Analysis for 4-D Fast Polarization MUSIC Algorithm
Guo Ran, Mao Xing-peng, Li Shao-bin, Lin Zhi-min, Deng Wei-bo, Jiang Peng
School of Electronics and Information Engineering
Harbin Institute of Technology
Harbin, Heilongjiang Province, China
mxp@hit.edu.cn
Abstract—Fast Polarization MUSIC is an algorithm which
can utilize the advantage of the signals polarization state to
estimate their directions. Its capacity in resolving close signals is
considerably enhanced, with acceptable computation complexity.
In this paper, analysis of this algorithm's performance in
estimation precision, resolution of close signals and computation
complexity are presented. Relating simulations are also provided.
I. INTRODUCTION
The estimation of the direction of arrivals (DOA) is of
great importance in various fields. Many algorithms have been
devised, and a class known as noise subspace algorithms are
being most widely studied, for their excellent performance and
acceptable computation complexity. These include multiple
emitter location and signal parameter estimation (MUSIC) [1],
estimation of signal parameters via rotational invariance
techniques (ESPRIT)[2], and some other algorithms.
However, in applications, the space for the array are often
limited, so ESPRIT which must exert on a uniform linear
array (ULA), usually cannot be adopted. As the resolution of
MUSIC largely depends on the aperture of arrays, small arrays
may well fail to resolve close signals. To obtain better
resolution, we need assistance from somewhere else. The
polarization diversity of signals have been utilized to separate
them in radar system, and is also expected to enhance the
resolution of noise subspace algorithms. In [3], [4], Jian Li
managed to embed ESPRIT algorithm in polarization sensitive
ULA. But to explore polarization in MUSIC, which provides
estimation for both azimuth and elevation, a 4-D peak search
is required, which incurs vast amount of computation.
Reference [8] proposed a fast algorithm, called 4-D Fast
Polarization MUSIC (FP-MUSIC) here. It reduces the amount
of calculation considerably, while still benefits from the
polarization. In this article, we offer the analysis on its
performance. In the simulation, the performance of FP-
MUSIC is compared with that of conventional MUSIC.
II. B
ASIC PRINCIPLE OF MUSIC AND FP-MUSIC
Consider an array consists of
M
non-polarization-
sensitive sensors.
N
narrow-band signals of the same central
frequency arrive from various directions. The output is:
() () ()
ttt=+XASN
(1)
The covariance matrix of
()
tX
is
1
M
H
iii
i
λ
=
=
∑
Ree
(2)
Here
i
λ
is the ith eigen value of
R
, which is arranged as
2
1+1
==
NMn
λλλσ
≥⋅⋅⋅≥ ⋅⋅⋅ =
, and
2
n
σ
is the power of the white
noise.
i
e
is the corresponding eigen vector of
i
λ
. In MUSIC,
the space spectrum is defined as follows:
2
=+1
1=1
M
HH
iN
iN
P =
∑
ae aU
(3)
The token
•
denotes the Frobenius norm, and
()
12
, ,...,
SN
=Ueee
,
()
12
, ,...,
NNN M++
=Uee e
(4)
The vector
a
is called steering vector. In the conventional
MUSIC algorithm, only the direction parameters, such as the
azimuth
and the elevation
θ
, are involved in
a
. A peak
search through
and
θ
will locate directions of the arrivals.
To enhance the performance in resolution, the polarization
sensitive array can be adopted. Here we use the polarization
angle
and polarization phase difference
to determine the
signals' polarization state, and the steering vector becomes
()()()
iiii P i iii S ii
,, ,,, ,
θ
ηθ
ηθ
=⊗aa a
(5)
where
⊗
means the Kronecker product. The column vector
()
Sii
θ
a
is the component of spatial direction and the effect
of polarization is manifested in the vector
()
Piiii
,,
θ
η
a
.
Using the
()
iiii
,,
θ
η
a
in expression (5) to substitute the
steering vector
a
in expression (3), a new form of joint space
spectrum will be obtained, which can be written as
() ()
2
,
,,, 1 ,,,
H
JSPN
P
θϕγη θϕγη
= aU
(6)
Theoretically, by the positions of the 'peaks' in
()
,,,
J
P
θ
η
, the direction and polarization parameters of the
arrivals can be estimated. A straightforward method is to exert
a 4-D search through the ranges of all parameters, which is,
however, extremely computation exhaustive.
To reduce the amount of calculation, the geometric
characters of
()
,,,
J
P
θ
η
need to be studied. Its expression is:
()() ()
1
1
,,, ,,, ,,,
J
PZ MU
θϕγη θϕγη θϕγη
−
−
==−
⎡⎤
⎣⎦
(7)
Here
()
,,,Z
θ
η
is called zero spectrum, and
This project is sponsored by the National Natural Science Foundation o
China (No. 61171180 and No.61001093), and Natural Scientific Researc
Innovation Foundation in Harbin Institute of Technology (HIT.
SRIF.2011117).
2013 IEEE Radar Conference (RadarCon13)
978-1-4673-5794-4/13/$31.00 ©2013 IEEE