# A Novel Generalized Group-Sparse Mixture Adaptive Filtering Algorithm
## 1 Introduction  
Adaptive filters (AF) is an important field of signal processing, which has been utilized for system identification (SI) [1; 2]. For the AF applications, the least mean square (LMS) and its variants based on the two-order error cost function have been used widely since they are low in complexity and simple in practical implementations [1; 2; 3; 4; 5]. However, these algorithms might diverge, especially in the presence of outliers. The least mean fourth (LMF) has been reported by using the four-order error, resulting in improved performance at low signal-to-noise ratio (SNR) [6; 7]. The LMF algorithm becomes unstable unless a small step size is used, resulting in that it is less accurate and has higher misalignment than the LMS algorithm. Then, mixed AF algorithms have been proposed by introducing a weighted factor to combine the two-order and higher-order errors to construct new cost function for improving the performance without sacrificing the simplicity and stability of the LMS [8; 9; 10; 11; 12]. Specifically, the least mean square/fourth (LMS/F) and the least mean mixed norm (LMMN) algorithms have been presented [8; 11; 12]. However, the early reported algorithms mentioned above cannot use the prior sparse-structure information in practical systems, and their performance might be reduced in the presence of the impulsive noise because of the use of the squared error criterion. In practical engineering applications, many systems have inherent sparse features, such as the channel impulse response (CIR) of the digital television transmissions, acoustic echo paths and satellite linked echo paths which are typical sparse systems [13; 14; 15]. In these CIRs, most of the channel coefficients, which are inactive, are equal to zero, while a few active coefficients have larger amplitudes [13; 14; 15]. In these sparse systems, the convergence speed rate can be accelerated by takingadvantage of the prior sparse information structures. Then, proportional update scheme has been proposed to implement the sparse SI, such as proportionate normalized LMS (PNLMS) which yields better convergence at the initial stage by assigning larger gain to the larger taps in comparison to the inactive taps, and the convergence rate for the gain allocated PNLMS algorithm can even be slower than that of the classical normalized LMS (NLMS) algorithm at the later adaptation stage [15]. To further enhance the behavior of the gain assigned PNLMS, several enhanced proportionate-type (Pt) algorithms have been reported, including the improved PNLMS (IPNLMS) algorithm [16]. Additionally, in recent years, zero-attraction (ZA) techniques have been carried out to use the sparse system characteristics to optimize the behavior of the classical LMS algorithm by considering both the convergence and steady-state mean-square-error (MSE). Then, the zero-attraction LMS (ZA-LMS) and its reweighted form (RZA-LMS) were presented by exerting the \(l_{1}\)-norm on the estimated filter coefficient vector to form a penalty to modify the basic LMS's cost function [17]. However, the ZA-based AF algorithms are not good enough to deal with the group sparse systems. According to the active coefficient distribution in these sparse systems, the sparse systems are divided into three types [18; 19; 20; 21; 22; 23]: (1) General sparse systems; (2) one-group sparse systems, and (3) multi-group sparse systems. It is known to us that the network echo paths are modeled as a typical one-group system while the satellite link echo paths have been modeled as multi-group systems, owing to the bulk delays which are always extant in network encoding and jitter buffer delays [19; 21]. Then, a block sparse LMS (BS-LMS) was developed by introducing the hybrid \(l_{2,0}\)-norm into the basic LMS's cost function to identify group-sparse systems [18] and a block sparse PNLMS (BS-PNLMS) is also realized by integrating mixed \(l_{2,1}\)-norm of coefficient vector into the basic PNLMS algorithm to provide enhanced performance for dealing with group bulked system identification [19].