  In this manuscript we address the problem of deriving \emph{analytic}
expressions for calculating universal decoherence-induced errors in qubits
undergoing arbitrary, unitary, time-dependent quantum-control protocols. For a
qubit undergoing unitary decoherence the evolution of a qubit state in the
presence of time-varying semiclassical fields may be treated geometrically. We
show that the fidelity of an arbitrary control operation may then be expressed
to arbitrary order in terms of experimentally relevant spectral characteristics
of the noise and the control over all Cartesian directions and accounting for
noise cross-correlations. We formulate \emph{control matrices} in the time
domain to capture the effects of piecewise-constant control, and convert them
to generalized Fourier-domain filter functions. Such generalized filter
functions may therefore be derived for complex temporally modulated control
protocols, accounting for susceptibility to rotations of the qubit state vector
in three dimensions. Taken together, this framework provides a computationally
efficient means to calculate the effects of universal noise on arbitrary
quantum control protocols without the need for time-consuming simulations of
Bloch vector evolution. As a concrete example, we apply our method to treating
the problem of dynamical decoupling incorporating realistic control pulses of
arbitrary duration or form, including the replacement of simple $\pi$-pulses
with complex dynamically corrected gates.
