Fourier-transform infrared (FTIR) spectra can show artifacts, for example, in the vicinity of the overtone when light is doubly modulated. Technical approaches, i.e., modifications to the spectrometer setup, have been devised in order to reduce those artifacts. Elimination of the artifacts was achieved only partly but at the expense of a loss of intensity or an increase in the noise level. We devised a computational demodulation scheme that is capable of almost fully reducing the artifacts with neither a loss of spectroscopic information nor an increase in the noise level. This has been demonstrated for the FTIR absorption spectra in the overtone regions of HCl(g) and CH(g).