Fourier transforms general principles

An important corollary of the principle of superposition is that a wave of any shape can be described mathematically as a sum of a series of simple sine and cosine terms, which is the basis of the mathematical procedure called the Fourier transform (see Section 4.2). Thus the square wave, frequently used in electronic circuits, can be described as the sum of an infinite superposition of sine waves, using the general equation ... 

The situation, however, is different for the infrared spectroscopic measurements with opposed anvil cells. The source beam in commercial Fourier transform infrared spectrometers is generally focused to about 1 cm diameter at the sample, whereas the diameter of the gasket hole in the high pressure cell is only about 0.3 mm. Therefore, a source beam condensing system is required in order to obtain infrared spectra with a good signal-to noise ratio. Commercial beam condensers (4X, 6X) could, in principle, be adapted for these purposes. In practice, however, the mirrors of the... 

To benefit general readers, the discussion has been limited to methodologies that are accessible to nonspecialists and that can be carried out on commercially available spectrometers without special modifications. The chapter illustrates the principles of mass spectrometry by demonstrating how various techniques [MALDI, ESI, Fourier transform ion cyclotron resonance (FT-ICR), ion traps, and tandem mass spectrometry (MS-MS)] work. It also provides examples of utilizing mass spectrometry to solve biological and biochemical problems in the field of protein analysis, protein folding, and noncovalent interactions of protein-DNA complexes. 

This book intends to supply the basic information necessary to apply the methods of vibrational spectroscopy, to design experimental procedures, to perform and evaluate experiments. It does not intend to provide a market survey of the instruments which are available at present, because such information would very soon be outdated. However, the general principles of the instruments and their accessories, which remain valid, are discussed. Details concerning sample preparation and the recording of the spectra, which is the subject of introductory courses, are assumed to be known. Special procedures which are described in monographs, such as Fourier transformation or chemometric methods, are also not exhaustively described. This book has been written for graduate students as well as for experienced scientists who intend to update their knowledge. 

Forcing function, 143 periodic, 144 transient, 143 Fourier transform, 170 Fractional time, 29 Fractionation factor, 301 Fraction theorem, general partial, 85 Frame, rotating, 170 Franck-Condon principle, 435 Free energy, 211 transfer, 418... 

The second rather general approach involves Fourier transforms, as has been largely elaborated by Logllo et al ). We shall not enter into this formalism in depth, but recall that the principles of Fourier transformations were outlined in appendix 10 of Volume I. A certain periodic function in the time domain /(t) can be transformed into its Fourier transform /(co) in the frequency domain, using [I.A10.301... 

Of course, the Fourier transforms of the correlation functions are calculated by applying the same principles. Moreover, the generalization to the polydis-perse case is trivial. Finally, we recall that 2 (9 NxS) can be easily expressed as a sum of contributions associated with anchored diagrams. 

Figure 5 The unit window function in frequency, W rn) and its Fourier transform in the time domain, W, (/). The widths of the transform pair are inverse to one another, and this mathematical result is true in general (51). (The physical implication of this theorem is, of course, the time-frequency uncertainty principle. In the present context, this theorem implies that short time dynamics determines the broad features of the spectrum, and vice versa). Convoluting the spectrum with the unit window function is the simplest form of coarse graining cf. Eq. (10).

Time-domain techniques record the intensity of the signal as a function of time, frequency-domain techniques record the phase and the amplitude of the signal as a function of frequency. Time domain and frequency domain are connected via the Fourier transform. Therefore, the time domain and the frequency domain are generally equivalent. However, this does not imply an equivalence between time-domain and frequency-domain recording techniques or the instruments used for each. An exhaustive comparison of the techniques is difficult and needs to include a number of different electronic design principles and applications. 

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