Four-parameter sine wave fitting by Gram–Schmidt orthogonalization

Four-parameter sine wave fitting based on the least square error principle is nonlinear. An iterative approach to these nonlinear equations requires solving matrix equations in each round. The Gram–Schmidt orthogonalization (GSO) is employed here to speed up the solving of these matrix equations and to ensure the robustness of the procedure. The factorization of the normal matrix by GSO can be achieved nearly analytically. Its computation burden is significantly reduced compared to that of singular value decomposition. Moreover, by utilizing Clenshaw’s recurrence formula, the total computation burden in each round is down by an order of O(3N), where N is the sampling size. Two numerical tests have been carried out to validate the proposed procedure.