International Journal of Antennas and Propagation

Volume 2018, Article ID 9237321, 5 pages

https://doi.org/10.1155/2018/9237321

## Worst-Case Performance Optimization Beamformer with Embedded Array’s Active Pattern

^{1}School of Electronic Science and Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China^{2}University of Technology Sydney, Ultimo, NSW, Australia

Correspondence should be addressed to Yuyue Luo; moc.361@ctseu_lyy

Received 15 December 2017; Revised 13 March 2018; Accepted 22 April 2018; Published 24 June 2018

Academic Editor: Francesco D'Agostino

Copyright © 2018 Yuyue Luo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper proposes an adaptive array beamforming method by embedding antennas’ active pattern in the worst-case performance optimization algorithm. This method can significantly reduce the beamformer’s performance degradation caused by inconsistency between hypothesized ideal array models and practical ones. Simulation and measured results consistently demonstrate the robustness and effectiveness of the proposed method in dealing with array manifold mismatches.

#### 1. Introduction

The assumption of ideal array elements in conventional adaptive beamforming technologies can cause severe performance degradation in real implementations due to ignored array imperfections (e.g., gain and phase mismatches and mutual coupling between elements), particularly for increasingly widely used small-profile arrays. Robust beamforming algorithms have been proposed to deal with these imperfections by treating an array’s response inconsistencies as nonspecific manifold mismatches. There are some classic algorithms, such as the diagonal loading (DL) method (also called loaded sample matrix inversion (LSMI) beamformer) [1] and the worst-case performance optimization robust beamformer (WCRB) [2], summarized in [3, 4]. The WCRB approach [2] is also extended and applied to several specific scenarios [5–8]. However, most robust beamforming methods solve uncertain problems based on simplified array models, without considering the array’s electromagnetic characteristics, which are actually essential to the manifold mismatches and are critical for the performance of the methods in practice.

The problem of array modelling mismatches is typically studied by antenna researchers. An earlier work exploiting the gain and frequency properties of practical antennas was reported in [9], without considering mutual coupling effect. In [10], improvement to [9] is made by incorporating the antenna’s active pattern (AP) introduced in [11], which calculates an elements’ radiation and its impact on the array environment (both mutual coupling between elements and workspace radiation) [12]. However, these methods rely on the exact knowledge of antennas’ electromagnetic characteristics and are quite sensitive to measurement mismatches.

In this paper, by creatively integrating antenna mismatch modelling into beamforming design, we propose a robust worst-case performance optimization beamformer with an embedded array’s AP. We call it as active pattern worst-case (APWC) method which can significantly improve the beamformer robustness under various mismatches. The APWC method essentially introduces the AP method [10] into the WCRB algorithm [2]. Via both simulation and experiments with real measurements, we demonstrate that the APWC beamformer can achieve significantly better performance (e.g., higher signal-to-interference-and-noise ratio (SINR)) than many existing schemes. It has better tolerance to both engineering and electromagnetic mismatches caused by elements’ modelling, manufacturing, aperture assembling, and channel debugging.

#### 2. Problem Formulation

We consider an *M*-element two-dimension antenna array and a narrowband system. Without considering any imperfections, its steering vector can be represented as
where is the wavenumber; and denote the frequency and the speed of the electromagnetic wave, respectively; and are the angles; and is the ()th sensor’s location vector.

Assume omnidirectional antenna elements. Let be the transmitted data symbol at time . The signal received at the array is given by where , , and are the steering vectors of signal and multiuser interferences, respectively, and is an vector denoting the combined self-interference from mismatches and noise components.

The well-known sample matrix inversion (SMI) beamformer solves a constrained minimization problem where denotes Hermitian transpose, is the complex beamforming vector, and is the sample covariance matrix of .

#### 3. Improved Model for Array Steering Vector

There are always mismatches, that is, the fluctuation of array parameters during design, processing, measuring, and assembling, between the ideal steering vector and the actual one . Mismatches generally have a minor impact on the electromagnetic characteristics of a single antenna element, such as current distribution and boundary conditions and, hence, cause small changes to each element’s basic radiation structure, as well as the radiation pattern and directivity. Such small changes, however, when beamforming is formed, can cause large beamforming gain variations, due to gain, sometimes phase, misalignment between different antenna elements [13]. In practice, an array’s radiation performance can be significantly affected by such array mismatches.

In this paper, we use an improved array steering vector , by taking into consideration the array aperture’s radiation property [9].
where is the known active gain response of the *i*th antenna. It can be obtained during antenna design using electromagnetic simulation software or through actual measurements.

This new steering vector in (4) is a closer approximate to the real steering vector , compared to the one without considering mismatches. This can be seen from the simulation results as will be presented in Section 5.

#### 4. Proposed APWC Algorithm

Define the approximation error for the radiation pattern expressions with and without considering mismatches as respectively.

Assume that the norm of is bounded by a known constant . We can formulate the APWC problem as a constrained minimization problem

It can be rewritten as

The APWC method belongs to the class of DL method. Similar to the WCRB method [2], the weight solution to (7) can be derived to be where , is a Lagrange multiplier, and . Equation (7) can also be converted to a convex second-order cone problem and finally solved via interior point method. The computational cost of the APWC algorithm is per iteration.

#### 5. Simulation Results

We refer to a practical 4-element uniform circular microstrip array shown in Figure 1 as a standard model in our simulation. This array was developed for an anti-interference subsystem in the BeiDou Navigation System. The commercial electromagnetic simulation software HFSS is used for all antenna simulations.