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A Fourier-based approach is presented for the investigation of multilayer superpositions of periodic structures and their moire effects. This approach fully explains the properties of the superposition of periodic layers and of their moire effects, both in the spectral domain and in the image domain. This approach provides also a full explanation of the various phenomena that occur because of phase shifts in one or more of the superposed layers. We show how such phase shifts influence the superposition as a whole and, in particular, how they affect each moire in the superposition individually. We show that each moire in the superposition undergoes a different shift, in its own main direction, whose size depends both on the moire parameters and on the shifts of the individual layers. However, phase shifts in the individual layers do not necessarily lead to a solid shift of the whole superposition, and they may rather cause modifications in its microstructure. We demonstrate our results by several illustrative figures