  In collisional ionization equilibrium (CIE), the X-ray spectrum from a plasma
depends on the distribution of emission measure over temperature (DEM). Due to
the well-known ill conditioning problem, no precisely resolved DEM can be
inverted directly from the spectrum, so often only a gross parametrization of
the DEM is used to approximate the data, in hopes that the parametrization can
provide useful model-independent constraints on the heating process. However,
ill conditioning also introduces ambiguity into the various different
parametrizations that could approximate the data, which may spoil the perceived
advantages of model independence. Thus, this paper instead suggests a single
parametrization for both the heating mechanism and the X-ray sources, based on
a model of impulsive heating followed by complete cooling. This approach is
similar to a ``cooling flow'' approach, but allows injection at multiple
initial temperatures, and applies even when the steady state is distribution of
different shock strengths, as for a standing shock with a range of obliquities,
or for embedded stochastic shocks that are only steady in a statistical sense.
This produces an alternative parametrization for X-ray spectra that is
especially streamlined for higher density plasmas with efficient radiative
cooling, and provides internal consistency checks on the assumption of
impulsive heating followed by complete cooling. The result is no longer model
independent, but the results are more directly interpretable in terms of useful
physical constraints on the impulsive heating distribution.
