  Computations of the strong field generation of gravitational waves by black
hole processes produce waveforms that are dominated by quasinormal (QN)
ringing, a damped oscillation characteristic of the black hole. We describe
here the mathematical problem of quantifying the QN content of the waveforms
generated. This is done in several steps: (i) We develop the mathematics of QN
systems that are complete (in a sense to be defined) and show that there is a
quantity, the ``excitation coefficient,'' that appears to have the properties
needed to quantify QN content. (ii) We show that incomplete systems can (at
least sometimes) be converted to physically equivalent complete systems. Most
notably, we give a rigorous proof of completeness for a specific modified model
problem. (iii) We evaluate the excitation coefficient for the model problem,
and demonstrate that the excitation coefficient is of limited utility. We
finish by discussing the general question of quantification of QN excitations,
and offer a few speculations about unavoidable differences between normal mode
and QN systems.
