Fourier Transform - Basic Properties

FT of a transient signal, i.e., a signal vanishing to zero amplitude at the end of the observation time span, delivers a Lorentzian curve in the frequency domain (Fig. 4.55c,d). In accordance with the energy-time uncertainty principle, the frequency domain output becomes increasingly sharper the longer the observation time span, because the frequency is more accurately determined the more cycles are recorded in the transient. 

Note Jean-Baptiste Joseph Fourier, French mathematician (1768-1830). Besides Fourier transformation, his influential work concerns the mathematical description of the conduction of heat in solids and the development of infinite series (Fourier series). He witnessed the French revolution and accompanied Napoleon on his expedition to Egypt. 

To reduce the side band-creating effect of truncation, the transient is normally subjected to apodization prior to FT. Apodization means the multiplication of the signal by a mathematical function that causes the values to smooth out to zero. Some sophisticated apodization methods are in use to deliver the ultimate resolution, nonetheless, single sin (x) or sin (x) functions work well. 

Another feature of FT is to yield better resolved frequency spectra from larger data sets, which not necessarily need to contain real data. Simply adding a row of zeros to the end of the experimental transient is beneficial in that it smoothes the peak shape by increasing the number of data points per m/z interval. This trick is known as zero-filling. The number of attached zeros normally equals the number of data points, sometimes even twice as many are filled in (double zero-filling). 

---

Before we can confront the problem of undoing the damage inflicted by spreading phenomena, we need to develop background material on the mathematics of convolution (the function of this chapter) and on the nature of spreading in a typical instrument, the optical spectrometer (see Chapter 2). In this chapter we introduce the fundamental concepts of convolution and review the properties of Fourier transforms, with emphasis on elements that should help the reader to develop an understanding of deconvolution basics. We go on to state the problem of deconvolution and its difficulties. 

Some very important surface properties of solids can be properly characterized only by certain wet chemical techniques, some of which are currently under rapid improvement. Studies of adsorption from solution allow determination of the surface density of adsorbing sites, and the characterization of the surface forces involved (the energy of dispersion forces, the strength of acidic or basic sites and the surface density of coul-ombic charge). Adsorption studies can now be extended with some newer spectroscopic tools (Fourier-transform infra-red spectroscopy, laser Raman spectroscopy, and solid NMR spectroscopy), as well as convenient modern versions of older techniques (Doppler electrophoresis, flow microcalorimetry, and automated ellipsometry). 

Metal oxides have surface sites which are acidic, basic, or both and these characteristics control important properties such as lubrication, adhesion, and corrosion. Some of the newer infrared techniques such as lazer-Raman and Fourier transform infrared reflection spectroscopy are important tools for assessing just how organic acids and bases interact with the oxide films on metal surfaces. Illustrations are given for the adsorption of acidic organic species onto aluminum or iron surfaces, using Fourier transform infrared reflection spectroscopy. 

Many heterocycles, for instance NH-azoles, have acid-base properties of either or both proton gain (basicity) and proton loss (acidity) [85], The possibility of studying such processes in the gas-phase (mass spectrometry and Fourier transform ion cyclotron resonance - FTICR - spectroscopy) provides theoreticians with values... 

The basic effects responsible for the properties of electrolyte solutions are ion solvation, ion 2issociation to ion pairs and higher ion aggregates with and without inclusion of solvent molecules. FTIR (Fourier transform infrared) and MW (microwave) spectra are a valuable source of information on ion-solvent and ion-ion interactions and yield factual knowledge on the structure and dynamics of electrolyte solutions. The efficiency of these methods is exemplified for solvation in aptotic and protic solvents, hydrophobic solvation, association to charged and neutral ion aggregates, and stability of ion pairs. 

FIGURE 1.9 The basic X-ray diffraction experiment is shown here schematically. X rays, produced by the impact of high-velocity electrons on a target of some pure metal, such as copper, are collimated so that a parallel beam is directed on a crystal. The electrons surrounding the nuclei of the atoms in the crystal scatter the X rays, which subsequently combine (interfere) with one another to produce the diffraction pattern on the film, or electronic detector face. Each atom in the crystal serves as a center for scattering of the waves, which then form the diffraction pattern. The magnitudes and phases of the waves contributed by each atom to the interference pattern (the diffraction pattern) is strictly a function of each atom s atomic number and its position x, y, z relative to all other atoms. Because atomic positions x, y, z determine the properties of the diffraction pattern, or Fourier transform, the diffraction pattern, conversely, must contain information specific to the relative atomic positions. The objective of an X-ray diffraction analysis is to extract that information and determine the relative atomic positions. 

Fourier transform of the data from the basic MQMAS scheme has the following properties (i) When the antiecho contribution is long lived, a linear combination of the echo and antiecho signals may be accomplished leading to a time domain data set that is amplitude modulated (AM). This leads to a pure phase absorptive spectrum with ambiguity in the sign of the frequencies. It was shown that if the + 3 — 1 transfer... 

In the last three decades, nuclear magnetic resonance has become a powerful tool for investigating the structural and physical properties of matter. Today, nuclear magnetic resonance is the physical method most widely used in analytical chemistry. For special applications, e.g. relaxation time measurements, there is available a variety of modifications of the basic nuclear magnetic resonance experiments such as pulse and spin-echo methods. In the course of this development and when electronic computers were provided at a reasonable price, Fourier transform spectroscopy was applied to nuclear magnetic resonance in the middle of the sixties. At that time, Fourier methods were already used to a large extent in far infrared spectroscopy (see Refs. and references cited therein). 

The best way to prove the uniqueness of the Fourier transform is to exploit the orthogonality property of sinusoids, combined with some basic linear algebra. We can express the DFT as a matrix multiplication ... 

Thirdly, in order to improve the dispersion of platinum catalysts deposited on carbon materials, the effects of surface plasma treatment of carbon blacks (CBs) were investigated. The surface characteristics of the CBs were determined by fourier transformed-infrared (FT-IR), X-ray photoelectron spectroscopy (XPS), and Boehm s titration method. The electrochemical properties of the plasma-treated CBs-supported Pt (Pt/CBs) catalysts were analyzed by linear sweep voltammetry (LSV) experiments. From the results of FT-IR and acid-base values, N2-plasma treatment of the CBs at 300 W intensity led to a formation of a free radical on the CBs. The peak intensity increased with increase of the treatment time, due to the formation of new basic functional groups (such as C-N, C=N, -NHs, -NH, and =NH) by the free radical on the CBs. Accordingly, the basic values were enhanced by the basic functional groups. However, after a specific reaction time, Nz-plasma treatment could hardly influence on change of the surface functional groups of CBs, due to the disappearance of free radical. Consequently, it was found that optimal treatment time was 30 second for the best electro activity of Pt/CBs catalysts and the N2-plasma treated Pt/CBs possessed the better electrochemical properties than the pristine Pt/CBs. 

Because linear algebra and vector calculus are so basic Secs. 23.6.8 and 23.6.9 are devoted to their summaries. The last section presents the various Fourier transforms and their properties in table form. Throughout the chapter various references are provided. 

Conversely, the problem of the infinite summation can be overcome by rewriting Eq. 10.44 in a time-dependent formulation. The basic idea is to switch from the frequency domain to the time domain by exploiting the properties of the Fourier transform of the delta function. After some mathematical manipulation, the general experimental observable can be rewritten as the Fourier transform of a time-dependent function, the transition dipole moment autocorrelation function. 

These algorithm, that is, DFR or direct Fourier imaging, are based on the basic properties of Fourier transform. Direct Fourier imaging is therefore vaUd only for NMR imaging, because the intrinsic nature of NMR imaging lies in the Fourier transform. 

Spectral studies such as Fourier transform infrared/attenuated total reflectance (FTIR/ATR) analysis, solid-state or pulse field gradient (PFG) nuclear magnetic resonance (NMR), x-ray diffraction (XRD) or small angle x-ray scattering (SAXS) analysis, and Raman study are basically the characterization methods for hybrid/ composite membranes. Sometimes, the multiple spectral analyses are necessary to better understand the structure-property relationship for a kind of composite polyelectrolytes. 

Basic elastic and geometric stiffness properties of the individual supporting columns are synthesized into a stiffness matrix compatible with an axisymmetrical shell element by a series of transformations. These are to be used in conjunction with a finite element representation of the cooling tower, where the displacements are decomposed into Fourier... 

---