The synthesis of soft pulses with a specified frequency response

We propose a new approach to synthesizing shaped pulses, by first synthesizing a hard pulse sequence and then converting it to a soft pulse. In previous papers, we developed a new approach to synthesizing hard pulse sequences by exactly inverting the Bloch equation, which yields the optimal frequency response. These results can now be applied to shaped pulses. Specifically, one can specify (1) the total duration of the pulse, (2) the frequency range it is desired to perturb, (3) the perturbation desired and, (4) the frequency range it is desired not to perturb. One can then synthesize a shaped pulse which has the desired effect with the least possible error. This enables us to obtain very sharp inversion and pi/2 pulses.