Compression Fourier

A disadvantage of Fourier compression is that it might not be optimal in cases where the dominant frequency components vary across the spectrum, which is often the case in NIR spectroscopy [40,41], This leads to the wavelet compression [26,27] method, which retains both position and frequency information. In contrast to Fourier compression, where the full spectral profile is fit to sine and cosine functions, wavelet compression involves variable-localized fitting of basis functions to various intervals of the spectrum. The... 

For data in which the variables are expressed as a continuous physical property (e.g. spectroscopy data, where the property is wavelength or wavenumber), the Fourier transform can provide a compressed representation of the data. The Fourier compression method can be applied to one analyzer profile (i.e. spectrum) at a time. 

Like PCA compression, Fourier compression involves the reduction of an analyzer response profile into a simpler representation that uses basis functions that are linear combinations of the original variables. However, in the case of Fourier compression, these basis functions are pre-defined trigonometric functions of the original variables, whereas in PCA they (the PC loadings) are application-specific and must be determined through an analysis of the entire data set. 

In Fourier compression, each profile (x ) is essentially decomposed into a linear combination of sine and cosine functions of different frequency. If the spectrum x is considered to be a continuous function of the variable number m, then this decomposition can be expressed as ... 

Although the Fourier compression method can be effective for reducing data into frequency components, it cannot effectively handle situations where the dominant frequency components vary as a function of position in the spectrum. For example, in Fourier transform near-infrared (FTNIR) spectroscopy, where wavenumber (cm-1) is used as the x-axis, the bandwidths of the combination bands at the lower wavenumbers can be much smaller than the bandwidths of the overtone bands at the higher wavenumbers.31,32 In any such case where relevant spectral information can exist at different frequencies for different positions, it can be advantageous to use a compression technique that compresses based on frequency but still preserves some position information. The Wavelet transform is one such technique.33... 

In general, wavelet functions are chosen such that they and their compressed representations are orthogonal to one another. As a result, the basis functions in Wavelet compression, like those in PCA and Fourier compression, are completely independent of one another. Several researchers have found that representation of spectral data in terms... 

Unlike PCA compression or Fourier compression, where spectra are described in terms of abstract PCs or trigonometric functions respectively, CLS uses estimated pure component spectra as the basis for explaining each spectrum. 

The compressibility equation can also be written in tenns of the direct correlation fiinction. Taking the Fourier transfomi of the Omstein-Zemike equation... 

The frill width at half maximum of the autocorrelation signal, 21 fs, corresponds to a pulse width of 13.5 fs if a sech shape for the l(t) fiinction is assumed. The corresponding output spectrum shown in fignre B2.1.3(T)) exhibits a width at half maximum of approximately 700 cm The time-bandwidth product A i A v is close to 0.3. This result implies that the pulse was compressed nearly to the Heisenberg indetenninacy (or Fourier transfonn) limit [53] by the double-passed prism pair placed in the beam path prior to the autocorrelator. 

If the second term on the right-hand side of the equation is omitted, the latter is transformed into Eq. (2.76). As the omission is possible only for t < tj, Fourier transformation of the reduced equation holds for co-tj 1 only. Consequently, the equality (2.75) is of asymptotic character, and may not be utilized to find full g(co) or its Fourier-transform Kj(t) at any times. When it was nevertheless used in [117], the rotational correlation function turned out to be alternating in sign. The oscillatory behaviour of Kj(t) occured not only in a compressed gas, but also at normal pressure, when Kj(t) should vanish monotonically, if not exponentially. The origin of these non-physical oscillations is easily... 

Fig. 40.31. Data compression by a Fourier transform, (a) A spectrum measured at 512 wavelengths (b) spectrum after reconstruction with 2, 4,..., 256 Fourier coefficients.

It is worth to be noted that (1 /a)h (rja) is the result of the dilation of h (r) by the factor a in which the area under the curve is conserved. The result in reciprocal space is a compressed function H. This property of the Fourier transform is the generalization of Bragg s law. 

The valence-electron wave functions of atoms, compressed beyond their ionization limits are Fourier sums of spherical Bessel functions corresponding to step functions (Compare 6.3.1) of the type... 

After several cycles of the compression and expansion, the dynamic jc-A curve becomes a single closed loop, somewhat distorted from a genuine ellipsoid. In order to analyze the forms of the hysteresis loop under stationary conditions, we have measured the time trace of the dynamic surface pressure after five cycles of the compression and expansion, and then Fourier-transformed it to the frequency domain. The Fourier-transformation was adapted to evaluate the nonlinear viscoelasticity in a quantitative manner. The detailed theoretical consideration for the use of the Fourier transformation to evaluate the nonlinearity, are contained in the published articles [8,43]. 

T. Fearn and A.M.C. Davies, A comparison of Fourier and wavelet transforms in the processing of near-infrared spectroscopic data part 1. Data compression, J. Near Infrared Spectrosc., 11, 3-15 (2003). 

Analysts. It has been our objective to determine criteria for resin, curative or formulation which would permit prediction of sucess prior to potting tests. Many tests, both chemical and physical in nature, have been executed on commercial resin systems. These have included high pressure liquid chromatography (HPLC), Fourier Transform infrared spectrometry (FTIR), gel permeation chromatography, compressive tensile tests by Instron on resin plaques in air and under various aqueous solutions and heat distortion temperature. 

Fourier transform for different, chemically very similar halomethanes and a mixture thereof. The time-domain data in Figure 7.11 can be directly interpreted as an observation of molecular motion in real time, made possible by the compressed ultrashort pulses in the microscope. From the presence of different oscillatory patterns and beatings, it already becomes clear that the different molecules can be discriminated with high resolution. Correspondingly, the Fourier spectra in Figure 7.11 show markedly different vibrational resonances, which can also be discriminated in the ternary mixture of all components. 

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