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Phase retrieval in dynamical sampling is a novel research direction, where anunknown signal has to be recovered from the phaseless measurements with respectto a dynamical frame, i.e. a sequence of sampling vectors constructed by therepeated action of an operator. The loss of the phase here turns the well-poseddynamical sampling into a severe ill-posed inverse problem. In the existingliterature, the involved operator is usually completely known. In this paper,we combine phase retrieval in dynamical sampling with the identification of thesystem. For instance, if the dynamical frame is based on a repeatedconvolution, then we want to recover the unknown convolution kernel in advance.Using Prony's method, we establish several recovery guarantees for signal andsystem, whose proofs are constructive and yield analytic recovery methods. Therequired assumptions are satisfied by almost all signals, operators, andsampling vectors. Moreover, these guarantees not only hold for thefinite-dimensional setting but also carry over to infinite-dimensional spaces.Studying the sensitivity of the analytic recovery procedures, we also establisherror bounds for the applied approximate Prony method with respect to complexexponential sums.

Phase Retrieval and System Identification in Dynamical Sampling via Prony's Method