Dynamic spectrum discretely sampled

Detailed description of the mathematical background of 2D correlation theory is provided in Appendix F of this book. Here we only briefly go over the correlation treatment of a set of discretely observed spectral data commonly encountered in practice. Let us assume spectral intensity x u, u) described as a function of two separate variables spectral index variable v of the probe and additional variable u reflecting the effect of the applied perturbation. Typically, spectral intensity x is sampled and stored as a function of variables v and u at finite and often constant increments. For convenience, here, we refer to the variable v as wavenumber v and the variable u as time t. For a set of m spectral data x(v, f,) with f = 1,2,. .., m, observed during the period between and we define the dynamic spectrum y(v, t ) as... 

The explicit analytical expressions given by Equations (FI 6) and (F24), obtained for the synchronous and asynchronous spectrum, are well suited for the efficient machine computation of correlation intensities from discretely sampled and digitized spectral data. If a discretely sampled dynamic spectrum y(v -, t,) with the total of n points of wavenumber Vj is obtained for m times at each point of time tj, with a constant time increment, that is, t,+i - t, i = At, the integrations in Equations (FI6) and (F24) ean be replaced with summations. 

The sampling of dynamic spectrum does not always have to occur with a constant increment of t. For unevenly sampled discrete data [10], some modifications are then required for Equations (F30), (F31), and (F33). The modified form of correlation spectra are given by... 

In microscopic approaches the solvent molecules are described as true discrete entities but in some simplified form, generally based on fotee-field methods (Allinger, 1977). These theories may be of the semicontinuum type if the distant bulk solvent is accounted for, or of the fully discrete type if the solvent description includes a large number of molecules. As an example, the spectrum of formaldehyde in water has been examined using a combination of classical molecular dynamics and ab initio quantum chemical methods and sampling the calculated spectrum at different classical conformations (Blair et al., 1989 Levy et al., 1990). These calculations predict most of the solvent shift as well as the line broadening. 

Static and dynamic scattering techniques are spectroscopic characterisation methods in the sense of Sect. 2.2. These techniques evaluate the functional dependency of measurement signals on a spectral parameter, i.e. on time, space, or classically on wavelength or frequency. The major advantage of spectroscopic methods is the reduced sample preparation (no fractionation), but they involve the inversion problem. That is, the spectrum is a—most frequently incomplete and discrete— nonlinear projection of the size distribution. Beside the scattering techniques, there are further spectroscopic methods which are based on the extinction of radiation or on any other response of the particle system to an external field. This section describes optical, acoustic, and electroacoustic methods that have gained relevance for the characterisation of colloidal suspensions. 

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