Generalized combined nonlinear adaptive filters: From the perspective of diffusion adaptation over networks

Abstract

Combination of nonlinear adaptive filters (CNAF) is gaining popularity as an effective solution to enhance the filter performance by addressing compromises with respect to parameter and structure settings. However, most existing algorithms study the optimal choice of different filters while neglecting the internal structure optimization of filters. This limitation restricts the performance of CNAF. In this work, we propose a new scheme of the CNAF from the perspective of diffusion adaptive over network. Instead of combining the filters in a parallel manner, we consider to organize the candidate filters in a network manner. Specifically, a network with two subnetworks is constructed, and nodes in each subnetwork serve as linear adaptive filters and nonlinear adaptive filters respectively. A generalized CNAF (GCNAF) is obtained by linking the nodes of the network. The proposed GCNAF allows information exchange and sharing among nodes, so as to enhance the performance of the combined filters. As a result, search direction of each filter is also adjusted by combining those of other filters via the diffusion over networks. We show some representative, state-of-the-art combined adaptive filters are special cases of the proposed framework. Simulations with an acoustic problem demonstrate the effectiveness of the proposed GCNAF.

Introduction

Although linear models have obviously practical advantages, there are many situations in which they are not appropriate and need to be replaced by a nonlinear one [1], [2]. As a result, many structures have been investigated in order to model nonlinear systems in practical applications, including trigonometric expansion [2], [3], [4], neural networks [5], [6], block-based Wiener-Hammerstein models [7], [8], [9], polynomial models [1], [10], [11], etc. Every structure has their own pros and cons. To deal with the tradeoff between convergence rate and convergence accuracy, which is a limitation inherent to adaptive filters, combination of nonlinear adaptive filters (CNAF) [12], [13], [14] is gaining interest in nonlinear signal processing applications such as nonlinear echo cancellation and channel equalization. For example, collaborative functional link adaptive filters (CFLAF) were proposed in [2] based on the adaptive combination of filters in order to improve their robustness against different degrees of nonlinearity. A combination of Volterra filters (CVF) and a combination of kernel (CK) filters were presented in [10]. While both approaches achieve similar performance that is superior to a single VF, the latter is significantly more efficient. An improved solution to the CK scheme was subsequently developed in [11], named D-NLAEC-AZK. However, the performance improvement is somehow limited as the design method in [11] neglects the internal structure optimization and information interaction between linear filters and nonlinear filters.

Recently, distributed adaptation has emerged as an attractive and challenging research area with the advent of multi-agent networks. In adaptive networks, the interconnected nodes continually learn and adapt, as well as perform assigned tasks, e.g., parameter estimation, from observations collected by the dispersed agents. There are several useful distributed strategies for data processing over networks including consensus strategies [15], [16], [17], [18], [19], [20], [21], [22], [23], incremental strategies [24], [25], [26], [27], [28], and diffusion strategies [29], [30], [31], [32], [33], [34], [35], [36]. The convergence rate of distributed optimization algorithm via diffusion strategies is enhanced and performs better than a non-cooperative strategy under certain conditions [30]. Nodes in an adaptive network may approach the centralized solution through a continuous process of cooperation and information sharing with neighbors. This strategy has been applied to both linear and nonlinear collaborative parameter estimation problems [37], [38], [39].

Upon investigating the above structures of filter combination strategies and distributed network models, some important questions arise: is it possible to devise a scheme of generalized CNAF through the optimization over networks? does the scheme have the capacity to unify existing CNAF and to provide better performance in terms of the convergence rate and steady-state accuracy? This paper answers these questions.

Inspired by the work in [30], we propose a new and generalized scheme of CNAF from the perspective of the distributed optimization based on the diffusion strategy over networks. The proposed GCNAF performs the filtering task via a linear filtering subnetwork and a nonlinear filtering one. In the linear filtering subnetwork, a combination of linear filters is obtained by optimally linking each node that is associated with a linear filter. Likewise, in the nonlinear filtering subnetwork, a combination of nonlinear filters is constructed by the topological linking of each node that is associated with a nonlinear filter. A new scheme is then established by combining the two subnetworks under the guidance of a unified optimization objective function. Although a couple of existing algorithms [40], [41], [42], [43] also consider the information exchange between component filters through diffusion networks, these algorithms are focused on the linear system model. In contrast to the existing CNAF methods, the proposed scheme makes full use of the diffusion ability of the network, and hence a more efficient nonlinear filter algorithm is obtained by sharing the information among nodes.

Theoretical performance analysis is conducted to characterize the stability condition of the proposed algorithm. Simulations with an acoustic echo cancellation problem are conducted to validate the effectiveness of the developed GCNAF.

The major contributions of this work are as follows:

• A scheme of GCNAF is proposed, which links the filters (nodes) via the topological structure of the network. This scheme facilitates information exchange and facilitates nodes to exchange intermediate estimates among nodes in order to enhance the performance of different filters.

• In the proposed GCNAF, the search direction of each filter is optimally combined and different filters are optimally combined via the diffusion ability over networks. The GCNAF is completely characterized by the combination matrices of the network topological structure. Typical state-of-the-art combination filtering methods, such as the ones in [2], [10], [11] are particular cases of the presented scheme with different combination matrices.

• An adaptive algorithm is derived to update the linear and nonlinear filters as well as the mixing parameters in GCNAF. Stability analysis is conducted and the stability conditions are specified.

The rest of the paper is organized as follows. Section 2 presents the data model and motivation of this work. Section 3 presents the network structure of the proposed GCNAF and derives the scheme of GCNAF through the distributed optimization based on the diffusion strategy over networks. We investigate a particular case of the developed scheme, i.e, CK in Section 4. Section 5 studies the stability condition in the mean sense via a theoretical analysis. The proposed scheme is validated in Section 6 through numerical simulations with a white Gaussian process and a speech signal as the system inputs. Finally, important conclusions are given in Section 7.