  Transition edge sensor microcalorimeters can measure x-ray and gamma-ray
energies with very high energy resolution and high photon-collection
efficiency. For this technology to reach its full potential in future x-ray
observatories, each sensor must be able to measure hundreds or even thousands
of photon energies per second. Current "optimal filtering" approaches to
achieve the best possible energy resolution work only for photons well isolated
in time, a requirement in direct conflict with the need for high-rate
measurements. We describe a new analysis procedure to allow fitting for the
pulse height of all photons even in the presence of heavy pulse pile-up. In the
limit of isolated pulses, the technique reduces to the standard optimal
filtering with long records. We employ reasonable approximations to the noise
covariance function in order to render multi-pulse fitting computationally
viable even for very long data records. The technique is employed to analyze
x-ray emission spectra at 600 eV and 6 keV at rates up to 250 counts per second
in microcalorimeters having exponential signal decay times of approximately 1.2
ms.
