Fourier series transform

Figure 5.10 Radio frequency variation of My(t) transverse magnetisation observed, acquired and stored digitally with time is known as a Free Induction Decay (FID). Stored FID either singly or averaged, are processed by fourier series transformation (FT) from time domain signal information, SnoCti), into frequency domain (spectral) information, /nmr( i)- Only chemically equivalent nuclei without spin-spin coupling and with an equivalent Lamor frequency, V, are being observed here hence only a single signal will result of frequency Vi.

Fourier series transformation of time domain data, Snoffi), into frequency domain signal intensity data Inmr(Ei) then yields a characteristic classical ID NMR spectrum. 

Figure 5.11 Alternative diagrammatic representation of general FT ID NMR experiment. There is a single 90° pulse then signal observation and acquisition in time domain tj prior to fourier series transformation of time domain signal information SFiD(ti) into frequency domain (spectral intensity) information, 7nmr(Fi).

Figure 5.17 Cartoon diagram to represent general structure of 4D correlation experiments. This is the same as for 3D correlation experiments (Fig. 5.14) except that an extra resonant population of heteroatom nuclei are involved in generation of transverse magnetisation (in time ts) and magnetisation transfer (during M3). Final pulse sequence generates transverse magnetisation in the Destination Nuclei S that is observed, acquired and digitised in time t/,. Fourier series transformation is used to transform time domain signal information Sfid (ti, ta, ts, 4) into frequency domain (spectral intensity) information, /NMR(fi, F2,...

Using separation of variables and Fourier series transformation, the solution of the partial differential Eq. 2 is... 

The Fourier Series, Fourier Transform and Fast Fourier Transform... 

Hg. 1.14 The connection between the Fourier transform and the Fourier series can be established by gradually increasing the period of the function. When the period is infinite a continuous spectrum is obtained. (Figure adapted from Ramirez R W, 1985, The FFT Fundamentals and Concepts. Englewood Cliffs, NJ, Prenhce Hall.)... 

XI. Complex Numbers, Fourier Series, Fourier Transforms, Basis Sets... 

If f is a function of several spatial coordinates and/or time, one can Fourier transform (or express as Fourier series) simultaneously in as many variables as one wishes. You can even Fourier transform in some variables, expand in Fourier series in others, and not transform in another set of variables. It all depends on whether the functions are periodic or not, and whether you can solve the problem more easily after you have transformed it. 

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... 

The breakdown of a given signal into a sum of oscillatory functions is accomplished by application of Fourier series techniques or by Fourier transforms. For a periodic function F t) with a period t, a Fourier series may be expressed as... 

Comparison of the previous equations with Equations (16-6) and (16-7) reveal that the Fourier transform is really just a Fourier series constructed over a finite interval. 

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). 

For the special case l = 0 this expansion reduces to a Fourier series. The ground-state series is the Fourier transform of sin ka/ka, which is the box function... 

A similar approach, also based on the Kubo-Tomita theory (103), has been proposed in a series of papers by Sharp and co-workers (109-114), summarized nicely in a recent review (14). Briefly, Sharp also expressed the PRE in terms of a power density function (or spectral density) of the dipolar interaction taken at the nuclear Larmor frequency. The power density was related to the Fourier-Laplace transform of the time correlation functions (14) ... 

The one-dimensional diffusion equation in isotropic medium for a binary system with a constant diffusivity is the most treated diffusion equation. In infinite and semi-infinite media with simple initial and boundary conditions, the diffusion equation is solved using the Boltzmann transformation and the solution is often an error function, such as Equation 3-44. In infinite and semi-infinite media with complicated initial and boundary conditions, the solution may be obtained using the superposition principle by integration, such as Equation 3-48a and solutions in Appendix 3. In a finite medium, the solution is often obtained by the separation of variables using Fourier series. 

The Fourier series, which has a discrete spectrum but periodic spatial function, is actually a special case of the Fourier transform. (Note that an equally spaced discrete spectrum necessarily implies a periodic function having a finite period given by the wavelength of the lowest frequency.) See Bracewell (1978) to see how the explicit form of the Fourier series may be obtained from the Fourier transform. Taking discrete, equally spaced... 

The coefficients of the sines and cosines will be real for real data. Restoring a high-frequency band of c (unique complex) discrete spectral components to a low-frequency band of b (unique complex) spectral components will be the same (when transformed) as forming the discrete Fourier series from the high-frequency band and adding this function to the series formed from the low-frequency band. When applying the constraints in the spatial domain, the Fourier series representation will be used. 

We discovered in Chapter 9 that the spatial function as given by the discrete Fourier transform (DFT) is a discrete Fourier series. Letting u(k) denote the (known) series consisting of only low-frequency terms and v(k) the series consisting of only high-frequency terms, we want to determine the unknown coefficients in v(k) that best satisfy the constraints. Expressing deviations of the total function forbidden by the constraints as some function of u(k) + v(k), we shall try to determine the coefficients of v(k) that minimize these deviations. Sum-of-squares expressions for these measures of the error have been found to result in the most efficient computational schemes. 

Eq. (2.11) can be solved by developing it as a complex series of sines and cosines according to relation (2.9). This is a Fourier series [14]. Thus, an exponential in the time domain, F ((), and a Lorentzian in the frequency domain, f (op, are Fourier transforms of each other [15-17],... 

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