## 1 Introduction  
System identifications, including channel estimation, seismic system identification, noise or echo cancellation, and image recovery have been successfully realized by adaptive filters (AFs) [1]. Among conventional AF algorithms, the least-mean-square (LMS) and normalized LMS (NLMS) algorithms play important roles due to their low computational complexity and mathematical tractability [2; 3]. However, such LMS-type algorithms suffer from filtering performance degradation under the heavy-tailed impulsive interference and heavy-colored input signal [4]. To remit the negative influence of heavy-colored input data on the convergence performance [5], derived the affine projection algorithm (APA), which is based on the idea of affine subspace projection. After that, the family of APA is developed by various methods [5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16]. Rather than a single current data vector used in LMS and NLMS, APA updates weight coefficients through multiple, most recent input vectors. Increasing the projection order of APA accelerates the convergence speed at the expense of computational complexity. However, APA is derived by the minimization of \(\ell_{2}\)-norm, which results in dramatic performance degradation under the non-Gaussian noise, especially for impulsive interference. It is known that lower-order norms are robust against impulsive disturbances. From the perspective of computational simplicity, a great number of algorithms are derived by minimizingthe \(\ell_{1}\)-norm to achieve robust filtering [10; 17; 18]. Moreover, [19] combined the AP method with \(\ell_{1}\)-norm optimization to derive a robust AP-like algorithm, which is termed the affine projection sign algorithm (APSA). To satisfy the contradictory requirements between fast convergence speed and small steady-state misalignment, the variable step-size APSA algorithms (VSS-APSAs) were proposed [20; 21; 22]. However, such VSS-APSAs reduce the tracking performance when the system abruptly changes. Therefore, to enhance the tracking performance of VSS-APSA, two convex combination methods are applied to APSA from different viewpoints. One is called the combination of APSA (CAPSA), and the other is named the combined step-size APSA (CSS-APSA). The CSS-APSA uses two different step sizes--the large one is related to faster convergence, and the small one is related to a lower steady-state misadjustment. In addition, the convex combination of affine projection-type algorithms using an impulsive noise indicator was developed [23]. However, [23] needs more calculations to realize good convergence performance. After that, the cost function based on the sigmodial active function was applied to a family of robust adaptive filtering algorithms, and it can be further simplified by the first-order Taylor series expansion [24]. In this paper, applying the first-order Taylor series expansion to the sigmodal active function (SAF) of CSS-APSA, we were inspired by [24] and will propose a simplified CSS-APSA (SCSS-APSA), which has lower computational complexity. However, although CSS-APSA realizes fast convergence rates and small steady-state misadjustment, there is still an open problem of enhancing convergence performance [25; 26; 27; 28]. Therefore, we modify the formation of SAF in CSS-APSA. Using the modified SAF to improve the performance of traditional CSS-APSA, we propose a modified CSS-APSA (MCSS-APSA). In MCSS-APSA, the two different step sizes work in the same way as CSS-APSA; however, the extra parameter of modified SAF can enable flexible adjustment of filtering performance. In addition, similar operations of SCSS-APSA can also be used for MCSS-APSA, and then a simplified MCSS-APSA (SMCSS-APSA) can be derived. Simulation results demonstrate that the proposed algorithms can achieve good filtering performances in terms of the convergence speed in system identification. The rest of this paper is organized as follows. Section 2 introduces the APSA algorithm based on step-size adjustment.