Fourier analysis techniques

When time-domain errors are additive, Fourier analysis techniques provide statistical properties that are intrinsic to transfer-function measurements. 

Fig. 3.47 is comparable to Fig. 3.41 for sinusoidal ac polarography if the tilted shape provides a net compensation of the charging current one obtains a symmetric bell-shaped curve of I in the square-wave polarogram, similar to that depicted in Fig. 3.42. In fact, virtually all of the statements made before on the sinusoidal technique are valid for the square-wave mode except for the rigid shape of its wave this conclusion is according to expectation, especially as Fourier analysis reveals the square wave to be a summation of a series of only...

Such a function exhibits peaks (Fig. 9C) that correspond to interatomic distances but are shifted to smaller values (recall the distance correction mentioned above). This finding was a major breakthrough in the analysis of EXAFS data since it allowed ready visualization. However, because of the shift to shorter distances and the effects of truncation, such an approach is generally not employed for accurate distance determination. This approach, however, allows for the use of Fourier filtering techniques which make possible the isolation of individual coordination shells (the dashed line in Fig. 9C represents a Fourier filtering window that isolates the first coordination shell). After Fourier filtering, the data is back-transformed to k space (Fig. 9D), where it is fitted for amplitude and phase. The basic principle behind the curve-fitting analysis is to employ a parameterized function that will model the... 

Future development of spectroscopic structure-determination methods will depend on the availability of more powerful photon and particle sources as well as advances in photon and particle detectors. Impressive progress has been made in molecular structure determinations based on advances in computation power and in computational algorithms, such as fast Fourier-transform techniques, for nearly every form of spectroscopy and diffraction analysis. Hajdu and co-work-... 

Most chemists tend to think of infrared (IR) spectroscopy as the only form of vibrational analysis for a molecular entity. In this framework, IR is typically used as an identification assay for various intermediates and final bulk drug products, and also as a quantitative technique for solution-phase studies. Full vibrational analysis of a molecule must also include Raman spectroscopy. Although IR and Raman spectroscopy are complementary techniques, widespread use of the Raman technique in pharmaceutical investigations has been limited. Before the advent of Fourier transform techniques and lasers, experimental difficulties limited the use of Raman spectroscopy. Over the last 20 years a renaissance of the Raman technique has been seen, however, due mainly to instrumentation development. 

NMR has been used comparatively little for quantitative analysis although peak areas are directly proportional to concentration. The principal drawbacks are the expensive instrumentation and a lack of sensitivity. The latter can be improved with the aid of computers to accumulate signals from multiple scans or by using a pulsed (Fourier transform) technique. Relative precision lies in the range 3-8%. 

The applicability of the ESE envelope modulation technique has been extended by two recent publications115,1161. Merks and de Beer1151 introduced a two-dimensional Fourier transform technique which is able to circumvent blind spots in the one-dimensional Fourier transformed display of ESE envelope modulation spectra, whereas van Ormondt and Nederveen1161 could enhance the resolution of ESE spectroscopy by applying the maximum entropy method for the spectral analysis of the time domain data. 

The final pair of methods for reduction of dimensionality which will be tackled in this chapter are Fourier analysis and the life table analysis. Fourier analysis seeks to identify cyclic patterns in data and then either analyze the patterns or the residuals after the patterns are taken out. Life table analysis techniques are directed to identifying and quantitating the time course of risks (such as death, or the occurrence of tumors). 

Fourier analysis (Bloomfield, 1976) is most frequently a univariate method used for either simplifying data (which is the basis for its inclusion in this chapter) or for modeling. It can, however, also be a multivariate technique for data analysis. 

FTIR is a natural for HPLC in that it (FTIR) is a technique that has been used mostly for liquids. The speed introduced by the Fourier transform technique allows, as was mentioned for GC, the recording of the complete IR spectrum of mixture components as they elute, thus allowing the IR photograph to be taken and interpreted for qualitative analysis. Of course, the mobile phase and its accompanying absorptions are ever present in such a technique and water must be absent if the NaCl windows are used, but IR holds great potential, at least for nonaqueous systems, as a detector for HPLC in the future. 

Since we deal with a periodic pattern, it is possible to apply a technique that was originally invented by the French physicist and mathematician Jean Baptiste Joseph Fourier (1768-1830). Fourier was the first who showed that every periodic process (or an object like in our case) can be described as the sum (a superposition) of an infinite number of individual periodic events (e.g. waves). This process is known as Fourier synthesis. The inverse process, the decomposition of the periodic event or object yields the individual components and is called Fourier analysis. How Fourier synthesis works in practice is shown in Figure 4. To keep the example most simple, we will first consider only the projection (a shadow image) of the black squares onto the horizontal a-axis in the beginning (Figure 3). 

The analysis of the SEXAFS data is basically identical to the analysis of conventional EXAFS data We will simply recall the basic ideas that sustain the conventionally used Fourier analysis of the SEXAFS data, making reference to Fig. 2, which will be further discussed below. The reason for being brief is that excellent reviews are available in widely diffused journals that were written by the promoters of the technique and warrant exhaustivity on the subjects The (S)EXAFS signal is defined as ... 

The entire analysis of synchronous detection, or lock-in amplification as it is sometimes called, can be conveniently analyzed by straightforward application of the Fourier transform techniques, transform directory, and convolution theorem developed in Section IV of Chapter 1. 

These studies stimulated the investigation of the use of TOMS data for the determination of the tropospheric ozone column amount by related techniques. Hudson, Thompson and co-workers have developed and refined a technique called the tropical tropospheric ozone (TTO) method (Hudson et al., 1995 Kim et al., 1996 Hudson and Thompson, 1998). This technique utilises a Fourier analysis to identify the range of latitudes for which the method is applicable by using the recognition of a planetary wave pattern to estimate stratospheric and background tropospheric ozone. 

The dHvA measurements were carried out using a field modulation technique at liquid helium temperatures and in magnetic fields up to 6 T. Second harmonic frequency signals of the pick-up coil were detected and analyzed by fast-Fourier analysis. 

The Phase Vocoder. The Phase Vocoder [Flanagan and Golden, 1966][Gordon and Strawn, 1985] is a common analysis technique because it provides an extremely flexible method of spectral modification. The phase vocoder models the signal as a bank of equally spaced bandpass filters with magnitude and phase outputs from each band. Portnoff s implementation of the Short Time Fourier Transform (STFT) provides a time-efficient implementation of the Phase Vocoder. The STFT requires a fast implementation of the Fast Fourier Transform (FFT), which typically involves bit addressed arithmetic. 

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