Fourier decompositions and pulse sequence design algorithms for nuclear magnetic resonance in inhomogeneous fields

In this paper, we introduce algorithms based on Fourier synthesis for designing phase compensating rf pulse sequences for high-resolution nuclear magnetic resonance (NMR) spectroscopy in an inhomogeneous B0 field. We show that using radio frequency pulses and time varying linear gradients in three dimensions, it is possible to impart the transverse magnetization of spins phase, which is a desired function of the spatial (x,y,z) location. Such a sequence can be used to precompensate the phase that will be acquired by spins at different spatial locations due to inhomogeneous magnetic fields. With this precompensation, the chemical shift information of the spins can be reliably extracted and high resolution NMR spectrum can be obtained.