The optical difference variation of the photoelastic modulate fourier transform spectrometers (PEM-FTS) is sinusoidal and high-speed, so more than 105 interferograms per second is acquired. In order to fleetly and accurately recover the spectrums of the signals which are sampled by equal interval, the cubic spline interpolation algorithm, nonuniform discrete Fourier transform (NDFT) with phase compensation algorithm, and accelerating nonuniform fast Fourier transform (NUFFT) algorithm are applied in the data processing system of PEM-FTS. It is found that the accelerating NUFFT is fast, and its speed is an order of magnitude higher than that of NDFT with phase compensation, and two times of that of the FFT with cubic spline interpolation algorithm. The advantage of the method is obvious when the number of sampling points is increased. And the precision of the method is high with its deviation less than 0.00054. The method can be applied in PEM-FTS.