###### Abstract
More specifically, the accuracy of the ANC control filter fitting affects the accuracy of the error signal used to update the SPM filter. This situation is more complex for an ANC control filter. The accuracy of the SPM filter not only affects the accuracy of the error signal, but also affects the accuracy of the reference signal used to update the ANC filter. Many researchers have conducted detailed analyses and discussions on important issues, such as convergence speed, steady-state mean square error, and system stability for ANC systems. [PERSON] analyzed the performance of FxLMS on the mean and mean square using a spherical invariant process. It was found that the delay in the coefficient update had only a minor effect on the stability bounds and steady-state behavior of a Gaussian white noise input [21]. Without using the independence assumption, [PERSON] et al. conducted a comprehensive analysis of the convergence performance of the FxLMS on the mean. Comprehensive research was conducted on the mean weight behavior of the FxLMS, and the impact of the error in SPM was studied [22]. [PERSON] et al. analyzed the convergence of the ANC system with band-limited white noise and the moving average secondary path. In addition, the pure delay secondary path was discussed in detail. Through a linear model, the upper bound of the adaptive step length was derived [23]. [PERSON] et al. analyzed the influence of auxiliary noise power using the mean square criterion and determined the optimal step size of the ANC system [24]. [PERSON] et al. conducted a comprehensive theoretical analysis of the narrowband ANC system fitted online using the secondary path. For the narrowband ANC system, the difference equation determines the dynamic performance of the system and derives the closed expression of the system in the steady state [25]. [PERSON] et al. proposed a strategy for scaling auxiliary noise based on the properties of the narrowband ANC system to improve its convergence and steady-state performance [26]. [PERSON] et al. used Gaussian white noise as the input signal and found that the difference equation describes the convergence performance of the system in terms of the mean and mean square. In addition, the steady-state error was used to decouple the difference equations to obtain the stability conditions of the system, and an auxiliary noise adjustment method was proposed [9]. [PERSON] et al. used statistical mechanics methods to analyze the mean square error of the FxLMS in the steady and transient states. Under the actual primary path condition, the theoretical model above can analyze the input signal of non-Gaussian white noise [27, 28]. [PERSON] et al. used an enhanced weighted vector to analyze the performance of any input reference signal and derived its stability bound [10]. However, the calculation of the eigenvalues of the correlation matrix is complicated. The above analysis of ANC filter convergence performance is based on the condition that the secondary path tends to be stable. In other words, the upper bound of the step size of the ANC filter can only be obtained if the relative error of the secondary path is small. To ensure the stability of the system, most online SPM methods have to use offline modeling for a period of time in advance so that the relative error of the secondary path is less than that of \(-5\) dB [14; 15; 17; 18]. This cannot guarantee the convergence of the system in the start-up stage, and sometimes the system does not have the conditions for offline modeling. Moreover, a sudden change in the secondary path can occur when the system is running. The change in the power amplification gain may cause a sizeable relative error in the secondary path and divergence of the entire system. In this paper, to study the influence of secondary path error on the ANC system, especially in the start-up stage and the sudden change in the secondary path, we analyze the performance of the ANC system with online SPM in detail.