  We consider the stability of front-type modulated waves in the complex
Ginzburg-Landau equation (CGL). The waves occur in the bistable regime (e.g. of
the quintic CGL) and connect the zero state to a spatially homogenous state
oscillating in time. For initial perturbations that decay at a certain
algebraic rate, we prove convergence to the wave with asymptotic phase. The
convergence holds in algebraically weighted Sobolev norms and with an algebraic
rate in time, where the asymptotic phase is approached by one order less than
the profile. On the technical side we use the theory of exponential
trichotomies to separate the spatial modes into growing, weakly decaying, and
strongly decaying ones. This allows us to derive resolvent and semigroup
estimates in weighted Sobolev norms and to close the argument with a Gronwall
lemma involving algebraic weights.
