  We study the extreme-ultraviolet (EUV) emissions modulated by leaky fast
sausage modes (FSMs) in solar active region loops and examine their
observational signatures via spectrometers like EIS. After computing fluid
variables of leaky FSMs with MHD simulations, we forward-model the intensity
and spectral properties of the Fe X 185~\AA~and Fe XII 195~\AA~lines by
incorporating non-equilibrium ionization (NEI) in the computations of the
relevant ionic fractions. The damping times derived from the intensity
variations are then compared with the wave values, namely the damping times
directly found from our MHD simulations. Our results show that in the
equilibrium ionization cases, the density variations and the intensity
variations can be either in phase or in anti-phase, depending on the loop
temperature. NEI considerably impacts the intensity variations but has only
marginal effects on the derived Doppler velocity or Doppler width. We find that
the damping time derived from the intensity can largely reflect the wave
damping time if the loop temperature is not drastically different from the
nominal formation temperature of the corresponding emission line. These results
are helpful for understanding the modulations to the EUV emissions by leaky
FSMs and hence helpful for identifying FSMs in solar active region loops.
