Distributed LCMV beamformer design by randomly permuted ADMM

Abstract

In recent years, distributed beamforming has attracted a lot of attention. Since each node has its own processing power, one significant advantage is the capability of distributed computing. In general, almost all distributed beamforming approaches are solving certain multi-block optimization problems. However, additional conditions are usually required to ensure convergence. In this paper, a new distributed beamforming algorithm is proposed. We first introduce the augmented Lagrangian method to implement the centralized LCMV beamformer design. Then, we propose an effective blockwise optimization method for the design of distributed LCMV beamformer based on the randomly permuted alternating direction method of multiplier (RP-ADMM). The expected convergence is obtained for distributed LCMV beamformer design without additional conditions. Numerical experiments are conducted to illustrate the performance of the proposed method.

Introduction

In traditional beamforming techniques, a dedicated device (the “fusion center”) is assumed to gather all the sensor observations for further processing. This approach is often referred to as the centralized beamforming and it requires large communication bandwidths and significant computing power at the fusion center. However, for many applications, the availability of a fusion center could be a major problem. Therefore, the distributed computing where each node has its own processing unit to update data independently and exchange compressed signal with other nodes, has been widely studied in the last few years [1], [2], [3], [4], [5], [6], [7], [8], [9].

In more recent works [10], [11], [8], [12], [13], a family of distributed beamformer design methods have been investigated based on convex optimization with different structures. For example, in [10], a distributed minimum variance (DMV) algorithm was proposed for robust linearly constrained minimum variance (LCMV) beamforming by casting it as a distributed convex optimization problem. In [11], a sparse distributed beamformer that trades off SNR performance against reduced inter-node power consumption was designed by using the bi-alternating direction method of multipliers (Bi-ADMM). In [8], a distributed MVDR beamforming technique was developed based on the primal-dual method of multipliers (PDMM) [12]. A stochastic ADMM approach for coordinated multicell beamforming (CMBF) was developed for a smart grid powered coordinated multicell downlink system in [13]. It is noted that most of the distributed approaches for speech enhancement are based on blockwise optimization techniques. However, the theoretical convergence results on ADMM-like algorithms (Bi-ADMM, stochastic ADMM) for such multi-block optimization problems have remained unclear for a very long time, except for the two-block optimization model.

The important proof of convergence for extension of ADMM beyond two-block convex minimization problems were unclear until 2016. In [14] it was shown that ADMM may actually fail to converge for blocks greater than or equal to three via a simple three block counterexample. To resolve the issue, a popular method called the randomized block-coordinate descent (RBCD) method was first developed for a class of unconstrained composite minimization problems [15], [16]. Subsequently, a variety of randomized strategies have been investigated to tackle multi-block optimization problems. More recent attempts can be found in [17], [18], [19], [20]. At the same time, many blockwise techniques and splitting methods for solving the constrained composite optimization problem have also been developed for separable optimization problems with applications in signal and imaging processing, machine learning, statistics, and engineering [21], [14], [22], [23], [24], [25], [26], [27], [28]. Indeed, numerous experiments have demonstrated that they are powerful for solving large-scale optimization problems arising in machine learning [15], [16], [29]. Among different randomized techniques, the randomly permuted alternating direction method (RP-ADMM) of multipliers developed in [30], [31] is noticeably important, which is convergent in expectation for cases with blocks greater than or equal to three for nonseparable quadratic programming problems, under the condition that the block diagonal matrix related to the coefficient matrices of objective and constraint is positive definite.

In this paper, we propose a new distributed LCMV beamformer design method based on augmented Lagrangian method together with the blockwise technique. The LCMV beamformer design is formulated into a blockwise optimization problem with each block related to a specific node. Then a simple but efficient alternative iteration algorithm is proposed to solve the problem by employing the RP-ADMM. This method transforms the original optimization problem into a class of sequential subproblems, and then distributes the computational workload to each node of the sensor array. It obtains a great reduction in the computational complexity and makes parallel computing achievable. Another contribution of this paper is to show that the proposed method is applicable to cases with blocks bigger than 3. In establishing the expected convergence of the proposed algorithm, we prove that the necessary condition for convergence is satisfied without further assumption. Finally, the complexity of the proposed distributed LCMV is discussed and compared with the centralized LCMV via both theoretical and numerical analysis.

The outline of this paper is as follows. In Section 2, we describe the LCMV beamformer design problem and introduce an iteration algorithm based on augmented Lagrangian method. In Section 3, we present the blockwise framework for the D-LCMV beamformer design, as well as the algorithm and the convergence analysis. In Section 4, we describe the application of the proposed algorithm for wireless acoustic sensor network and provide experimental results for evaluation. Conclusions are drawn in Section 5.