## 2 The Whole Stage Stability of the ANC System with Online SPM
### Mean Weight Behavior
#### 2.1.2 ANC Filter
\right]\.\]  
By using expressions (9) and (2), subtracting \(\mathbf{w}(\infty)\) from both sides of expression (2), \(\Delta\mathbf{w}(n+1)\) can be computed as:  
\[\Delta\mathbf{w}(n+1)= \Delta\mathbf{w}(n)-\mu_{w}\mathbf{G}(n)\Delta\mathbf{W}(n)+\mu_ {w}\mathbf{b}(n), \tag{21}\]  
where:  
\[\mathbf{G}(n)=[\mathbf{G}_{0}(n),\mathbf{G}_{1}(n),\cdots,\mathbf{G}_{L-1}(n)], \tag{22}\]  
\[\mathbf{G}_{j}(n) =(\sum\limits_{i=0}^{L-1}s_{L-1-i}s_{L-1-j}\mathbf{x}(n-i)) \mathbf{x}^{\mathsf{T}}(n-j) \tag{23}\] \[+(\sum\limits_{i=0}^{L-1}\Delta s_{L-1-i}(n)s_{L-1-j}\mathbf{x}( n-i))\mathbf{x}^{\mathsf{T}}(n-j)\,\] \[\mathbf{b}(n) =\mathbf{X}(n)(\mathbf{s}(n)+\Delta\mathbf{s}(n))\times\] (24) \[(d(n)-\mathbf{s}^{\mathsf{T}}\mathbf{X}(n)\mathbf{w}(\infty)+u( n)+e_{v}(n))\] \[=\mathbf{X}(n)(\mathbf{s}(n)+\Delta\mathbf{s}(n))(\eta(n)+u(n)+e_{v}(n))\] \[\eta(n)=d(n)-\mathbf{s}^{\mathsf{T}}\mathbf{X}(n)\mathbf{w}(\infty),\]  
where \(\eta(n)\) denotes the optimal error of the ANC filter. We assume that the update \(\Delta\mathbf{w}(n)\) is very slow and the value of \(\mu_{w}\) is so small that \(\Delta\mathbf{w}(n)\approx\Delta\mathbf{w}(n-1)\approx\cdots\approx\Delta \mathbf{w}(n+1-L)\), and we can obtain:  
\[\Delta\mathbf{w}(n+1) =\Delta\mathbf{w}(n)-\mu_{w}\sum\limits_{j=0}^{L-1}\mathbf{G}_{j }(n)\Delta\mathbf{w}(n)+\mu_{w}\mathbf{b}(n) \tag{26}\] \[=\Delta\mathbf{w}(n)-\mu_{w}(\mathbf{A}(n)+\mathbf{B}(n))\Delta \mathbf{w}(n)+\mu_{w}\mathbf{b}(n)\]  
where:  
\[\mathbf{G}(n)=[\mathbf{G}_{0}(n),\mathbf{G}_{1}(n),\cdots,\mathbf{G}_{L-1}(n)], \tag{22}\]0  
\[\mathbf{G}(n)=[\mathbf{G}_{0}(n),\mathbf{G}_{1}(n),\cdots,\mathbf{G}_{L-1}(n)], \tag{22}\]1  
The sum \(\sum\limits_{i=0}^{L-1}\Delta s_{L-1-i}^{2}\) can be estimated by (5) and (6). \[ \begin{array}{ll}E(e_{v}^{2}(n))&=\sigma_{v}^{2}\sum\limits_{i=0}^{L-1}\Delta s _{L-1-i}^{2}\\ &\leq E\big{(}f^{2}(n)\big{)}=E\big{(}e_{v}^{2}(n)+e_{x}^{2}(n)+u^{2}(n)\big{)} \end{array} . \tag{40}\]  
Let \(P_{f}(n)=E(f^{2}(n))\) and \(P_{x}(n)=E\big{(}x^{2}(n)\big{)}\) represent the power of the error signal \(f(n)\) and reference signal \(x(n)\), respectively. \(P_{f}(n)\) and \(P_{x}(n)\) are estimated using an exponentially smoothed estimator. \[P_{f}(n+1)=\lambda P_{f}(n)+(1-\lambda)f^{2}(n), \tag{41}\]  
\[P_{x}(n+1)=\lambda P_{x}(n)+(1-\lambda)x^{2}(n), \tag{42}\]  
where \(\lambda\) is a forgetting factor close to one. Because the specific value of each \(s_{i}\) is not known, it seems difficult to estimate \(\sum\limits_{i=0}^{L-1}s_{L-1-i}^{2}\). However, the sum shows the power amplification ability of the system to a certain extent. Here, it is assumed that the upper bound \(R\) of the signal power amplification gain is known. Even if the value of \(R\) is unknown, the measurement of \(K\) is relatively easy. \[\sum\limits_{i=0}^{L-1}s_{L-1-i}^{2}\leq R. \tag{43}\]  
Using (39)-(43), the upper bound of \(\mu_{w}\) can be expressed as:  
\[\mu_{w}\leq\frac{2}{NP_{x}(n)(P_{f}(n)/\sigma_{v}^{2}+3R)}. \tag{44}\]  
This suggests that the stability condition changed with the relative error of the SPM filter. As the ANC and SPM filters converge, \(P_{f}(n)/\sigma_{v}^{2}\ll 3R\), the stability condition can be replaced by:  
\[\mu_{w}\leq\frac{2}{3 NRP_{x}(n)}, \tag{45}\]  
which is the stability condition for the mean square behavior in [9]. This is only acceptable when the relative error of the secondary path is small. The stability condition in Equation (44) is more robust. The upper bound given above is applicable to all periods of ANC system operation.