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 (Mz) is expressed as a function of the off-resonance frequency as an Nth order complex Fourier series. We then form a consistent Fourier series for (Mxy). As many as 2(2)N different pulse sequences may be directly generated all of which produce a different Mxy(f), but the same Mz(f). A pulse sequence is then generated which will yield the desired Mz(f) and Mxy(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 Mz which is potentially realizable by a pulse sequence.