  In this note, we prove the profile decomposition for hyperbolic Schr\"odinger
(or mixed signature) equations on $\mathbb{R}^2$ in two cases, one
mass-supercritical and one mass-critical. First, as a warm up, we show that the
profile decomposition works for the ${\dot H}^{\frac12}$ critical problem,
which gives a simple generalization of for instance one of the results in
Fanelli-Visciglia (2013). Then, we give the derivation of the profile
decomposition in the mass-critical case by proving an improved Strichartz
estimate. We will use a very similar approach to that laid out in the notes of
Killip-Visan (2008), but we are forced to do a double Whitney decomposition to
accommodate an extra scaling symmetry that arises in the problem with mixed
signature.
