  We introduce a toy model for the time-frequency structure of fast radio
bursts (FRB), in which the observed emission is produced as a narrowly-peaked
intrinsic spectral energy distribution sweeps down in frequency across the
instrumental bandpass as a power-law in time. Though originally motivated by
emission models which invoke a relativistic shock, the model could in principle
apply to a wider range of emission scenarios. We quantify the burst's
detectability using the frequency bandwidth over which most of its
signal-to-noise ratio (SNR) is accumulated. We demonstrate that by varying just
a single parameter of the toy model-the power-law index \beta of the frequency
drift rate-one can transform a long (and hence preferentially time-resolved)
burst with a narrow time-integrated spectrum into a shorter burst with a broad
power-law time-integrated spectrum. We suggest that burst-to-burst diversity in
the value of \beta could generate the dichotomy between burst duration and
frequency-width recently found by CHIME. In shock models, the value of \beta is
related to the radial density profile of external medium, which in light of the
preferentially longer duration of bursts from repeating sources may point to
diversity in the external environments surrounding repeating versus one-off FRB
sources.
