  The dynamics of quantum systems are unavoidably influenced by their
environment and in turn observing a quantum system (probe) can allow one to
measure its environment: Measurements and controlled manipulation of the probe
such as dynamical decoupling sequences as an extension of the Ramsey
interference measurement allow to spectrally resolve a noise field coupled to
the probe. Here, we introduce fast and robust estimation strategies for the
characterization of the spectral properties of classical and quantum dephasing
environments. These strategies are based on filter function orthogonalization,
optimal control filters maximizing the relevant Fisher Information and
multi-qubit entanglement. We investigate and quantify the robustness of the
schemes under different types of noise such as finite-precision measurements,
dephasing of the probe, spectral leakage and slow temporal fluctuations of the
spectrum.
