## 1 Introduction  
Sparse adaptive filtering (AF) algorithms are becoming a boosted topic in recent decades and they are discussed and used for sparse system identifications, multi-path channel estimations, echo cancelation and underwater acoustic communications [1; 2; 3; 4; 5; 6]. Additionally, the AF has been used for improving the electrocardiogram (ECG) detection during magnetic resonance imaging (MRI) [7]. Furthermore, most of the in-nature signals are sparse or can be regarded as sparse such as underwater channels, a multi-path channel in complex terrain environments, echoes in a high-definition television channel, in which only a few coefficients are dominant, while most of the coefficients are unimportant [8; 9; 10; 11]. In this case, the insignificant coefficients are usually inactive, which are usually zeros or near zeros in a system. For such sparse signal processing, traditional least mean square (LMS), and normalized LMS (NLMS) might perform poorly by considering only the convergence, and steady-state error (SSE) since these conventional adaptive filters did not use the prior sparse information of the sparse systems. Thus, the demand for sparse signal processing has been becoming a hot topic. The fast convergence algorithms and sparse AF are desired in the sparse finite impulse response (FIR) environments, resulting in the rapid development of proportionate-type (Pt) adaptive filters, including the PNLMS algorithm [2]. The PNLMS algorithm exploits the sparsity characteristics in the sparse systems via proportionally assigning different gains to the magnitudes of the estimated coefficients [2; 3]. By using this technique, the convergence speed at the initial stage can be improved. However, the PNLMS's convergence speed may be slower than the traditional NLMS algorithm for estimating long sparse signals because the small coefficients are assigned small gains, which increases the convergence time for achieving a steady-state mean square error (MSE) [3]. Another drawback of the typical PNLMS algorithm is that the convergence speed rate may be slower in comparison with NLMS algorithm when the sparse systems are less sparse or dispersive. In the sequel, several improved PNLMS algorithms have been proposed to enhance its performance [3; 4; 12; 13; 14]. These PNLMS algorithms with low complexity were realized by using the squared error criterion to get optimal for processing Gaussian data. All of these PNLMS algorithms mainly update their favored adaptive filtering coefficients, which render these PNLMS algorithms suitable for sparse signal processing [12; 13; 14; 15]. Moreover, the PNLMS algorithm is also an NLMS algorithm if all the coefficients are assigned the same gains. However, the PNLMS algorithm may perform worse when the data are non-Gaussian noises such as the low frequency atmospheric noise [16]. The PNLMS algorithm and its variants just take the second-order statistics into consideration to construct the cost function [2; 3; 4], which is not enough to capture all the information from the data. Therefore, the performance of these PNLMS algorithms will be poor if they are used in the non-Gaussian environments. With the on-going AF algorithms, a minimum error entropy (MEE) was reported within information theoretic learning framework for handling the non-Gaussian signals [16; 17; 18; 19; 20; 21; 22; 23]. In the information theoretic learning, a quadratic Renyi's entropy has been used for data classification, channel modeling and data fusion [24; 25; 26] and it is used to provide an alternative method instead of MSE [16]. As a result, the entropy can access the samples well by using a non-parametric Parzen window method. Then, the MEE adaptive filters were used for identifying systems, which can obtain a better estimating performance in non-Gaussian environments [23]. This is because the entropy employs a high-order statistic and utilizes the contents of the information instead of the power of the signal power.