# Variable Tap-Length Algorithm with Mixed Parameter
## 4 Performance Analysis  
### Parameter Analysis  
In this section, the analysis of the proposed algorithm and general guidelines for the parameters \(\rho_{1}\) and \(\rho_{2}\) are exhibited. Given the information provided by [PERSON] et al. [8] and [PERSON] et al. [13], we deduced the reasonable values of \(\rho_{1}\) and \(\rho_{2}\). Based on the analysis provided by [PERSON] et al. [8], the low bound and the up bound of \(\gamma\) are as follows:  
\[\gamma_{l}=\frac{\alpha}{(\Delta-k\delta)\sigma_{x}^{2}\sigma_{c}^{2}-\frac{ \Delta\mu^{\prime}\sigma_{v}^{2}}{(2-\mu^{\prime})\mathrm{L}_{opt}}} \tag{22}\]  
\[\gamma_{u}=\frac{-\frac{K_{2}\alpha}{K_{3}}-\sqrt{\frac{K_{2}^{2}\alpha^{2}}{K _{3}^{2}}+\frac{4(\eta^{2}\beta^{2}+\alpha^{2})}{K_{3}}}}{2} \tag{23}\]  
\[K_{1}=\frac{\Delta\mu^{\prime}\sigma_{v}^{2}}{(2-\mu^{\prime})\mathrm{L}_{opt}} \tag{24}\]  
\[K_{2}=2+4\sigma_{v}^{2}+\frac{4\mathrm{L}_{opt}K_{1}}{\Delta} \tag{25}\]  
\[K_{3}=2K_{1}K_{2}-2K_{1}\sigma_{c}^{2}\Big{(}(\Delta-1)\sigma_{x}^{2}+3\Big{)} \tag{26}\]  
where \(\sigma_{c}^{2}\) is the variance of the tail sequence of the optimal weight. \(k\) and \(\eta\) are small positive numbers which can be selected according to the requirement of the steady tap-length oscillation. The values of \(\gamma_{l}\) and \(\gamma_{u}\) are also the guidelines of ST in all FT algorithms. These two values are coincident in different FT algorithms. The fixed \(\gamma\) is selected by these two parameters in FT algorithms. Then, Equation (15) can also be expressed as:  
\[q_{1}(n)=\lambda^{n}q_{1}(0)+(1-\lambda)\sum_{i=1}^{n}\lambda^{n-i}|\mathrm{ FE}(i)| \tag{27}\]  
Taking the expectation on both sides, Equation (27) becomes:  
\[E\{q_{1}(n)\}=\lambda^{n}q_{1}(0)+(1-\lambda)\sum_{i=1}^{n}\lambda^{n-i}E\{| \mathrm{FE}(i)|\}\approx\lambda^{n}q_{1}(0)+(1-\lambda^{n})E\{|\mathrm{FE}(n)|\} \tag{28}\]  
In the steady-state, \(n\rightarrow\infty\), as \(\lambda\) is smaller than one, and Equation (28) becomes:  
\[E\{q_{1}\{\infty\}\}=E\{|\mathrm{FE}(\infty)|\} \tag{29}\]  
The values of \(|\mathrm{FE}(n)|\) and \(\mathrm{FE}(n)\) can be expressed from Equation (10). \[|\mathrm{FE}(n)|=\left|e_{L(n)}^{(L(n))}(n)^{2}-e_{L(n)-\Delta}^{(L(n))}(n)^{2 }\right|=\left|\frac{L_{f}(n+1)-L_{f}(n)+\alpha}{-\gamma(n)}\right| \tag{30}\]  
\[\mathrm{FE}(n)=e_{L(n)}^{(L(n))}(n)^{2}-e_{L(n)-\Delta}^{(L(n))}(n)^{2}=\frac{ L_{f}(n+1)-L_{f}(n)+\alpha}{-\gamma(n)} \tag{31}\]  
In addition, in the steady-state, \(n\rightarrow\infty\), the mean values of Equations (30) and (31) are expressed as follows:  
close to the optimal solution. Similarly, we infer that ST has the same characteristic. Thus, ST and error width should be large enough to speed up convergence in the transient state. In the steady-state, ST and error width should be reduced to improve the steady-state performance. Given these characteristics, their changing trends should be the same. The proposed algorithm uses the same FE to control ST and error width so that they are relatively consistent. The main theoretical difference between the proposed algorithm and algorithm [13] is that the proposed algorithm adds the analysis and changing strategy of the other important parameter ST. ST and error width of the proposed algorithm all have good adaptive ability. They can meet the corresponding values at different stages. In the next step, we analyzed the characteristic of FE at different stages according to [PERSON] et al. [13]. In the transient state, \(\textbf{W}_{opt}\), \(\textbf{X}_{L_{opt}}\), and \(\textbf{P}\) are defined as follows:  
\[\textbf{W}_{opt}=\left[ \begin{array}{c}\textbf{W}_{L(n)-\Delta}^{*}\\ \overline{\textbf{W}}_{L(n)-\Delta+1:L(n)}^{*}\\ \overline{\textbf{W}}_{L(n)+1:L_{opt}}^{*}\end{array} \right] \tag{42}\]  
\[\textbf{X}_{Lopt}=\left[ \begin{array}{c}\textbf{x}_{L(n)-\Delta}^{*}\\ \overline{\textbf{x}}_{L(n)-\Delta+1:L(n)}^{*}\\ \overline{\overline{\textbf{x}}}_{L(n)+1:L_{opt}}^{*}\end{array} \right] \tag{43}\]  
\[\textbf{P}=\left[