Avoiding Bias in Circular Arrays Using Optimal Beampattern Shaping and EADF

Abstract

In this paper we show the impact of the complex array sensor beampattern on the Effective Aperture Distribution Function (EADF). For our purposes we focus on planar array, such as Uniform Circular Array (UCA). Here, differences between ideal omnidirectional and real-world UCA are discussed through examples. Several algorithms for DoA estimation on UCA exploit the Beamspace Transform (BT) because it allows fast computation. However, it introduces an error. In this work we show that by forming an array of sensors with optimal shaped beampattern, the BT will then perform a practically error- free mapping between UCA and ULA. Simulation results show that we can obtain unbiased DoA estimates even without using techniques for bias removal when the complex beampattern of the array sensors are optimally designed. I. INTRODUCTION Uniform Circular Arrays (UCA) are of interest in a variety of applications, e.g. in multiantenna transceivers. Moreover, UCA's have almost uniform performance regardless of the angle of azimuth. Sev- eral DoA estimators for UCA, such as root-MUSIC and ESPRIT (1) employ the Beamspace Transform (BT). This transform essentially maps the steering vectors of UCA into the steering vectors of a ULA- like array, called virtual array, with approximately Vandermonde structure (3). In (3)-(4) it has been shown that when the BT is applied to certain array configurations (number of elements, interelement spacing,...), a bias and an excess variance may appear in the DoA estimates which severely degrade the algorithm performance. Recently, we proposed a novel iterative approach for removing the bias (3). Moreover, a new criterion based on the Effective Aperture Distribution Function (EADF), for reducing the excess variance (4) in the DoA estimates was introduced. We have shown that by sampling the array aperture in the spatial harmonics domain, we get infinite number of replicas of the discrete aperiodic EADF in the excitation mode domain. This leads to aliasing in the mode domain. In this paper we introduce an different approach for avoiding the influence of the aliasing terms. Here, we show that the bias may be avoided by optimally designing the sensors beampattern and performance close to the Cramer-Rao Lower Bound (CRB) is achieved. We propose a study where starting from the compression of the EADF for aliasing cancellation, we will come up with optimal shaping for the array sensor beampattern. Hence, even in presence of spatial sampling, the EADF's are practically non-aliased. The design can be formulated as a Total Least Squares (TLS) problem (2). The paper is organized as follows. First, the system model is presented. In Section III, we introduce the EADF. In Section IV, we consider a real-world antenna array. Here some of the observations which have motivated our work are also given. In Section V we introduce the idea of optimal EADF with compressed support. In Section VI the complex directional beampattern of a array sensors which could meet the specification imposed by the desired EADF is shown. In Section VII we show a numerical example showing the benefits of having an antenna with optimally designed directional sensors. Finally, in Section VIII, conclusions are drawn.