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NMR Imaging of Materials$
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Bernhard Blumich

Print publication date: 2003

Print ISBN-13: 9780198526766

Published to Oxford Scholarship Online: January 2010

DOI: 10.1093/acprof:oso/9780198526766.001.0001

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Transformation, convolution, and correlation

Transformation, convolution, and correlation

Chapter:
(p.125) 4 Transformation, convolution, and correlation
Source:
NMR Imaging of Materials
Author(s):

Bernhard Blümich

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198526766.003.0004

Transformation, convolution, and correlation are used over and over again in nuclear magnetic resonance (NMR) spectroscopy and imaging in different contexts and sometimes with different meanings. The transformation best known in NMR is the Fourier transformation in one or more dimensions. It is used to generate one- and multi-dimensional spectra from experimental data as well as ID, 2D, and 3D images. Furthermore, different types of multi-dimensional spectra are explicitly called correlation spectra. These are related to nonlinear correlation functions of excitation and response. This chapter discusses convolution in linear and nonlinear systems, along with the convolution theorem, linear system analysis, nonlinear cross-correlation, correlation theorem, Laplace transformation, Hankel transformation, Abel transformation, z transformation, Hadamard transformation, and wavelet transformation.

Keywords:   nuclear magnetic resonance spectroscopy, Fourier transformation, convolution, correlation, imaging, nonlinear correlation functions, Laplace transformation, wavelet transformation, Hankel transformation

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