Fourier transform Numerical examples

To get a feeling of Fourier transformation, in this section we will go through a small numerical example. 

Suppose we want to transform signals that are 9 points long. We need to set up a 9-by-9 transformation matrix. The matrix is constructed following Eq. (15)  

Note The values have been rounded to two decimals merely to keep the above table small. For future calculations, more decimals may be needed. 

The columns of this matrix, i.e. the Fourier basis functions for signals that are 9 points long, are plotted in Fig. 13. The functions are more ragged than those of Fig. 9. 

Now let us take a very simple signal all zeros except for a one in the middle, i.e. on position 5. This may seem a bit artificial, but it is a very important signal in system analysis. It is the impulse we can use to perturb a system in order to obtain its impulse response. The calculation of the Fourier coefficients following Eq. (11) is given below  

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First, the application should involve symbolic reasoning. There is no point in trying to develop an expert system to perform numerical calculations, for example, Fourier transforms. 

Numerous analyses in the quality control of most kinds of samples occurring in the flavour industry are done by different chromatographic procedures, for example gas chromatography (GC), high-pressure liquid chromatography (fiPLC) and capillary electrophoresis (CE). Besides the different IR methods mentioned already, further spectroscopic techniques are used, for example nuclear magnetic resonance, ultraviolet spectroscopy, mass spectroscopy (MS) and atomic absorption spectroscopy. In addition, also in quality control modern coupled techniques like GC-MS, GC-Fourier transform IR spectroscopy, HPLC-MS and CE-MS are gaining more and more importance. 

Selected topics in Fourier-Transform Ion Cyclotron Resonance Mass Spectrometry instrumentation are discussed in depth, and numerous analytical application examples are given. In particular, optimization ofthe single-cell FTMS design and some of its analytical applications, like pulsed-valve Cl and CID, static SIMS, and ion clustering reactions are described. Magnet requirements and the software used in advanced FTICR mass spectrometers are considered. Implementation and advantages of an external differentially-pumped ion source for LD, GC/MS, liquid SIMS, FAB and LC/MS are discussed in detail, and an attempt is made to anticipate future developments in FTMS instrumentation. 

FTs are best understood by a simple numerical example. For simplicity we will give an example where there is a purely real spectrum and bodi real and imaginary time series - die opposite to normal but perfectly reasonable in the case of Fourier selfconvolution (Section 3.5.2.3) diis indeed is die procedure. We will show only the real half of the transformed time series. Consider a spike as pictured in Figure 3.19. The spectrum is of zero intensity except at one point, m = 2. We assume there are M(=20) points numbered from 0 to 19 in the spectrum. 

The beauty of the linear viscoelastic analysis lies in the fact that once a viscoelastic function is known, the rest of the functions can be determined. For example, if one measures the comphance function J t), the values of the components of the complex compliance function can in principle be determined from J(t) by using Fourier transforms [Eqs. (6.30)]. On the other hand, the components of the complex relaxation moduh can be obtained from those of / (co) by using Eq. (6.50). Even more, the real components of both the complex relaxation modulus and the complex compliance function can be determined from the respective imaginary components, and vice versa, by using the Kronig-Kramers relations. Moreover, the inverse of the Fourier transform of G (m) and/or G"(co) [/ (co) and/or /"(co)] allows the determination of the shear relaxation modulus (shear creep compliance). Finally, the convolution integrals of Eq. (5.57) allow the determination of J t) and G t) by an efficient method of numerical calculation outlined by Hopkins and Hamming (13). 

Initially packed columns were used, but capillary columns are most frequently used today. A variety of polar and non-polar column phases have been used to determine VOCs depending on the particular compound(s) analyzed. Numerous examples were summarized by Wille and Lambert.5 Similarly, a range of detection devices have been utilized. Flame ionization detection (FID) and mass spectrometry (MS) in the selected-ion monitoring mode (SIM) are most frequently used, but other techniques include electron-capture detection (ECD) and Fourier transform infrared (FTIR) detection. 

In using Fourier transformation one should be aware that there are several conflicting conventions. For example, many engineering texts (as well as the Numerical Recipes) use conventions for forward and inverse transforms that are the opposite from those used in mathematics and physics the latter convention is used here. Likewise, there are different conventions for how to distribute the scale factors between the forward and inverse transform, and for whether the time and frequency axes should be centered around zero (as used here) or start at zero. (The Fourier transform macro will accept either input format). 

The following is the forward Fourier transform routine FOUR1 from J. C. Sprott, "Numerical Recipes Routines and Examples in BASIC", Cambridge University Press, Copyright (C) 1991 by Numerical Recipes Software. Used by permission. Use of this routine other than as an integral part of the present book requires an additional license from Numerical Recipes Software. 

Now, let us suppose that the system is in a black-box, and that all we can know of it is the observable Q, sampled up to a finite time interval. This is the typical situation occurring in Physics, where one obtains information on some system on the basis of the output of an experiment. The amplitudes and frequencies (6) can be numerically computed from g(t), for example, by means of the frequency analysis method (Laskar 1990, Laskar et al. 1992). However, if we are interested mainly in recognizing the quasi-periodic nature of the solution, it is not necessary to use a refined frequency analysis, but it is sufficient to compute the Fast Fourier transform of time interval [—T,T], where >( ) is a suitable analytic window on [—T, T] (see Section 4 for details). Figure 1 shows an example of such an analysis. Within the precision of our computation (a line is identified with an error of about 10-5 in frequency) we can easily recognize that the spectrum of g(t) is a line spectrum. Now, we consider the more interesting quasi-integrable Hamiltonian ... 

Whereas in sequential spectroscopy one wavelength after the other is selected by moving the spectrum across the exit slit of the monochromator, multiplex spectroscopy measures the whole spectrum at a time. In these instruments the white light of the measurement source is resolved with respect to wavelength either by a numerical approach as for example Fourier transform spectroscopy (preferable in the IR) or by imaging the dispersed spec-... 

When the spectral content of a SR signal is unknown, or when a signal of a certain type is expected but its frequency is not known, one has no choice but to begin the analysis with a Fourier transform. Even after the initial identification of signals, Fourier transform /ASR spectroscopy is the most economical and convenient way to characterize data with numerous frequencies, examples of which will be mentioned in the following section. However, it is usually a mistake to try to extract final results from frequency spetra. This is a somewhat contentious statement, especially in view of the fact... 

Modification is performed by separating the harmonics from the spectral envelope, but this is achieved in a way that doesn t perform explicit source/filter separation as with LP analysis. The spectral envelope can be found by a number of numerical techniques. For example, Kain [244] transforms the spectra into a power spectrum and then uses an inverse Fourier transform to find the time domain autocorrelation function. LP analysis is performed on this to give an allpole representation of the spectral envelope. This has a number of advantages over standard LP analysis in that the power spectrum can be weighted so as to emphasise the perceptually important parts of the spectrum. Other techniques use peak picking in the spectrum to determine the spectral envelope. Once the envelope has been found, the harmonics can be moved in the frequency domain and new amplitudes found from the envelope. From this, the standard synthesis algorithm can be used to generate waveforms. 

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