Decomposition time-domain data

A similar zone folding also occurs at Cs/Pt(lll) and the phonon mode appears as a small dip in the Fourier spectrum in Figure 19.4. A detailed analysis of the time domain data by linear prediction singular value decomposition has been performed and a decomposition of the time-domain data to phonon modes and alkali-substrate stretching modes has been carried out. Coherent nuclear motions have been observed on substrates other than Pt. Figure 19.5a shows time-resolved SHG traces... 

The comparison of 2D spectra is often simplified by the decomposition or splitting of a 2D data matrix into a series of ID spectra. Time domain data can be used to optimize weighting functions prior to processing the 2D data matrix whilst frequency domain data can be used in the evaluation and development of modified pulse sequence. ID spectra can also be used to optimize the phase correction in a phase sensitive experiment. 

The original linear prediction and state-space methods are known in the nuclear magnetic resonance literature as LPSVD and Hankel singular value decomposition (HSVD), respectively, and many variants of them exist. Not only do these methods model the data, but also the fitted model parameters relate directly to actual physical parameters, thus making modelling and quantification a one-step process. The analysis is carried out in the time domain, although it is usually more convenient to display the results in the frequency domain by Fourier transformation of the fitted function. 

Filters which combine measurements of two types of relaxation lead to 2D parameter sets which characterize the evolution of magnetization for each type of relaxation along one axis and correlate both processes. In case of multi-component relaxation such time-domain correlation maps can be analysed for accurate identification of individual components, where discrimination of components is improved by the 2D nature of the available data as compared to the decomposition of ID relaxation curves [Olel, Peel, Wegl],... 

Radial sampling was the first sparse sampling scheme introduced into NMR. Apart from FT [44, 55, 56], other data processing techniques were proposed. These include reduced dimensionality [57], projection-reconstruction [41], and multiway decomposition [58]. Radial sampling scheme consists of points placed on a set of lines in the time domain (see Fig. 6c, d). The coordinates of the th point, lying on the /th line, can be described in a polar coordinate system as... 

Data compression is another application of WT that has shown remarkable results (Artursson and Holmberg 2002). The mathematical treatment for data compression by WT is similar to that for denoising and smoothing (fetter et al. 2000). Chemical data is treated with WT and transformed to the scale-time domain where its spectral content is reduced by elinunating coefficients belonging to high frequency content. Compression with this teclmique is highly efficient since a one level decomposition and... 

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