Fourier transform theorems

We now go into pulse NMR in more detail after learning a few Fourier transform theorems. The second section considers the behavior of the magnetization in the rotating frame under pulsed irradiation and how the spectrum is affected, a very important topic. The third section is a discussion of quadrature detection with an educational as well as a practical aim. 

Moving averages as described above are special cases of convolution as defined in the section (II.A.l.) on Fourier transform theorems. The digital equivalent of the definition stated therein for the convolution of f(t) with g(t) is... 

The proof of the low-pass uniform sampling theorem (hereafter referred to simply as the sampHng theorem) is based on the following Fourier transform theorems (Ziemer and Tranter, 2002) ... 

Typical tomographic 2D-reconstruction, like the filtered backprojection teelinique in Fan-Beam geometry, are based on the Radon transform and the Fourier slice theorem [6]. 

Showing that T(p) is the proper fourier transform of T(x) suggests that the fourier integral theorem should hold for the two wavefunetions T(x) and T(p) we have obtained, e.g. 

It is a well known fact, called the Wiener-Khintchine Theorem [gardi85], that the correlation function and power spectrum are Fourier Transforms of one another ... 

This is the autocorrelation and by the Wiener-Khintchine theorem the power spectrum of the disturbance is given by its Fourier transform,... 

Thus the mutual intensity at the observer is the Fourier transform of the source. This is a special case of the van Cittert-Zernike theorem. The mutual intensity is translation invariant or homogeneous, i.e., it depends only on the separation of Pi and P2. The intensity at the observer is simply / = J. Measuring the mutual intensity will give Fourier components of the object. 

We are now ready to derive an expression for the intensity pattern observed with the Young s interferometer. The correlation term is replaced by the complex coherence factor transported to the interferometer from the source, and which contains the baseline B = xi — X2. Exactly this term quantifies the contrast of the interference fringes. Upon closer inspection it becomes apparent that the complex coherence factor contains the two-dimensional Fourier transform of the apparent source distribution I(1 ) taken at a spatial frequency s = B/A (with units line pairs per radian ). The notion that the fringe contrast in an interferometer is determined by the Fourier transform of the source intensity distribution is the essence of the theorem of van Cittert - Zemike. 

The fundamental quantity for interferometry is the source s visibility function. The spatial coherence properties of the source is connected with the two-dimensional Fourier transform of the spatial intensity distribution on the ce-setial sphere by virtue of the van Cittert - Zemike theorem. The measured fringe contrast is given by the source s visibility at a spatial frequency B/X, measured in units line pairs per radian. The temporal coherence properties is determined by the spectral distribution of the detected radiation. The measured fringe contrast therefore also depends on the spectral properties of the source and the instrument. 

Since the function A k) is the Fourier transform of (x, t), the two functions obey Parseval s theorem as given by equation (B.28) in Appendix B... 

Favorable properties of the Fourier transform itself provide general means either to split the general problem of data analysis into sub-problems or even to obtain structure parameters without much modeling work. In this respect the Fourier slice theorem must be pointed out because of its superior impact on scattering (Bonart [16] ... 

In combination with other theorems of Fourier transformation theory many of the fundamental structural parameters in the field of scattering are readily established. Because the corresponding relations are not easily accessible in textbooks, a synopsis of the most important tools is presented in the sequel. 

From the definition of Fourier transform the derivative theorem... 

Then it follows from the slice theorem Eq. (2.38) for the integral breadth of the Fourier transformed function H (5)... 

In order to deduce Scherrer s equation first an infinite crystal is considered that is, second, restricted (i.e multiplied) by a shape function (cf. p. 17). Thus from the Fourier convolution theorem (Sect. 2.7.8) it follows that in reciprocal space each reflection is convolved by the Fourier transform of the square of the shape function - and Scherrer s equation is readily established. 

This property is readily established from the definition of Fourier transform and convolution. In scattering theory this theorem is the basis of methods for the separation of (particle) size from distortions (Stokes [27], Warren-Averbach [28,29] lattice distortion, Ruland [30-34] misorientation of anisotropic structural entities) of the scattering pattern. 

Desmearing. In practice, there are two pathways to desmear the measured image. The first is a simple result of the convolution theorem (cf. Sect. 2.7.8) which permits to carry out desmearing by means of Fourier transform, division and back-transformation (Stokes [27])... 

The usefulness of the Fourier transforms lies in the fact that the following convolution theorem can be established.4 The sum over all configurations of n defects in a chain ... 

The 3D reconstruction of an object is performed more conveniently in reciprocal (Fourier) space. The 2D Fourier transform of a projection of an object is identical to a plane of 3D Fourier transform of the original object normal to the projection direction (electron beam). The origin of each 2D Fourier transform of a projection is identical to the origin of the 3D Fourier transform of an object, provided that the projections are aligned so that they have the same (common) phase origin. This is known as the Fourier slice theorem or the central projection theorem. 

By applying a variant of the extremely powerful convolution theorem stated above, computing the overlap integral of one scalar field (e.g., an electron density), translated by t relative to another scalar field for all possible translations t, simplifies to computing the product of the two Fourier-transformed scalar fields. Furthermore, if periodic boundary conditions can be imposed (artificially), the computation simplifies further to the evaluation of these products at only a discrete set of integral points (Laue vectors) in Fourier space. 

According to the Fourier convolution theorem, further discussed in section 5.1.3, the Fourier transform of the convolution in expression (2.14) is the product of the Fourier transforms of the individual functions, or... 

The topological analysis of the total density, developed by Bader and coworkers, leads to a scheme of natural partitioning into atomic basins which each obey the virial theorem. The sum of the energies of the individual atoms defined in this way equals the total energy of the system. While the Bader partitioning was initially developed for the analysis of theoretical densities, it is equally applicable to model densities based on the experimental data. The density obtained from the Fourier transform of the structure factors is generally not suitable for this purpose, because of experimental noise, truncation effects, and thermal smearing. 

In Eq. (8.18), we wrote the potential as a convolution of the total density and the operator 1 /r. Similarly, the integrals encountered in the evaluation of the peripheral electronic contributions to Eqs. (8.35) (8.37) are convolutions of the electron density p(r) and the pertinent operator. They can be evaluated with the Fourier convolution theorem (Prosser and Blanchard 1962), which implies that the convolution of /(r) and p(r) is the inverse transform of the product of their... 

Easily proved from the definition of the Fourier transform, this theorem states that convolving two functions is equivalent to finding the product of their Fourier transforms. Specifically, if a(x), h(x), and g(x) have transforms A(oo), B(co), and G(co), then... 

The entire analysis of synchronous detection, or lock-in amplification as it is sometimes called, can be conveniently analyzed by straightforward application of the Fourier transform techniques, transform directory, and convolution theorem developed in Section IV of Chapter 1. 

Brief reflection on the sampling theorem (Chapter 1, Section IV.C) with the aid of the Fourier transform directory (Chapter 1, Fig. 2) leads to the conclusion that the Rayleigh distance is precisely two times the Nyquist interval. We may therefore easily specify the sample density required to recover all the information in a spectrum obtained from a band-limiting instrument with a sine-squared spread function evenly spaced samples must be selected so that four data points would cover the interval between the first zeros on either side of the spread function s central maximum. In practice, it is often advantageous to place samples somewhat closer together. 

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