Results of systematic numerical simulations of the interaction between a low-intensity solitary signal pulse belonging to an anomalous-dispersion channel in a nonlinear optical fiber and a simple Periodic Support Structure (PSS) launched by a sinusoidal beating signal in the mate channel, that may have either anomalous or normal dispersion, are reported. If the PSS is generated in the normal-dispersion channel, the purely sinusoidal form of PSS-generating input is sufficient to stabilize the signal pulse, provided that the powers of the signal pulse and PSS belong to specially selected intervals. If the PSS is launched in the anomalous-dispersion wavelength region, we find that the purely sinusoidal PSS shape cannot stabilize the pulse, and it quickly decays into radiation. However, mixing the strong sinusoidal input with additional small-amplitude harmonics provides for a nearly stable propagation of the pulse over distances ≥50 pulse dispersion lengths. A drastic difference between the cases of the normal-dispersion and anomalous-dispersion PSS is that, in the former case, the signal pulse with a too small intensity is unstable, while in the latter case the pulse is an essentially linear one and the increase of its intensity only deteriorates the stability.