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