Fourier transform analytic signal

However, in order to be able to apply the inverse Fourier transformation, we need to know the dependence of the signal not only for a particular value of k (one gradient pulse), but as a continuous function. In practice, it is the Fast Fourier Transform (FFT) that is performed rather than the full, analytical Fourier Transform, so that the sampling of k-space at discrete, equidistant steps (typically 32, 64, 128) is being performed. 

Fourier transform spectroscopy technology is widely used in infrared spectroscopy. A spectrum that formerly required 15 min to obtain on a continuous wave instrument can be obtained in a few seconds on an FT-IR. This greatly increases research and analytical productivity. In addition to increased productivity, the FT-IR instrument can use a concept called Fleggetts Advantage where the entire spectrum is determined in the same time it takes a continuous wave (CW) device to measure a small fraction of the spectrum. Therefore many spectra can be obtained in the same time as one CW spectrum. If these spectra are summed, the signal-to-noise ratio, S/N can be greatly increased. Finally, because of the inherent computer-based nature of the FT-IR system, databases of infrared spectra are easily searched for matching or similar compounds. 

NMR spectroscopy is relatively insensitive compared to optical spectroscopies such as Fourier transform infrared, requiring acquisitions of several minutes to obtain high signal-to-noise ratio spectra adequate for process analytical applications. The linear response and low sensitivity of NMR also prevents its use for observing very low level contaminants (<1000ppm) in complex mixtures. 

By construction, the Fourier transform of the analytical signal contains only positive frequencies. We further assume that the field comprises a finite number of pulses with wave vectors kj. We then have... 

Fourier transform infrared spectroscopy (FT—IR) has been developing into a viable analytical technique (56). The use of an interferometer requires a computer which increases the cost of the system. The ability of IR to differentiate geometrical isomers is still an advantage of the system, and computer techniques such as signal averaging and background subtraction, improve capabilities for certain analyses. 

One of the principal contributions of electronic data processing over the past several years in terms of chemical analysis is the saving of manual effort in interpreting analytical data. Special techniques, such as Fourier transform, have increased speed (as well as sensitivity) by orders of magnitude in connection with infrared, nuclear magnetic resonance, and mass spectroscopy, Of course, for on-line process analyses, essentially instantaneous interpretation is required to provide the proper error signal that is used to position the final control element (valve, feeder, damper, etc.). 

The Fourier transform (FT) has revolutionised spectroscopy such as NMR and IR over the past two decades. The raw data are not obtained as a comprehensible spectrum but as a time series, where all spectroscopic information is muddled up and a mathematical transformation is required to obtain a comprehensible spectrum. One reason for performing FT spectroscopy is that a spectrum of acceptable signal to noise ratio is recorded much more rapidly then via conventional spectrometers, often 100 times more rapidly. This has allowed the development of, for example, 13C NMR as a routine analytical tool, because the low abundance of 13 C is compensated by faster data acquisition. However, special methods are required to convert this time domain ... 

As discussed previously with regard to sampling theory, real analytical signals are band-limited. The Fourier equations therefore should be modified for practical use as we cannot sample an infinite number of data points. With this practical constraint, the discrete forward complex transform is given by... 

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