Fourier series and transformation

The original literature on Fourier series and transforms involved applications to continuous datasets. However, in chemical instrumentation, data are not sampled continuously but at regular intervals in time, so all data are digitised. The discrete Fourier transform (DFT) is used to process such data and will be described below. It is important to recognise that DFTs have specific properties that distinguish them from continuous FTs. 

Appendix B contains a brief introduction to Fourier series and transforms. Equation (23.3-14) is analogous to a linear combination of basis functions, but with an integration instead of a sum. The variable co is sometimes called the circular frequency of the... 

SeeL.Glasser,7. Chem.Educ., 64, Pi261 (1987) and7. Cfem. if c.,64,A306(1987)fa-aninlroduction to Fourier series and transforms. 

Besides the intrinsic usefulness of Fourier series and Fourier transforms for chemists (e.g., in FTIR spectroscopy), we have developed these ideas to illustrate a point that is important in quantum chemistry. Much of quantum chemistry is involved with basis sets and expansions. This has nothing in particular to do with quantum mechanics. Any time one is dealing with linear differential equations like those that govern light (e.g. spectroscopy) or matter (e.g. molecules), the solution can be written as linear combinations of complete sets of solutions. 

Time domains and frequeney domains are related through Fourier series and Fourier transforms. By Fourier analysis, a variable expressed as a funetion of time may be deeomposed into a series of oseillatory funetions (eaeh with a eharaeteristie frequeney), whieh when superpositioned or summed at eaeh time, will equal the original expression of the variable. This... 

References Brown, J. W., and R. V. Churchill, Fourier Series and Boundary Value Problems, 6th ed., McGraw-Hill, New York (2000) Churchill, R. V, Operational Mathematics, 3d ed., McGraw-Hill, New York (1972) Davies, B., Integral Transforms and Their Applications, 3d ed., Springer (2002) Duffy, D. G., Transform Methods for Solving Partial Differential Equations, Chapman Hall/CRC, New York (2004) Varma, A., and M. Morbidelli, Mathematical Methods in Chemical Engineering, Oxford, New York (1997). 

First I will discuss Fourier series and the Fourier transform in general terms. I will emphasize the form of these equations and the information they contain, in the hope of helping you to interpret the equations — that is, to translate the equations into words and visual images. Then I will present the specific types of Fourier series that represent structure factors and electron density and show how the Fourier transform inter con verts them. 

As stated previously, with most applications in analytical chemistry and chemometrics, the data we wish to transform are not continuous and infinite in size but discrete and finite. We cannot simply discretise the continuous wavelet transform equations to provide us with the lattice decomposition and reconstruction equations. Furthermore it is not possible to define a MRA for discrete data. One approach taken is similar to that of the continuous Fourier transform and its associated discrete Fourier series and discrete Fourier transform. That is, we can define a discrete wavelet series by using the fact that discrete data can be viewed as a sequence of weights of a set of continuous scaling functions. This can then be extended to defining a discrete wavelet transform (over a finite interval) by equating it to one period of the data length and generating a discrete wavelet series by its infinite periodic extension. This can be conveniently done in a matrix framework. 

Wilson, R.G. Fourier Series and Optical Transform Techniques in Contemporary Optics An... 

The discrete-time waveform counterparts of the Fourier series and Fourier transform provide viable alternatives for estimating the frequency content of signals if discrete measurements of the desired waveforms are available. This avenue of analysis, which is popular in benchtop instrumentation, is considered in further detail in Sec. 20.6.7. 

Continuous-time waveform A waveform, herein represented by x(t), that takes on values over the continuum of time t, the assumed independent variable. The Fourier series and Fourier transform apply to continuous-time waveforms. Compare with discrete-time waveform. 

The two extensions of the Fourier series and Fourier transform are the Lorentz line shape and the autocorrelation function. 

Fourier s law is named for Jean Baptiste Joseph Fourier, 1768-1830, a famous French mathematician and physicist who also invented Fourier series and Fourier transforms. 

Inverse Mellin transform returns fit)f in the domain and looks as a Fourier series, and F iyk)Hb are the (complex) coefficients of the series. 

Feldman (2007) Discrete-time Fourier Series and Fourier Transforms , by J. Feldman. Accessed December 2014 at http //www.math.ubc.ca/ feldman/m267/dft.pdf... 

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