Theory of stochastic NMR spectroscopy. Application of the ITÔ and Stratonovich calculus

ArticleinChemical Physics 18(1-2):57-84 · December 1976with7 Reads
DOI: 10.1016/0301-0104(76)87037-1
The theory of stochastic differential equations is used to give a new description of a stochastic NMR experiment. It replaces an earlier approach, which was based on Wiener's orthogonal expansion of the stochastic response. For the first time, the saturation behaviour in cross and power spectra is predicted correctly. A numerical experiment confirms the theoretical results. The relative signal intensities in a stochastic resonance spectrum are calculated and compared with those obtained in a slow passage experiment. Conditions for equal relative intensities are given for various experimental situations.
    • "Reasons have been presented various places to the effect that when developing physical applications it is simpler to start with the Stratonovich form and then switch to the Itô for developing properties of the process. See the discussions in: Bartholdi et al. (1976), Mortensen (1969, Sussman (1978), Karlin and Taylor (1981). To start, some of the previous work on the planar case with drift and the spherical case without drift will be presented. "
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  • [Show abstract] [Hide abstract] ABSTRACT: It is shown how 1D and 2D NMR spectra can be derived from one NMR experiment. The method is based on the evaluation of the non-linear response of the nuclear spin system to stochastic excitation the strength of which corresponds to values close to the maximum of the saturation curve. Thus, optimum sensitivity is obtained at the expense of slight line broadening in the 1D spectrum. Crosspeaks in the 2D plane indicate the existence of saturation transfer between two transitions at the respective frequencies. This is demonstrated with simulated (AB) as well as experimental spectra (AB, chemical exchange).Es wird gezeigt, wie 1D- und 2D-NMR-Spektren aus einem NMR-Experiment erhalten werden können. Die Methode basiert auf der Auswertung der nichtlinearen Antwort des Spinsystems auf eine stochastische Anregung, deren Stärke Werten nahe dem Maximum der Sättigungskurve entspricht. Es resultiert daraus optimale Empfindlichkeit auf Kosten einer gewissen Linienverbreiterung im 1D-Spektrum. Kreuzsignale in der 2D-Ebene weisen auf die Existenz von Sättigungsübertragung zwischen zwei NMR-Übergängen der respektiven Frequenzen hin. Das wird demonstriert anhand simulierter (AB) als auch experimenteller Spektren (AB, chemischer Austausch).
    Article · Nov 1980
  • [Show abstract] [Hide abstract] ABSTRACT: In broadband NMR spectroscopy excitation with pseudorandom binary amplitude or phase modulation permits the distribution of the excitation power over the entire data acquisition time while peak power requirements are kept low. For sufficiently low excitation power, the magnetization is the linear response of the spin system to its input. The transfer function of the linearly driven system is recovered with the fast Hadamard transform. It is identical to the FID signal in FT NMR. Increasing excitation levels produce distorted lineshapes resulting from linear processing of a nonlinear spin response. Spectra measured for different degrees of saturation are reproduced faithfully by a numerical solution of the Bloch equations including relaxation during excitation. The origin of the lineshape distortions is discussed on the basis of an expansion of the nonlinear response in terms of the linear response. This expansion is in good agreement with the Bloch equations for limited excitation levels. Its nonlinear response terms are generalized to account for connectivities in coupled spin systems.
    Article · Feb 1982
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