Hierarchical recursive signal modeling for multifrequency signals based on discrete measured data

This paper studies the problem of parameter estimation for the multifrequency sine signals, which have multiple characteristic parameters such as the amplitudes, phases, and frequencies. It is noted that the signal output is nonlinear with respect to the phase and frequency parameters while it is linear with respect to the amplitude parameters. This feature inspires us to separate all of the characteristic parameters into a linear parameter set and a nonlinear parameter set, where the linear set is composed of the amplitude parameters and the nonlinear set is composed of the phase parameters and the frequency parameters. After the parameter separation, two identification submodels are constructed for optimizing the linear parameter set and the nonlinear parameter set. Then the nonlinear identification model becomes a linear identification submodel and a nonlinear identification submodel. Therefore, the nonlinear optimization for minimizing the objective function is converted into the combination of the quadratic optimization and nonlinear optimization. Based on the separable identification submodels, a recursive least squares subalgorithm and a recursive gradient subalgorithm are proposed for identifying the linear parameters and nonlinear parameters, respectively. Moreover, an interactive estimation algorithm is designed to remove the related parameter sets between the subalgorithms and a hierarchical identification method is presented by combining the subalgorithms. For the purpose of tracking the time-varying, a forgetting factor is introduced to improve the convergence speed. The numerical examples are provided to qualify the performance of the proposed method based on some performance measures.