Stability analysis of the FBANC system having delay error in the estimated secondary path model

The Derivation of the Stability Bound of the Feedback ANC System That Has an Error in the Estimated Secondary Path Model

“…The frequency response of the adaptive filter W(e jω ) can be specified as a simple formation in assumption that the primary noise d(n) is a single-tone sinusoid d(n) = cos(ωn + φ), and the coefficients of the ANC filter W(e jω ) converge to the optimum state [8]. Where, ω is frequency of the single-tone primary noise d(n) and φ is phase.…”

“…In this section, the closed-form stability bound equation is derived by utilizing the magnitude response and phase response of the two terms of the open-loop frequency response, i.e., converge to the optimum state [8]. Where,  is frequency of the single-tone primary noise d(n) and  is phase.…”

“…This paper is considered as a major extension of the authors' previous work [8], which can be considered as an incomplete work to reflect only the magnitude response of the system's open loop response in deriving the system's stability bound. This paper provides a more realistic stability bound by exploiting both the magnitude and phase of the FBANC system's open loop response in deriving the stability bound equation.…”

“…Particularly, the delay of the acoustic path, which is relatively long compare to that of the electric path, makes the order of the characteristic equation even higher. Therefore, many literatures on the stability analysis of feedback ANC systems have been analyzed using the frequency domain analysis approach [3,7,8]. However, the stability bound result is shown in the transfer function model domain, which reflects, indirectly, the effect of the physical domain parameters such as delay length of an acoustic system, ANC filter length, and center frequency of target noise.…”

“…Theoretically, if the secondary path model is perfectly estimated, the feedback ANC scheme has no stability problem [2][3][4]. However, in practice, it is not available to exactly estimate them as a linear filter model, since the electro-acoustic devices and the acoustic path are a non-linear and time-varying system [3,[7][8][9]. If the estimated secondary path model is not perfect, its adverse effects considering both the stability aspect of the feedback loop and the performance issues, such as convergence and stability, of the FXLMS-based adaptive algorithm must be analyzed.…”