# Convergence Analysis of the Phase-Scheduled-Command FXLMS Algorithm with Phase Error
## 3 Performance Analysis
### Convergence Performance Analysis
In other words, the adaptive vector \(\mathbf{w}_{non}(n,\phi)\) converges to a time-varying objective with every updating step. Then, the iteration of Equation (18) can be regarded as a one-step steepest descent update. In cases far from convergence, and when the changing speed of \(\mathbf{w}_{non}^{\theta}(n,\phi)\) is much slower than the adaption speed of \(\mathbf{w}_{non}(n,\phi)\), it can be assumed that \(\mathbf{w}_{non}^{\theta}(n+1,\phi)\approx\mathbf{w}_{non}^{\theta}(n,\phi)\), and the MSE also, can be expressed by Equation (25). Under these assumptions, the following approximation can be made based on Equations (16)-(19), (23)-(24):  
\[\xi(n+1,\phi) \approx\xi_{\min}+\widetilde{\mathbf{w}  
which implies that the convergence rate of a one-step adaption may have some influence from the phase error. In one-step adaption, the phase error leads to a slight increase in convergence rate. However, as the adaptive vector updates, the optimum of \(\mathbf{w}_{non}^{o}(n,\phi)\) is time-varying, and there may be an influence of the adaptive filter reaching the optimum at every step. This may make the estimation of the influence difficult, or even impossible. However, as with the stationary case, due to the domination of \(\mu\) for the convergence rate, the influence on the convergence rate due to the phase error is minimal, and even negligible. Consequently, the phase error \(\phi\) may result in an inevitable increase in the convergence time constant.  
To investigate the effect of the phase error on a nonstationary system more clearly, a ratio \(\beta\) can be defined by  
\[\beta=\frac{\xi(n,\phi)}{\xi(n,0)}=\frac{\xi_{\min}+\sum\limits_{l=0}^{L-1} \lambda_{l}(n)[1-\mu\lambda_{l}(n)]^{2n}\widetilde{w}_{non,l}^{2}(n,\phi)}{ \xi_{\min}+\sum\limits_{l=0}^{L-1}\lambda_{l}(n)[1-\mu\lambda_{l}(n)]^{2n} \widetilde{w}_{non,l}^{2}(n,\phi)} \tag{50}\]  
Assuming that the change in the optimum of \(\mathbf{w}_{non}^{o}(n,\phi)\) is linear or quasi-linear, the ratio \(\beta\) can be rewritten similarly to Equation (46):  
\[\beta=\frac{\xi_{\min}+\sum\limits_{l=0}^{L-1}\lambda_{l}(n)[1-\mu\lambda_{l} (n)]^{2n}\widetilde{w}_{non,l}^{2}(n,\phi)}{\xi_{\min}+\sum\limits_{l=0}^{L-1} \lambda_{l}(n)[1-\mu\lambda_{l}(n)]^{2n}\widetilde{w}_{non,l}^{2}(n,\phi)} \approx\frac{\xi_{\min}+\alpha^{2}\sum\limits_{l=0}^{L-1}\lambda_{l}(n)[1-\mu \lambda_{l}(n)]^{2n}}{\xi_{\min}+\sum\limits_{l=0}^{L-1}\lambda_{l}(n)[1-\mu \lambda_{l}(n)]^{2n}} \tag{51}\]  
Similarly, at the far-from-steady state, the \(\xi_{\min}\) is always negligibly small, and the ratio given by Equation (51) can be expressed as follows:  
\[\beta=\frac{\xi_{\min}+\alpha^{2}\sum\limits_{l=0}^{L-1}\lambda_{l}(n)[1-\mu \lambda_{l}(n)]^{2n}}{\xi_{\min}+\sum\limits_{l=0}^{L-1}\lambda_{l}(n)[1-\mu \lambda_{l}(n)]^{2n}}\approx\alpha^{2} \tag{52}\]  
therefore, the phase error has nonnegligible influence on convergence performance at the far-from-steady state. This conclusion is identical to that of stationary cases. As the adaptive filter converges, the term \([1-\mu\lambda_{l}(n)]^{2n}\) converges to zero, and the ratio \(\beta\) converges to one. Consequently, the MSE performance at the steady state is slightly affected by the phase error \(\phi\).