In this paper, a new modulation method defined as Ramanujan Periodic Subspace Division Multiplexing (RPSDM) is proposed using Ramanujan subspaces. Each subspace contains an integer valued Ramanujan Sum (RS) and its circular downshifts as a basis. The proposed RPSDM decomposes the linear time-invariant wireless channels into a Toeplitz stair block diagonal matrices, whereas Orthogonal Frequency Division Multiplexing (OFDM) decompose the same into diagonal. Advantages of such structured subspaces representation are studied and compared with an OFDM representation in terms of Peak-Average Power Ratio (PAPR) and Bit-Error-Rate (BER). Zero Forcing (ZF) and Minimum Mean Square Error (MMSE) detectors are applied to evaluate the performance of OFDM and RPSDM techniques. Finally, the simulation results show that the proposed design (with an additional receiver complexity) outperforms OFDM under both detectors.