## 3 Stability Analysis of Feedback ANC System  
A new stability analysis method is proposed to obtain the stability bound of the feedback ANC system as a form of a closed-form equation. The closed-form stability bound equation is derived in terms of the length of the delay error of secondary path, the ANC filter length, and the frequency of a single-tone primary noise. In the proposed method, the stability bound equation is obtained by separating the magnitude and phase response equations of the open loop function \(W(e^{i\omega})[e^{-i\omega\Delta}-e^{-i\omega\Delta}]\) within the Nyquist stability bound and exploiting their linear approximation. The magnitude and phase response equations are approximated as linear equations, assuming that the primary noise is a single-tone sinusoid and the secondary path can be modeled as a pure delay.  
The stability definition of the Nyquist criterion is that the polar plot of the open loop frequency response must not enclose the Nyquist point (\(-1\), \(0\)) for all frequency range [12; 13]. Instead of this original Nyquist definition, we introduce an alternative stability definition that the magnitude of the open loop frequency response must be less than one at the specific frequency point \(\omega_{180}\), which makes the phase of the open loop frequency response equal \(\pm\pi\). From the proposed stability definition, the FBANC system's stability can be guaranteed when the following formula is satisfied.  
\[|W(e^{i\omega})|\cdot|e^{-i\omega\Delta}-e^{-i\omega\Delta}|_{\,\omega=\omega _{180}}<1.0, \tag{4}\]  
in which the frequency point \(\omega_{180}\) can be expressed as  
\[\angle W(e^{i\omega})[e^{-j\omega\Delta}-e^{-j\omega\hat{\Delta}}]\Big{|}_{ \omega=\omega_{180}}=\pm\pi.\]  
At the frequency point \(\omega_{180}\), the open-loop frequency response \(W(e^{i\omega})[e^{-j\omega\Delta}-e^{-j\omega\hat{\Delta}}]\) is on the real-axis of z-plane.  
In Figure 2, an example Nyquist plot of the feedback ANC system (solid line) is drawn within unit circle (dotted line) on z-plane. The proposed stability definition is well-described with this example figure. If the Nyquist plot's most left cross point on the real-axis does not reach the Nyquist point \((-1,0)\), the open loop frequency response never encloses the Nyquist point so that the Nyquist stability criterion always is guaranteed.  
With the assumption that the adaptive algorithm converges to a stationary point and the primary noise is one of the specific types, i.e., pure-tone sinusoid or auto-regressive process model, each of the two magnitude responses, i.e., \(|W(e^{i\omega})|\) and \(|e^{-j\omega\Delta}-e^{-j\omega\Delta}|\), can be represented as a combined pattern using typical functions, i.e., a single sinusoid, a sinc function, or a quadratic function.  
To derive the close-form equation of the stability bound, each term of the magnitude responses is linearized at the point where the magnitude response of \(|e^{-j\omega\Delta}-e^{-j\omega\Delta}|\) alone becomes \(1.0\). Such approximation guarantees conservative bound of stability. However, the approximated stability bound is not significantly different from the true bound as verified through simulations in Section 5, which results from the fact that the term \(|e^{-j\omega\Delta}-e^{-j\omega\Delta}|\) always dominates with faster varying characteristics of magnitude response than \(|W(e^{j\omega})|\). Then, the overall magnitude response, i.e., the product of the two linearized terms, becomes a quadratic function. By finding the range for the quadratic equation to be less than \(1.0\), the stability bound is readily derived.  
A significant feature of the proposed stability determination method is that it enables the analytic derivation of the stability bound equation without obtaining the roots of the characteristic equation, which is generally high-order. The proposed stability analysis method can be also utilized for other feedback control applications' stability bound derivation.