# Variable Tap-Length Algorithm with Mixed Parameter
## Mixed Parameter Fractional Variable Tap-Length Algorithm  
Equation (10) is the updating equation of tap-length in the transient state. As \(\alpha\) is a small positive leakage parameter, the core method to increase or decrease the tap-length is \(\gamma\left[\left(e_{L(n)}^{(L(n))}(n)\right)^{2}-\left(e_{L(n)-\Delta}^{(L(n) )}(n)\right)^{2}\right]\). A large value of the core part is needed to quickly change the tap-length. Thus, large ST \(\gamma\) and error width \(\Delta\) are required to increase the convergence speed of tap-length in the transient state. In the steady-state, the updating equation of tap-length can be expressed as follows [8]:  
\[L_{f}(\infty)=L_{opt}+\Delta-\frac{\alpha}{\sigma_{x}^{2}\sigma_{c}^{2}}-\frac {\Delta\mu\sigma_{v}^{2}}{(2-\mu)L_{opt}\sigma_{x}^{2}\sigma_{c}^{2}} \tag{13}\]  
where \(\sigma_{x}^{2},\sigma_{v}^{2}\) are the variances of the input signal and noise signal, respectively. \(\sigma_{c}^{2}\) is the variance of the tail elements of the optimal weight \(\mathbf{W}_{opt}\). For the determined noise signal, a small ST and errorwidth are needed to eliminate the last three items in the steady-state. Thus, small ST and error width allow tap-length to avoid the large fluctuations in the steady-state. From the above, we determined that ST and error width have similar trends. In the beginning, ST and error width should be large to increase convergence speed and avoid the under-modeling problem. In the steady-state, small ST and error width are needed to improve steady-state performance and avoid overestimation of tap-length. In other words, to improve performance, the FT algorithm should simultaneously have the same time-varying strategies of ST and error width. Therefore, a novel method which adjusts ST and error width based on the fragment-full error (FE) is proposed in this study. The proposed algorithm reasonably combines ST and error width so that they coordinate and cooperate with each other. The complete iteration formula is:  
\[FE(n)=\left(e_{L(n)}^{(L(n))}(n)\right)^{2}-\left(e_{L(n)-\Delta}^{(L(n))}(n) \right)^{2} \tag{14}\]  
\[q_{1}(n)=\lambda q_{1}(n-1)+(1-\lambda)|FE(n)| \tag{15}\]  
\[q_{2}(n)=\lambda q_{2}(n-1)+(1-\lambda)FE(n) \tag{16}\]  
\[\gamma(n)=\rho_{1}q_{1}(n) \tag{17}\]  
\[\Delta(n)=\min(\Delta_{max},\rho_{2}|q_{2}(n)|) \tag{18}\]  
where \(q_{1}(0)\) and \(q_{2}(0)\) are zero. \(\lambda\) is a smoothing parameter and should be close to 1. \(\rho_{1}\) and \(\rho_{2}\) are the factors and their values can be referenced in Section 4. As the changing trends of ST and error width are the same, we used the same variable FE to make their changes consistent. Full error and fragment error all contain the noise signal. Subtracting them to obtain FE can eliminate the influence of the noise signal. In other words, FE is noise-free. Thus, FE reflects the state accurately and avoids external interference. Once the tap-length is far from the optimum, FE will be large. In the steady state, FE is small. Thus, FE acts as an effective method to adjust ST and error width. In the initial stage, a large FE brings a large ST and error width to avoid the suboptimal value of tap-length and improve the convergence speed. In the steady-state, a small FE produces a corresponding ST and error width to avoid fluctuation and improve the steady-state performance. We also add the restricted function introduced by [PERSON] et al. [13] into the proposed algorithm. \[\zeta(n)=\frac{\text{C}*\left(\left(e_{L(n)}^{(L(n))}(n)\right)^{2}-\left(e_{L (n)-\Delta}^{(L(n))}(n)\right)^{2}\right)}{\left[1+\left(\left(e_{L(n)}^{(L(n) )}(n)\right)^{2}-\left(e_{L(n)-\Delta}^{(L(n))}(n)\right)^{2}\right)^{2} \right]^{0.5}} \tag{19}\]  
\[LFF(n)= \begin{cases}\lambda_{1}*\zeta(n)+(1-\lambda_{1})*FE(n)&\text{if }e(n)*e(n)<\text{EST}\\ \lambda_{2}*\zeta(n)+(1-\lambda_{2})*FE(n)&\text{else}\end{cases}  \tag{20}\]  
\[L_{f}(n+1)=\left(L_{f}(n)-\alpha\right)-\gamma(n)*LFF(n) \tag{21}\]  
where EST is the estimated value of MSE in deficient and sufficient tap-length situations.