Context. Since CoRoT observations unveiled the very low amplitude modes that form a flat plateau in the power spectrum structure of δ Scuti stars, the nature of this phenomenon, including the possibility of spurious signals due to the light curve analysis, has been a matter of long-standing scientific debate. Aims: We contribute to this debate by finding the structural parameters of a sample of four δ Scuti stars, CID 546, CID 3619, CID 8669, and KIC 5892969, and looking for a possible relation between these stars' structural parameters and their power spectrum structure. Methods: For the purposes of characterization, we developed a method of studying and analysing the power spectrum with high precision and have applied it to both CoRoT and Kepler light curves. Results: We obtain the best estimates to date of these stars' structural parameters. Moreover, we observe that the power spectrum structure depends on the inclination, oblateness, and convective efficiency of each star. Conclusions: Our results suggest that the power spectrum structure is real and is possibly formed by 2-period island modes and chaotic modes.