REAL POLYNOMIAL FORM OF MUSIC FOR UNIFORM LINEAR ARRAY
Xiang Cao
∗
, Jingmin Xin
Xi’an Jiaotong University
Institute of Artificial Intelligence and
Robotics
Xi’an 710049, China
Yoshifumi Nishio
The University of Tokushima
Department of Electrical and Electronic
Engineering
Tokushima 770-8506, Japan
ABSTRACT
This paper presents real polynomial form of multiple signal
characterization (MUSIC) with uniform linear array (ULA).
Firstly, the proposed method can reduce the computational
burden of spectral music by taking the place of a large num-
ber of search points by several real roots of polynomial.
Secondly, to transform Root-MUSIC algorithm into poly-
nomial with real coefficients, a higher order Root-MUSIC
whose variable is only defined on the unit upper semicircle is
presented. Furthermore, this paper also improves the results
obtained by J.Selva in 2005.
Index Terms— Parameter estimation, uniform linear ar-
ray (ULA), array signal processing
1. INTRODUCTION
The multiple signal characterization (MUSIC) algorithm for
direction finding [1] has been advanced for many years. To
a certain degree, the more search points one use, the more
accurate results one will obtain. The main computational bur-
den locate the process of spectral search. Thus, many effi-
cient methods are proposed to reduce the search complexity
of spectral music.
A called C-MUSIC recently appears in [2]. The approach
firstly divided the total interested observation field into sev-
eral equal segments. The virtual source positions can be gen-
erated by spectral search over one segment and the final esti-
mation values will be selected from these candidates. It shows
from Figs. 4 and 5 of [2] that the smaller number of segments
may result in the better estimation quality. However, it means
that C-MUSIC needs more search points to achieve a certain
accuracy.
To avoid the tremendous computation load of spectral
search, Root-MUSIC [3],[4] is proposed. The search-free
method has an improved resolution threshold. Taking advan-
tage of conformal transformation, the author of [5] translated
the Root-MUSIC into an univariate polynomial with real
∗
Xiang Cao is also with the Department of Electrical and Electronic En-
gineering, The University of Tokushima,Tokushima 770-8506, Japan
coefficients. Here, we have to point out that the interested
spatial frequency in Root-MUSIC lies in the interval [−π, π]
instead of [0,π]. Since only half of field-of-view is consid-
ered, the real coefficients polynomial in [5] has the same
degree with standard Root-MUSIC. In this paper, we improve
the result and give the real polynomial form of Root-MUSIC
in [−π,π]. The same mapping idea also appears in [6] to
estimate single target. The interested range is mapped into
real line by a tangent function, while the transformation func-
tion is not monotonic within target range. That is to say,
the target may not be uniquely determined from the inverse
transformation.
In this paper, a polynomial form with real coefficients
of spectral music is presented. Our method treats the whole
range of interest in uniform linear array. The complexity of
spectral music can be reduced by replacing a large number
of search points by real roots belonging to [−1, 1]. Further-
more, we propose a higher order Root-MUSIC whose vari-
able is only defined on the unit upper semicircle. The sim-
ilar Root-MUSIC then is transformed into real polynomial
form. The process of solving this polynomial just involves
real arithmetic [7]. Our method also can be used in unitary
Root-MUSIC [8] and unitary MUSIC [9].
2. DATA MODEL
Assume p far-field narrowband signals {s
k
} impinging on a
Uniform linear array (ULA) with M (M>p) sensors. The
output vector y(t) of the array at time t can be written as
y(t)=A(θ)s(t)+n(t), (1)
where s(t) is the vector of incident signals and n(t) is the
vector of additive noises. The so-called array steering matrix
and steering vector have the following form:
A(θ)=[a(θ
1
), a(θ
2
), ··· , a(θ
p
)] (2)
a(θ
i
)=
1,e
j
2π
λ
d sin(θ
i
)
, ··· ,e
j
2π
λ
(M−1)d sin(θ
i
)
T
, (3)
366
2013 IEEE Workshop on Si
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