Discrete Inverse Scattering Theory for NMR Pulse Design

Abstract

We introduce a discrete analogue of the scattering theory for the Zakharov-Shabat (ZS) system, and use it to continue the work of C.L. Epstein by deriving an efficient, recursive algorithm for generating RF-pulses for nuclear magnetic resonance (NMR). In the process, we present a straightforward derivation of the standard Gel'fand-Levitan-Marchenko (Marchenko) equations, and we derive similar equations which apply to the discrete framework. In addition, we prove that the potentials obtained by solving the Marchenko equations produce the correct scattering data. We explain how the generally accepted Shinnar-Le Roux (SLR) technique fits into the more general framework of discrete inverse scattering, and we show, using examples, how inverse scattering theory can be used to produce pulses which are, in some ways, superior to standard SLR pulses.