## 2 The Whole Stage Stability of the ANC System with Online SPM
### Mean Square Behavior
#### 2.2.3 Steady-State MSE  
From (4) and \(e(n)=d(n)-y^{\prime}(n)+v^{\prime}(n)+u(n)\), the steady-state MSE is given by:  
\[E(e^{2}(\infty))=E(e_{x}^{2}(\infty))+\sigma_{v}^{2}\mathbf{s}^{\mathsf{T}} \mathbf{s}+\sigma_{u}^{2}, \tag{56}\]  
where:  
\[E(e_{x}^{2}(\infty))=\sigma_{\eta}^{2}+tr(E(\mathbf{A}(\infty)\mathbf{c}_{\mu }(\infty))), \tag{57}\]  
\(\sigma_{\eta}^{2}\) is the variance of \(\eta(n)\). The second term on the right side of (57) is the steady-state excess MSE (EMSE) of the ANC filter. Using (50) the steady-state EMSE of the ANC filter can be expressed as:  
\[\begin{array}{ll}tr(E(\mathbf{A}(\infty)\mathbf{c}_{w}(\infty)))&=\frac{\mu _{w}}{2}tr(E(\mathbf{A}(\infty)\mathbf{c}_{w}(\infty)\mathbf{A}(\infty))\\ &+E(\mathbf{B}(\infty)\mathbf{c}_{w}(\infty)\mathbf{B}^{\mathsf{T}}(\infty)) \\ &+\frac{\mu_{w}}{2}tr(E(\mathbf{b}^{\mathsf{T}}(\infty)\mathbf{b}(\infty))) \end{array}. \tag{58}\]  
Similar to \(n\rightarrow\infty\), \(\Delta\mathbf{s}(n)\) tends to a zero vector, which results in \(E(\mathbf{B}(\infty)\mathbf{c}_{w}(\infty)\mathbf{B}^{\mathsf{T}}(\infty))\) being small compared to \(E(\mathbf{A}(\infty)\mathbf{c}_{w}(\infty)\mathbf{A}(\infty)\). Hence, \(E(\mathbf{B}(\infty)\mathbf{c}_{w}(\infty)\mathbf{B}^{\mathsf{T}}(\infty)\) can be neglected. Substituting (24) into (58) yields:  
\[E(\mathbf{b}^{\mathsf{T}}(\infty)\mathbf{b}(\infty))=(\sigma_{\eta}^{2}+ \sigma_{u}^{2}+\sigma_{e_{v}}^{2}(\infty))(E(\mathbf{A}(\infty))+E(\mathbf{C }(\infty))), \tag{59}\]  
where:  
\[\sigma_{e_{v}}^{2}(\infty)=tr(E((\mathbf{R}_{vv}\mathbf{c}_{s}(\infty))), \tag{60}\]  
is the steady-state EMSE of SPM filter. The definition of \(\mathbf{C}(n)\) is:  
\[\mathbf{C}(n)=\sum_{j=0}^{L-1}\sum_{i=0}^{L-1}(\Delta s_{L-1-i}(n)\Delta s_{L -1-j}(n)\mathbf{x}(n-i)\mathbf{x}^{\mathsf{T}}(n-j)). \tag{61}\]  
Using assumption 1, it can be derived that:  
\[E(\mathbf{C}(\infty)))=\sigma_{x}^{2}E(\sum_{i=0}^{L-1}\Delta s_{L-1-i}^{2}( \infty))=\sigma_{x}^{2}\frac{\sigma_{e_{v}}^{2}(\infty)}{\sigma_{v}^{2}} \mathbf{I}. \tag{62}\]  
From (58), (59), and (62), the steady-state EMSE of ANC can be expressed as:  
\[\begin{array}{ll}tr(E(\mathbf{A}(\infty)\mathbf{c}_{w}(\infty)))&=\frac{\mu _{w}}{2}tr(E(\mathbf{A}(\infty)\mathbf{c}_{w}(\infty)\mathbf{A}(\infty))\\ &+\frac{\mu_{w}}{2}(\sigma_{\eta}^{2}+\sigma_{u}^{2}+\sigma_{e_{v}}^{2}( \infty))\sigma_{x}^{2}\frac{\sigma_{e_{v}}^{2}(\infty)}{\sigma_{v}^{2}}\\ &\times tr(E(\mathbf{A}(\infty))+\mathbf{I})\end{array}. \tag{63}\]  
Using (46), the steady-state EMSE of the SPM filter can be computed as:  
\[\sigma_{e_{v}}^{2}(\infty)=\frac{\mu_{s}L\sigma_{v}^{2}(\sigma_{e_{v}}^{2}( \infty)+\sigma_{u}^{2})}{(2-\mu_{s}\sigma_{v}^{2}(L+2))}. \tag{64}\]  
As shown above, we cannot find the closed form of the steady-state EMSE of the ANC filter for the existence of \(E(\mathbf{A}(\infty)\mathbf{c}_{w}(\infty)\mathbf{A}(\infty)\). Moreover, the SPM and the ANC filters' steady-state EMSE are coupled with each other. Both steady-state EMSEs can be controlled by step size \(\mu_{w}\) and \(\mu_{s}\). Therefore, when both filters tend to be stable, a smaller step-size factor can be used to reduce the excess mean square error of the system.