Ramanujan subspace pursuit for signal periodic decomposition

The period estimation and periodic decomposition of a signal represent long-standing problems in the field of signal processing and biomolecular sequence analysis. To address such problems, we introduce the Ramanujan subspace pursuit (RSP) based on the Ramanujan subspace. As a greedy iterative algorithm, the RSP can uniquely decompose any signal into a sum of exactly periodic components by selecting and removing the most dominant periodic component from the residual signal in each iteration. In the RSP, a novel periodicity metric is derived based on the energy of the exactly periodic component obtained by orthogonally projecting the residual signal into the Ramanujan subspace. The metric is then used to select the most dominant periodic component in each iteration. To reduce the computational cost of the RSP, we also propose the fast RSP (FRSP) based on the relationship between the periodic subspace and the Ramanujan subspace and based on the maximum likelihood estimation of the energy of the periodic component in the periodic subspace. The fast RSP has a lower computational cost and can decompose a signal of length N into a sum of K exactly periodic components in O(KNlogN). In short, the main contributions of this paper are threefold: First, we present the RSP algorithm for decomposing a signal into its periodic components and theoretically prove the convergence of the algorithm based on the Ramanujan subspaces. Second, we present the FRSP algorithm, which is used to reduce the computational cost. Finally, we derive a periodic metric to measure the periodicity of the hidden periodic components of a signal. In addition, our results show that the RSP outperforms current algorithms for period estimation.