We show a new way of compressing the description of a quantum system by coarse graining. Given an arbitrary time dependent unitary system and a set of POVM elements that model a saturated detector, we describe the quantum channel and the effective state that describes all possible measurement statistics of any experiment done with the given POVM set. As an application we investigate the relationship between the coarse graining of a spin system and memory effects (Non-Markovianity. Quantum state tomography on OAM qudits.