  We study the performance of composite pulses in the presence of time-varying
control noise on a single qubit. These protocols, originally devised only to
correct for static, systematic errors, are shown to be robust to time-dependent
non-Markovian noise in the control field up to frequencies as high as ~10% of
the Rabi frequency. Our study combines a generalized filter-function approach
with asymptotic dc-limit calculations to give a simple analytic framework for
error analysis applied to a number of composite-pulse sequences relevant to
nuclear magnetic resonance as well as quantum information experiments. Results
include examination of recently introduced concatenated composite pulses and
dynamically corrected gates, demonstrating equivalent first-order suppression
of time-dependent fluctuations in amplitude and/or detuning, as appropriate for
the sequence in question. Our analytic results agree well with numerical
simulations for realistic $1/f$ noise spectra with a roll-off to $1/f^2$,
providing independent validation of our theoretical insights.
