# Variable Tap-Length Algorithm with Mixed Parameter
## 1 Introduction  
The adaptive algorithm involves considerable investigation into signal processing, being widely applied to diverse fields including active noise control, beamforming, and echo canceling [1; 2]. The tap-length is a special parameter of the adaptive algorithm. It plays a decisive role in the performance of transient and steady-state. Too small a tap-length will cause divergence [3; 4]. To avoid this situation, the tap-length is usually assigned a large value in general algorithms. Nevertheless, a large tap-length also results in a large amount of computation and poor steady-state performance. Thus, finding a method to adjust tap-length is necessary. Many researchers have proposed variable tap-length algorithms. [PERSON] et al. [5] presented a gradient descent algorithm that adjusts tap-length according to the negative gradient direction of squared estimation error. A segmented filter algorithm [6] separated the entirety into some parts and the tap-length updated in one part based on the difference of the error in the previous part. Among these variable tap-length algorithms, the fractional variable tap-length (FT) algorithm [7] is the most favored strategy.  
The step-size of tap-length (ST) is important to the performance of the FT algorithm. Large ST causes fluctuations and fast convergence speed. Conversely, small ST causes the under-modeling problem and good steady-state performance [8]. Generally, there is a tradeoff between stability and convergence speed. However, ST is usually fixed in the FT algorithm. The error width of the FT algorithm also influences convergence rate and steady-state performance. Large error widthimproves convergence rate but results in poor steady-state performance. Small error width causes little fluctuation but decreases convergence performance [9]. Thus, many researchers proposed time-varying strategies [10; 11]. Nevertheless, these algorithms cannot avoid the interference of system error or indicate the exact parameter at the same time. More importantly, they are only concerned with either ST or error width and ignore the effect of the other parameters. Coordination of these two parameters is lacking. These algorithms cannot provide the good performance.  
Algorithms in [12; 13] improved a certain performance compared with algorithms in [10; 11]. However, they only concerned with either ST or error width and also ignored the effect of the other one. There is no reasonable analysis or adjustment strategy for the other parameter in algorithms [12; 13]. Their performance may be affected by this. Thus, a mixed variable tap-length algorithm that uses the time-varying strategies of ST and error width is proposed in this paper. The proposed algorithm combines algorithms in [12; 13] and balances them reasonably. ST and error width of the proposed algorithm all have good adaptive ability. The significant difference between the proposed algorithm and the algorithms in [12; 13] is the performance in high noise environment. As ST and error width can meet the corresponding values at different stages, the proposed algorithm can perform well even in the high noise environment. By using a suitable and unified strategy to control ST and error width, the proposed algorithm improves convergence rate and steady-state performance. The accurate tap-length of the proposed algorithm makes the accurate estimate of the corresponding weight coefficients. Moreover, the proposed algorithm avoids interference of the system error and provides a good level of robustness. Some experiments were performed to compare the performance.  
The rest of this paper is organized as follows: Section 2 introduces the least mean square (LMS) algorithm, the influence of tap-length and the FT-LMS algorithm. The proposed mixed variable tap-length algorithm and its performance analysis are introduced in Sections 3 and 4, respectively. The detailed simulation experiments and results are outlined in Section 5. The conclusions are summarized in Section 6.