Symmetric Fourier Transformation

The symmetric Fourier transformation is used if a phase correction is not necessary, because the data contains one half of a symmetric or antisymmetric interferogram. This may occur for example, if the interferogram is generated by an inverse FT. 

it is advised that, if the interferogram is perfectly symmetric or perfectly antisymmetric, you should select the symmetric transformation. Define the data range used for the transformation on the Frequency Range page and start the transformation. The result is a single channel spectrum. In the case of an antisymmetric transformation only the real part of the single channel spectrum will be saved. 

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The Fourier transform of a convolution integral can be found by substituting the symmetric Fourier transforms Fand G and using the deltafunction formula (7.106). The result is the convolution theorem, which can be expressed very compactly as... 

As mentioned, we also carried out IR studies (a fast vibrational spectroscopy) early in our work on carbocations. In our studies of the norbornyl cation we obtained Raman spectra as well, although at the time it was not possible to theoretically calculate the spectra. Comparison with model compounds (the 2-norbornyl system and nortri-cyclane, respectively) indicated the symmetrical, bridged nature of the ion. In recent years, Sunko and Schleyer were able, using the since-developed Fourier transform-infrared (FT-IR) method, to obtain the spectrum of the norbornyl cation and to compare it with the theoretically calculated one. Again, it was rewarding that their data were in excellent accord with our earlier work. 

Fig. la-c. Theoretical 2H NMR line shapes for axially symmetric FGT (r = 0) in rigid solids, cf. Equ. (1). a Line shapes for the two NMR transitions b 2H spectrum (Pake diagram) in absorption mode as obtained by Fourier transform methods c 2H spectrum in derivative mode as obtained by wide line methods... 

At the end of the 2D experiment, we will have acquired a set of N FIDs composed of quadrature data points, with N /2 points from channel A and points from channel B, acquired with sequential (alternate) sampling. How the data are processed is critical for a successful outcome. The data processing involves (a) dc (direct current) correction (performed automatically by the instrument software), (b) apodization (window multiplication) of the <2 time-domain data, (c) Fourier transformation and phase correction, (d) window multiplication of the t domain data and phase correction (unless it is a magnitude or a power-mode spectrum, in which case phase correction is not required), (e) complex Fourier transformation in Fu (f) coaddition of real and imaginary data (if phase-sensitive representation is required) to give a magnitude (M) or a power-mode (P) spectrum. Additional steps may be tilting, symmetrization, and calculation of projections. A schematic representation of the steps involved is presented in Fig. 3.5. 

Another resolution-enhancement procedure used is convolution difference (Campbell et ai, 1973). This suppresses the ridges from the cross-peaks and weakens the peaks on the diagonal. Alternatively, we can use a shaping function that involves production of pseudoechoes. This makes the envelope of the time-domain signal symmetrical about its midpoint, so the dispersionmode contributions in both halves are equal and opposite in sign (Bax et ai, 1979,1981). Fourier transformation of the pseudoecho produces signals... 

Ifourth(fd, 2 Q) was multiplied with a window function and then converted to a frequency-domain spectrum via Fourier transformation. The window function determined the wavenumber resolution of the transformed spectrum. Figure 6.3c presents the spectrum transformed with a resolution of 6cm as the fwhm. Negative, symmetrically shaped bands are present at 534, 558, 594, 620, and 683 cm in the real part, together with dispersive shaped bands in the imaginary part at the corresponding wavenumbers. The band shapes indicate the phase of the fourth-order field c() to be n. Cosine-like coherence was generated in the five vibrational modes by an impulsive stimulated Raman transition resonant to an electronic excitation. 

The Fourier transform of the spherical atomic density is particularly simple. One can select S to lie along the z axis of the spherical polar coordinate system (Fig. 1.4), in which case S-r = Sr cos. If pj(r) is the radial density function of the spherically symmetric atom,... 

The way to ensure a clean extraction of an experimental reference signal is thus to zero-fill the experimental free induction decay s it) once before Fourier transformation, zero completely the imaginary part of the resultant spectmm, and zero all but the reference region to wr of the real part [7], Inverse Fourier transformation then gives a symmetric time-domain signal, the first half of which is the required experimental reference signal Sr t) ... 

Note that (2.17) and (2.18) are not unique all complex functions could be replaced by their complex conjugates, and the factor 1 /2it could appear either in (2.17) or (2.18). If we want our expressions to appear more symmetrical, both integrals can have the common multiplicative factor 1 / flfr. There is no universally accepted convention for Fourier transforms. However, once the form of the Fourier transform has been specified, the corresponding expression for the inverse Fourier transform is uniquely determined. 

Scale factors can be used in various ways to define Fourier transform pairs. We adopt the symmetrical convention... 

An alternative symmetrical convention that has gained popularity specifies the Fourier transform pair g(x) and G( ) to be related by... 

In the operating mode customarily used, which is to determine the existence, location, and intensity of the spectral lines, the interferometer produces an interferogram that is symmetric about the zero displacement position. If the zero displacement position (the maximum point on the central fringe ) is taken as the origin of the interferogram function, the Fourier transform of this will produce an infrared spectrum that is real and symmetric about... 

Fourier transform infrared spectroscopy (FTIR) was also used to study the anisotropic structure of polyimide films. This work was based on the fact that there are characteristic absorptions associated with in-plane and out-of-plane vibrations of some functional groups, such as the carbonyl doublet absorption bands at 1700-1800 cm . The origin of this doublet has been attributed to the in-phase (symmetrical stretching) and out-of-phase (asymmetrical stretching) coupled... 

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