# Convergence Analysis of the Phase-Scheduled-Command FXLMS Algorithm with Phase Error
## 3 Performance Analysis
### Convergence Time Constant  
From Equation (25), it can be seen that \(1-\mu\lambda_{l}(n)<\mathbf{1}\), meaning that when \(n\rightarrow\infty\), then \(\left(1-\mu\lambda_{l}(n)\right)^{2n}\to 0\); from Equation (23), it can be easily obtained that when \(n\rightarrow\infty\), then \(\widetilde{\mathbf{w}}(n,\phi)\rightarrow\mathbf{0}\); that is, \(\mathbf{w}(n,\phi)\rightarrow\mathbf{w}^{o}(n,\phi)\). Based on the basic knowledge about the sequence limit, it holds that \(\widetilde{\mathbf{w}}(n,\phi)\rightarrow\mathbf{0}\) is equivalent to \(\left\|\widetilde{\mathbf{w}}(n,\phi)\right\|_{2}\to 0\). The norm \(\left\|\ \right\|_{2}\) of Equation (23) is given by  
\[ \begin{array}{l}\left\|\widetilde{\mathbf{w}}(n,\phi)\right\|_{2}=\left\| \left[\mathbf{I}-\mu\mathbf{\Lambda}(n)\right]^{n}\widetilde{\mathbf{w}}(0, \phi)\right\|_{2}\\ \begin{array}{l}\leq\left\|\left[\mathbf{I}-\mu\mathbf{\Lambda}(n)\right] ^{n}\right\|_{2}\left\|\widetilde{\mathbf{w}}(0,\phi)\right\|_{2}\\ \begin{array}{l}=\left[1-\mu\lambda_{\min}(n)\right]^{n}\left\|\widetilde{ \mathbf{w}}(0,\phi)\right\|_{2}\\ \begin{array}{l}\leq\left[1-\mu\lambda_{\min}(n)\right]^{n}\left\|\mathbf{ Q}^{T}(n)\right\|_{2}\left[\left\|\mathbf{w}^{o}(n,\phi)\right\|_{2}+\left\| \mathbf{w}(0,\phi)\right\|_{2}\end{array}\end{array}\end{array}\end{array}  \tag{32}\]  
Further, based on the knowledge about the LIMIT, the problem \(\left\|\widetilde{\mathbf{w}}(n,\phi)\right\|_{2}\to 0\) can be defined as follows. For any \(\varepsilon>0\), there exists an \(N\) such that when \(n>N\), it holds that \(\left\|\widetilde{\mathbf{w}}(n,\phi)\right\|_{2}<\varepsilon\). In practice, a real constant \(\varepsilon_{0}\ll 1\) can be defined, and when \(n>N\), then \(\left\|\widetilde{\mathbf{w}}(n,\phi)\right\|_{2}<\varepsilon_{0}\), which means that \(\left\|\widetilde{\mathbf{w}}(n,\phi)\right\|_{2}\) has converged. That is, when  
\[\left[1-\mu\lambda_{\min}(n)\right]^{n}\left\|\mathbf{Q}^{T}(n)\right\|_{2} \left[\left\|\mathbf{w}^{o}(n,\phi)\right\|_{2}+\left\|\mathbf{w}(0,\phi) \right\|_{2}\right]<\varepsilon_{0} \tag{33}\]  
the norm \(\left\|\widetilde{\mathbf{w}}(n,\phi)\right\|_{2}\) has converged. From the Equations (2) and (4), The required \(Z\)-transform of \(e^{\prime}(n)\) is denoted as \(E^{\prime}(z)\) and is expressed as  
\[E^{\prime}(z)=D(z)+Y(z)S(z)-C(z) \tag{34}\]  
where \(D(z)\), \(Y(z)\) and \(C(z)\) are the \(Z\)-transformations of the disturbance signal, output signal and command signal, respectively. Substituting the \(Z\)-transformations of Equation (3) into Equation (6), the \(E^{\prime}(z)\) can be expressed as  
\[E^{\prime}(z)=D(z)+W(z)X(z)S(z)-C(z) \tag{35}\]  
It can be seen that when the pseudo-error signal converges to zero, [i.e., \(E^{\prime}(z)=0\)], which indicates that \(W(z)\) realized the optimal transfer function  
\[W^{o}(z)=\frac{C(z)-D(z)}{X(z)S(z)}=\frac{C(z)-D(z)}{X^{\prime}(z)} \tag{36}\]  
where the superscript \"\(o\)\" indicates the optimum value. Then the magnitude of the optimal weight vector is expressed as  
\[\left\|\mathbf{w}^{o}(n,\phi)\right\|_{2}=\frac{\sqrt{C^{2}+D^{2}-2 CD\cos\phi }}{\left|X^{\prime}\right|} \tag{37}\]  
where \(\left|X^{\prime}\right|\) is the magnitude of the filtered reference signal.