Fourier transform equations

Any continuous sequence of data h(t) in the time domain can also be described as a continuous sequence in the frequency domain, where the sequence is specified by giving its amplitude as a function of frequency, H(f). For a real sequence h(t) (the case for any physical process), H(f) is series of complex numbers. It is useful to regard h(t) and 11(f) as two representations of the same sequence, with h(t) representing the sequence in the time domain and H( f) representing the sequence in the frequency domain. These two representations are called transform pairs. The frequency and time domains are related through the Fourier transform equations... 

Analogous to the previous treatments /69/ the Fourier transformed equation of motion of the dilated electron propagator may now be written as... 

The diffraction pattern of the scattered x-rays, including intensities, can be calculated from the electron density distribution of the sample by means of Fourier transforms. Equation 1 shows the calculation of the structure factors F(hkl) for an orthorhombic unit cell of dimensions axbxc from the electron density p(x,y,z) and the exponential phase-factor term. The integers h, k, and / specify the order of the diffraction peaks and are related to the Miller indices that specify the reflecting planes within the crystal producing the diffraction spots (vide infra). An observed intensity I is calculated from I = F hkl)F hkl) = F hkl) where... 

The Fourier transform equations show that the electron density is the Fourier transform of the structure factor and the structure factor is the Fourier transform of the electron density. Examples are worked out in Figures 6.14 and 6.15. If the electron density can be expressed as the sum of cosine waves, then its Fourier transform corresponds to the sum of the Fourier transforms of the individual cosine waves (Figure 6.16). The inversion theorem states that the Fourier transform of the Fourier transform of an object is the original object, hence the opposite signs in Equations 6.12.1 and 6.12.2. This theorem provides the possibility of using a mathematical expression to go back and forth between reciprocal space (structure factors) and real space (electron density), so that the phrase and vice versa is applicable here. 

FIGURE 6.13. Graphical examples of the Fourier transforms of (a) a cosine and (b) a sine function. Note that the Fourier transform contains information on phase, but that this information is lost when intensities (which involve the square of the displacement) are measured. The designation real and imaginary derives from the presence of i = in the Fourier transform Equation 6.14.1. 

In deriving the structure of a molecule, or distribution of atoms using X-ray crystallography, we do not directly obtain the x, y, z coordinates of the atoms. That is, we don t solve some system of linear equations whose solution is the set of numerical values for jc, y, z. We employ a Fourier transform equation that incorporates the diffraction data, the structure factors, and yields the value of the electron density p(x, y, z) at any and all points x, y, z within the crystallographic unit cell. From the peaks and features of this continuous electron density distribution in the unit cell we then infer the locations of the atoms, and hence their coordinates. This will be described as it is done in practice, in Chapter 10. Following this, the coordinates are improved by applying refinement procedures, as also outlined in Chapter 10. Here, however, our objective is to understand this Fourier transform equation, namely the electron density equation. 

Finally, we see from the Fourier transform equations, for the structure factor Fhu and the electron density p x, y, z), that any change in real space (e.g., the repositioning of an atom) affects the amplitude and phase of every reflection in diffraction space. Conversely, any change in the intensities or phases in reciprocal space (e.g., the inclusion of new reflections) affects all of the atomic positions and properties in real space. There is no point-to-point correspondence between real and reciprocal space. With the Fourier transform and diffraction phenomena, it is One for all, and all for one (Dumas, The Three Musketeers, 1844). 

If a homodyne method is used, the measured autocorrelation function (g,x) can be interpreted by using the Siegert relation. Equation 5.454. The translational and rotational diffusion coefficients for several specific shapes of the particles are given in Table 5.9. The respective power spectrum functions can be calculated by using the Fourier transform. Equation 5.449b. 

We define a set of renormalized potentials through the matrix Fourier transform equation... 

FTIR takes a completely different approach. The spectral data are acquired as an Interferogram (Figure 1) which must be transformed Into a plot of Intensity versus wavenumber or wavelength through the application of Fourier transform equations. Thus, the computer Is an Integral part of the system without which little useful Information could be obtained. FTIR has the following advantages over computerized dispersive Infrared spectroscopy ... 

Fourier transforming Equation 16 gives a different expression for H(ju) ... 

For a continuous signal, we can move from the time domain to the frequency domain via the Fourier transform, but as we are now dealing with digital signals, this formula cannot be used directly. We will now derive an equivalent transform for digital signals. Starting with the previously defined Fourier transform (Equation 10.21)... 

Applying the Fourier transformation (Equation 84) and the convolution theorem, we obtain... 

Applying the Fourier transform (Equation 13), wo calculate the moment (Equation 24) from... 

Then, using Fourier transforms. Equation 6.134 becomes ... 

Derivative spectra in wavelength space may be produced by applying the inverse Fourier transform (Equation (6.9) to Equation (6.22) and Equation (6.23)). Note diat the position of first two terms in Equation (6.22) must be switched before doing the inverse transformation. 

Once Equations [8] and [9] have been Fourier transformed. Equations [6] and [7] can be used to isolate the VCD spectrum although both ratios now also include the exponential function of the lock-in time constant. However, this function vanishes when the calibration curve is divided into the ratio of the AC and DC intensities and does not enter the final VCD spectrum. 

From the properties of the Fourier transform, equation (8.8), we see that the shape of the line profile at a frequency separation A(o=a)Q-(o from line centre is determined by the wavetrain emitted during the time interval At, where At 1/Aco. If we imagine the atom to emit a wavetrain until perturbed by a strong collision, the time of interest. At,... 

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