# Convergence Analysis of the Phase-Scheduled-Command FXLMS Algorithm with Phase Error
## 3 Performance Analysis
### Convergence Performance Analysis  
As mentioned above, in the process of \(\left\|\widetilde{\mathbf{w}}(n,\phi)\right\|_{2}\to 0\), the iteration focuses on the MSE difference given by Equation (25). The difference of \(\xi(n,\phi)\) can be defined as follows:  
\[ \begin{array}{rl}\Delta\xi(n,\phi)&=\xi(n+1,\phi)-\xi(n,\phi)\\ &=\sum\limits_{l=0}^{L-1}\lambda_{l}(n)[1-\mu\lambda_{l}(n)]^{2n+2}\widetilde {w}_{l}^{2}(0,\phi)-\sum\limits_{l=0}^{L-1}\lambda_{l}(n)[1-\mu\lambda_{l}(n)] ^{2n}\widetilde{w}_{l}^{2}(0,\phi)\\ &=-2\mu\sum\limits_{l=0}^{L-1}\lambda_{l}^{2}(n)[1-\mu\lambda_{l}(n)]^{2n} \widetilde{w}_{l}^{2}(0,\phi)+\mu^{2}\sum\limits_{l=0}^{L-1}\lambda_{l}^{3}(n )[1-\mu\lambda_{l}(n)]^{2n}\widetilde{w}_{l}^{2}(0,\phi)\end{array} \] (43) when the iteration is far from the convergence, \(\Delta\xi(n,\phi)<0,\forall n\), and according to the Lyapunov stability theory, the smaller the difference of the Lyapunov function is, the faster the system convergence speed is. Therefore, according to \(\left\|\mathbf{w}^{o}(n,\phi)\right\|_{2}>\left\|\mathbf{w}^{o}(n,0)\right\|_ {2}\), the phase error \(\phi\) satisfies the condition of \(\Delta\xi(n,\phi)<\Delta\xi(n,0)\), leading to a faster convergence speed when the command and disturbance signals are out of phase. However, due to the slow adaptation and the fact that \(\mu\ll 1\), \(\Delta\xi(n,\phi)\) is mostly determined by \(\mu\) and slightly by \(\widetilde{w}_{l}^{2}(0,\phi)\). Therefore, the increase in the convergence rate due to the phase error is minimal, and even negligible. Consequently, the command signal and disturbance signal being out of phase may lead to an inevitable increase in the convergence time constant. From Equation (25), it can be seen that the MSE is influenced by the phase error \(\phi\), and a ratio can be defined by  
\[\eta=\frac{\xi(n,\phi)}{\xi(n,0)}=\frac{\xi_{\min}+\sum\limits_{l=0}^{L-1}\lambda_ {l}(n)[1-\mu\lambda_{l}(n)]^{2n}\tilde{w}_{l}^{2}(0,\phi)}{\xi_{\min}+\sum \limits_{l=0}^{L-1}\lambda_{l}(n)[1-\mu\lambda_{l}(n)]^{2n}\tilde{w}_{l}^{2}(0,0)} \tag{44}\]  
Without loss of generality, the initial value of weight vector can be set as \(\mathbf{w}(0,\phi)=0\); then, from Equation (37), it can be imprecisely obtained:  
\[\frac{\mathbf{w}_{l}^{\theta}(n,\phi)}{\mathbf{w}_{l}^{\theta}(n,0)}=\sqrt{ \frac{C^{2}+D^{2}-2 CD\cos\phi}{\left(C-D\right)^{2}}}=\alpha \tag{45}\]  
where the \(\alpha\) is defined as a ratio related to the phase error. The Equation (44) can be rewritten as  
\[\eta=\frac{\xi(n,\phi)}{\xi(n,0)}=\frac{\xi_{\min}+\alpha^{2}\sum\limits_{l=0 }^{L-1}\lambda_{l}(n)[1-\mu\lambda_{l}(n)]^{2n}}{\xi_{\min}+\sum\limits_{l=0 }^{L-1}\lambda_{l}(n)[1-\mu\lambda_{l}(n)]^{2n}} \tag{46}\]  
When the algorithm is far from the steady state, assuming that \(\xi_{\min}\) is negligibly small, the ratio given by Equation (46) can be expressed as follows:  
\[\eta=\frac{\xi_{\min}+\alpha^{2}\sum\limits_{l=0}^{L-1}\lambda_{l}(n)[1-\mu \lambda_{l}(n)]^{2n}}{\xi_{\min}+\sum\limits_{l=0}^{L-1}\lambda_{l}(n)[1-\mu \lambda_{l}(n)]^{2n}}\approx\frac{\alpha^{2}\sum\limits_{l=0}^{L-1}\lambda_{l }(n)[1-\mu\lambda_{l}(n)]^{2n}}{\sum\limits_{l=0}^{L-1}\lambda_{l}(n)[1-\mu \lambda_{l}(n)]^{2n}}=\alpha^{2} \tag{47}\]  
Therefore, the phase error has nonnegligible influence on convergence performance when adaption is far from the steady state. This coincides with the conclusion that the phase error may increase the convergence time constant by resulting in a \(\mathbf{w}^{\theta}(n,\phi)\) that is distant from \(\mathbf{w}(0,\phi)\). The term \([1-\mu\lambda_{l}(n)]^{2n}\) converges to zero, as the adaptive filter converges. As given in Equation (44), the \(\xi(n,\phi)\) converges to \(\xi_{\min}\), and the ratio \(\eta\) converges to one. Consequently, the MSE performance at the steady state is slightly affected by the phase error \(\phi\). For nonstationary signals, according to Equation (14), the optimum weight vector \(\mathbf{w}_{non}^{\theta}(n,\phi)\) is time-varying.