Continuous Wavelet Transformation Continuous Wavelet Transformation Institute for Advanced Studies in Basic Sciences – Zanjan Mahdi Vasighi

Table of content Introduction Fourier Transformation Short-Time Fourier Transformation Continuous Wavelet Transformation Applications of CWT

Most of the signals in practice, are TIME- DOMAIN signals in their raw format. It means that measured signal is a function of time. In many cases, the most distinguished information is hidden in the frequency content of the signal. Introduction Why do we need the frequency information?

Frequency content of stationary signals do not change in time. All frequency components exist at all times Stationary signal 20Hz 80Hz120Hz

Transformation Fourier Transformation (FT) is probably the most popular transform being used (especially in electrical engineering and signal processing), There are many other transforms that are used quite often by engineers and mathematicians:  Hilbert transform  Short-Time Fourier transform (STFT)  Radon Transform,  Wavelet transform, (WT) Every transformation technique has its own area of application, with advantages and disadvantages.

Fourier Transformation In 19th century, the French mathematician J. Fourier, showed that any periodic function can be expressed as an infinite sum of periodic complex exponential functions.

Raw Signal (time domain) x(t) cos(2  ft) 5Hz 10Hz

1 Hz  x(t).*cos(2  ft) = -8.8e-15

2 Hz  x(t).*cos(2  ft) = -5.7e-15

3 Hz  x(t).*cos(2  ft) = -4.6e-14

4 Hz  x(t).*cos(2  ft) = -2.2e-14

4.8 Hz  x(t).*cos(2  ft) = 74.5

5 Hz  x(t).*cos(2  ft) = 100

5.2 Hz  x(t).*cos(2  ft) = 77.5

6 Hz  x(t).*cos(2  ft) = 1.0e-14

Frequency (X) Amplitude 20, 80, 120 Hz FT

Frequency content of stationary signals change in time. Non-Stationary signal Magnitude 20 Hz80 Hz120 Hz

Frequency (X) Amplitude FT

So, how come the spectrums of two entirely different signals look very much alike? Recall that the FT gives the spectral content of the signal, but it gives no information regarding where in time those spectral components appear. Once again please note that, the FT gives what frequency components (spectral components) exist in the signal. Nothing more, nothing less.

Almost all biological signals are non-stationary. Some of the most famous ones are ECG (electrical activity of the heart, electrocardiograph), EEG (electrical activity of the brain, electroencephalogram), and EMG (electrical activity of the muscles, electromyogram). ECG EEG EMG

Can we assume that, some portion of a non-stationary signal is stationary? Short-Time Fourier Transformation The answer is yes. In STFT, the signal is divided into small enough segments, where these segments (portions) of the signal can be assumed to be stationary. For this purpose, a window function "w" is chosen. ?

FT X

X

X

X

X

X

X

X

X

X

X

Time step Frequency Amplitude time-frequency representation (TFR) Window width = 0.05 Time step = 100 milisec

FT STFT  

Time stepFrequency Amplitude Time step Frequency Amplitude Narrow windows give good time resolution, but poor frequency resolution. Window width = 0.02 Time step = 10 milisec

Time stepFrequency Amplitude Time step Frequency Amplitude Wide windows give good frequency resolution, but poor time resolution; Window width = 0.1 Time step = 10 milisec

What kind of a window to use ? The answer, of course, is application dependent: If the frequency components are well separated from each other in the original signal, than we may sacrifice some frequency resolution and go for good time resolution, since the spectral components are already well separated from each other. The Wavelet transform (WT) solves the dilemma of resolution to a certain extent, as we will see.

Multi Resolution Analysis MRA, as implied by its name, analyzes the signal at different frequencies with different resolutions. Every spectral component is not resolved equally as was the case in the STFT. MRA is designed to give good time resolution and poor frequency resolution at high frequencies and good frequency resolution and poor time resolution at low frequencies. This approach makes sense especially when the signal at hand has high frequency components for short durations and low frequency components for long durations.

Continuous Wavelet Transformation The continuous wavelet transform was developed as an alternative approach to the short time Fourier transform to overcome the resolution problem. The wavelet analysis is done in a similar way to the STFT analysis, in the sense that the signal is multiplied with a function, wavelet, similar to the window function in the STFT, and the transform is computed separately for different segments of the time domain signal. wavelet

Mexican hat Morlet

X t = 0 Scale = 1  (s,t) x(t) × Inner product

X t = 50 Scale = 1  (s,t) x(t) × Inner product

X t = 100 Scale = 1  (s,t) x(t) × Inner product

X t = 150 Scale = 1  (s,t) x(t) × Inner product

X t = 200 Scale = 1  (s,t) x(t) × Inner product

X t = 200 Scale = 1  (s,t) x(t) × Inner product 0

X t = 0 Scale = 10  (s,t) x(t) × Inner product

X t = 50 Scale = 10  (s,t) x(t) × Inner product

X t = 100 Scale = 10  (s,t) x(t) × Inner product

X t = 150 Scale = 10  (s,t) x(t) × Inner product

X t = 200 Scale = 10  (s,t) x(t) × Inner product

X Scale = 10  (s,t) x(t) × Inner product 0

X Scale = 20  (s,t) x(t) × Inner product

X Scale = 30  (s,t) x(t) × Inner product

X Scale = 40  (s,t) x(t) × Inner product

X Scale = 50  (s,t) x(t) × Inner product

As seen in the above equation, the transformed signal is a function of two variables,  and s, the translation and scale parameters, respectively.  (t) is the transforming function, and it is called the mother wavelet. If the signal has a spectral component that corresponds to the value of s, the product of the wavelet with the signal at the location where this spectral component exists gives a relatively large value.

Magnitude 20 Hz50 Hz120 Hz Translation increment=50 milisecond Scale inc.=0.5

10 Hz 20 Hz60 Hz120 Hz

CWT Applications Identifying time-scale (time-frequency) scheme Frequency filtering (Noise filtering) Wavelet Synthesis Reconstructing signal using selected range of scales

CWT result for non-stationary signal (10 & 20 Hz )

CWT Applications Solving peak overlapping problem in different analytical techniques (simultaneous determination) Journal of Pharmaceutical and Biomedical Analysis (2007) in press Continuous wavelet and derivative transforms for the simultaneous quantitative analysis and dissolution test of levodopa–benserazide tablets Erdal Dinc et.al. Simultaneous analyses of levodopa–benserazide tablets were carried out by continuous wavelet transform (CWT) without using any chemical separation step. The developed spectrophotometric resolution is based on the transformation of the original UV spectra.

S=20

S=50

S=100

S=150

S=200

S=250

Conc. A Conc. B CWT A CWT B

CWT sym6 (s = 128) Unknown mixture spectrum CWT sym6 (s = 128) Calibration model Prediction

Determination of bismuth and copper using adsorptive stripping voltammetry couple with continuous wavelet transform Shokooh S. Khaloo, Ali A. Ensafi, T. Khayamian Talanta 71 (2007) 324–332 A new method is proposed for the determination of bismuth and copper in the presence of each other based on adsorptive stripping voltammetry of complexes of Bi(III)-chromazorul-S and Cu(II)-chromazorul-S at a hanging mercury drop electrode (HMDE). Copper is an interfering element for the determination of Bi(III) because, the voltammograms of Bi(III) and Cu(II) overlapped with each other. Continuous wavelet transform (CWT) was applied to separate the voltammograms. The method was used for determination of these two cations in water and human hair samples. The results indicate the ability of method for the determination of these two elements in real samples.

The combination of both continuous wavelet and chemometrics techniques Spectrophotometric Multicomponent Determination of Tetramethrin, Propoxur and Piperonyl Butoxide in Insecticide Formulation by Principal Component Regression and Partial Least Squares Techniques with Continuous Wavelet Transform Canadian Journal of Analytical Sciences and Spectroscopy 49 (2004) 218 A continuous wavelet transform (CWT) followed by a principal component regression (PCR) and partial least squares (PLS) were applied for the quantitative determination of tetramethrin (TRM), propoxur (PPS) and piperonil butoxide (PPR) in their formulations. A CWT was applied to the absorbance data. The resulting CWT-coefficients (xblock) and concentration set (y-block) were used for the construction of CWT-PCR and CWT-PLS calibrations. The combination of both continuous wavelet and chemometrics techniques indicates good results for the determination of insecticide in synthetic mixtures and commercial formulation.

References Mathworks, Inc. Wavelet Toolbox Help Robi Polikar, The Wavelet Tutorial Multiresolution Wavelet Analysis of Event Related Potentials for the Detection of Alzheimer's Disease, Iowa State University, 06/06/1995 Robi Polikar An Introduction to Wavelets, IEEE Computational Sciences and Engineering, Vol. 2, No 2, Summer 1995, pp Continuous wavelet and derivative transforms for the simultaneous quantitative analysis and dissolution test of levodopa–benserazide tablets, Journal of Pharmaceutical and Biomedical Analysis (2007) In press. Determination of bismuth and copper using adsorptive stripping voltammetry couple with continuous wavelet transform, Talanta 71 (2007) 324–332 Canadian Journal of Analytical Sciences and Spectroscopy 49 (2004) 218

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