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Chemometrics from basick to wavelet transform - Chau F.T

Chau F.T Chemometrics from basick to wavelet transform - Wiley publishing , 2004. - 333 p.
ISBN 0-471-20242-8
Download (direct link): chemometricsfrombasics2004.pdf
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Example 2.2. A signal in the time domain can be represented by a combination of periodic sine and cosine functions. Usually, any time-dependent or continuous signal can be considered as a combination of sine and cosine functions. This explains why Fourier transformation has wide application.
Figure 2.10 illustrates how Fourier transformation works. The plot shown in Figure 2.10a is the sum of three trigonometric functions (Fig. 2.10c) with two sine functions with the periods of 1n and 2n as well as one cosine function with the period of 3n. Through applying Fourier transformation to the plot, the dependence of the intensity on frequency from the calculation is depicted in Figure 2.10b. It can be seen from the figure that the three component functions are definitely identified.
48
one-dimensional signal processing techniques in chemistry
Time
Figure 2.10. Fourier transformation: the composite signal (a) contains two sine functions with the periods of 1n and 2n and one cosine function with the period of 3n (b); the dependence of intensity on frequency after Fourier transformation (c).
2.2.3.2. Fast Fourier Transformation
In this section we briefly describe how the fast Fourier transformation can be used to carry out inverse Fourier transformation. For more detail, readers can refer to Brigham’s treatise [11]. Here a simple case of N = 4 is considered. Let us define w to be a complex number
w = e-2nJ/4 (2.46)
Then, the expression of DFT can be written as
N -1
f (n) = J2 f(k)wnk (2.47)
k=0
The basic idea of fast Fourier transform (FFT) is to decompose this formula so as to reduce the calculation burden. When N is equal to a power of 2, that is, N = 2a where a is an integer, the computation is very simple. Now,
N = 22 = 4 is utilized as an example to illustrate the FFT decomposition
transformation methods of analytical signals
49
procedure [18,19]. In this case, Equation (2.47) becomes
4-1
f (n) = ? fo(k)
w
nk
n = 0,1,2,3
(2.48)
k=0
A slight change in the notation of f (k) to f0(k) is made here so as to indicate the signal before Fourier transformation. This equation can be written as follows:
f (0) = fo(0) w0 + fo(1)w0 + fo(2)w0 + /0(3) w0
f (1) = f0(0) w0 + /0(1)w 1 + /0(2)w 2 + f0(3) w3
f (2) = f0(0)w0 + /0(2)w 2 + /0(2)w 4 + f0(3) w6
f (3) = f0(0) w0 + /0(1)w 3 + /0(2)w 6 + f0(3) w9
(2.49)
or
f(0) w0 w0 w0 w0 ff0(0)l
f (1) w0 w1 w2 w3 f0(1)
f (2) w0 w2 w4 w6 f0(2)
f(3) w0 w3 w6 w9 Lf0(3)J
(2.50)
In matrix form, we have
f (n) = Wnk f0(k)
where
f(n) =
f(0)
f (1) f (2)
f(3)
W
nk
w0 w0 w0 w0
w0 w1 w2 w 3
w0 w2 w4 w 6
w0 w3 w6 w9
and f0(k) =
f0(0)
f0(1)
f0(2)
,f0(3).
It should be noted that wnk = e-2nj(nk/4). Here the symbol f is used to denote the remainder of the division of (kn) by 4. Then, only f is needed to be considered in the following calculation. For nk = 6, we have
w 6 = e -j2n 6/4 = e-j 2n (4/4)e-j2n 2/4 = e-( j 2n/4)2 = w 2
and f = 2. Equation (2.50) can now be written as
\f (0)1 '11 1 1 ? ff0(0)l
f (1) 1 w1 w2 w3 f0(1)
f (2) 1 w2 w0 w2 f0(2)
Lf (3)J 1 w3 w2 w1 Lf0(3)J
(2.51)
50
one-dimensional signal processing techniques in chemistry
The matrix containing 1 and w% terms can be factorized into two matrices as
\f (0)1
f (2)
f (1)
Lf (3)J
0
w' w 2 0 0
0
0
w2
w 3
0
w
0
w2
0
0
w0
0
w
/0(0)-
/0(1)
/0(2)
./0(3).
(2.52)
Attention should be paid to the interchange of f (1) and f (2) in the matrix on the left-hand side of the equation. Computing f(n) via Equation (2.52) requires four multiplication and eight addition operations of complex numbers. In contrast, finding the value of the same elements through Equation (2.50) requires 16 complex multiplications and 12 complex additions. Hence, the total number of mathematical operations is reduced significantly with the use of Equation (2.52), instead of Equation (2.50). It is obvious that the reduction in operations becomes more dramatic when N is much greater than 4.
In MATLAB software, the fast Fourier transformation (FFT) has its specific statement. One can simply use one command to obtain the FFT result:
x = fft (y)
This makes Fourier transformation very simple using MATLAB. Here, fft(x) is the discrete Fourier transformation (DFT) of vector x. If the length of x is a power of 2, a fast radix-2 fast Fourier transform algorithm is utilized. If not, a slower non-power-of 2 algorithm is employed. For matrices, the FFT operation is applied to each column separately.
2.2.3.3. Fourier Transformation as Applied to Smooth Analytical Signals
The major feature of Fourier transformation is that it transforms analytical signals from the time or space domain into the frequency domain. So it is not strange that it can be applied to smooth noisy analytical signals. The reasoning behind is quite simple. In chemical study, noises are usually generated in instrumental measurement and are called white noises that obey the normal distribution of zero mean and equal variance. In general, noises are of high frequency while analytical signals are of low frequency in the time domain. Hence, after transforming the analytical signals into the frequency domain, if one discards the high-frequency part but keeps the low-frequency part, it is possible to eliminate the white noises present in the signals. An example is provided here to illustrate the treatment.
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