The translation of climate signals into sediments is a complex and poorly understood process, and one that is likely to be inherently nonlinear. A great deal of debate concerns the mismatch between the amplitude spectrum of astronomical insolation forcing and the amplitude spectrum of climatic signals derived from geological archives. Many ancient shallow-water successions, such as peritidal carbonates, are widely suggested to show evidence for eccentricity-band forcing, despite eccentricity having a negligible effect on insolation. Instead, eccentricity manifests itself primarily as the amplitude modulator of precession, which dominates the insolation signal at low to mid latitudes. The appearance and dominance of eccentricity-band frequencies in stratigraphic data can arise, however, without the need to appeal to complex nonlinearities in the climate system or depositional environment. Instead, eccentricity frequencies will appear as a consequence of signal-driven nondeposition that results in a rectification of the precession-dominated insolation signal. Furthermore, it can be shown that this rectification effect is capable of inducing a signal-driven modulation of cycle thickness (i.e., a frequency modulation) that can lead to a similar range of cycle bundling patterns documented from ancient shallow-water successions. Numerical experiments using an astronomical insolation solution as an input signal emphasize how eccentricity can dominate climate records that have undergone only partial rectification, and that are stratigraphically complete at astronomical time scales of resolution. These experiments, supported by geological observations, thus describe a viable mechanism capable of explaining the eccentricity paradox in ancient shallow-water successions, and also emphasize how measures of stratigraphic completeness at astronomical time scales furnish no clue as to the likely fidelity of the actual climate forcing records or their spectra.