Decomposition of a Periodic Perturbed Signal with Unknown Perturbation Frequency by the Method of Variable Projection

This paper presents a new approach to separate a signal into its periodic and aperiodic components; whereby, the exact frequency of the periodic component is unknown. In other words, it is shown how to determine the underlying trend of a periodic perturbed signal and simultaneously identify the shape of the periodic perturbation. Therefore the signal is modeled by a nonlinear design matrix containing periodic basis functions, which depend on the unknown frequency, and aperiodic basis functions, more precisely discrete orthogonal polynomials (DOP). The nonlinear least squares problem of computing the model coefficients is solved by the method of variable projection. A well chosen partitioning of the design matrix enables an orthogonal residualization corresponding to a generalized Eckart-Young-Mirsky matrix approximation, which yields an efficient implementation of the variable projection method. This implementation is thoroughly tested using Monte Carlo simulations and the results are compared with those obtained by the classical implementation of the method of variable projection.