  We present a Fourier-domain approach to modulations and delays of
gravitational wave signals, a problem which arises in two different contexts.
For space-based detectors like LISA, the orbital motion of the detector
introduces a time-dependency in the response of the detector, consisting of
both a modulation and a varying delay. In the context of signals from
precessing spinning binary systems, a useful tool for building models of the
waveform consists in representing the signal as a time-dependent rotation of a
quasi-non-precessing waveform. In both cases, being able to compute transfer
functions for these effects directly in the Fourier domain may enable
performance gains for data analysis applications by using fast frequency-domain
waveforms. Our results generalize previous approaches based on the stationary
phase approximation for inspiral signals, extending them by including delays
and computing corrections beyond the leading order, while being applicable to
the broader class of inspiral-merger-ringdown signals. In the LISA case, we
find that a leading-order treatment is accurate for high-mass and low-mass
signals that are chirping fast enough, with errors consistently reduced by the
corrections we derived. By contrast, low-mass binary black holes, if far away
from merger and slowly-chirping, cannot be handled by this formalism and we
develop another approach for these systems. In the case of precessing binaries,
we explore the merger-ringdown range for a handful of cases, using a simple
model for the post-merger precession. We find that deviations from leading
order can give large fractional errors, while affecting mainly subdominant
modes and giving rise to a limited unfaithfulness in the full waveform.
Including higher-order corrections consistently reduces the unfaithfulness, and
we further develop an alternative approach to accurately represent post-merger
features.
