We develop a new way of synthesizing pulse sequences with desirable frequency characteristics. By combining our previous results with techniques from the theory of finite impulse response filters, we can specify (1) the total duration of the pulse sequence, (2) the frequency ranges it is desired to perturb, (3) the desired perturbation, and (4) the frequency range it is desired not to perturb. We can then synthesize a hard pulse sequence which will yield that desired perturbation with the minimal possible error. The minimum error is global, in the sense that no pulse sequence can do better at achieving the specifications, and is not just a local minimum, around the pulse sequences close to the derived pulse.