Sparse array design for multiple switched beams using iterative antenna selection method

Abstract

Digital beamforming (DBF) arrays with a large number of small antennas are extensively employed in millimeter-wave (mmWave) sensing systems. Sparse arrays have been posed as an attractive solution to mmWave systems due to their capability of striking the best compromise between performance and complexity. We propose two iterative antenna selection strategies, referred to as deterministic selection and adaptive selection, to design sparse DBF arrays in this paper. The first strategy assumes a common sparse array associated with different beamforming weights for multiple switched beams, while the other exploits switching networks to adaptively change the sparse array configuration with different beams. To counteract hardware-related issues that arise in practical realizations, we then propose guided adaptive selection and regularized adaptive selection to impose additional constraints for different switching networks. The optimality of sparse DBF arrays is defined in terms of both transmit and receive patterns. Taking that into account, an iterative re-weighted $l_1$-norm is modified to promote boolean sparsity of the selection vector in this work. Simulation results validate the effectiveness of proposed antenna selection methods.

Introduction

In recent years, there is a spurt of novel sensing systems in the millimeter-wave (mmWave) band, such as 5G communications and mmWave radar systems, which usually employ digital beamforming (DBF) arrays with a large number of small antennas for achieving unprecedented spatial resolutions and for compensating for the high signal attenuations encountered at high frequencies [1], [2], [3], [4], [5]. The mmWave radar systems employing DBF arrays find extensive applications, including autonomous vehicles [6], gesture recognition [7], cloud observation [8], RF identification [9], indoor localization [10], and health monitoring [11], to list a few. DBF arrays enable signal digitalization in element level, and thus are the most generically flexible and favorable phased array architecture in mmWave radar systems, in turn offering prominent advantages in terms of performance, versatility, and cost. One obvious advantage is that DBF arrays are capable of electronically forming and flexibly switching both transmitting and receiving beams, thus performing spatial filtering with beampattern characterizing their spatial responses [12], [13].

A promising solution to narrow beams and sharp transition is to enlarge array aperture length, which inevitably results in prohibitively high cost and burdensome complexity for uniform antenna arrays due to a dedicated Radio Frequency (RF) processing chain per antenna, which usually comprises amplifiers, AD/DA converters, mixers, etc. In the advent of cheaper, smaller antennas and faster switching networks, sparse arrays through antenna selection have been posed as an attractive solution to mmWave systems due to their capability of striking the best compromise between performance and complexity [14], [15]. The fundamental idea behind antenna selection is to configure a sparse array by judiciously activating a subset of antennas that maximizes performance [16], [17]. The beampatterns of antenna arrays are not only dependent on the excitation coefficients but also on the array configuration. For the same number of antennas, different array structures yield differed beampatterns. Antenna selection allows for increased flexibility in both the design of the array's geometry and the individual control of current excitation amplitudes and phases. From the perspective of pattern synthesis, this would provide many more degrees of freedom (DoFs) to achieve the synthesis performance improvement, such as better shape control of synthesized beampatterns. In this regard, optimum sparse array design should fully utilize both the array structure and excitation weights toward synthesizing the desired beampattern.

Sparse DBF arrays can be designed using iterative antenna selection strategies, which can be broadly classified into deterministic selection and adaptive selection. As shown in Fig. 1 in section 2, the sparse array configuration by the deterministic strategy, once designed, is fixed and multiple switched beams are realized by assigning different sets of excitation weights to the already designed sparse array, each corresponding to one beam. The second strategy, however, is cast as selecting a subset from a large set of uniformly spaced antennas to connect with a smaller number of RF processing channels. The RF switching network is employed to adaptively change the array structure such that one beam is synthesized by one sparse array, steering towards the direction of interest with suppressed sidelobes. The RF switching networks represent the hardware components fulfilling the job of interconnecting the RF chains with the selected antennas for further digital signal processing. In this work, we consider two general classes of switching networks, those are fully-flexible switching networks (SN-FF) that can facilitate the inter-connection of any input port to all output ports, and partially-connected switching networks (SN-PC) that can only connect each RF chain to a predefined subset of antennas for reducing the complexity. The schematic of two switching networks, SN-FF and SN-PC, are illustrated in Fig. 3, Fig. 4 in section 3, respectively. For a detailed performance analysis and design procedure of different RF switching networks, interested readers can refer to [18] and references therein.

A large portion of the works in the literature examined the problem of sparse array design by deterministic antenna selection in the case of single receiving pattern [19], [20], [21], [22], [23], [24], [25], [26]. In these works and references therein, the sparse array design in the receiver mode is formulated as the problem of beampattern synthesis with sparsity constraints imposed on the excitation vector, with zero entries implying discarded antennas. There are variant algorithms proposed to promote sparsity, such as reweighted $l_1$-norm [20], Bayesian inference [21], soft-thresholding shrinkage method [25], and other compressive sensing methods [27], [28]. However, the sparse array in those works is usually designed to steer towards one specific target direction with suppressed sidelobes. When the beam is switching, the beampattern is not controlled. Thus, it is preferred that a common sparse array can be designed with a well-controlled beampattern regardless of the switching direction. However, few works conduct a comprehensive study of a critical aspect: the design of sparse arrays for multiple switched beams. It is unclear whether these aforementioned methods can be easily extended to the multi-pattern case since the best element positions usually change with different patterns.

The design of sparse arrays for multiple different-shaped beams has been previously considered in [26], [29], [30]. Yet, all of the designed beams point towards the same direction. Moreover, only receiving pattern synthesis is considered so far. For DBF arrays, transmit pattern, although not as important as receive pattern, would deteriorate the overall filtering performance if poorly shaped. Thus, the design of sparse DBF arrays requires the consideration of both transmitting and receiving modes. As most radar transmitters operate at saturated power levels for efficiency reasons, excitation amplitude variation at the transmitter is not desirable. Thus, the optimum sparse DBF array should maintain a well-controlled conventional transmit beampattern [12], which is only relevant to array configuration and cannot be achieved by the aforementioned sparsity-promoting algorithms. Therefore, in addition to a sparse receiving excitation vector w, a selection vector z with the same sparsity support as w is entailed. In order to promote boolean sparsity of antenna selection vector z, a modified iterative re-weighting scheme is proposed in this work. Both large and small coefficients are penalized more heavily than the in-between coefficients. Moreover, the small coefficients are penalized positively while the large entries are penalized negatively.

The design and development of sparse arrays for multiple switched beams by iterative antenna selection methods, including both common sparse array design by deterministic antenna selection and separate sparse array design by adaptive antenna selection strategy, constitute the focus of this work.

The contributions of this paper are three-fold:

(1) We develop a solution to the problem of common sparse array design, where the array is shared by multiple switched beams and considering both transmit and receive pattern constraints.

(2) We present a solution to the problem of separate sparse array design for different switched beams by both guided and regularized antenna selection with the aim of increased practical feasibility.

(3) We propose a new update formula for the weight vector that promotes the boolean sparseness of the antenna selection vector using the iterative reweighted $l_1$-norm minimization.

The remaining part of this paper is organized as follows: Common sparse array design shared by multiple switched beams is delineated in section 2. Separate sparse array design, including both guided and regulated antenna selection, is introduced in section 3. Simulation results and conclusions are provided in sections 4 and 5, respectively.