Quad-Quaternion MUSIC for DOA Estimation Using Electromagnetic Vector Sensors
EURASIP Journal on Advances in Signal Processing
Hindawi Publishing Corporation
Quad-Quaternion MUSIC for DOA Estimation Using Electromagnetic Vector Sensors
Xiaofeng Gong 0
Zhiwen Liu 0
Yougen Xu 0
0 Department of Electronic Engineering, Beijing Institute of Technology , Beijing 100081 , China
A new quad-quaternion model is herein established for an electromagnetic vector-sensor array, under which a multidimensional algebra-based direction-of-arrival (DOA) estimation algorithm, termed as quad-quaternion MUSIC (QQ-MUSIC), is proposed. This method provides DOA estimation (decoupled from polarization) by exploiting the orthogonality of the newly defined “quadquaternion” signal and noise subspaces. Due to the stronger constraints that quad-quaternion orthogonality imposes on quadquaternion vectors, QQ-MUSIC is shown to offer high robustness to model errors, and thus is very competent in practice. Simulation results have validated the proposed method.
1. INTRODUCTION
A “complete” electromagnetic (EM) vector sensor
comprises six collocated and orthogonally oriented EM sensors
(e.g., short dipole and small loop), and provides complete
electric and magnetic field measurements induced by an
EM incidence [
1–3
]. An “incomplete” EM vector sensor
with one or more components removed is also of high
interest in some practical applications [
4, 5
]. Numerous
algorithms for direction-of-arrival (DOA) estimation using
one or more EM vector sensors have been proposed.
For example, vector sensor-based maximum likelihood
strategy was addressed in [
6–9
], multiple signal
classification (MUSIC [
10
]) was extended for both incomplete
and complete EM vector-sensor arrays in [
11–16
],
subspace fitting technique was reconsidered for incomplete
EM vector sensors in [
17, 18
], and estimation of signal
parameters via rotational invariance techniques (ESPRIT
[
19
]) was revised for EM vector sensor(s) in [
20–26
].
The identifiability issue of EM vector sensor-based DOA
estimation has been discussed in [
27–29
]. Some other
related work can be found in [
30–34
]. In all the
contributions mentioned above, complex-valued vectors are
used to represent the output of each EM vector sensor
in the array, and the collection of an EM vector-sensor
array is arranged via concatenation of these vectors into a
“long vector.” Consequently, the corresponding algorithms
somehow destroy the vector nature of incident signals
carrying multidimensional information in space, time, and
polarization.
More recently, a few efforts have been made on
characterizing the output of vector sensors within a
hypercomplex framework, wherein hypercomplex values, such
as quaternions and biquaternions, are used to retain the
vector nature of each vector sensor [
35–37
]. In particular,
singular value decomposition technique was extended for
quaternion matrices in [
35
] using three-component vector
sensors. Quaternion-based MUSIC variant (Q-MUSIC) was
proposed in [
36
] by using two-component vector sensors.
Biquaternion-based MUSIC (BQ-MUSIC) was proposed in
[
37
] by employing three-component vector sensors. The
advantage of using quaternions and biquaternions for vector
sensors is that the local vector nature of a vector-sensor
array is preserved in multiple imaginary parts, and thus
could result in a more compact formalism and a better
estimation of signal subspace [
36, 37
]. More importantly,
from the algebraic point of view, the algebras of quaternions
and biquaternions are associative division algebras using
specified norms [38], and therefore are convenient to use in
the modeling and analysis of vector-sensor array processing.
However, it is important to note that quaternions and
biquaternions deal with only four-dimensional (4D) and
(8D) algebras, respectively, while a full characterization of
the sensor output for complete six-component EM vector
sensors requires an algebra with dimensions equal to 12 or
more.
Unfortunately, not all algebras having 12 or more
dimensions are associative division algebras. For example,
sedenions, as a well-known 16D algebra, are neither an
associative algebra nor a division algebra [
39
], and thus
are not suitable for the modeling and analysis of vector
sensors. In this paper, we use a specific 16D algebra—
quad-quaternions algebra [
40–42
] to model the output of
six-component EM vector sensor(s) [
3
]. This 16D
quadquaternion algebra can be proved to be an associative
division algebra, and thus is well adapted to the
modeling and analysis of complete EM vector sensors. More
precisely, We redefine the array manifold, signal subspace,
and noise subspace from a quad-quaternion perspective,
and propose a quad-quaternion-based MUSIC variant
(QQMUSIC) for DOA estimation by recognizing and
exploiting the quad-quaternion orthogonality between the
quadquaternion signal and noise subspaces. QQ-MUSIC here
is shown to be more attractive in the presence of two
typical model errors, that is, s (...truncated)