Quaternion-MUSIC for vector-sensor array processing

This paper considers the problem of direction of arrival (DOA) and polarization parameters estimation in the case of multiple polarized sources impinging on a vector-sensor array. The quaternion model is used, and a data covariance model is proposed using quaternion formalism. A comparison between long vector orthogonality and quaternion vector orthogonality is also performed, and its implications for signal subspace estimation are discussed. Consequently, a MUSIC-like algorithm is presented, allowing estimation of wave's DOAs and polarization parameters. The algorithm is tested in numerical simulations, and performance analysis is conducted. When compared with other MUSIC-like algorithms for vector-sensor array, the newly proposed algorithm results in a reduction by half of memory requirements for representation of data covariance model and reduces the computational effort, for equivalent performance. This paper also illustrates a compact and elegant way of dealing with multicomponent complex-valued data.

I. Introduction

Hamilton's quaternions , as a nontrivial generalization of complex numbers , have been considered for a long time of pure theoretical interest. In signal processing, it is only in the last decade that quaternion-based algorithms were proposed [1]. More recently, hypercomplex spectral transformations and color images processing techniques have been introduced by Ell and Sangwine in [2]–[5] and Pei and Cheng in [6]. A hypercomplex version of the multidimensional complex signals [7] was also proposed by Bülow and Sommer [8]. In seismic data processing, as multicomponent acquisitions fit perfectly with the quaternion model, quaternion algebra has been used to extract seismic attributes [9], [10], to enhance signal-to-noise ratio (SNR) and to separate sources on multicomponent ¹

In this paper, the terms multicomponent data and polarized data are used to designate the data set recorded on a vector-sensor array.

seismic data set [11], [12]. Most of these methods encode a real-valued three-component signal on the three imaginary parts of a pure quaternion (see Subsection II-A). In this paper, we propose a data model allowing to deal with multicomponent modulus-phase information by means of quaternions. The resulting data model is then used to illustrate an eigenstructure-based algorithm for vector-sensor array processing yielding direction of arrival (DOA) and polarization parameters estimation.