The synthesis of pulse sequences yielding arbitrary magnetization vectors

A new procedure and algorithm are presented to allow the synthesis of a pulse sequence which will generate an arbitrary frequency‐dependent spin excitation. This procedure is a generalization of our previous paper, where this was done subject to the restriction that the spin excitation was symmetric about zero offset frequency, and pulses were restricted to being about a fixed axis. The required final z‐magnetization vector ( M z ) is expressed as a function of the off‐resonance frequency as an N th order complex Fourier series. We then form a consistent Fourier series for ( M xy ). As many as 2 2N different pulse sequences may be directly generated all of which produce a different M xy (f ), but the same M z (f). A pulse sequence is then generated which will yield the desired M z (f ) and M xy (f). This is done by an analytic inversion of the Bloch equation, not by the classical Fourier approximation. This technique enables us to generate any M z which is potentially realizable by a pulse sequence. © 1989 Academic Press, Inc.