  Observations find that some fast radio bursts (FRBs) have extremely
narrow-band spectra, i.e., $\Delta\nu/\nu_0 \ll 1$. We show that when the
angular size of the emission region is larger than the Doppler beaming angle,
the observed spectral width ($\Delta\nu/\nu_0$) exceeds 0.58 due to the {\it
high latitude effects} for a source outside the magnetosphere, even when the
spectrum in the source's comoving frame is monochromatic. The angular size of
the source for magnetospheric models of FRBs can be smaller than the Doppler
beaming angle, in which case this geometric effect does not influence the
observed bandwidth. We discuss various propagation effects to determine if any
could transform a broad-spectrum radio pulse into a narrow-spectrum signal at
the observer's location. We find that plasma lensing and scintillation can
result in a narrow bandwidth in the observed spectrum. However, the likelihood
of these phenomena being responsible for the narrow observed spectra with
$\Delta\nu/\nu_0 < 0.58$ in the fairly large observed sample of FRBs is
exceedingly small.
