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obtained by wavelet transform method is obviously superior to that of the cubic spline method. The reason for this is that the cubic spline method performs the background removal according to the points selected by the operator. However, there is no operator interference in the WT method.
5.3.3. Baseline Correction
Baseline drift is caused mainly by continuous variations of experiment conditions, such as temperature, solvent programming in liquid chromatography, or temperature programming in gas chromatography. Therefore, baseline drift is a very common problem in chromatographic studies.
Figure 5.28 shows an example of the separation of the drifting baseline from a chromatogram with gradient elution by method B. Curve (a) is the experimental chromatogram. From the figure, it can be seen that there is a strong baseline drift caused by the gradient elution in the chromatogram. Curve (b) is the 8th-scale discrete approximation c8 decomposed by WT with Symmlet (S5) wavelet. Apparently, it resembles the baseline. Figure 5.29 shows the result obtained by subtracting curve (b) from curve (a) of Figure 5.28 with a factor f of 0.93. From the result, it is clear that the removal of baseline by this method is complete and satisfactory.
5.3.4. Background Removal Using Continuous Wavelet Transform
As stated in Chapter 4, the CWT of a signal s(t) with an analyzing wavelet f (t) is the convolution of s(t) with a scaled and conjugated wavelet
Figure 5.28. An experimental chromatogram (a) and its eighth discrete approximation obtained by WT decomposition (b).
application of wavelet transform in chemistry
Figure 5.29. Baseline-corrected chromatogram obtained by subtracting the eigth discrete approximation from the experimental chromatogram.
fa(t) = fa(- t):
1 f +x /1 - b \
Wf(a, b) = fa * s(b) = — J f ^f(t)dt (5.49)
In Fourier domain, the equation takes the form
1 n +x -------
Wf (a, b) = — f (av)s(M)eimbdM (5.50)
where f and S are the Fourier transforms of the wavelet f and the signal s, respectively. Equations (5.49) and (5.50) show clearly that the wavelet analysis is a time--frequency analysis, or, more properly, a timescale analysis because the scale parameter a behaves as the inverse of a frequency. In particular, Equation (5.50) shows that the CWT of a signal is a filter with a constant relative bandwidth Am/m. Therefore, the CWT should be used for separating the smooth background and the sharp peaks. In the following paragraphs, a method for removal of large spectral line from NMR spectrum is introduced.
Let s(t) be a signal of the form
s(t) = ? s, (t) (5.51)
where si(t) = Ai(t)exp(iM/t) is the lth spectral line, which has a constant frequency fi = mi/2n, and N is the number of the spectral lines. Its CWT
is given by
Wf (a, b) = J2 Wf!(a, b) (5.52)
and the Wf i is
Wfi(a, b) = — f (a®)A,(® - ®i)elmbd®
1 ( +?
— e1®'b f (a(® + ®i))Ai(®)elMbd® (5.53)
= 7T- e 2n
Using the Taylor expansion of the Fourier transform of the analyzing wavelet f around the pulsation ®i
we obtain the following expansion for Wf :
f (a(?+®,)) = f (aM,) + ^ (a®,) (5.54)
( ia)k dkf dkAi
Wf )(a, b) = f (a®)s,(b) + elM/b ? ±—L-®L(a®,) ' (b) (5.55)
k >1 '
Therefore, we have
Wf (a, b) « f (a®, )s, (b ) (5.56)
Wf (a, b) « f (a®, )s, (b) (5.57)
If the values of frequency ®i are sufficiently far away from each other, the factor f (a®) will allow us to treat each spectral line independently. In this case, the contribution of the fth spectral line to the Wf (a, b ) is localized on the scale a, = ®0/®t, where ®0 is the frequency of the analyzing wavelet. Therefore, we have
Wft®07®', b) « s,(b) (5.58)
Using this equation, we can easily separate the large spectral line and small peaks. However, in many cases, especially when the frequency of
application of wavelet transform in chemistry
the each component are close to each other, we cannot obtain satisfactory results using this equation because
the second term in the equation is a sum over the other spectral lines, with the amplitudes attenuated by the exponential factor.
Therefore, in practical applications, we can define
certain number of iterations, the second term in Equation (5.59) will become negligible.
Example 5.9: Large Spectral Line Removal of a Simulated NMR Spectrum. The NMR signal in Fourier domain can be simulated by
Figure 5.30 shows three simulated signals s1(f), s2(t), and s(t) = s1(t) + s2(t) and their Fourier transforms, where s1(t) and s2(t) were simulated by
where t is sampled by 2048 data points. It can be seen that the large spectral line s1(t) can be viewed as the baseline or the background of the small peak s2(t).
In order to separate the signals s1(t) and s2(t) from the mixed signal, we can use the iterative procedure described by Equation (5.60). Figure 5.31 shows the results at the number of the iteration k = 100, 200, and 300. In the calculation, Morlet wavelet, which is defined by