# Iterative Parameter Estimation Algorithms for Dual-Frequency Signal Models
## 1 Introduction  
Parameter estimation is used widely in system identification [1; 2; 3] and signal processing [4; 5]. The existing parameter estimation methods for signal models can be classified into the following basic categories: the frequency-domain methods and the time-domain methods. The frequency-domain methods based on the fast Fourier transform (FFT) mainly include the Rife method, the phase difference method, etc. The accuracy of the Rife method is high in the case of noiseless or higher signal-to-noise ratio with adaptive sampling points; however, the error given by the Rife method is large if the signal frequency is near the DFT quantization frequency point. Therefore, some improved methods were proposed [6; 7]. For example, [PERSON] and [PERSON] used three largest spectral lines of the FFT spectrum to calibrate the frequency estimation [8]. [PERSON] et al. proposed a modified Rife method for the frequency shift of signals [9]. However, there is a risk that the shift direction of the frequency may be error. The time-domain methods mainly include the maximum likelihood algorithm, the subspace method and the self-correlation phase method. The maximum likelihood method is effective estimation to minimize the average risk though imposing significant computational costs. The methods in [10; 11; 12] used multiple autocorrelation coefficients or multi-steps autocorrelation functions to estimate the frequency so the amount of computations increase. In practice, actual signals are usually disturbed by various stochastic noise, and the time series signals like vibration signals or biomedical signals are subjected to dynamic excitations, including nonlinear and non-stationary properties. In order to solve the difficulties, time-frequency representations (TFR) provide a powerful tool since a TFR can give information about the frequencies contained in signal over time, such as short-time Fourier transform, wavelet transform, and Hilbert-Huang transform [13]. Recently, [PERSON] and [PERSON] presented an adaptive multiple signal classification-empirical wavelet transform methodology for accurate time-frequencyrepresentation of noisy non-stationary and nonlinear signals [14]. [PERSON] et al. used an empirical mode decomposition-like tool to decompose into the blocks functions, with slowly varying amplitudes and frequencies [15]. These existing approaches can obtain signal models indirectly, because they are realized based on the transform techniques like the Fourier transform and wavelet transform. In this paper, we propose the direct parameter estimation algorithms for signal modeling. The iterative methods and/or the recursive methods play an important role not only in finding the solutions of nonlinear matrix equations, but also in deriving parameter estimation algorithms for signal models [16; 17; 18; 19; 20; 21; 22]. Furthermore, the iterative algorithms can give more accurate parameter estimates because of making full use of the observed data. [PERSON] utilized the iterative methods to find the roots of nonlinear equations [23]. [PERSON] and [PERSON] presented an iterative algorithm for solving the generalized coupled Sylvester matrix equations over the generalized centro-symmetric matrices [24]. [PERSON] et al. proposed an iterative method for a class of complex symmetric linear systems [25]. [PERSON] derived a Newton iterative algorithm to the parameter estimation for dynamical systems [26]. [PERSON] et al. used a monotone iterative technique to get the existence of positive solutions and to seek the positive minimal and maximal solutions for a Hadamard type fractional integro-differential equation [27]. As an optimization tool, the [PERSON] method is useful for solving roots of nonlinear problems or deriving parameter estimation algorithms from observed data [28; 29; 30]. For a long time, the [PERSON] method has been utilized in much literature, such as transcendental equations, minimization and maximization problems, and numerical verification for solutions of nonlinear equations.