International Journal of Antennas and Propagation

Volume 2019, Article ID 1523469, 12 pages

https://doi.org/10.1155/2019/1523469

## On Calibration and Direction Finding with Uniform Circular Arrays

Correspondence should be addressed to Stephan Häfner; ed.uanemli-ut@renfeah.nahpets

Received 24 February 2019; Revised 15 May 2019; Accepted 27 May 2019; Published 3 July 2019

Academic Editor: Chien-Jen Wang

Copyright © 2019 Stephan Häfner 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

Antenna array calibration methods and narrowband direction finding (DF) techniques will be outlined and compared for a uniform circular array. DF is stated as an inverse problem, which solution requires a parametric model of the array itself. Because real arrays suffer from mechanical and electrical imperfections, analytic array models are per se not applicable. Mitigation of such disturbances by a global calibration matrix will be addressed, and methods to estimate this calibration matrix will be recapped from literature. Also, a novel method will be presented, which circumvents the problem of a changed noise statistic due to calibration. Furthermore, local calibration, where array calibration measurements are incorporated in the DF algorithm, is considered as well. Common DF algorithms will be outlined, their assumptions regarding array properties will be addressed, and required preprocessing steps such as the beam-space transformation will be presented. Also, two novel DF techniques will be proposed, based on the Capon beamformer, but with reduced computational effort and higher resolution for bearing estimation. Simulations are used to exemplary compare calibration and DF methods in conjunction with each other. Furthermore, measurements with a single and two coherent sources are considered. It turns out that global calibration enables computational efficient DF algorithms but causes biased estimates. Furthermore, resolution of two coherent sources necessitates array calibration.

#### 1. Introduction

Direction finding (DF) is a task which occurs in several applications of surveillance, reconnaissance, radar, or sonar. Basically, DF can be defined as estimation of the bearing of one or multiple signal sources with respect to (w.r.t.) a reference point in space. Typically, an array of spatially distributed sensors is placed at this reference point and the array output is exploited for DF. Hence, DF estimation is an inverse problem. Solving the inverse problem requires a parametric model of the array output in terms of the parameters of interest: azimuth of arrival (AoA) and elevation of arrival (EoA) , which together define the direction of arrival (DoA).

In order to derive a parametric model of the array output, a model of the sensor array itself is necessary. The array model highly depends on the array geometry and the characteristic of each sensor. Theoretical array models typically assume omnidirectional sensors and an ideal array geometry, which cannot be assured for real arrays. Apart from these assumptions, real arrays suffer from disturbances as, e.g., mutual coupling between the sensors or the support structure of the array, unknown sensor gain, and phase or mechanical imperfections [1]. Consequently, DoA estimation performance degrades, because the assumed array model does not coincide with the real array characteristics. Hence, calibration is necessary to mitigate these imperfections.

All investigations are subject to a uniform circular array (UCA). UCAs feature a very attractive geometry, because their aperture covers the whole azimuth range and hence ambiguous free AoA estimates are ensured, on the contrary to, e.g., uniform linear array (ULA). Also, UCAs can be employed to estimate elevation too, but generally not unambiguous. For simplification, only AoA estimation and copolarised sources w.r.t. the array sensors are considered. For the conducted investigations it is not necessary to consider elevation and arbitrarily polarised sources. However, neglecting source polarisation and assuming fix elevation may result in biased estimates in real DF applications [2].

The goal of this paper is to jointly investigate array calibration methods and narrowband DF techniques. Global calibration, where a direction independent calibration matrix is used, will be considered. Methods to estimate the global calibration matrix from array calibration measurements are reviewed and a new method is proposed, which accounts for the change of the noise statistic due to the application of the calibration matrix. Also, local calibration, where the array calibration data are incorporated in the DF method, is considered. Known DF techniques will be outlined and two novel DF techniques based on the Capon beamformer will be proposed, featuring a reduced computational effort and better resolution in case of multiple sources. Some of the considered DF techniques take advantage of special array structures, which are not provided by UCAs. Hence, beam-space transformation will be briefly recapped. Simulations and measurements are employed for the investigations. Measurements with two coherent sources, hence sources with a fix phase relation, will be considered. Resolution of coherent sources is crucial in DF [3], because of the rank-degeneration of the spatial covariance matrix. Hence, the coherent source case will be used as benchmark to justify calibration necessity and to investigate the DF accuracy.

The reminder of the paper is organised as follows: a parametric model of the array output is derived in Section 2. In Section 3 the DF techniques are presented. The beam-space transformation for UCA is described in Section 4. The problem of array calibration and its influence on the sensor characteristic is presented in Section 5. Simulation based comparison of calibration and DF methods is presented in Section 6. In Section 7, the DF methods are compared using measurements with a single source and two coherent sources. Section 8 concludes the paper.

Mathematical notation is as follows: scalars are italic letters. Vectors are in column format and written as boldface, lower-case, italic letters. Matrices correspond to boldface, upper-case letters. The matrix operations , , , and are defined as the transpose, conjugate transpose, inverse, and Moore-Penrose pseudo inverse of a matrix, respectively. The Frobenius norm of a matrix is stated as . The imaginary unit is defined as .

#### 2. Measurement Data Model

DoA estimation requires a parametric model of the measurement data in terms of the DoAs. Consider an array of sensors, having its reference point in the origin of a spherical coordinate system. Directions of impinging waves are defined w.r.t. this origin in terms of AoA and EoA ; see Figure 1. Consider plane waves, emerging from far field sources and impinging at the array. The waves are assumed to impinge in the azimuth plane, hence holds. Under narrowband assumption [1, 4], the array output in the complex baseband can be approximated aswith vector denoting the narrowband array response w.r.t. the impingement angle . The narrowband array response is commonly denoted as steering vector. The vectors , , and matrix contain the complex envelope of the source signals, the AoAs of all sources, and the steering vectors, respectively. Measurement noise and uncertainties due to, e.g., model errors are accounted for by an additive error term . This error term is modelled as a zero-mean and proper complex normal distributed random process, which is spatially white and homogeneous, and uncorrelated with the source signal: . In summary, the model for the observations is [1]In practice snapshots are taken from the sensors. Accordingly, the model of the array output becomes