# An Improved Set-Membership Proportionate Adaptive Algorithm for a Block-Sparse System
## 1 Introduction  
Echo cancellation is one of the most typical applications in adaptive filtering scenarios [1; 2]. Among the many adaptive algorithms, the normalized least mean square (NLMS) is a classical algorithm due to its good performance and easy implementation [3; 4]. The current research on echo cancellation focuses on network echo cancellation (NEC) and acoustic echo cancellation (AEC). Both of these applications have long and sparse impulse responses, which means that the filter needs many more taps and there are mostly zero or close to zero responses [5]. Additionally, the input signal of these two systems is usually a speech signal rather than white noise. Therefore, the performance of the traditional NLMS algorithm may reduce, and the convergence speed will slow down when it is used for dealing with acoustic and network echo cancellation. To further exploit the sparse characteristic of the echo path, the proportionate NLMS (PNLMS) and zero-attracting algorithms were proposed [6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16]. The proportionate adaptive filtering algorithms assign corresponding step sizes to each coefficient according to its magnitude, in contrast to the NLMS algorithm whose step sizes are unified. The result of the proportionate assignment is that the large coefficients obtain large step sizes while the small coefficients obtain small step sizes. Thus, proportionate adaptive filtering algorithms are more suitable for sparse systems. However, the convergent performance of the PNLMS algorithm decreases sharply following the initial fast convergence, and the PNLMS may even behave more poorly than the NLMS. We can draw the conclusion that we can benefit from the PNLMS algorithm only in the case when the system is quite sparse. To address this problem, many related improved algorithms have been proposed [17; 18; 19; 20; 21]. The improved PNLMS (IPNLMS) algorithm is the most classical of the variation algorithms [17]. The IPNLMS algorithm fully exploits the \"proportionate\" characteristic and has betterperformance than the NLMS and PNLMS when the channel response is neither sparse nor extremely dispersive. The sparse system responses can usually be divided into three types based on their sparse features: generalized sparse response; single-clustering block-sparse response; and two-cluster or multi-cluster block-sparse response, as shown in Figure 1[22]. In the generalized sparse system, the nonzero coefficients distribute randomly. In the clustering block-sparse systems, the nonzero coefficients distribute in clumps of 1-2 or even more. If there is more than one cluster, we generally consider the system as a multi-clustering block-sparse system. The echo response is a kind of single-clustering block-sparse system, and the satellite communication is a kind of multi-clustering block-sparse system. To take further advantage of the prior knowledge of the block-sparse feature, many corresponding algorithms were proposed in [22; 23; 24; 25]. The algorithm proposed in [22] is named the BS-LMS, in which the main idea is to insert a mixed \(l_{2,0}\) norm as a penalty to the cost function of the conventional LMS with same group partition sizes. The PB-IPNLMS algorithm proposed in [23] divides the channel response into two parts, the sparse part and the dispersive part. The proportionate algorithm is used for the sparse part while the non-proportionate algorithm is used for the dispersive part. However, the PB-IPNLMS has a notable unpleasant property in that the assumption of just one cluster exists, and its location must be known ahead of time. The IAF-PNLMS algorithm devised in [24] uses individual factors for each coefficient rather than a overall one. As a result, the IAF-PNLMS algorithm can get outstanding performance only when the impulse response exhibits high sparseness. The IIPNLMS algorithm proposed in [25] divides the echo path response into an active region and inactive region. The NLMS algorithm is used for the former, while the PNLMS is used for the latter.