A Student's Guide to Fourier Transforms

Abstract

Fourier transform theory is of central importance in a vast range of applications in physical science, engineering and applied mathematics. Providing a concise introduction to the theory and practice of Fourier transforms, this book is invaluable to students of physics, electrical and electronic engineering, and computer science. After a brief description of the basic ideas and theorems, the power of the technique is illustrated through applications in optics, spectroscopy, electronics and telecommunications. The rarely discussed but important field of multi-dimensional Fourier theory is covered, including a description of Computer Axial Tomography (CAT scanning). The book concludes by discussing digital methods, with particular attention to the Fast Fourier Transform and its implementation. This new edition has been revised to include new and interesting material, such as convolution with a sinusoid, coherence, the Michelson stellar interferometer and the van Cittert–Zernike theorem, Babinet's principle and dipole arrays.

A Conspicuous Clay Ovoid in Nakhla: Evidence for Subsurface Hydrothermal Alteration on Mars with Implications for Astrobiology

“…Complementary selected area electron diffraction (SAED) imaging was performed both at Liverpool University with the JEOL 2100FS and at Manchester with the FEI Tecnai F30 and the use of selected area apertures with diameters of between 100 nm and 1 lm. To analyze the repeat distances present within our HRTEM data, the images were Fourier transformed in a standard manner ( James, 2011). Regular lattice spacings in the images then appear as discrete spots or rings in the Fourier transforms produced.…”

Low-Loss Complex Permittivity and Permeability Determination in Transmission/Reflection Measurements With Time-Domain Smoothing

“…The reflection and transmission parameters, S 11 and S 21 , as measured at reference planes 1 and 2 (or S 22 and S 12 at reference planes 2 and 1), are given by where Γ is the reflection coefficient of the interface, and P is the phase factor that represents the phase shift of the wave traveling through the material slab. Because in this paper we use an APC-7 coaxial line as a sample fixture that supports transverse electromagnetic (TEM) propagation waves, Γ and P are expressed as…”

“…First, we acquire the real and imaginary parts Z r and Z r of the time-domain wave impedance, by truncating the frequency range from f A to f B to include the λ/2 resonance (see Fig. 2) and by performing the inverse discrete Fourier transform (DFT) (for example, see [22]) for the frequency responses of the real and imaginary parts ς r and ς r , individually, such as…”

“…Note also that since oscillations in ς r and ς r are not necessarily the same, w p and w p do not always need to be the identical window functions, and the window widths p for each w p , w p = 1 may be optimally chosen. In the final stage, we do the forward DFT (see [22]) to get frequency responses of the smoothed ς r and ς r :…”

Processo para produção e caracterização de rochas sintéticas com porosidade controlada para aplicações em Petrofísica por RMN de alto e baixo campo.

“…Assim como podemos tomar a transformada de Fourier de um sinal f (t), que possua amplitude representada dentro de um intervalo de tempo qualquer, para obtermos o espectro de frequências deste mesmo sinal sob a forma de uma nova função g(ω) [21], podemos utilizar, de forma análoga, a transformada de Laplace inversa do sinal de M xy (t) para obter uma distribuição dos tempos de relaxação transversal que compõe aquele sinal [25,26]. A inversão dos dados de decaimento oferece uma nova abordagem matemática para o problema da relaxação em meios porosos, diferente do ajuste direto das curvas de decaimento (fitting).…”