  Weyl functions conveniently describe the evolution of wave coherences in
periodic or quadratic potentials. In this work we use Weyl functions to study
the ``Talbot-Lau effect'' in a time-domain matter-wave interferometer. A
``displacement diagram'' is introduced to analyze and calculate the matter-wave
interference for an atomic cloud in a quadratic potential that interacts with a
sequence of short optical standing wave pulses producing an atomic grating
echo. Unlike previous treatments, this new approach allows the atomic ensemble
to have an arbitrary initial phase-space distribution, and the standing wave
grating vectors to span three dimensions. Several examples are discussed to
illustrate the convenience of the diagrammatic technique including the
following: a two-dimensional Talbot-Lau effect, the shift in the echo time and
the recoil phase for the interferometer perturbed by a quadratic potential; and
the realization of a time-domain ``Lau effect'' using a pulsed harmonic
potential. The diagrammatic technique is applicable to diffraction gratings
with arbitrary grating transmission functions. We conclude the paper with a
general discussion on the Weyl function representations of matter-wave
coherence, and relate the conservation of matter-wave coherence with the
conservation of purity that distinguishes decoherence effects from dephasing
effects.
