## 1 Introduction  
A class of adaptive filtering (AF) algorithms are extensively considered in use in channel estimation (CE), echo cancellation, noise elimination, etc. [1; 2; 3; 4; 5; 6; 7; 8; 9; 10]. For example, the well-known least mean square (LMS), normalized LMS (NLMS) and recursive least square (RLS) algorithms were used in various systems. Although the LMS algorithm has a simple principle and a small amount of computation in practice, it might converge slowly in low signal-to-noise ratio (SNR) scenes. In contrast, the RLS converges faster than the basic LMS. However, it is proved to have high cost of increased computational complexity, which will use more computing resources when the order of the AF is large. In addition, if the input signal is a speech signal, the convergence speed for the basic LMS algorithm becomes very slow as the eigenvalue distribution range for the input signal autocorrelation matrix is large [11]. To enhance the identification behaviors of the RLS and NLMS algorithms in practical engineering and to obtain high accuracy and fast convergence, the affine projection (AP) algorithm is proposed by reusing latest input signals to improve the NLMS's performance [12]. The computation burden of the AP is between the LMS and RLS algorithms, and the AP algorithm has a fast convergence, especially for colored or speech signal input signals [13]. In many engineerings, such as speech signal processing and real-time traffic predictions, noise often exhibits strongly impulsive characteristics [14; 15]. Traditional NLMS and AP algorithms, which use the minimum-mean-square-error (MMSE) criterion to construct an expected cost function, will suffer from performance degradation in those impulsive noise environments. To find the solution for handling these problems, the maximum correntropy criterion (MCC) and the minimum error entropy criterion (MEEC) have been proposed to give resistance to the impulsive noise [16; 17]. Although the MEEC is a robust criterion, its computational complexity is very high, while the MCC algorithm whose computational complexity is comparable to the LMS has been widely used to resist the impulsive noise [18; 19; 20; 21]. On the other hand, scholars found that the sparse characteristics are existing in a great number of scenarios such as network echo channels and underwater acoustic communication channels [22; 23; 24; 25]. However, classical LMS, AP and MCC algorithms cannot take advantage of the sparse structures of these sparse channels. Then, the proportionate AF algorithms have been proposed to make use of the sparse information in the mentioned channels [26]. For example, the proportionate NLMS (PNLMS) combines the proportionate scheme into the NLMS to reassign the gains to each channel coefficients [26]. Then, proportionate-type AF algorithms were widely realized and utilized for channel estimation as well as the echo cancellation [27; 28; 29; 30]. For the sake of comparison with the traditional NLMS, the PNLMS suffers from slow convergence if the input signal is driven by colored or speech signals, resulting in that steady-state error might be worse than that of the NLMS. Inspired by the PNLMS, the proportionate AP (PAP) algorithm has been proposed by using the idea in PNLMS to fully use the sparse structure-information of the echo channels [31] based on the data reusing principle. Then, various proportionate-type AP algorithms have been proposed [32; 33; 34; 35; 36; 37]. However, the PAP-type algorithms have performance degradation in impulsive noise environments because of the MMSE scheme. Thus, the sign algorithms, such as affine projection sign (APS) algorithm and proportionate APS (PAPS) algorithm [38; 39], are successfully used for dealing with impulsive noise. Additionally, another kind of sparse-aware APs have been reported and analyzed by taking the consideration of the compressed sensing (CS) theory [40].