  We consider the evolution of scalar perturbations in a class of non-singular
bouncing universes obtained with higher-order corrections to the low-energy
bosonic string action. We show that previous studies have relied on a singular
evolution equation for the perturbations. From a simple criterium we show that
scalar perturbations cannot be described at all times by an homogeneous
second-order perturbation equation in pre-big bang type universes if we are to
regularise the background evolution with higher-order curvature and string
coupling corrections, and we propose a new system of first-order coupled
differential equations. Given a bouncing cosmological background with inflation
driven by the kinetic energy of the dilaton field, we obtain numerically the
final power spectra generated from the vacuum quantum fluctuations of the
metric and the dilaton field during inflation. Our result shows that both
Bardeen's potential, $\Phi(\eta,k)$, and the curvature perturbation in the
uniform curvature gauge, ${\mathcal R}(\eta,k)$, lead to a blue spectral
distribution long after the transition.
