Ramanujan cooperative sparse decomposition: a method for extracting periodic impulses

This paper proposes a feature extraction method based on Ramanujan collaborative dictionaries, termed Ramanujan cooperative sparse decomposition (RCSD), which aims to integrate number theory with sparse representation for more effective signal feature extraction. The core of the RCSD method lies in constructing a Ramanujan collaborative dictionary to enhance the generalization capability of sparse representation. This dictionary incorporates Ramanujan’s number theory, providing a rigorous mathematical foundation for accurately representing periodic components, while also enabling precise capture of periodic impulse information in signals. To solve the problem of sparse coding in this redundant dictionary, this paper uses the alternating direction multiplier method (ADMM) to solve the sparse coefficients, which ensures the convergence and efficiency of the algorithm. The experimental results show that compared with the traditional dictionaries, the proposed RCSD method has excellent performance in atomic correlation, calculation efficiency and comprehensive reconstruction error, and is especially suitable for extracting periodic impact features from signals, which is used in engineering applications such as fault diagnosis.