Fabio Belloni

DoA Estimation Via Manifold Separation for Arbitrary Array Structures

In this paper, we consider the manifold separation technique (MST), which stems from the wavefield modeling formalism developed for array processing. MST is a method for modeling the steering vector of antenna arrays of practical interest with arbitrary 2-D or 3-D geometry. It is the product of a sampling matrix (dependent on the antenna array only) and a Vandermonde structured coefficients vector depending on the wavefield only. This allows fast direction-of-arrival (DoA) algorithms designed for linear arrays to be used on arrays with arbitrary configuration. In real-world applications, the calibration measurements used to determine the sampling matrix are corrupted by noise. This impairs the performance of MST-based algorithms. In particular, we study the effect of noisy calibration measurements on subspace-based DoA algorithms using MST. Expressions describing the error in the DoA estimates due to calibration noise and truncation are derived. This allows predicting the performance of MST-based algorithms in real-world applications. The analysis is verified by simulations. We established a link between the optimal number of selected modes and the statistics of calibration noise. We analyze the modeling error when MST is used for 1-D (azimuth) DoA estimation.

Beamspace Transform for UCA: Error Analysis and Bias Reduction

In this paper, we analyze the error caused by the beamspace transform (BT) when it is applied to uniform circular array (UCA) configuration. Several algorithms for direction of arrival (DoA) estimation exploit this modal transform because it allows using computationally efficient techniques such as polynomial rooting and dealing with coherent sources. The BT is based on the phase-mode excitation principle. The performance of such DoA estimators is degraded if the array has a small number of elements. We introduce a modified beamspace transform (MBT) that performs mapping from element-space to beamspace domain taking into account the error caused by the transform. Justification of the difference in the statistical performances of MUSIC and root-MUSIC algorithms for UCA is also given. Moreover, we show that there is a significant difference in the performance of the UCA root-MUSIC technique depending on whether an even or odd number of elements is used. We derive an expression approximating the bias in the DoA estimates that is caused by the beamspace transform. Some design guidelines are provided for choosing the key UCA configuration parameters such as number of sensors, array radius, and interelement spacing in order to reduce the error. Finally, we propose a novel technique for bias removal. It allows practically bias-free DoA estimation

Extension of root-MUSIC to non-ULA Array Configurations

In this paper we introduce a method for modelling the steering vector of an arbitrary array such that its steering vector can be expressed as the product of a characteristic matrix of the array itself and a vector with a Vandermonde structure containing the unknown parameter. We call this technique manifold separation. By exploiting this concept, we developed a novel version of the root-MUSIC algorithm for direction of arrival (DoA) estimation of sources. It can be applied to arbitrary 2-D array configurations. The proposed algorithm processes the data in element-space domain and does not require any transformation or array interpolation. The novel algorithm, named element-space root-MUSIC, provides computationally low complexity (search-free) DoA estimation and has close to CRB performance already at low SNRs

Unitary root-MUSIC technique for uniform circular array

In this paper, unitary root-MUSIC algorithm for direction of arrival estimation is proposed for uniform circular array. Uniform circular array provides uniform performance in any direction and simultaneous azimuth and elevation angle estimates. The proposed algorithm has low computational complexity because eigenvalue decomposition for real number may be used. It also provides low variance estimates because the number of observations is doubled in comparison to conventional MUSIC algorithm. Also, a robust extension to the method is introduced based on multivariate extensions of nonparametric statistics. It gives highly reliable results in the face of heavy-tailed noise and interference. The additional computational cost is negligible.

DoA and Polarization Estimation using Arbitrary Polarimetric Array Configurations

In this paper we propose an algorithm for angle and polarization estimation of non-coherent sources using real-world polarimetric antenna arrays. In order to construct an accurate model of an antenna array, we exploit its effective aperture distribution function (EADF). This model contains information on the array response for a vertical and horizontal excitation, the directional characteristic of each element, and array imperfections. Here, we propose a method stemming from the RARE (rank-reduction estimator) algorithm. It performs fast (search-free) DoAs and polarization coefficients estimation. Throughout simulation results, we verify that the algorithm has close to optimal statistical performance with polarimetric real-world arrays. We present the statistical performance for three different array configurations

Reducing Excess Variance in Beamspace Methods for Uniform Circular Array

In this paper we present a procedure for reducing the excess variance introduced by beamspace transform when it is applied to uniform circular array (UCA). Several algorithms for direction of arrival (DoA) estimation exploit modal transforms which are based on the phase-mode excitation principle (D.E.N. Davies, 1983). Here we analyze the inverse Fourier series of the array impulse response, called effective aperture distribution function (EADF), for defining an efficient criterion for selecting the number of virtual array elements. The proposed criterion is optimum in the sense that maximizes the aperture of the virtual array while satisfying certain constraints on the maximum number of virtual array elements. Simulation results show that we can obtain DoA estimates with a lower variance such that they are closer to the CRB

Reducing bias in beamspace methods for uniform circular array [DoA estimation applications]

In this paper, we characterize the error introduced by the beamspace transform when it is applied to a uniform circular array (UCA). Several algorithms for direction of arrival (DoA) estimation employ this modal transform. In particular, we focus on the UCA unitary root-MUSIC algorithm. The performance of such an estimator is degraded and bias occurs especially if the array has a small number of elements. Here we propose a novel technique for reducing the bias. This leads to practically bias-free DoA estimates.

Analysis of beamspace transform on uniform circular array

In this paper we analyze the error introduced by beamspace transform when it is applied to uniform circular array (UCA). Several algorithms for direction of arrival (DoA) estimation exploit this modal transform. The performance of such estimators is degraded if the array has a small number of elements. The beamspace transform is based on the phase-mode excitation principle (Davies, DEN, 1983). Here we introduce a generalized beamspace transform that performs mapping from element-space to beamspace domain taking into account the error caused by the transform. Justification of the difference in the statistical performances of MUSIC and root-MUSIC algorithms for UCA is also given. Finally, we introduce a first order approximation of the bias in the DoA estimates that arises after the beamspace transform. An expression for the bias is provided.

Avoiding Bias in Circular Arrays Using Optimal Beampattern Shaping and EADF

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, UCAs 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 EADFs 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.

Polynomial rooting-based direction finding for arbitrary array configurations

In this paper, we propose angle of arrival estimation algorithms for arbitrary array geometries. The proposed methods extend the root-WSF and modified variable projection (MVP) algorithms to arbitrary array configurations. This is accomplished by employing the recently introduced manifold separation technique (MST), which stems from wavefield modelling. The algorithms process the data in the element-space domain, i.e. no mapping of the data that introduces errors is required. Moreover, coherent sources can be handled. The proposed MST-based MVP algorithm shows a statistical performance close to the Cramer-Rao lower bound (CRLB). The performance is illustrated using calibration data from two real-world arrays.