## 1 Introduction  
With the rapid rise of various wireless technologies, wireless transmissions have been widely developed in various fields such as mobile communications and satellite communication systems [1; 2]. However, the signal might be distorted due to the diffraction, refraction, reflection or deviation from obstacles such as buildings, mountains and so on, resulting in transmission delays and other adverse effects. In the wireless communication transmission channels, the selective fading will lead to delays, which is also known as multi-path effects. In fact, the multi-path channel is always sparse [3; 4; 5], which means that most of the channel impulse response (CIR) coefficients are small while only a few of them are large in magnitude [6]. Since the CIR is sparse, many channel estimation algorithms have been presented to utilize this characteristic to improve the communication quality [2; 7]. It is known that the adaptive filtering (AF) algorithm can be used for implementing channel estimation. Thus, various AF algorithms have been reported and used for channel estimation. Among these AF algorithms, the most typical algorithm is the least mean square (LMS) which is invented by [PERSON]. The LMSalgorithm has been extensively investigated in channel estimation and noise cancellation owing to its simple implementation, high stability and fast convergence speed [4; 8]. However, its performance is not satisfactory for sparse channel estimation with a low signal to noise ratio (SNR). Recently, the compressive sampling (CS) concept has been introduced into AF algorithms to handle the sparse signals [9; 10]. After that, [PERSON] et al. put forward to the zero attracting LMS (ZA-LMS) and reweighting ZA-LMS (RZA-LMS) algorithms [11]. ZA-LMS and RZA-LMS algorithms are implemented by integrating \(l_{1}\)-norm and reweighting \(l_{1}\)-norm into the LMS's cost function, respectively. These two algorithms achieve lower steady-error and faster convergence speed than that of the basic LMS algorithm for handling sparse signals owing to the constructed zero attractors. Moreover, \(l_{0}\)-norm and \(l_{p}\)-norm have also been employed and introduced into the LMS's cost function to improve the performance of the ZA- and RZA-LMS algorithms in the sparse signal processing area [12; 13; 14; 15; 16]. All of those norm-constrained LMS algorithms can effectively exploit sparse characteristics of the in-nature sparse channels. However, they have a common drawback, i.e., their sensitivity to the input signal scaling (ISS) and noise interferences. In order to reduce the effects of the ISS, several improved AF algorithms have been presented by using high order error criteria or mixed error norms such as least mean fourth (LMF), least mean squares-fourth (LMS/F) and so on [17; 18; 19; 20]. Similarly, their related sparse forms have also been developed based on the above mentioned norm penalties [17; 21; 22; 23; 24; 25; 26; 27]. However, those AF algorithms and their related sparse forms are not good enough for dealing with the sparse channel under non-Gaussian or mixed noise environments. In recent years, information theoretic quantities were used for implementing cost function in adaptive systems. The effective entropy-based AF algorithms include the maximum correntropy criterion (MCC) and the minimum error entropy (MEE) [28; 29; 30; 31; 32]. In [28], it is shown that the MEE is more complex than the MCC algorithm in the computational overhead. Therefore, the MCC algorithm has been extensively developed in non-Gaussian environments [29; 31; 32]. Furthermore, the MCC has low complexity which is nearly the same as that of the LMS-like algorithms. However, the performance of the MCC algorithm may be degraded for sparse signal processing. In order to enhance the MCC algorithm for handling sparse signal and sparse system identification, \(l_{1}\)-norm and reweighting \(l_{1}\)-norm constraints have been exerted on the channel coefficient vector and integrated into the MCC's cost function.