多非相干子阵列下的非圆光源DOA估计优化与MUSIC算法

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本文是一篇发表在2017年8月《IEEE通信快报》上的研究论文，标题为“具有多个非相干子阵列的非圆形光源的DOA估计”。文章关注的核心问题是针对非圆周源（noncircular sources）的DOA（方向到达角）估计，这在众多应用中，如信号处理、无线通信和分布式传感器网络中具有重要意义。首先，作者探讨了该问题的Cramér-Rao下界（Cramer-Rao Lower Bound, CRLB），这是评估任何DOA估计方法性能的基础标准。CRLB给出了在给定条件下，估计误差的最小可能方差，是衡量估计精度的理论极限。接下来，研究者针对多个非相干子阵列提出了两种情况下的二次均方误差（Mean Squared Error, MSE）最小化估计器：一是针对圆周源（circular sources）的情况，另一种则是更为复杂且实际的非圆周源情况。这两种估计器旨在提供更精确的方向信息，特别是在非线性源特性的影响下。此外，为了实现计算效率并利用非圆周源的特性，作者还提出了一种类似于MUSIC算法（Multiple Signal Classification, MUSIC）的新方法。MUSIC算法通常用于处理线性阵列的信号处理，而这个改进版的算法则扩展到了非线性阵列，允许在复杂环境中进行有效的DOA估计。最后，通过数值例子展示了新方法的有效性和性能优势，对比了其与传统方法在估计精度、复杂度和适应性方面的改进。这些例子证实了在处理非圆形光源时，利用非相干子阵列和考虑信号的非圆周特性能够显著提升DOA估计的准确性和实用性。关键词包括非圆周信号（noncircular signals）、DOA估计、MUSIC算法、时间变化阵列（time-varying arrays）以及分布式传感器阵列网络（distributed sensor array networks）。这篇文章不仅深化了对非线性源环境下DOA估计的理解，也为实际系统设计提供了有价值的理论支持和技术路线。

IEEE COMMUNICATIONS LETTERS, VOL. 21, NO. 8, AUGUST 2017 1783

DOA Estimation for Noncircular Sources with

Multiple Noncoherent Subarrays

Fei Wen, Member, IEEE, Wei Xie, Xin Chen, Member, IEEE, and Peilin Liu, Member, IEEE

Abstract—This letter addresses the direction-of-arrival

estimation issue for noncircular sources with multiple noncoher-

ent subarrays. First, we present the Cramer–Rao lower bound

for this problem. Then, for multiple noncoherent subarrays,

we propose the asymptotically minimum variance second-order

estimators for the two cases of circular and noncircular sources,

respectively. Furthermore, we propose a computationally efﬁcient

MUSIC-like algorithm for such arrays which can exploit the

noncircularity of the sources. Finally, numerical examples have

been provided to demonstrate the performance of the new

methods.

Index Terms— Noncircular signals, direction of arrival (DOA)

estimation, MUSIC algorithm, time varying arrays, distributed

sensor array networks.

I. INTRODUCTION

IRECTION-OF-ARRIVAL (DOA) estimation of multiple

sourcesusinganarrayisanessentialtaskinmanyappli-

cations, such as radar, sonar, communications, geophysics,

tracking and localization [1]. In this work, we consider

the DOA estimation problem using multiple noncoherent

subarrays. Such arrays arise in the many important applica-

tions, such as time-varying arrays [5]–[11], subarray sam-

pling [12], [13] and partially coherent arrays. It is also the

case of distributed sensor array networks, in which each

array is perfectly coherent locally but the correlation between

different arrays is irrelevant [2]–[4].

All of the above works [2]–[13] focused on either determin-

istic or stochastic circular signals, while this work considers

the case of noncircular signals. Noncircular signals are usually

encountered in the context of communications, such as AM,

MASK, BPSK or UQPSK signals. It has been shown that,

the noncircularity of complex signals can be exploited to

improve the performance of DOA estimation [14]–[19], [27],

beamforming [20]–[22] and time delay estimation [23], [24].

Existing DOA estimation methods for noncircular sig-

nals (e.g., [14]–[19] and the references therein) commonly

assume a single coherent array, and none of them applies

to multiple noncoherent subarrays. The goal of this work

is to develop DOA estimation methods for noncircular sig-

nals applicable to generalized multiple noncoherent subarrays.

Manuscript received March 10, 2017; revised April 12, 2017; accepted

May 7, 2017. Date of publication May 9, 2017; date of current version

August 10, 2017. This work was supported in part by the National Natural

Science Foundation of China (NSFC) under grants 61401501 and 61472442.

The associate editor coordinating the review of this paper and approving it

for publication was J. Prieto. (Corresponding author: Fei Wen.)

F. Wen, X. Chen, and P. Liu are with the Department of Electronic

Engineering, Shanghai Jiao Tong University, Shanghai 200240, China (e-mail:

wenfei@sjtu.edu.cn; xin.chen@sjtu.edu.cn; liupeilin@sjtu.edu.cn).

W. Xie is with the Department of Electronic Engineering, University

of Electronic Science and Technology of China, Chengdu 610054, China

(e-mail: raysheik@foxmail.com).

Digital Object Identiﬁer 10.1109/LCOMM.2017.2703649

The main contributions of this work are as follows. First,

we present the CRB of DOA estimation for noncircular

complex Gaussian (NCG) sources with multiple noncoher-

ent subarrays. Then, the GLS estimator [10] is improved

to be an asymptotically minimum variance (AMV) second-

order (SO) estimator for circular sources, and the AMV

SO estimator for noncircular sources [15] is extended for

multiple noncoherent subarrays. For multiple sources, these

methods require multi-dimensional nonlinear optimization,

which prompts us to design a computationally simple MUSIC-

like method subsequently. Speciﬁcally, we propose a weighted

MUSIC algorithm for noncircular sources, which fuses the

spatial spectrum of the subarrays in a weighted summation

manner.

Notations: (·)

and (·)

denote the transpose and Hermitian

transpose, respectively. δ

l,k

is the Kronneck function. || · ||

(·) and (·) stand for the Frobenius norm, real and imaginary

part operators, respectively. vec(·) is the “vectorization” oper-

ator stacking the columns of the matrix one below another, and

v(·) denotes the operator obtained from vec(·) by eliminating

all supradiagonal elements of the matrix. ⊗ denotes the

Kronecker product. I

stands for an M × M identity matrix.

II. P

ROBLEM STATEMENT,MLESTIMATOR AND CRB

A. Signal Model

Consider an array consisting of L noncoherent subar-

rays receiving d narrowband far-ﬁeld signals impinging with

unknown DOAs θ

, ··· ,θ

.Thel-th subarray has M

sensors.

The output of the l-th subarray at time t

at some local

reference point can be expressed as

) = A

()x(t

) + n

), t

= 1, ··· , N

(1)

where A

() =[a

(θ

) ··· a

(θ

)] is the full column rank

steering matrix with a

(θ) represents the complex manifold of

the l-th subarray and  =[θ

··· θ

]

denotes the vector

of the unknown DOAs; x(t

) =[x

) ··· x

)]

and

) are the complex signal amplitudes and additive white

sensor noise, respectively; x(t

) and n

) are multivariate

independent zero-mean wide-sense stationary processes; n

)

is assumed spatially white circular complex Gaussian (CG)

with E{n

)}=σ

, whereas x(t

) is NCG with

P = E{x(t

)} and Q = E{x(t

)}.

From (1), the two covariance matrices corresponding to the

l-th subarray are

() = A

+ σ

and 

() = A

(2)

where  =[



]

denotes the unknown para-

meter vector with  being a real vector made from the

free real parameters P(i, i), {(P(i, j)), (P(i, j))}

j>i

and

{(Q(i, j)), (Q(i, j))}

j≥i

See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

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