast decays in time and in frequency are required for the wavelets and its Fourier Transform to have finite values for the numerator integrals in equations 6 through 9.

The time-frequency resolution is governed by the uncertainity principle given by equation

ψ ψ ψ ψ

ψ ∆ω ∆ ∆ω

∆t (a) (a)= t =c (10)

Where ∆tψ(a) and ∆ωψ(a) are the RMS duration and bandwidth respectively of the wavelet ψ(t),dilated by the scale a and cψ is a constant. Equation 10 indicates that decreasing ∆tψ(a) results in an increase in ∆ωψ(a) and vice-versa. The smaller the value of, ∆tψ(a) the better the ability of the CWT to resolve events closely spaced in time. Similarly, the smaller the value of ∆ωψ(a), the better the ability of the CWT to resolve events closely spaced in frequency.

3. Experimental Results and Discussions

We have developed a computer simulation method to illustrate the time frequency resolution trade off, a non-stationary sinusoid corresponding to ECG during ventricular fibrillation is simulated. The signal comprises of a 5Hz sinusoid from 0-30 min., a 6Hz sinusoid from 27-33.6 min and a 7Hz sinusoid from 31.2-6 min. The wavelet transforms corresponding to Paul, Morlet and Gaussian Derivative wavelets for different wave numbers are shown in Figures(1) through (3) respectively.

The wavelet peak has been zoomed in to illustrate this time frequency trade off. The RMS duration

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and RMS bandwidth, of each of the three wavelets for different wave numbers, m, is computed.

The result is recorded in Table 1.

Table 1: RMS durations and bandwidths of Paul , Morlet and Gaussian Derivative wavelets of different orders

Paul Morlet Gaussian

Derivative

m ∆t ∆ω m ∆t ∆ω m ∆t ∆ω

4 0.377964 4.74342 6 0.707107 6.0415 2 1.08012 1.58114 16 0.179605 16.748 12 0.707107 12.0208 12 1.01081 3.53554 40 0.112509 40.7492 20 0.707107 20.0125 20 1.00000 4.5277

All the three wavelet basis is compactly supported which means that the oscillations are effectively localized in time by rapid decay. Thus all three of the wavelets offer very good time localization. The Paul wavelet localizes most efficiently in the time domain but frequency resolution is not appreciable. Morlet offers better time resolution than the Gaussian Derivative but greater differences tend to be in the frequency domain. For a given count of evident oscillations in the wavelet, the Gaussian derivative offers the best frequency localization.

Figure1: CWT using Paul Wavelet with Different Orders

Figure2: CWT using Morlet Wavelet with Different Orders

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Figure3: CWT using Gaussian Derivative Wavelet with Different Orders

4. Conclusion

In this paper properties of Paul, Morlet, and Gaussian derivative wavelets have been studied in both time and frequency domain. Comparative performance of these wavelets for time-frequency analysis, selecting time and frequency localization as the performance metrics is carried out.

Finally, it is observed that if in analysis of any signal, interest lies in fine resolution in time domain then Paul wavelet is the best option, while if higher precision is required in frequency domain then Gaussian Derivative wavelet of higher orders gives nice results.

References

[1] Bertrand, O., Bohorquez, J. and Pernier, J. (1994) Time frequency digital filtering based on an invertible wavelet transform: An application to evoked potentials, IEEE Transactions on Bio Medical Engineering, vol.41, no.1., pp.77-88.

[2] Cohen, L., (1989) Time frequency distributions- A review, Proceedings of the IEEE, vol.77, no.7, pp. 941-981

[3] Georgiou E.F.and Kumar P., (1995) Wavelets in Geophysics Academic Press

[4] Goldstein,S., Landis,J.R., Leighton,R., Ritter,G.,Vasu,C.M.,Lantis,A.and Serokman ,R.,(

1981) Characteristics of the resuscitated out-of-hospital cardiac arrest victim with coronary disease,Circulation, vol. 64, no. 5, pp. 977-986.

[5] Hlawatsch F.& Boudeaux- Bartels, G.F.,(1992) Linear and quadratic time frequency signal representations, IEEE Signal Processing Magazine, pp 21-67

[6] Huynh,Q.Q., Cooper,L.N., Intrator N and Shouval, H.,(1998) Classification of underwater mammals using feature extraction based on time frequency analysis and BCM theory, IEEE Transaction on Signal Processing, vol. 46, no.5, pp.1202-1207.

[7] Jacobs,I.G. and Oxer,H.F.,(1990) A review of prehospital defibrillation by ambulance officersin Perth, Western Australia, Med. J. Aust., vol.153, pp. 662-664.

[8] Jianhai,Y., Zheng ding,Q. and Boshi,C.,(2002) Application of wavelet transform to defect detection of wheel flats of railway wheels, Proceedings of International Conference on Signal Processing, Beijing, China, IEEE Press, pp.92-95.

[9] Rao,R.M. &.Bopardikar, A.S.,(2002)Wavelet Transform, Introduction to Theory and

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Applications, Pearson Education

[10] Rioul,O. and Vetterli,M.,(1991)Wavelets and signal processing,IEEE Signal Processing Magazine.pp14-38.

[11] Serrano, E.P., and Fabio,M.A.,(1996) Application of wavelet transform to acoustic emission signal processing, IEEE Transaction on Signal Processing, vol. 44, no 5, pp. 1270-1275 [12] Wang,B. and Johnson,D.H.,(2002)Computing linear transforms of symbolic signals,IEEE

Transactions on Signal Processing, vol.50, no.3, pp.628-634.