## 2 The Whole Stage Stability of the ANC System with Online SPM
### Mean Weight Behavior
#### 2.1.2 ANC Filter  
First, we show the Wiener solution of the ANC filter. The signal from the output \(y(n)\) of the ANC filter to the noise control point is:  
\[ \begin{array}{ll}y^{\prime}(n)&=\sum\limits_{i=0}^{L-1}s_{L-1-i}y(n-i)\\ &=\sum\limits_{i=0}^{L-1}w^{\rm T}(n-i)x(n-i)s_{L-1-i}\\ &=X_{s}^{\rm T}(n)W(n)\end{array} . \tag{9}\]  
Combining (9) with (6):  
\[ \begin{array}{ll}E(f^{2}(n))&=E(e_{2}^{2}(n))+E(u^{2}(n))\\ &+2\sum\limits_{i=0}^{L-1}s_{L-1-i}E(d(n)\mathbf{x}^{\rm T}(n-i))\mathbf{w}\\ &+\mathbf{w}^{\rm T}\sum\limits_{j=0}^{L-1}\sum\limits_{i=0}^{L-1}s_{L-1-j}s_ {L-1-i}E(\mathbf{x}(n-i)\mathbf{x}^{\rm T}(n-j))\mathbf{w}\end{array} . \tag{10}\]  
In (10), by taking the derivative of the vector \(\mathbf{w}\) and setting the derivative to zero, the Wiener solution \(\mathbf{w}^{*}\) of the ANC filter can be presented as:  
\[\mathbf{w}^{*}=-\mathbf{R}_{s\rm S}^{-1}\mathbf{P}_{s}, \tag{11}\]  
where:  
\[ \begin{array}{ll}\mathbf{R}_{s\rm S}&=\sum\limits_{j=0}^{L-1}\sum\limits_{ i=0}^{L-1}s_{L-1-j}s_{L-1-i}E(\mathbf{x}(n-i)\mathbf{x}^{\rm T}(n-j))\\ &=\sum\limits_{j=0}^{L-1}\sum\limits_{i=0}^{L-1}s_{L-1-j}s_{L-1-i}\mathbf{R}_ {j-i}\\ &\mathbf{P}_{s}=\sum\limits_{i=0}^{L-1}s_{L-1-i}E(d(n)\mathbf{x}^{\rm T}(n-i) ).\end{array} , \tag{12}\]  
The correlation matrix of the reference signal is defined as:  
\[\mathbf{R}_{i}=E(\mathbf{x}(n)\mathbf{x}^{\rm T}(n-i)). \tag{14}\]  
It is evident that \(\mathbf{R}_{-i}=\mathbf{R}_{i}^{\rm T}\). Next, the steady-state solution of the ANC filter is derived. Combining Equations (2), (3) and (9), and taking the expectations on both sides of (2), we obtain:  
\[ \begin{array}{ll}E(\mathbf{w}(n+1))&=E(\mathbf{w}(n))+\mu_{w}E(\mathbf{X}(n )\hat{\mathbf{s}}(n)(d(n)\\ &-\mathbf{X}_{s}^{\rm T}(n)\mathbf{W}(n))+e_{v}(n)+u(n)))\end{array} , \tag{15}\]  
where \(\mathbf{X}(n)=[\mathbf{x}(n),\mathbf{x}(n-1),\cdots,\mathbf{x}(n+1-L)]\) is the reference matrix. As \(n\rightarrow\infty\), \(E(\mathbf{w}(n+1))=E(\mathbf{w}(n))\), we obtain:  
\[E(\mathbf{X}(n)\hat{\mathbf{s}}(n)\mathbf{X}_{s}^{\rm T}(n))\mathbf{W}(\infty )=-E(d(n)\mathbf{X}(n)\hat{\mathbf{s}}(n)), \tag{16}\]  
where \(\mathbf{W}^{\rm T}(\infty)=[\mathbf{w}^{\rm T}(\infty),\mathbf{w}^{\rm T}( \infty),\cdots,\mathbf{w}^{\rm T}(\infty)]^{\rm T}\) is the steady-state augmented weight vector. Using (16), the steady state can be derived as follows:  
\[\mathbf{w}(\infty)=-\mathbf{R}_{s\rm S}^{-1}\mathbf{P}_{s}, \tag{17}\]where:  
\[\mathbf{R}_{s\delta} =\sum\limits_{j=0}^{L-1}\sum\limits_{i=0}^{L-1}s_{L-1-j}(\infty)s_{ L-1-i}E(\mathbf{x}(n-i)\mathbf{x}^{\mathsf{T}}(n-j)) \tag{18}\] \[=\sum\limits_{j=0}^{L-1}\sum\limits_{i=0}^{L-1}s_{L-1-j}(\infty)s _{L-1-i}\mathbf{R}_{j-i}\] \[\mathbf{P}_{s}=\sum\limits_{i=0}^{L-1}\hat{s}_{L-1-i}(\infty)E(d( n)\mathbf{x}^{\mathsf{T}}(n-i)). \tag{19}\]  
As shown in (17), the accuracy of modeling the secondary path influences the steady state of the ANC filter. As in \(n\to\infty\), if the SPM filter converges to the actual secondary path, the ANC filter will converge to its Wiener solution. We begin by analyzing the stability conditions of the ANC filter. The coefficient vector of the ANC filter is defined as \(\Delta\mathbf{w}(n)=\mathbf{w}(n)-\mathbf{w}(\infty)\). The augmented coefficient vector of the ANC filter is:  
\[\Delta\mathbf{W}(n) =\mathbf{W}(n)-\mathbf{W}(\infty) \tag{20}\] \[=\left[ \begin{array}{c}\Delta\mathbf{w}(n)\\ \Delta\mathbf{w}(n-1)\\ \vdots\\ \Delta\mathbf{w}(n+1-L)\end{array}