Regularized Parallel Factor Analysis for
Estimation of DOA and Polarization with a
Single Electromagnetic Vector-sensor
Xiaofeng Gong Zhiwen Liu Yougen Xu
Department of Electronic Engineering, Beijing Institute of Technology, Beijing 100081, China
E-mail: xiaofenggong@live.cn
Abstract: This paper considers the problem of direction-of-arrival (DOA) and polarization estimation based
on a single six-component electromagnetic vector-sensor. A regularized PARAFAC model that fuses both
second- and fourth-order statistics of the sensor signal is established within the regularized framework. The
steering vectors can be uniquely identified by exploiting the link between this trilinear model and parallel
factor analysis (PARAFAC), from which unambiguous estimation of 2-D DOAs and polarization states can
be further obtained. The proposed method combines the nice variance-property of second-order statistics and
the intrinsic multi-invariance structure of fourth-order cumulant in a tensorial manner, and offers better
performance than regularized ESPRIT and fourth-order cumulant based PARAFAC in the presence of noise
and finite data length. Simulations are provided to illustrate the performance of the proposed method.
1
Introduction
Estimation of direction-of-arrival (DOA) and polarization with six-component electromagnetic (EM) vector-sensors has attracted growing interests in the past decades. A six-component EM vector-sensor is
composed of six spatially collocated and orthogonally oriented EM sensors (for example, short dipole and
small loop), measuring all six EM field components induced by an EM incidence [1-3]. Due to the inherent
amplitude-phase relationship among all six EM field components that is related to the DOAs and polarizations of incident signals, it is possible to estimate the arrival angles and polarizations with only one EM
vector-sensor. The identifiability of an EM vector-sensor is investigated in [4], and numerous algorithms for
DOA and polarization estimation with a single six-component EM vector-sensor were proposed [2, 5-7]. For
example, a simple cross-product based method was proposed in [2], and was revised for source direction
tracking in [5]. A temporal-ESPRIT (Estimation of Signal Parameters via Rotational Invariance Techniques)
algorithm was proposed in [6] for a six-component EM vector-sensor. It was shown that the number of
resolvable signals for the temporal ESPRIT method could increase up to five, given that all the signals have
distinct center frequencies.
In addition, the ESPRIT-based method in [7, 8] jointly exploits the virtual rotational-invariance present in
the fourth-order cumulant and the nice variance-property to finite samples of second-order statistics for a
single EM vector-sensor within a regularization framework, and offers better performance than fourth-order
cumulant based ESPRIT. However, the regularized ESPRIT method uses only a small proportion of the
fourth-order cumulant information (two cumulant slices plus two columns of the signal covariance matrix)
for DOA and polarization estimation, and the intrinsic multiple rotational-invariance present in fourth-order
cumulant is not adequately utilized. In this paper, we extend the idea of regularization to the tensor domain.
More precisely, a new tensorial model that fuses the second- and fourth-order statistics is established in a
regularized manner, and a PARAFAC (Parallel Factor Analysis) based DOA-polarization estimator is
proposed. The advantage of the proposed method over regularized ESPRIT is due to the more sufficient
exploitation of the fourth-order cumulant and the second-order statistics, which would in return results in an
improved estimation accuracy. When compared with fourth-order cumulant based PARAFAC [9], the
proposed method takes the second-order statistics into consideration, and thus is less sensitive to the errors
caused by noise and finite data length.
The rest of the paper is organized as follows. Section 2 introduces the data model for a single EM vector-sensor and some used statistics. In Section 3, we set up the regularized PARAFAC model, analyze its
identifiability, and propose to obtain the DOA and polarization estimates via its decomposition. Simulation
results to illustrate the potential advantage of our proposed algorithm are provided in Section 4, and Section
5 summarizes the conclusions.
2
Data model and statistics
Let (θ , φ) and (γ, η) be the azimuth-elevation two-dimensional DOA (see Figure 1) and polarization of an
EM signal, respectively, where 0  θ  2π denotes the azimuth angle, 0  φ  π is the elevation angle,
0  γ  π / 2 is the polarization amplitude angle, and π  η  π denotes the polarization phase difference.
The steering vector of a six-component EM vector-sensor then is given by:
aθ ,φ, γ , η
 sin θ
cos φ cos θ


cos θ
cos φ sin θ


0
 sin φ

  cos φ cos θ  sin θ
  cos φ sin θ
cos θ

sin φ
0




  cos γ 

jη 
  sin γe 

hγ ,η


(1)
Θθ ,φ
Assume that there are M completely polarized incident signals, the sensor output then can be expressed as
M
x (t )   aθm ,φm , γm , ηm sm (t )  n(t )
m 1
(2)
am
where x(t )  C 6 , (θm , φm , γm , ηm ) are the DOA-polarization parameters of the m th signal sm (t ) , a m 
aθm ,φm , γm , ηm  C 6 is the m th steering vector, and n(t )  C 6 is the additive noise term.
We assume that all the incident signals are kurtic signals (a signal is said to be kurtic if has a non-zero
kurtosis [9]) and are independent of each other. Then we can further define the following cumulant slice: