A novel algorithm to get the Fourier power spectra of a real sequence

For a real sequence of length of m = nl, we may deduce its congruence derivative sequence with length of l. The discrete Fourier transform of original sequence can be calculated by the discrete Fourier transform of the congruence derivative sequence. Based on the relation of discrete Fourier transforms between the two sequences, the features of Fourier power spectra of the integer and fractional periods for a real sequence have been investigated. It has proved mathematically that after calculating the Fourier power spectrum at an integer period, the Fourier power spectra of the fractional periods associated this integer period can be easily represented by the computational result of the Fourier power spectrum at the integer period for the sequence. A computational experience using a protein sequence shows that some of the computed results are a kind of Fourier power spectra corresponding to new frequencies which can't be obtained from the traditional discrete Fourier transform. Therefore, the algorithm would be a new realization method for discrete Fourier transform of the real sequence.