Properties of Fourier Transform

One of the most important properties of Fourier transforms and, consequently, of characteristic functions, is their invertibility. Given a characteristic function M, one can calculate the probability density function p by means of the inversion formula... 

The reader is assumed to be familiar with the elementary properties of Fourier transforms. 

It is a known property of Fourier transforms that given a convolution product in the reciprocal space, it becomes a simple product of the Fourier transforms of each term in the real space. Then, as the peak broadening is due to the convolution of size and strains (and instrumental) effects, the Fourier transform A 1) of the peak profile I s) is [36] ... 

We note that the wave packet (x, t) is the inverse Fourier transform of A k). The mathematical development and properties of Fourier transforms are presented in Appendix B. Equation (1.11) has the form of equation (B.19). According to equation (B.20), the Fourier transform A k) is related to (x, t) by... 

This uncertainty relation is also a property of Fourier transforms and is valid for all wave packets. 

Again, this relation arises from the representation of a particle by a wave packet and is a property of Fourier transforms. 

This reciprocal relationship is a general property of Fourier transforms. The -function of the previous paragraph and its transform demonstrate the same reciprocity. To characterise this property more precisely a bracket function, called a scalar or inner (dot) product, is used to define an overlap integral... 

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. 

The interpretation of the Patterson function is based on a specific property of Fourier transformation (denoted as 3[...]) when it is applied to convolutions (<8>) of functions ... 

An important property of Fourier transforms that we did not emphasize in the previous chapter is that spatial relationships in one space are maintained in the corresponding transform space. That is, specific relationships between the orientations in real space of the members of a set of objects are carried across into reciprocal space. This is particularly important in terms of crystallographic symmetry, and we will encounter it again when we consider the process known as molecular replacement (see Chapter 8). 

The problem is therefore to determine the nature and the density of the stractural defects from the measurement of the experimental profile h(x) which contains the contribution from the instrament. There are two ways to go about solving this problem. The first method consists of deconvoluting this equation by using, in particular, the properties of Fourier transforms and extracting the pure profile which induced only by the defects. The second approach is described as convolutive . This time, the stractural defects are described without extracting the pure profile, but instead by taking into account the instrument s contribution, which is assumed to be an analytical function, either known or directly calculated from the characteristics of the diffractometer. This instrumental function is then convoluted with the functions expressing the contributions from the various microstructural effects that are assumed to be present. 

In order to imderstand this important property of Fourier transform spectroscopy, let us consider a broad spectrum over a wide wave number range with one narrow absorption line in it (cf. Fig. 12). Then, according to the rules of Fourier transformation, the broad spectrum produces an interferogram with highly damped oscillation. The maximum amplitude of the oscillation and also the mean value of the interferogram are equal to the total intensity or to the area under the spectral distribution [see Appdx 1 and Eqs. (A 1.1) and (A 1.2)] ... 

Fourier transformation has found important applications in many branches of science here we mention especially its use in various analytical instruments (such as nuclear magnetic resonance, infrared, and mass spectrometry), and in signal processing. Below we will illustrate some properties of Fourier transformation in the latter context. 

It is one of the main properties of Fourier transforms that they allow us to view the individual components of a complex mixture of sinusoidal signals. In Fig. 7.1-3 we see the cosine in the real part of the transform, at /= 0.25 Hz, and the sine in its imaginary component, with a frequency of 0.5 Hz and a three times larger amplitude. 

One of the most useful properties of Fourier transformation is that it converts a convolution into a multiplication. (Convolution is one of many correlations, i.e., mathematical operations between functions, that can be greatly simplified by Fourier transformation.) Since convolution is a rather involved mathematical operation, whereas multiplication is simple, convolutions are often performed with the help of Fourier transformation. We will explore this property in more detail in sections 7.5 and 7.6. 

In the sections above, various properties of Fourier transforms were discussed first with reference to the one-dimensional Fourier transform, which is simpler and easier to understand. With the three-dimensional transform, expressions analogous to those discussed above hold and can be obtained from the one-dimensional versions by exchanging x - r,s 5, and /v, as has been done in obtaining (B.45) from... 

Explicit examples of the use of models will be given in chapter 8. It has been pointed out that, even in the absence both of Fourier transformation and of model fitting, important information about the arrangement of the molecule at the interface can be obtained by this kind of approach, namely the centre-to-centre separation between the components (Simister et al. 1992). The ability to do this stems from the properties of Fourier transforms and some observations concerning the nature of the number distribution of each species. Thus components A and B are probably symmetrical about the centres of their distributions since when z is large the number densities are zero, i.e. they are even functions ... 

Additional properties of Fourier transforms can be fotmd in most advanced engineering texts [2,7,8]. 

In photoacoustic spectroscopy, signal amplitude increases only when absorption occurs. Thus, photoacoustic spectra will be absorption spectra as opposed to FTIR spectra which are transmission spectra. FTPAS interferograms should then show modulation at high values of retardation as well as near zero retardation. This comes about as a consequence of the properties of Fourier transforms which cause very broad general feature information to be collected at low retardation, with narrow and sharp features ("high frequency") to be collected at higher retardation. 

Because of the properties of Fourier transforms, the labels real and reciprocal are interchangeable, that is a one-dimensional row of points of separation a will give rise to periodic sheets of scattering extending normal to the row, of separation 2 nia. 

It is a property of Fourier transform mathematics that multiplication in one domain is equivalent to convolution in the other. (Convolution has already been introduced with regard to apodization in Section 2.3.) If we sample an analog interferogram at constant intervals of retardation, we have in effect multiplied the interferogram by a repetitive impulse function. The repetitive impulse function is in actuality an infinite series of Dirac delta functions spaced at an interval 1 jx. That is,... 

Secondly, we consider the asymptotic behavior S q oo), looking first at the layer system. Due to the reciprocity property of Fourier transforms, E q oo) relates to the limiting behavior K z 0). Therefore, using... 

The techniques exploit properties of Fourier transforms of the integrand. 

In studies of isotropic and anisotropic liquids, key to relating the diffraction pattern to the scatterers in real space are two properties of Fourier transforms additivity and rotation [1]. Thus the transform of a sum is equal to the sum of the transforms also rotation in real space causes an equal rotation in reciprocal space. The reciprocal space, i.e. F(Q), will therefore accurately reflect the averaging over the irradiated volume of the specimen. For a spherically averaged rigid molecule the scatto-ed intensity is... 

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. 

These boundary gradients can be accounted for with only a slight modification to Eq. (6.17) [39]. Using the convolution property of Fourier Transforms, it can be shown that the scattering intensity (at high q) for an isotropic system can then be approximated as ... 

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