  The filter function formalism quantitatively describes the dephasing of a
qubit by a bath that causes Gaussian fluctuations in the qubit energies with an
arbitrary noise power spectrum. Here, we extend this formalism to account for
more general types of noise that couple to the qubit through terms that do not
commute with the qubit's bare Hamiltonian. Our approach applies to any power
spectrum that generates slow noise fluctuations in the qubit's evolution. We
demonstrate our formalism in the case of singlet-triplet qubits subject to both
quasistatic nuclear noise and $1/\omega^\alpha$ charge noise and find good
agreement with recent experimental findings. This comparison shows the efficacy
of our approach in describing real systems and additionally highlights the
challenges with distinguishing different types of noise in free induction decay
experiments.
