  Gravitational wave astronomy provides an unprecedented opportunity to test
the nature of black holes and search for exotic, compact alternatives. Recent
studies have shown that exotic compact objects (ECOs) can ring down in a manner
similar to black holes, but can also produce a sequence of distinct pulses
resembling the initial ringdown. These "echoes" would provide definite evidence
for the existence of ECOs. In this work we study the generation of these echoes
in a generic, parametrized model for the ECO, using Green's functions. We show
how to reprocess radiation in the near-horizon region of a Schwarzschild black
hole into the asymptotic radiation from the corresponding source in an ECO
spacetime. Our methods allow us to understand the connection between distinct
echoes and ringing at the resonant frequencies of the compact object. We find
that the quasinormal mode ringing in the black hole spacetime plays a central
role in determining the shape of the first few echoes. We use this observation
to develop a simple template for echo waveforms. This template preforms well
over a variety of ECO parameters, and with improvements may prove useful in the
analysis of gravitational waves.
