This paper describes in detail a new method to calculate the integral that appears in the expression of cross-section Doppler broadening. This method is based on a Fourier transform and seems quite promising as our tests suggest it is able to reach any required precision within a reasonable amount of calculation time. Another method to calculate Doppler-broadened cross sections based on Gauss quadrature will be presented, even if it requires too much computation time to be of practical use. Both methods have been implemented in a new nuclear data-processing software called GAIA, which is currently under development at the French Institut de Radioprotection et de Sûreté Nucléaire. Results of the comparison of the broadened cross sections obtained with the GAIA methods and with the NJOY processing system are discussed in the paper.