Legendre Ramanujan Sums transform

In this paper, Legendre Ramanujan Sums transform(LRST) is proposed and derived by applying DFT to the complete generalized Legendre sequence (CGLS) matrices. The original matrix based Ramanujan Sums transform (RST) by truncating the Ramanujan Sums series is nonorthogonal and lack of fast algorithm, the proposed LRST has orthogonal property and O(Nlog2N) complexity fast algorithm. The LRST transform matrix is a sparse matrix and can be calculated with only additions and multiplications with more improvement in efficiency. It is suitable for image compression and transform coding. Meanwhile the LRST is useful to analyze to periodic signal especially for already known periodic sequences.