# Iterative Parameter Estimation Algorithms for Dual-Frequency Signal Models
## 1 Introduction
[PERSON] noted that the [PERSON] method can be used to give the generalization to systems of two equations and to solve optimization problems [31]. [PERSON] provided [PERSON]'s method or quasi-Newton methods for multidimensional unconstrained optimization and nonlinear equation problems [32]. [PERSON] studied the accelerated convergence of the Newton method by molding a given function into a new one that the roots remain unchanged, but it looks nearly linear in a neighborhood of the root [33]. [PERSON] et al. presented a guided recursive Newton method involving inverse iteration to solve the transcendental eigenproblems by reducing it to a generalised linear eigenproblem [34]. [PERSON] described the numerical solution of large-scale Lyapunov equations, Riccati equations, and linear-quadratic optimal control problems [35]. [PERSON] et al. used a quasi-Newton search technique and a barrier modification to enforce closed-loop stability for the H-infinity control problem [36]. [PERSON] et al. presented an iterative identification algorithm for Wiener nonlinear systems using the Newton method [37]. The gradient method with the search directions defined by the gradient of the function at the current point has been developed for optimization problems. For instance, [PERSON] used the gradient descent method for minimizing a nonlinear function of \(n\) real variables [38]. [PERSON] et al. studied the development, convergence theory and numerical testing of a class of gradient unconstrained minimization algorithms with adaptive step-size [39]. [PERSON] proposed a gradient-based iterative algorithm to find the solutions of the general Sylvester discrete-time periodic matrix equations. The moving data window with a fixed length is moved as time, which is a first-in-first-out sequence. When a new observation arrives, the data in this moving window are updated by including the new observation and eliminating the oldest one. The length of the moving window remains fixed. The algorithm computes parameter estimates using the observed data in the current window. Recently, [PERSON] et al. presented a moving-window second order blind identification for time-varying transient operational modal parameter identification of linear time-varying structures [41]. [PERSON] and [PERSON] developed a multiple-window moving horizon estimation strategy that exploits constraint inactivity to reduce the problem size in long horizon estimation problems [42]. This paper focuses on the parameter estimation problems of dual-frequency signal models. The main contributions of this paper are twofold. The basic idea is to present a gradient-based iterative (GI) algorithm and to estimate the parameters for signal models. Several estimation errors obtained by the Newton iterative and the moving data window based GI algorithms are compared to the errors given by the GI algorithm. To close this section, we give the outline of this paper. Section 2 derives a GI parameter estimation algorithm. Sections 3 and 4 propose the Newton and moving data window gradient based iterative parameter estimation algorithms. Section 5 provides an example to verify the effectiveness of the proposed algorithms. Finally, some concluding remarks are given in Section 6.