Improving four-parameter sine wave fitting by normalization

Abstract

Matrix equations need to be solved in each iterative round of four-parameter sine wave fitting. To ensure the robustness, normalizing factors rendering the minimum condition number (MCN) of the normal equation coefficient matrix was studied for the practical scenario (large sample size, and frequency well within [0, Nyquist Frequency]). Two parameter models, effective parameter model (EPM) and peak parameter model (PPM) were examined. The study shows that, firstly, with ignoring secondary entries of the coefficient matrix, the condition numbers depend solely on the amplitude of the four parameters. Secondly, for both models, there exists the optimal amplitude that renders the MCN. As amplitudes deviate from this optimal value, the condition number increases quickly. Thirdly, both the EPM and PPM possess a MCN close to 14. One numerical example shows that, if the realization data is normalized to the optimal amplitude, the condition numbers are indeed small, if the frequency lies in the practical range. With the initial frequency obtained from the method of the fast Fourier transform, outcomes at the fourth round iteration arrive to the Cramer-Rao bound.