For the requirements of the discrete sampling of the single side interferogram beyond zero path difference in Fourier spectroscopy and the need of the fast real time data processing, the key problem of real time data processing in the spectrical reconstruction has been analyzed in this paper. In the experiment, Michelson interferometer is applied to acquire the interference fringes, the 3.5~5 μm atmospheric window is the measured object and the 632.8 nm laser is used as the referenced light source. The interferential signals are obtained by the dot HgCdTe detector.The interferential data is predisposed, apodized, phase corrected and calibrated to rebuild the spectrums based on the Matlab software. The experiment indicates that the interferential data of the atmospheric window of 4 096 points is phase corrected by the method of Mertz in which the amount of the little double sided data is 512 points and apodized by the symmetrical triangle function, the total data is apodized by the asymmetrical triangle function, Fourier transformed and calibrated to quickly and accurately rebuild the spectrum which is high resolution and the spectrial resolution is about 4 cm-1.