  Monitoring time-dependence with diffusion MRI yields observables sensitive to
compartment sizes (restricted diffusion) and membrane permeability (water
exchange). However, restricted diffusion and exchange have opposite effects on
the diffusion-weighted signal, which can confound parameter estimates. In this
work, we present a signal representation that captures the effects of both
restricted diffusion and exchange up to second order in b-value and is
compatible with gradient waveforms of arbitrary shape. The representation
features mappings from a gradient waveform to two scalars that separately
control the sensitivity to restriction and exchange. We demonstrate that these
scalars span a two-dimensional space that can be used to choose waveforms that
selectively probe restricted diffusion or exchange, in order to eliminate the
correlation between the two phenomena. We found that waveforms with specific
but unconventional shapes provide an advantage over conventional pulsed and
oscillating gradient acquisitions. We also show that parametrisation of
waveforms into a two-dimensional space can be used to understand protocols from
other approaches that probe restricted diffusion and exchange. For example, we
find that the variation of mixing time in filter-exchange imaging corresponds
to variation of our exchange-weighting scalar at a fixed value of the
restriction-weighting scalar. Numerical evaluation of the proposed signal
representation using Monte Carlo simulations on a synthetic substrate showed
that the theory is applicable to sizes in the range 2 - 7 micrometres and
barrier-limited exchange in the range 0 - 20 s$^{-1}$. The presented theory
constitutes a simple and intuitive description of how restricted diffusion and
exchange influence the signal as well as how to design a protocol to separate
the two effects.
