Discrete Inverse Scattering Theory for NMR Pulse Design

Contents Chapter 1. Introduction 5 1.1. The selective excitation transform 5 1.2. The problem of selective excitation pulse design 6 1.3. Examples 7 1.4. The discrete selective excitation transform (hard pulse approximation) 10 1.5. Main results: continuum theory 11 1.6. Main results: discrete theory 12 1.7. Energy Formulas 13 1.8. Algorithms 13 Chapter 2. Scattering theory 15 2.1. Preliminaries 15 2.2. The Zakharov-Shabat system 19 2.3. Continuum theory 20 2.4. Discrete Theory 22 2.5. Forward discrete scattering 25 2.6. Derivation of the right Marchenko equation 26 2.7. Derivation of the left Marchenko equation 27 2.8. Derivation of the discrete Marchenko equations 29 2.9. Derivation of the DIST recursion 31 2.10. The discrete energy formula 31 2.11. Proof of the main result: discrete case 32 2.12. Proof of the main result: continuum case 36 2.13. Applying the discrete algorithm to continuum scattering data 40 Chapter 3. Pulses with finite rephasing time and applications 42 3.1. A recursive algorithm for pulses of finite rephasing time 42 3.2.