Forward Fourier transform

Equation A. 1 is called the forward Fourier transform and Equation A.2 is called the inverse Fourier transform. If v is defined as the oscillation frequency, the angular frequency is co = 2nv. Therefore, the forward Fourier transform can be used to express the function F(v) in the frequency domain by the integral of function /(f) in the time domain, whereas the inverse Fourier transform can be used to express the function /(f) in the time domain by the integral of function F(v) the frequency domain. 

As suggested by Patterson in 1934, the complex coefficients in the forward Fourier transformation (Eqs. 2.129 and 2.132) may be substituted by the squares of structure amplitudes, which are real, and therefore, no information about phase angles is required to calculate the distribution of the following density function in the unit cell ... 

Finally, the generated phase angles are used in a forward Fourier transformation combined with the normalized structure amplitudes... 

In the first three chapters of this book, we considered the fundamentals of crystallographic symmetry, the phenomenon of diffraction from a crystal lattice, and the basics of a powder diffraction experiment. Familiarity with these broad subjects is essential in understanding how waves are scattered by crystalline matter, how structural information is encoded into a three-dimensional distribution of discrete intensity maxima, and how it is convoluted with numerous instrumental and specimen-dependent functions when projected along one direction and measured as the scattered intensity V versus the Bragg angle 20. We already learned that this knowledge can be applied to the structural characterization of materials as it gives us the ability to decode a one-dimensional snapshot of a reciprocal lattice and therefore, to reconstruct a three-dimensional distribution of atoms in an infinite crystal lattice by means of a forward Fourier transformation. 

Figure 3.5. Radial structure functions (RSFs) produced by forward Fourier transforms of Ni sorbed on pyrophyllite, kaolinite, gibbsite, and montmorillonite, compared to the spectrum of crystalline Ni(OH)2(s) and takovite. The spectra are uncorrected for phase shift. (From Scheidegger et al., 1997.)...

Call the forward Fourier transform, then the inverse transform. 

You are now ready to go. Highlight A13 C524 and call the forward Fourier transform. Then highlight G13 I524 and use the inverse Fourier transform instead. Voila. 

Below we show a macro for the forward Fourier transform, and then describe how it can be modified to perform an inverse Fourier transform. The macro is readily incorporated in the spreadsheet. It uses no input boxes, but instead requires that the input data are organized as an array of n data pairs, where n is an integer power of 2. The data should be entered in three... 

Forward Fourier transformation of an array of complex data,... 

The following is the forward Fourier transform routine FOUR1 from J. C. Sprott, "Numerical Recipes Routines and Examples in BASIC", Cambridge University Press, Copyright (C) 1991 by Numerical Recipes Software. Used by permission. Use of this routine other than as an integral part of the present book requires an additional license from Numerical Recipes Software. 

The macro performs a forward Fourier transformation on the highlighted data, which must be organized in a rectangular array of 2N rows by 3 columns, and are then read by the statement dataArray Selection. Value. At the end of the macro, the complementary instruction Selection. Value = dataArray writes the results of the calculation in the adjacent three columns. 

Figure 9 illustrates radial structure fimetions (RSFs) produced by forward Fourier transforms of the XAFS spectra represented in Figure 8 (76). The spectra were uncorrected for phase shift. All spectra showed a peak of R 1.8A, which represents the first coordination shell of Ni. A second peak representing the second Ni shell can be observed at R 2.8A in the spectra of the Ni sorption samples and takovite (Figure 9). These spectra also showed peaks beyond the second shell at R == 5-6A (Figure 9) these peaks resulted from multiple scattering among Ni atoms (72).

A major property from Equations 17-23 is that if a forward Fourier transform [e.g., conversion of f(t) to A(o)) and D( >)] is followed by an inverse transform, the successive integrals must be multiplied by a net factor of (l/2ir) in order to give back the original f(t). We have chosen to introduce the factor of (1/2 1 in Equation 19 another convention is to use a factor of (1/27t)1/2 for each of the forward and inverse transforms. Both conventions (and others) are in common use, as discussed in detail in Reference 3. 

Notation N, coordination number for absorber-backscatterer pair R, distance Ak, limits used for forward Fourier transformation (/ is the wave vector) Ar, limits used for shell isolation (r is distance) n, power of k used for Fourier transformation. 

As the next step, one must Fourier transform the dynamic spectra measured in the time-domain into the frequency domain (18). The forward Fourier transform Ti(spectral intensity fluctuations y(vi, t) observed at vi is given by... 

A forward Fourier transform back to the frequency domain yields the spectra shown in Figure 6.9, a complex electric field ... 

Inverse Fourier transforming each interferogram into the time domain to select only the signal whieh arrives near the time difference of ts - t o, then forward Fourier transforming back to the frequency domain. 

A model-free approach to analyze the complete SAXS of a dilute system of particles is Clatter s indirect transformation method. The isotropic form of the forward Fourier transform corresponding to the Fourier back transform [5] can be written as... 

---