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 [58]. 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 ith 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.

We simulate array mismatches using HFSS and then abstract and store the corresponding AP for each mismatch. Three types of common mismatches are studied, including (a) the element’s position mismatches, (b) size errors of the metal working platform, and (c) dielectric parameter errors of elements’ substrate, including relative dielectric permittivity (RDP) and loss tangent (LT). The sample covariance matrix , which reflects the varying electromagnetic environment, varies with different prestored AP data. Using some of the data, Figure 2 compares the norm and in (5) under different types of mismatches. The results clearly show that our adopted radiation expression with AP in (4) approximates the real radiation pattern much better than the one without considering mismatch.

We assume that the desired signal and interference have a plane wavefront with and , respectively. For each simulated point, we ran 200 implementations. The input signal-to-noise ratio (SNR) and interference-to-noise ratio (INR) in a single antenna element equal to 25 dB and 30 dB, respectively. Signals are in the training data cell (training data size unless stated otherwise). Additive noise in the array is modelled as spatially and temporally independent complex Gaussian noise with zero mean and unit variance. The mean output SINR for the following five algorithms is compared under mismatch situations: the SMI beamformer, the LSMI method [1], the AP beamformer [10], the WCRB method [2], and the APWC method using the SeDuMi convex optimization MATLAB toolbox. The optimal SINR, , is also shown.

Figure 3 shows how beamformers’ output SINR is affected by antenna elements’ position mismatch. Figure 4 illustrates how the performance changes with the size of the working platform, as an example of the working environment variation. It indicates that the size variation of the metal working platform can cause the change of edge scattering and affect the electromagnetic radiation, particularly for AP and SMI algorithms. Both figures show that the proposed APWC method outperforms other methods with improved SINR and robustness.

In Figures 5 and 6, we present how the output SINR changes with the substrate’s RDP and LT (the range of variation is determined by engineering experience). From the two figures, we can see that the proposed APWC method also achieves improved SINR and robustness under these two types of mismatches.

6. Measured Results

The array mentioned in Section 5 has been fabricated, debugged, and then measured in the microwave anechoic chamber. Due to the small mismatches introduced in the process of fabrication, we can expect a numerical difference of the AP between the designed and practical arrays. Such difference is shown in Table 1, which compares the active gain data of array elements at the direction of arrival (DOA) of signal and interference. With the measured data, we assessed the earlier mentioned algorithms’ performance using the same parameters (e.g., DOA and INR) with those in V.

Figure 7 compares the mean SINR for the six methods with a varying number of training snapshots. Figure 8 shows how the mean output SINR changes with different SNRs. Figures 7 and 8 demonstrate that the proposed APWC method consistently achieves better performance than other methods at varying input SNR values and snapshot numbers. We can also compare the simulation and measured results here, referring to the parameters  dB and that are available in both results. Compared to the simulation results that are impacted by a single type of mismatch, the SINR of the measured results is reduced by about 2 dB, which is the consequence of combined mismatches.

7. Conclusion

In this paper, we propose an improved adaptive beamforming algorithm APWC, which embeds the array’s electromagnetic characteristics in a robust beamformer. Mathematical analysis, computer simulation, and measured results illustrate the effectiveness and robustness of the proposed algorithm to array manifold mismatches. APWC is particularly suitable for systems with small and compact arrays, where serious mutual coupling and environment scattering could significantly influence antennas’ radiation and the performance of conventional beamforming algorithms.

Conflicts of Interest

The authors declare that they have no conflicts of interest.