A family of sparse group Lasso RLS algorithms with adaptive regularization parameters for adaptive decision feedback equalizer in the underwater acoustic communication system

Abstract

In this paper, we propose a family of sparse group Lasso (least absolute shrinkage and selection operator) Recursive Least Squares (RLS) algorithms for sparse underwater acoustic channel equalization. The proposed adaptive RLS algorithms employ a family of mixed norms, such as $l_1{l}_{2,1}$-norm, $l_1{l}_{\infty ,1}$-norm, $l_1{l}_{2,0}$-norm, $l_1{l}_{1,0}$-norm, $l_0{l}_{2,1}$-norm, $l_0{l}_{\infty ,1}$-norm, $l_0{l}_{2,0}$-norm, and $l_0{l}_{1,0}$-norm, as the sparsity constraint in the penalty function to exploit the sparsity of the underwater acoustic communication system. The proposed adaptive RLS algorithms can adaptively select the regularization parameters regardless of whether the channel of underwater acoustic channel is general sparse channel, group sparse channel or the mixed sparse channel consisting of general sparse channel and group sparse channel. Moreover, this paper presents a direct adaptive decision feedback equalizer (DA-DFE) that exploits any sparse channel structure with the proposed adaptive RLS algorithms in the lake and sea experiments. Experimental results verify that the DA-DFE receiver with the proposed family of sparse group Lasso RLS algorithms can achieve a better performance in terms of convergence rate, mean square deviation (MSD) and symbol error rate (SER) in the single-input single-output (SISO) single carrier underwater acoustic communication system.

Introduction

Some underwater acoustic channels are characterized by sparse time-varying structure with a long multipath spread. Given these limitations, reliable phase coherent underwater acoustic communication system requires the design of efficient decision feedback equalizer (DFE). Stojanovic and Proakis proposed a typical DFE receiver embedded with a second order digital phase-lock loop (DPLL) to resist inter-symbol interference (ISI) introduced by the long multipath spread in the underwater acoustic communication system  [1]. However, the sparsity of the underwater acoustic channel is not explored during the adaptive equalization in the DFE receiver in  [1]. Even though the sparse DFE have been proposed in  [2], [3], the proposed sparse DFE just exploit the sparsity of the UWA channel only by focusing on those channel impulse responses (CIR) which contain the significant portion of the signal energy and neglecting the small CIR. Nonetheless, the proposed sparse DFE in  [2], [3] did not take full advantage of the sparsity in the UWA channel by improving the performance of the DFE in the adaptive equalizer.

In the previous work of direct adaptive decision feedback equalizer (DA-DFE), either the sparsity of the UWA channel was not considered or the sparsity was not employed in the adaptive equalizer. However, for practical sparse UWA systems, some algorithms were used to exploit the sparsity of the channel to improve the performance of the adaptive equalizer  [4], [5], [6], [7], [8], [9], [10], [11]. The basic idea of such algorithms was to exploit the characteristics of unknown CIR by exerting the sparsity constraint on the penalty function. In  [5], the least mean square (LMS) variants were derived to exploit the sparsity within a framework of Natural Gradient (NG) algorithms. A modified proportionate updating method was used to exploit the sparsity in  [6]. In  [7], the authors developed a family of robust normalized least mean square (NLMS) algorithms to exploit the sparsity in the adaptive equalizers. In particular, $l_1$ norm had been used as an effective promotor for sparsity in  [8]. Compared with LMS algorithm, RLS algorithm is more popular because of its fast convergence rate and low misadjustment  [12]. Sparsity was also exploited as a significant prior condition in the RLS algorithms  [9], [10], [11]. Time-weighted Lasso (TWL) and time- and norm-weighted Lasso (TNWL) approaches were presented in RLS algorithm in  [9]. The authors of  [10] developed a recursive $l_1$-regularized least squares algorithm (SPARLS) in the sparse channel. SPARLS utilized an expectation maximization (EM) type algorithm to minimize the $l_1$-regularized least squares penalty function. The research on  [13] showed that sparsity can be best represented by $l_0$ norm, which were able to acquire the sparsity constraint in the sparse channel. Considering that $l_0$ norm minimization was an Non-deterministic Polynomial-time (NP) hard problem, an analytic approximation of the $l_0$ norm was introduced in the convex regularized RLS (CR-RLS) to improve the performance of the RLS algorithm in  [11]. Group sparsity was also used as a prior condition in adaptive equalization algorithms  [14], [15], [16], [17], [18], [19]. Mixed norms such as the $l_{2,1}$ norm  [14], [15], [16] and $l_{1,\infty }$ norm  [17], [18], [19] had been recognized as an effective method to promote group-level sparsity of the channel. In  [16], the authors demonstrated that the performance of $l_{2,1}$-regularized LMS was superior to the $l_1$-regularized LMS in the group sparse channel. In  [18], $l_{2,1}$ norm and $l_{1,\infty }$ norm were used as the penalty function in the RLS algorithm to promote group sparsity. The authors of  [19] developed a new analytic approximation for $l_{p,0}$-penalized RLS algorithm to further improve the performance in the group sparse channel.

In order to solve problems brought by the mixed sparse channel consisting of general sparse channel and group sparse channel, the sparse group Lasso method was proposed in  [20], [21]. However, the regularization parameters were set to be fixed constants in  [20], [21], which would limit the performance of the adaptive RLS algorithm for adaptive equalization in the sparse UWA channel. In this paper, we propose a family of adaptive sparse group Lasso RLS algorithms, which can adaptively select the regularization parameters in the DA-DFE receiver for the SISO underwater acoustic communication system. These algorithms guarantee provable dominance for not only the general sparse channel and the group sparse channel, but also a mixed sparse channel consisting of the general sparse channel and group sparse channel. Therefore, the proposed family of sparse group Lasso RLS algorithms can adaptively select the regularization parameters for penalizing a more extensive sparse channel in the underwater acoustic communication system.

In summary, the main contributions of this paper are listed as follows:

• Employing a family of mixed norms, such as $l_1{l}_{2,1}$-norm, $l_1{l}_{\infty ,1}$-norm, $l_1{l}_{2,0}$-norm, $l_1{l}_{1,0}$-norm, $l_0{l}_{2,1}$-norm, $l_0{l}_{\infty ,1}$-norm, $l_0{l}_{2,0}$-norm, and $l_0{l}_{1,0}$-norm, as the sparsity constraint in the penalty function of the adaptive RLS algorithms in the DA-DFE receiver;

• Selecting the regularization parameters adaptively according to the structure of the channel (i.e., general sparse channel, group sparse channel or a mixed sparse channel consisting of general sparse channel and group sparse channel) in the DA-DFE receiver for the SISO underwater acoustic communication system.

The paper is organized as follows. In Section  2, we derive a family of adaptive sparse group Lasso RLS algorithms, which can adaptively select the regularization parameters in the DA-DFE receiver. In Section  3, we provide results of the lake and sea experiments. Finally, in Section  4, we summarize the conclusion about this work.

Notation: in this paper, matrices and vectors are denoted by boldface upper case letters and boldface lower case letters, respectively. ${(\cdot )}^T$ denotes the transposition. $\mathrm{sgn}(\cdot )$ is the component-wise sign function. ${\left(x\right)}_{+}=\max (0,x)$.