  Context. Wide-field spectrometers are needed to deal with current
astrophysical challenges that require multiband observations at millimeter
wavelengths. An example of these is the KIDs Interferometer Spectrum Survey
(KISS), which uses two arrays of kinetic inductance detectors (KIDs) coupled to
a Martin-Puplett interferometer (MPI). KISS has a wide instantaneous field of
view (1 deg in diameter) and a spectral resolution up to 1.45 GHz in the
120-180 GHz electromagnetic band. The instrument is installed on the 2.25 m
Q-U-I JOint TEnerife telescope in Teide Observatory (Tenerife, Canary Islands),
at an altitude of 2395 m above sea level. Aims. This work presents an original
readout modulation method developed to improve the sky signal reconstruction
accuracy for types of instruments for which a fast sampling frequency is
required both to remove atmospheric fluctuations and to perform full
spectroscopic measurements on each sampled sky position. Methods. We first
demonstrate the feasibility of this technique using simulations. Then, we apply
such a scheme to on-sky calibration. Results. We show that the sky signal can
be reconstructed to better than 0.5% for astrophysical sources, and to better
than 2% for large background variations such as in "skydip", in an ideal
noiseless scenario. The readout modulation method is validated by observations
on-sky during the KISS commissioning campaign. Conclusions. We conclude that
accurate photometry can be obtained for future KID-based MPI.
