RF pulse design using the inverse scattering transform.

The inverse scattering transform (IST) is a mathematical transformation that can be used to derive RF pulses from functions called continuous spectra describing the final state of the spin system. This paper reviews three seemingly unrelated numerical algorithms that have appeared in the literature, and shows that they are all derivable from the IST. When the continuous spectra are rational, the finite rank kernel method is used to convert the IST to a matrix equation that is easily solved. Another algorithm, equivalent to the so-called "layer stripping" algorithm used in seismology, is derived by assuming that the spectra are Fourier series. Finally, the Shinnar-Le Roux (SLR) algorithm is derived by assuming that the spectra are ratios of Fourier series. With proper interconversion between the rational, series, and ratio of series forms of the continuous spectra, these algorithms generate RF pulses with identical or nearly identical shapes and performance properties, and can be regarded as equivalent.