Abstract: By analyzing some properties of discrete Fourier transform (DFT), exploiting the symmetries of DFT and decomposing a complex sequence into four sequences, a few mistakes in the direct formulas for simultaneously calculating the DFTs of two real sequences and the DFT & inverse discrete Fourier transform (IDFT) of real sequences which were proposed by Gunther were corrected, the new direct formulas of simultaneously calculating DFT of a real sequence and IDFT of the DFT of another real sequence were proposed and proved.