# Convergence Analysis of the Phase-Scheduled-Command FXLMS Algorithm with Phase Error
## 2 Active Sound Profiling Algorithms
### Characteristic Equation With Phase Error  
According to Equations (1)-(3) and (6), the mean square error (MSE) of the pseudo-error, which is related to the phase error, can be expressed by  
\[ \begin{split}\xi(n,\phi)&=E\Big{\{}\big{[}e^{ \prime}(n)\big{]}^{2}\Big{\}}\\ &=E\Big{\{}\big{[}c(n)-d(n)\big{]}^{2}\Big{\}}-2\mathbf{p}(n,\phi )^{T}\mathbf{w}(n,\phi)+\mathbf{w}^{T}(n,\phi)\mathbf{R}(n)\mathbf{w}(n,\phi) \end{split}  \tag{9}\]  
where \(E\{\cdot\}\) denotes the expected value, the \(\mathbf{p}(n,\phi)\) is the desired-to-input cross-correlation vector at iterations \(n\), when the phase error of the command signal and disturbance signal is \(\phi\) and the \(\mathbf{R}(n)\) is the input autocorrelation matrix, which are defined as follows  
\[\mathbf{p}(n,\phi)=E\big{\{}\big{[}c(n)-d(n)\big{]}\mathbf{x}^{\prime}(n) \big{\}} \tag{10}\]  
Figure 2: Block diagram of the PSC-FXLMS algorithm. \[\mathbf{R}(n)=E[\mathbf{x}^{\prime}(n){\mathbf{x}^{\prime}}^{T}(n)] \tag{11}\]  
According to the concept of steepest descent, the iteration with the phase error \(\phi\) can be implemented into  
\[\mathbf{w}(n+1,\phi)=\mathbf{w}(n,\phi)-\frac{\mu}{2}\
abla\xi(n,\phi) \tag{12}\]  
where \(\mu\) is the step size and vector \(\
abla\xi(n,\phi)\) denotes the gradient of the error noise function. From Equation (9), the gradient can be easily calculated as follows  
\[\
abla\xi(n,\phi)=-2\mathbf{p}(n,\phi)+2\mathbf{R}(n)\mathbf{w}(n,\phi) \tag{13}\]  
when the gradient is zero [i.e., \(\
abla\xi(n,\phi)=0\)], the filter \(\mathbf{w}(n,\phi)\) minimizes the MSE cost function \(\xi(n,\phi)\) and the optimum of \(\mathbf{w}(n,\phi)\) can be expressed as  
\[\mathbf{w}^{o}(n,\phi)=\mathbf{p}(n,\phi)/\mathbf{R}(n) \tag{14}\]  
where the \(\mathbf{w}^{o}(n,\phi)\) is the expected optimum weight vector. Substituting Equation (14) into Equation (9), the minimum MSE \(\xi_{\min}\) can be obtained  
\[\xi_{\min}=E\Big{\{}[c(n)-d(n)]^{2}\Big{\}}-\mathbf{p}(n,\phi)^{T}\mathbf{w}^ {o}(n,\phi) \tag{15}\]  
then the MSE the pseudo-error can be calculated by  
\[\xi(n,\phi)=\xi_{\min}+\widehat{\mathbf{w}}(n,\phi)^{T}\mathbf{R}(n)\widehat{ \mathbf{w}}(n,\phi) \tag{16}\]  
where the \(\widehat{\mathbf{w}}(n,\phi)\) is the difference of the \(\mathbf{w}(n,\phi)\) and \(\mathbf{w}^{o}(n,\phi)\), and it is defined as  
\[\widehat{\mathbf{w}}(n,\phi)=\mathbf{w}(n,\phi)-\mathbf{w}^{o}(n,\phi) \tag{17}\]  
Substituting Equation (13) into Equation (12); the iteration is expressed as  
\[\mathbf{w}(n+1,\phi)=[\mathbf{I}-\mu\mathbf{R}(n)]\mathbf{w}(n,\phi)+\mu \mathbf{p}(n,\phi) \tag{18}\]  
Subtracting the optimum weight vector from both sides of the Equation (18) and according to the Equation (17), the process of the adaptive weight vector iteration can be equivalent to the iteration of the \(\widehat{\mathbf{w}}(n,\phi)\), and it can be rewritten from Equation (18) and expressed as  
Figure 2:0  
The autocorrelation matrix defined in Equation (11) can be decomposed into  
Figure 2:1  
where \(\mathbf{\Lambda}(n)=diag[\lambda_{0}(n),\lambda_{1}(n),\ldots,\lambda_{L-1}( n)]\), \(\mathbf{Q}(n)\) is the modal matrix formed by the eigenvectors of \(\mathbf{R}(n)\) and it satisfies the following relationship: \(\mathbf{I}=\mathbf{Q}(n)\mathbf{Q}^{T}(n)\). A rotated weight misalignment vector can be defined by  
Figure 2:2  
from Equation (19), it holds that:  
Figure 2:3  
and  
Figure 2:4Further, Equation (16) can be rewritten as follows  
Figure 2:5  
and  
Figure 2:6 Figure 2:7