Fourier ripples

Fourier ripples. Constantly spaced bumps in the frequency spectrum found on either side of a peak. In a 1 -D spectrum or a half-transform 2-D data matrix, these ripples are found when the apodized intensity has not faded to the level of the background noise by the time the digitization of the FID ceases. In a fully transformed 2-D spectrum, Fourier ripples parallel to the f, frequency axis are observed when an inappropriate t apodization function is used prior to conversion of the t, time domain to the f, frequency domain. 

As already pointed out by Jauch [30], the series appearing in the exponential factor that modulates m (x) in (6) has a finite number of terms, and can therefore give rise to series termination artefacts. In particular, although the exponentiation will ensure positivity of the resulting density, series termination ripples will be present in the reconstructed map whenever the spectrum of the modulation required by the observations extends significantly past the resolution of the series appearing in the exponential. This in turn will depend both on the true density whose Fourier coefficients are being fitted, and on the choice for the prior prejudice. 

Figure 1 shows the average strength of the Fourier coefficients of log( (x)/m(x)), with q(x) a multipolar synthetic density for L-alanine at 23 K, and two different prior-prejudice distributions mix). It is apparent that the exponential needed to modulate the uniform prior still has Fourier coefficients larger than 0.01 past the experimental resolution limit of 0.463 A. Any attempt at fitting the corresponding experimental structure factor amplitudes by modulation of the uniform prior-prejudice distribution will therefore create series termination ripples in the resulting MaxEnt distribution. 

Within the computational scheme described in the course of this work, the available information about the atomic substructure (core+valence) can be taken into account explicitly. In the simplest possible calculation, a fragment of atomic cores is used, and a MaxEnt distribution for valence electrons is computed by modulation of a uniform prior prejudice. As we have shown in the noise-free calculations on l-alanine described in Section 3.1.1, the method will yield a better representation of bonding and non-bonding valence charge concentration regions, but bias will still be present because of Fourier truncation ripples and aliasing errors ... 

Figure 4. Principle of Fourier synthesis in one dimension. In this simple example of a Fourier series with cosine waves we need to know the amplitude A and the index h for each wave. The index h gives the frequency, i.e. the number of full wave trains per unit cell along the a-axis. The left row of images shows how the intensity within the unit eell ehanges for each Fourier component. The last image at the bottom gives the result after superposition of the waves with index /z = 2 to 10 (areas with high potential are shown in black, brighter areas in the map indicate low potential). The corresponding intensity profiles along the a-axis for one unit cell are shown in the middle row. The ripples in the profile of the Fourier sum arise from the limited number of eomponents that have been used in the synthesis (termination errors). If the...

Due to termination of the series, however, p(r) is severely affected by ripples. In addition, especially in the case of non-centrosymmetric crystals, the phase of vector F(S) is not known with precision and this affects a correct reconstruction of the density. Therefore, Fourier summation cannot be used for precise and accurate mapping of electron density. On the other hand, a model is necessary to overcome these limitations and to produce a function that is sufficiently close to the real, quantum mechanical />(r) in all regions of the crystal. 

Since the Fourier transformation of equation (2.2) yields only a radial distribution function about the absorber, we note that information obtained from EXAFS is limited to an average, one-dimensional representation of structure. Furthermore, in order that the transform be comparatively free of ripples, the data should extend to at least... 

Filter out any fast ripple of period Aw/2 in V(z), due to interference with internal reverberations in the lens (Fig. 8.5(b)). This may be achieved most simply by convolving with a rectangular function of length Aw/2. This is known as a moving average filter it is equivalent to a sine filter in the Fourier domain, but is computationally somewhat more efficient. Because of its period the ripple removed at this stage is sometimes called water ripple. 

Fig. 8.5. Steps in the analysis of V(z) for fused quartz (Kushibiki and Chubachi 1985). (a) V(z) on a linear scale (b) V(z) filtered to remove short period ripple due to lens reverberations (c) V(z) for Teflon VL (d) Best value of V" after subtracting long period error in Vf (e) Fourier transform of (d) (f) Final Fourier transform from which the Rayleigh wave velocity and attenuation are found using eqns (8.36) and (8.37). 225 MHz, Ao = 6.6 m.

From equation 5, it is apparent that each shell of scatterers will contribute a different frequency of oscillation to the overall EXAFS spectrum. A common method used to visualize these contributions is to calculate the Fourier transform (FT) of the EXAFS spectrum. The FT is a pseudoradial-distribution function of electron density around the absorber. Because of the phase shift [< ( )], all of the peaks in the FT are shifted, typically by ca. —0.4 A, from their true distances. The back-scattering amplitude, Debye-Waller factor, and mean free-path terms make it impossible to correlate the FT amplitude directly with coordination number. Finally, the limited k range of the data gives rise to so-called truncation ripples, which are spurious peaks appearing on the wings of the true peaks. For these reasons, FTs are never used for quantitative analysis of EXAFS spectra. They are useful, however, for visualizing the major components of an EXAFS spectrum. 

FIGURE 9.12. Series-termination errors, (a) A normal atomic scattering factor curve and (b) the atomic peak obtained by Fourier transformation, (c) A truncated atomic scattering factor curve, such as that used for data that are measured to a lower sin 6/ value than advisable. The missing portion of the scattering curve is indicated, (d) The atomic peak obtained by Fourier transformation. Note the ripples caused by loss of the missing portion of the atomic scattering curve. 

Series-termination errors Errors that result from a limitation in the number of terms in a Fourier series. Ideally an infinite number of data is required to calculate a Fourier series. In practice, the number of data depends on the resolution (reciprocal radius or sin0/A) to which the data have been measured. Because of truncation of the Fourier series at the highest value of sin 0/X of the data, peaks in the resulting Fourier syntheses are surrounded by a series of ripples. These are especially noticeable around a heavy atom because its scattering factor is still appreciable at the highest values of sin 9/X measured. Difference maps (q-v.) can be used to obviate most of the effects of series-termination errors. 

Another difficulty with which we must deal with in diffraction experiments is that any instrumental setup has a maximum accessible momentum transfer, q ax, and the Fourier transformation of that finite pattern leads to peak broadening in real space as well as to non-physical oscillations in G(r) and its related functions. Since those ripples can be confused with the physical diffraction peaks, especially in the range of smaller distances, they must be avoided to obtain a reliable analysis. 

The intensity of diffraction by the single tiny crystallite depends on the square of the magnitude of Fxti, the Fourier transform of the crystallite density function pxti. This is shown schematically in Figure 2. The peak shape at each reciprocal lattice point depends on the square of the magnitude of T, the Fourier transform of the crystal shape function S. The width of the peak decreases, and the ratios of the heights of the ripples to that of the main peak also decrease, as the crystallite size increases. In the mosaic model of the real crystal the individual perfect crystallites scatter incoherently (i.e., with random phase) with respect to each other, and their intensities are additive owing to the tilts of the mosaic blocks with respect to one another, however, the maxima of the various blocks may appear at slightly different places so that the... 

We have introduced apodisation as a weighting of the signal, but we can just as well view it as a weighting of the Fourier basis functions. The sines and cosines become squeezed down at the ends, as illustrated by Fig. 24. To the left it shows a sine base function, a gaussian apodisation that is chosen narrow in order to amplify its effect, and the resulting apodised base function that has the shape of a ripple. 

The electron density function is given in the form of a Fourier summation (Equation 4.2). This means that electron density and hence the atoms in a structure are represented by a number of sine-waves, which are added up. The higher the number of sine-waves, the smoother and more accurate the electron density becomes. As with every Fourier summation, if terms are missing, ripples appear. Especially when some strong reflections are missing from the dataset (e.g. incomplete dataset or some reflections hidden behind the beamstop) artefactual electron density— negative or positive—can appear near heavy atom sites. The same effect can be observed with low-resolution data. An excellent description of the theory behind this effect can be found in an article by Cochran and Lipson (1966). 

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