## 2 The Stability Problem of the Feedback ANC System  
In this section, we address the stability issue caused by the secondary path modeling error of a feedback ANC system. A typical structure of an adaptive feedback ANC using the FXLMS system [1] is shown in Figure 1. The basic idea of the adaptive feedback ANC is to estimate the primary noise \(d(n)\) and use it as a reference signal \(x(n)\) for the ANC filter. The reference signal \(x(n)\) is synthesized using the estimate of secondary path model \(\hat{S}(z)\) as shown in (1). The feedback ANC system suppresses primary noise \(d(n)\) by generating anti-noise \(y(n)\) with a loud speaker. Signal \(e(n)\) denotes residual noise measured by an error microphone.  
\[\hat{d}(n)=\hat{y}^{\prime}(n)+e(n)=y(n)*\hat{s}(n)+e(n)\equiv x(n). \tag{1}\]  
The followings are the rest of definitions.  
\(W(z)\): N-tap adaptive filter,  
\(S(z)\): Secondary path, the electro-acoustic coupling path from the loudspeaker to the error microphone,  
\(\hat{S}(z)\): The internal estimated model of \(S(z)\).  
Stability of a linear feedback system can be determined by examining whether its overall transfer function's pole locations are within the unit circle. The FBANC system's closed-loop transfer function from desired input \(d(n)\) to error output \(e(n)\) can be obtained as,  
\[H(z)=\frac{E(z)}{D(z)}=\frac{1-W(z)\hat{S}(z)}{1+W(z)[S(z)-\hat{S}(z)]}. \tag{2}\]  
When there is no error in the estimated secondary path model, \(\hat{S}(z)=S(z)\), the denominator of (2), becomes one and the overall transfer function \(H(z)\) becomes  
\[H(z)=1-W(z)\hat{S}(z)=1-W(z)S(z).\]  
in which the FBANC system has no poles and becomes an all-zero system that guarantees its stability. However, when the estimated secondary path \(\hat{S}(z)\) has an error, the denominator of (2) becomes a high order equation. This is because the adaptive filter \(W(z)\) is generally tens to hundreds order, and the  
Figure 1: Block diagram of a typical adaptive feedback ANC system using the FXLMS.  
secondary path \(S(z)\) and its estimation \(\hat{S}(z)\) also contain a very long delay because of the acoustical propagation path.  
In order to examine the effect of the delay error explicitly, the transfer functions of the secondary path and its estimated model are modeled as a simple delay component as \(S(z)=z^{-\Delta}\) and \(\hat{S}(z)=z^{-\hat{\Delta}}\), respectively. The constant magnitude component of the both secondary path models can be accommodated in the adaptive filter model \(W(z)\). The delay \(\Delta\) corresponds to the total time, including the acoustic propagation time between the speaker and microphone, as well as the AD/DA converting processing, which generally ranges hundreds of microseconds to tens of milliseconds. By substituting the delay models for \(H(z)\), the transfer function can be expressed as  
\[H(z)=\frac{1-W(z)z^{-\hat{\Delta}}}{1+W(z)[z^{-\Delta}-z^{-\hat{\Delta}}]}. \tag{3}\]  
This result transfer function shows the full behavior of the system that has a delay error in its secondary path model. However, from this function, it is hard to recognize the effect of delay error intuitively, since the transfer function-based expression does not directly provide explicit and geometric interpretations in the domain of physical parameters, such as delay, noise characteristics parameters [4]. Moreover, the combined effect of the adaptive filter \(W(z)\) on the secondary path error term is determined depending on characteristics of the primary noise \(d(n)\). In the next section, we introduce a new stability analysis method to obtain a stability bound equation that intuitively represents the effect of the physical parameters on the system stability.