Practical pulse synthesis via the discrete inverse scattering transform

This paper provides the practical details required to use the inverse scattering (IST) approach to design selective RF-pulses. As in the Shinnar-Le Roux (SLR) approach, we use a hard pulse approximation to actually design the pulse. Unlike SLR, the pulse is designed using the full inverse scattering data (the reflection coefficient and the bound states) rather than the flip angle profile. We explain how to approximate the reflection coefficient to obtain a pulse with a prescribed rephasing time. In contrast to the SLR approach, we retain direct control on the phase of the magnetization profile throughout the design process. We give explicit recursive algorithms for computing the hard pulse from the inverse scattering data. These algorithms are quite different from the SLR recursion, being essentially discretizations of the Marchenko equations. We call our approach the discrete inverse scattering transform or DIST. Overall, it is as fast as the SLR approach. When bound states are present, we use both the left and right Marchenko equations to improve the numerical stability of the algorithm. We compute a variety of examples and consider the effect of amplitude errors on the magnetization profile.