Because of the advantages of high resolution etc, Fourier transform spectrometers have been used as a new generation of atmospheric sounders. Such a spectrometer will acquire a very large quantity of data. To reduce the amount of the data to be transmitted, the data should be processed on board. One of the methods is use of in-orbit spectral recovery. Since there exists phase difference in the interferograms sampled by non zero optical path difference sampling, the complex spectra obtained by using Fourier transformation should be corrected in phase so as to generate the real spectra.The Field-Programmable Gate Array (FPGA) is a promising on-board data processing hardware platform. Phase error sources and phase correction methods are presented. The square root method is suitable for a FPGA to implement. In view of the problem that a lot of hardware resources are needed and poor precision is obtained when the square root is calculated by using the traditional method, a Coordinate Rotation Digital Computer (CORDIC) algorithm is proposed for the calculation of complex modulus spectrum and phase angle. The FPGA implementation structure of the CORDIC algorithm is mainly discussed, and the required precision, speed and hardware resources are analyzed. The results show that the phase and modulus of complex spectra are good in precision and speed when the CORDIC algorithm is used. Less hardware resources are needed.