a review on the wavelet transform applications in...

A review on the wavelet transform applicationsin analytical chemistry

Erdal Dinc1 and Dumitru Baleanu2,3

1 Department of Analytical Chemistry, Faculty of Pharmacy, Ankara University,06100, Tandogan, Ankara, Turkey [email protected]

2 Cankaya University, Faculty of Arts and Science, Department of Mathematicsand Computer Sciences, Ankara, Turkey [email protected]

3 Institute of Space Sciences, P.O. BOX, MG-23,R 76900 Magurele-Bucharest,Romania [email protected]

In the spectral analysis, the continuous wavelet transform or very recentlydeveloped fractional wavelet transform are powerful tools for the data reduc-tion, de-noising, compressing and baseline correction of the analytical signalsand resolution of multicomponent overlapping signals. Recently, continuouswavelet transform in combination of zero-crossing approach and spectral ra-tio treatment has been used for the quantitative resolution and the predictionof multi-mixtures in the presence of the original overlapping signals. Thiscombined approach provides a short time analysis, accurate, precision, rapidand low cost for the quality control and routine analysis of the commercialproducts containing active compounds. This hybrid approach indicates thatthis technique is perfectly suitable for the multicomponent analysis of theoverlapping analytical signals in the various fields of the analytical chemistry.In addition, the wavelet transform method are an alternative and promisingsignal analysis approach for the elimination or reduction of the disadvanta-geous of the classical spectral derivative methods for the analytical purposes.This review presents briefly the theoretical basis of the applications of con-tinuous wavelet transform and fractional wavelet transform with the classicalanalytical approaches and reports some of their analytical applications.

1 Introduction

The new advancements in computer and information science, statistics andapplied mathematics in recent years have caused the major changes in thecontent of the analytical chemistry. All these developments together withnew combined analytical instrumentation devices offer a new possibility tochemists and pharmacists for their researches and analytical applications. Ascombined analytical instrumentations, a new sophisticated technology called

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Erdal Dinc and Dumitru Baleanu

hyphenated instrumentation such as the liquid chromatography-diode arraydetector system (LC-DAD), gas chromatography-mass spectrometry (GC-MS) and liquid chromatography-mass spectrometry (LC-MS), etc., has beenused for the multicomponent determination. The above sophisticated analyt-ical methods require chemical separation step such as derivation, extractionand other chemical processes during analysis. In some cases, these analysismethods may not give always better results for some of the multicomponentdetermination of active compounds in samples. At the same time, the relatedtechniques having complex components bring high cost and time consumption.We can clearly say that there is a need of new analytical approaches, tech-niques and methods to solve the above drawbacks and to provide alternativeresolutions for the complex problems of analytical chemistry.

In recent years, the developments of wavelet transform (WT) method andits applications in the analytical chemistry have significantly amplified thepotential power of various spectral techniques [Da92, Ma98, Le98, Wa00]. Inthe spectral analytical studies, continuous wavelet transform (CWT) is greatuseful for the simultaneous determination of active compounds in samples.In this context, CWT approach is a powerful signal processing tool for datareduction, de-noising, baseline correction and resolution of multi-componentoverlapping spectra.

In our previous studies, CWT method with zero-crossing technique andratio spectra procedure were directly applied to the multicomponent determi-nation of compounds in samples in presence of the strongly overlapping ab-sorption signals, without using any priory chemical treatment such as deriva-tion and extraction,( see Refs. [Wa00]-[AN05] for more details) without usingany chemical separation procedure, and successful results were obtained.

It was observed that CWT method in the combination with zero crossingtechnique and spectral ratio procedure is a new hybrid analytical approachwith very simple application for the higher resolution of the overlapping signalsand ratio signals, despite the difficult mathematical theory of these waveletfamilies.

Very recently, fractional wavelet transform (FWT) method, which is anew wavelet transform based on the fractional B-splines [UB99]-[BU02] wassuccessfully applied to image and signal analysis [UB00]. The presence of acontinuous order parameter makes FWT more powerful than other methods,especially when fractal signals or fractional Brownian motions are present.

In the spectral analysis studies, the usage of the usual derivative spec-trophotometry to the original absorption spectra give us several disadvantagessuch as: peak intensity diminished with higher order derivation, it requiredthe additional smooth function mode as well as the additional scaling factorprocesses. In the spectral derivation, when the above mentioned parameterswere used for obtaining the derivative spectrum, the form of obtained deriva-tive spectrum may suffer some deformations from the original one. As a resultthe usual derivative method produces several errors in the quantitative anal-ysis. Therefore, the drawbacks can be eliminated by applying FWT approach

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A review on the wavelet transform applications

to the original absorption spectra. In the cases of low concentration and cor-responding undetectable signals in the analytical chemistry, the combined useof FWT and CWT approaches give the better results than of the classicalanalytical approaches.

The aim of this paper is to demonstrate the analytical applications ofthe CWT and FWT to the multicomponent determination in the analyticalchemistry.

The plan of this study is as follows:In section two we briefly present some basic definitions of CWT. Section threeillustrates the basic definitions of FWT. Section four contains briefly the the-oretical information of the chemical applications of CWT and FWT in ana-lytical chemistry. Finally, section five is dedicated to our conclusions.

2 Continuous wavelet transform

Nowadays wavelets are popular as signal and image processing methods forvarious fields of science and engineering. Wavelets are mathematical func-tions generated from one basic function by the dilatation (scale parameter)(W (x) −→W (2x)) and a translation (shift parameter) (W (x) −→W (x+1)).Projection of a signal onto wavelet basic functions is called the wavelet trans-form (WT). Given a mother wavelet [Da92, Wa00] Ψ(λ) by scaling and shiftingof Ψ(λ) a set of functions denoted by Ψa,b is obtained as indicated below

Ψa,b(λ) =1√(|a|)Ψ(

λ − b

a), a �= 0, a, b ∈ R. (1)

Here a represents the scale parameter, which is a variable, used to controlthe scaling, b represents the translation parameter controlling the translationand R is the domain of real numbers.

The action of a given CWT on a function f(λ) is given below

CWT{f(λ); a, b} =∫ ∞

−∞

f(λ)ψ∗

a,b(λ)dλ = 〈f(λ),ψa,b〉, (2)

where the superscript ∗ denotes the complex conjugate and 〈f(λ),ψa,b〉 de-notes the inner product of function f(λ) onto the wavelet function Ψa,b(λ).

3 Fractional wavelet transform

Recently, a new wavelet transform based on the fractional B-splines wasinitiated [UB99],[UB00],[BU02]. The mathematical idea of fractional deriva-tives has represented the subject of interest for various branches of science[Po99]. As it is already known the splines play a significant role on the earlydevelopment of the theory of the wavelet transform. The generalization of

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the spline constructions was proposed in [UB99],[UB00],[Po99], namely newwavelet bases with a continuous order parameter was obtained. The new frac-tional splines have all properties of the polynomial splines with the exceptionof compact support when the order is non-integer. The main advantage ofthis construction is that we can build the wavelet bases parameterized by thecontinuously-varying regularity parameter α.

In the following we define the notion of B-spline. As it is already knowna B-spline is a generalization of the Bezier curve. Let a vector known asthe knot is defined T = {t0, t1, · · · tm} where T is a non-decreasing sequencewith ti ∈ [0, 1], and define control points P0,Pn. Let us define degree asp = m − n − 1. The knots tp+1, · · · , tm−p+1 are called internal knots. If wedefine the basis functional as

Ni,0(t) = 1 , ifti ≤ t ≤ ti+1, (3)0 , otherwise (4)

and Ni,p(t) = t−ti

ti+p−tiNi,p−1(t)+ i+p+l−t

ti+p+l−ti+lNi+l,p−l(t), then the curve defined

by

C(t) =n∑

i=0

PiNi,p(t) (5)

is a B-spline.

Fractional B-spline

As it was proved in the literature this family interpolates between the integerdegrees of polynomial B-splines and that they allow a fractional order ofapproximation [UB99],[UB00],[BU02]. The fractional B-spline is defined as

βα+ =

∑∞

k=0(−1)k(α + 1

k)(x − k)α

+

Γ (α + 1), (6)

where

Γ (α + 1) =∫ +∞

0

xαe−xdx (7)

and(x − k)α

+ = max(x − k, 0)α. (8)

The forward fractional finite difference operator of order α is defined as

∆α+f(x) =

∞∑k=0

(−1)k(αk

)f(x − k), | α + 1k

|| x − k |α∗ , (9)

where

(αk

) = − Γ (α + 1)Γ (k + 1)Γ (1 − k)

. (10)

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The above defined B-splines fulfill the convolution property

βα1+ ∗ βα2

+ = βα1+α2+ . (11)

The centered fractional B-splines of degree α are given by

βα∗ (x) =

1Γ (α + 1)

∑k∈Z

(−1)k|α + 1k

|x − k|α� , (12)

where | x |α∗

has the following form

| x |α∗

=|x|α

−2 sin( πα2 ) , α not even

x2nlogx(−1)1+n , α even

. (13)

Fractional B-spline wavelets

The definition of the fractional B-spline wavelets is given as follows

ψα+(

x

2) =

∑k∈Z

(−1)k

2α

∑l∈Z

(α + 1l

)β2∗α+1� (l + k − 1)βα

+(x − k). (14)

The fractional splines wavelets obey∫ +∞

−∞

xnΨα+(x)dx = 0 (15)

and the Fourier transform fulfills the following relations

ψα+(ω) = C(jω)α+1, as ω −→ 0 (16)

and

ψα+∗(ω) = C(jω)α+1, as ω −→ 0. (17)

Here ψα+∗(ω) is symmetric. The last formula indicates that the fractional

spline wavelets behave like fractional derivative operator.

4 Analytical applications of continuous wavelettransform and fractional wavelet transform

Especially, the transformation of original absorption spectra is widely used inthe analytical chemistry for the quantitative resolution of mixtures containingactive compounds. This method, which is called derivative spectrophotome-try, is based on the derivation of absorption spectra. However, one of the main

269


problems of the derivative method is that the signal-to-noise ratio (S/N) be-comes progressively worse for higher order. This influences the accuracy andprecision of the analysis results.

To eliminate the above mentioned analytical problem, CWT and FWTmethods are used as a powerful signal processing tool for analytical purpose.In addition, CWT and FWT signal processing methods are simultaneouslyused in the resolution of the complex analytical problems [DB06, DRIB06].The application of the FWT and CWT in this review is classified as follows:

-CWT-zero crossing method

-Ratio spectra-CWT method

-Ratio spectra-CWT method-zero crossing method

-FWT-zero-crossing method

-FWT- CWT method.

CWT and FWT methods as well as their above mentioned analytical ap-plications will be explained in the following sections.

4.1 Continuous wavelet transform-zero crossing approach

If a mixture of two analytes, M and N is considered and if the absorbancevalue of this binary mixture is measured at λi, the following equation can bewritten as

Amixλi= αM,λi

CM + βN,λiCN , (18)

where Amixλiis the absorbance of the binary mixture at wavelength λi,

and the coefficients αλiare βλi

absorptivities of M and N analytes. CM andCN represent the concentrations of analytes.

If CWT is applied to Eq. (18), the following equation is obtained as follows

CWT (Amixλi) = CWT (αM,λi

CM ) + CWT (βN,λiCN ). (19)

If CWT (αM,λiCM ) = 0, then we obtain the following result

CWT (Amixλi) = CWT (βN,λi

CN ). (20)

In Eq. (20), the CWT amplitudes of the N compounds in mixture dependonly on CN and are independent of the concentration of M in mixture. Afterthat, the calibration graphs can be obtained by measuring the CWT ampli-tudes corresponding to zero-crossing points. This procedure is repeated forM compound. Calibration functions obtained are used for the quantitativedetermination of the compounds in their mixtures.

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Application of continuous wavelet transform-zero crossingapproach

An example of the application of this CWT-zero crossing approach to binarymixtures is the quantitative resolution of the binary mixture of diminazeneaceturate (D) and phenazone (P) in the veterinary granules for injection with-out any chemical separation [DBK05].

In this study, the absorption spectra of two standard series of D and Pcomponents in the concentration range 4-20 µg/mL were plotted and storedin the range of 200-310 nm as shown in Figure 1.

Fig. 1. Absorption spectra of D (—) a1) 4µg/mL, a2) 8 µg/mL, a3) 12 µg/mL,a4)16 µg/mL, a5) 20 µg/mL and P (- - - -) b1) 4 µg/mL, b2) 8 µg/mL, b3) 12 µg/mL,b4) 16 µg/mL, b5) 20µg/mL in 0.01 M NaOH and methanol (50:50, V/V).

As it can be seen from Figure 1, the quantitative determination of D and Pcompounds is not possible by using the direct absorbance measurements dueto their overlapping spectra in the same spectral region. To solve this problem,several wavelet families with the various values of the scale parameter (a) weretested and Daubechies (db) wavelet approach was found to be the optimal.

In this study, the original spectral data vectors were processed by db4wavelet transform. A linear regression function for P was obtained by measur-ing the CWT signal amplitudes at 279.5 nm corresponding to a zero-crossingpoint for D. In the similar way, the calibration function for D was constructedby measuring the CWT signal amplitude at the 286.1 nm which correspondsto the zero-crossing point for P as shown in Figure 2.

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Fig. 2. The CWT spectra of D a1) 4 µg/mL, a2) 8 µg/mL, a3) 12 µg/mL, a4) 16µg/mL, a5) 20 µg/mL and P b1) 4 µg/mL, b2) 8 µg/mL, b3) 12 µg/mL, b4) 16µg/mL, b5) 20 µg/mL.

The linear regression functions and their statistical parameters for bothanalytical characteristics were given in Table 1.

When method was applied to the real veterinary samples, the obtainedexperimental results can be seen in Table 2.

Examples including the applications of the CWT-zero crossing approachcan be find for example in Ref. [DB03a].

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4.2 Ratio spectra- continuous wavelet transform approach

As in the CWT-zero crossing approach, if a mixture of M and N compoundsis considered and if the absorbance of this mixture is measured at λi, thefollowing equation can be written as

Amixλi= αM,λi

CM + βN,λiCN , (21)

where Amixλidenotes the absorbance of the mixture at the wavelength λi.

The coefficients αM,λiand βN,λi

are absorptivities of M and N analytes. CM

and CN represent the concentrations of M and N respectively.The standard spectrum of one of compounds in the same mixture is ex-

pressed by the following equation

AN,λi= βN,λi

C0N . (22)

If Eq. (21) is divided by Eq.(22), the ratio spectrum is obtained as

Amixλi

βN,λiC0

X

=αM,λi

CM

βN,λiC0

N

+βN,λi

CN

βN,λiC0

N

. (23)

The equation (8) can be simplified as follows

Amixλi

βN,λi

=αM,λi

βN,λi

CM + CN . (24)

The data vector of the ratio signal, corresponding to Eq.(23) is transferredinto wavelet domain and then if CWT is applied on Eq.(23) we obtain

CWT (Amixλi

βN,λi

) = CWT (αM,λi

βN,λi

)CM . (25)

In Eq.(25) the CWT signal, CWT (Amixλi

βN,λi), corresponding to minimum

and maximum depends only on the values of CM , and it independent of theCN value in the binary mixture.

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A calibration function for M is obtained by measuring the CWT ampli-tudes at the minimum or maximum points in the wavelet domain. By using thecalibration function, the amount of M in its binary mixture with N compoundis determined.

The determination of N compound is carried out by similar procedure.The modified version of this application is the ratio spectra-CWT-zero-

crossing method for the ternary mixture analysis. An example of this appli-cation can be find in the literature [DOB05a].

For example, when CWT approach is applied to the ratio spectra of pureB compound (16 µg/mLB(x)) and its binary mixture (m) containing 16µg/mLB and 20 µg/mL H, the transformed CWT spectrum is obtained asshown in Figure 3.

Fig. 3. Continuous wavelet transformed coincident spectrum of 16 µg/mLB(x)and its binary mixture (m) with 20 µg/mL H (the transformed coincident spec-trum of data signals obtained by applying a 6-level one-dimensional db10 com-pression wavelet and continuous wavelet transform to the ratio spectra data of16µg/mLBE+20µg/mLHCT

20µg/mLHCT(m) and 16µg/mLBE

20µg/mLHCT(x) in the wavelet domain).

The coincident points corresponding to the maximum and minimum of thewavelength is selected as working wavelength to obtain calibration function[DB04b]. The similar procedure is repeated for the determination of othercompound in the binary mixture.

The advantage of this proposed method versus the classical derivative andCWT-zero crossing is that there is no need of any critical point to obtaincalibration graph in the qualitative analysis.

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Application of ratio spectra-continuous wavelet transformapproach

An application of ratio spectra-CWT method to multicomponent analysis ofthe mixtures containing benazepril (B) and hydrochlorothiazide (H) in tabletscan be given as an example [DB04b]. In first step, the absorption spectra ofH and B were recorded as indicated in Figure 4.

Fig. 4. Absorption spectra of a1) 10 µg/mL , a2) 12 µg/mL, a3) 14 µg/mL, a4)16 µg/mL, a5) 18 µg/mL, a6) 20 µg/mL, a7) 22 µg/mL HTC and b1) 12 µg/mL,b2) 16 µg/mL, b3) µ16g/mL, b4) 16 µg/mL, b5) 16µg/mL, b6) 16 µg/mL, b7) 16µg/mL BE in 0.1 M NaOH.

In the ratio treatment, the absorption spectra of H were divided by thestandard spectrum of B (Figure 5A) and the obtained ratio spectra weretransferred into the wavelet domain (Figure 5B).

As it can be see the transferred signal (see Figure 5) is not smooth so, wedecided to make it smooth by using the discrete wavelet transform (DWT) toreduce the noise and to increase the spectral resolution.

275


Fig. 5. Ratio spectrum (A) , Analyzed spectrum (B), Compressed spectrum (C) ,Gauss4 CWT spectrum (D) and Coif2 CWT spectrum (E) of : a1) 10 µg/mL , a2)12 µg/mL, a3) 14 µg/mL, a4) 16 µg/mL, a5) 18 µg/mL, a6) 20 µg/mL, a7) 22µg/mL HTC using 16 µg/mL BE as a divisor.

In this study, all transferred ratio data signals of H and B compounds werecompressed by using a 6-level of Daubechies 10 discrete wavelet (see Figure6).

Fig. 6. Compressed spectrum (C) of: a1) 10 µg/mL , a2) 12 µg/mL, a3) 14 µg/mL,a4) 16 µg/mL, a5) 18 µg/mL, a6) 20 µg/mL, a7) 22 µg/mL HTC using 16 µg/mLB as a divisor.

276


After the above compression processing, to identify the optimal CWT fam-ilies, various CWT were tested and GAUSS (GAUS) and COIFLETS (COIF)CWT methods were found to be suitable to obtain the coincident CWT signalsin the wavelet domain. In these cases, the concentration of H in its mixturewere proportional to the transformed amplitudes of the maxima and minima.Other compound (B) in the mixture was processed by using the similar way.

In this study, the calibration graphs were obtained by measuring the CWTamplitudes of the maxima or minima. These calibration graphs were usedfor the determination of H and B compounds in their binary mixtures andcommercial pharmaceutical samples.

The calibration equations and their statistical results were presented inTable 3.

Two CWT methods were applied to two commercial tablet formulationsand satisfactory results were obtained as indicated in Table 4 [DBK05],[DB04b].

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4.3 Ratio spectra-continuous wavelet transform and zero crossingtechnique

Another interesting method is the combined application of CWT approachwith ratio signal and zero crossing technique to the ternary mixture analysis.This approach is based on the simultaneous use of ratio spectra-CWT andzero crossing technique for the overlapping absorption spectra of ternary mix-tures [DOB05a]. In the application of this hybrid approach, the method gavesuccessful results for the quantitative multiresolution of tablets and ternarymixtures consisting of paracetamol, acetylsalicylic acid and caffeine withoutany separation step.

4.4 Application of continuous wavelet transform in combinationwith multivariate calibration techniques

The experimental results indicate that the CWT method in combination withmultivariate calibration techniques is a promising mathematical too for themulticomponent determination of active compounds in complex mixtures.Some applications of this combined approach can be summarized in the fol-lowing references [DBUK05],[DOB05b]- [DBK04].

4.5 Fractional wavelet transform-derivative approach

One of the important applications of the CWT methods is the FWT to obtainhigher peak amplitude, less noise, and sharper peaks [DB06],[DRIB06]. Tocompare the proposed FWT approach, first derivative spectrophotometry andchemometric methods (CLS, PCR and PLS) were used for resolving the samesamples.

In this application, the spectra of A and S were processed by FWT ap-proach. After that the derivative technique was subjected to the FWT andoriginal spectra. The results obtained from two approaches were comparedwith each other.

We observed that the FWT method in combination with derivative tech-nique has a higher performance then the classical derivative technique. Figures7A and 7B indicate the original absorption spectra and their FWT spectra ofA and S.

278


Fig. 7. Compressed spectrum (C) of: a1) 10 µg/mL , a2) 12 µg/mL, a3) 14 µg/mL,a4) 16 µg/mL, a5) 18 µg/mL, a6) 20 µg/mL, a7) 22 µg/mL HTC using 16 µg/mLB as a divisor.

First derivative forms of original absorption spectra and their FWT spectraare presented in Figure 8A and 8B.

Fig. 8. First derivative transform of absorption spectra (A) and FWT spectra (B)of of 10 µg/mL , 30µg/mL, 50µg/mL, 70µg/mL AP ( , a1-a4) and of 10 µg/mL ,25µg/mL, 40µg/mL, 55µg/mL SB (., b1-b4) in acetonitrile and 0.1 M NaOH (50:50(v/v)).

279


4.6 Fractional and continuous wavelet transform

FWT was applied to the original absorption spectra of lacidipine (LAC) andits photo-degradation product (LACD) [DRIB06]. The resulting FWT spectrawere processed by CWT and multilinear regression calibration for the simulta-neous quantitative resolution of lacidipine and its photo-degradation productin their binary mixtures. These methods don’t require any chemical separa-tion step and chemical complex reaction to obtain a detectable signal for thedegradation product. By using the Mexican hat function, two calibration func-tions for LAC and LACD were obtained by measuring the CWT transformedsignals at 416.1 nm for LAC and 414.6 nm for LACD, after FWT process-ing of the original absorption spectra. For a comparison MLRC approach wasconstructed using the linear regression functions for the individual drug andits degradation product.

Fig. 9. Original absorption spectra (A) and their corresponding FWT spectra (B)of 5.08, 10.16, 20.32, 30.48 and 40.64 µg/mL LAC (· · ·) and 0.51, 1.02, 2.04, 4.08and 8.16 LACD (−).

The absorption spectra of lacidipine and its degradation product are shownin Figure 9 (A). The examination of this figure shows that the simultaneouslydetermination of LAC and LACD isn’t possible by using the classical spec-trophotometric method and its modified versions due to the low content ofthe degradation product. To bypass this difficulty FWT approach was applied

280


to the absorption spectra of LAC and LACD in the spectral range of 225-430nm.

To determine LAC and LACD in their mixture samples, FWT data vectorswere processed by Mexican hat function at the scaling factor a=40. CWT spec-tra were obtained by plotting the CWT Ca,b coefficients versus wavelengthsas seen in Figure 10.

405 410 415 420 425

-8

-6

-4

-2

0

2

Wavelength (nm)

CW

T-am

plitu

des

of th

e F

WT

sign

als

240 260 280 300 320 340 360 380 400 420

-15

-10

-5

0

5

10

15

20

Wavelength (nm)

CW

T-a

mpl

itude

s of

the

FW

T si

gnal

s

414.6 nm

416.1 nm

A)

B)

Fig. 10. The CWT spectrum (A) and its detailed form (B) of the FWT signals of5.08, 10.16, 20.32, 30.48 and 40.64 µg/mL LAC(· · ·) and 0.51, 1.02, 2.04, 4.08 and8.16 LACD (−).

4.7 Application of CWT to electrochemical and IR signals

Some of the main problems of analytical studies based on signal analysis aresome limitations such as noise, irrelevant information and signal baseline cor-rection to obtain accurate, precise and safe results. A promising tool to solvethese mentioned problems is to use wavelets having versatile mathematicalproperties. In some cases, the nature of electrochemical and IR manipulations

281


can give us the voltammetric and IR signals containing noise, irrelevant in-formation and signal baseline problems. For these reasons, wavelets functionsgenerated from one basic function by dilatation and translation has been usedin the areas of data compression, de-noising and data reduction in analyticalapplications [UB99]-[UB00].

5 Conclusions

Despite of the developments of the analytical instrumentations analyticalchemistry needs the advanced new analytical approaches for the resolution ofthe complex analytical problems. This study presents some of the combinedused of the wavelet method and the classical analytical approaches.

The hybrid approaches in this study offer new possibilities and alternativeways for the resolution of mixtures of the active compounds having overlap-ping spectra.

Contrary to classical derivative spectrophotometry, CWT approaches donot need any optimization such as smoothing function, scaling factor andsampling interval (∆λ). In addition, these combined CWT methods do notproduce any problem like diminishing peak intensity in higher order derivativecalculation as well as derivative spectrophotometry.

One of the main advantages of CWT approach is the simultaneous datareduction and de-noising for the signal analysis. Beside, CWT and FWT ap-proaches provide higher peak amplitude, less noise, and sharper peaks thenderivative spectroscopy.

All wavelet families fulfilling the optimal conditions can be used for themulticomponent determination of active compounds in real samples.

The combination of CWT, FWT and the classical approaches provide safe,reliable, accuracy and rapid analysis of the experimental results for the qualitycontrol, routine analysis in drug industry and related branches.

Acknowledgments

One of the authors (E.D.) would like to thank to the organizers of MME06symposium for giving him the financial support.

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[DB4] Dinc, E., Baleanu, D. : One-dimension continuous wavelet resolution forthe simultaneous analysis of binary mixture of benazepril and hydrochloroth-iazide in tablets using spectrophotometric absorbance data. Rom. Journ. Chem.,49(11), 917–925 (2004).

[DBUK05] Dinc, E., Baleanu, D., Ustundag, O., Tas, K.: Chemometric calibrationbased on the wavelet transform for the quantitative resolution of two-colorantsmixtures., Rom. Journ. Chem., 50(4), 283–290 (2005).

[DBK05] Dinc, E., Baleanu, D., Kanbur, M.: A comparative application of waveletapproaches to the absorption and ratio spectra for the simultaneous determina-tion of diminazene aceturate and phenazone in veterinary granules for injection.Die Pharmazie, 60(12) 892–896 (2005).

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[DB06] Dinc, E., Baleanu, D.: A new fractional wavelet approach for the simulta-neous determination of ampicillin sodium and sulbactam sodium in a binarymixture. Spectr. Acta Part A, 63(3), 631–638 (2006).

[DB05] Dinc, E., Baleanu, D. :Bivariate and multivariate spectral resolution of amixture of benazepril hydrochloride and hydrochlorothiazide in tablets by usinglinear regression lines. Rev. Chim., 56(9), 937–942 (2005).

[DB03a] Dinc, E., Baleanu, D. : A zero-crossing technique for the multidetermi-nation of thiamine HCl and pyridoxine HCl in their mixture by using one-dimensional wavelet transform. J.Pharm. Biomed. Anal., 31, 969–978 ( 2003).

[DBOH046] Dinc, E., Baleanu, D., Ustundag,O., Aboul-Enein, H.:Discrete and con-tinuous wavelet transforms for the multicomponent determination of sunset yel-low and tartrazine in their soft drink powders. Rev. Chim., 57(1), 29–35 (2006).

[DRIB06] Dinc, E., Ragno, G., Ioele, G., Baleanu, D.:Fractional wavelet analysis forthe simultaneous quantitative resolution of lacidipine and its photo-degradationproduct by continuous wavelet transform and multilinear regression calibration.J.AOAC Int., to appear (2006).

[DBUK05] Dinc, E., Ozdemir, A., Baleanu, D., Tas, K. :Wavelet transform withchemometric techniques for quantitative multiresolution analysis of a ternarymixture consisting of paracetamol, ascorbic acid and acetylsalicylic acid in ef-fervescent tablets, Rev. Chim., 57(5), 505–510 (2006).

[DB06] Dinc, E., Baleanu, D.:Continuous wavelet transform applied to the overlap-ping absorption signals and their ratio signals for the quantitative resolution ofmixture of oxfendazole and oxyclozanide in bolus. J.Food Drug Anal., in press,(2006).

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