非圆形源DOA估计：稳定噪声下的随机Cramér-Rao界

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"这篇研究论文探讨了在稳定噪声背景下，特别是α-稳定噪声环境中，非圆形源DOA（方向-of-arrival）估计的随机Cramér-Rao界限（SCRB）。作者扩展了这一理论，从Cauchy过程模型的噪声到更广泛的α-稳定噪声情况。文中还对EXC-NC-MUSIC算法在脉冲噪声环境中的性能进行了模拟实验，并通过SCRB进行评估。关键词包括：随机Cramér-Rao界限、稳定噪声、DOA估计、非圆形信号。" 在现代无线通信和雷达系统中，DOA估计是一项关键任务，它涉及确定发射信号的来源方向。非圆形信号，如BPSK和OQPSK调制信号，由于其相位特性，在DOA估计中提供了额外的信息，从而可能提高估计精度。Cramér-Rao界（CRB）是一种理论性能下限，用于衡量参数估计的最小可能误差，对于理解估计算法的最优性具有重要意义。在本研究中，作者首先关注的是在Cauchy过程噪声背景下的非圆形源DOA估计的SCRB。Cauchy噪声是一种典型的脉冲噪声模型，适用于描述具有重尾分布的随机过程，这种噪声在实际环境中普遍存在，例如雷达接收机中的干扰。SCRB的推导为评估在该特定噪声环境下的DOA估计算法提供了理论基础。进一步，作者将研究扩展到了更一般的α-稳定噪声模型。α-稳定分布是一类包含了高斯分布（α=2）和Cauchy分布（α=1）在内的广义分布，可以更好地描述各种极端值和非高斯噪声特性。这对于分析在各种复杂环境下的DOA估计性能至关重要。 EXC-NC-MUSIC（扩展循环非圆形音乐算法）是作者提出的一种针对非圆形信号的DOA估计方法。在本论文中，通过模拟实验，他们在各种脉冲噪声环境中测试了该算法的性能，并将其与SCRB进行比较。这不仅验证了EXC-NC-MUSIC在实际应用中的有效性，也为未来优化算法提供了指导。这篇论文为非圆形信号在稳定噪声环境下的DOA估计提供了一个深入的理论框架，并对一个具体算法的性能进行了实验验证。这些成果对于设计更高效、更稳健的DOA估计算法以及在实际系统中应对各种噪声挑战具有重要的参考价值。

880

Stochastic Cramér-Rao Bound for Noncircular Sources’ DOA Estimation in Alpha-

Stable Noise

Jinfeng Zhang

Shenzhen Key Laboratory of Antennas and Propagation

Shenzhen University

Shenzhen, China

e-mail: zhangjf@szu.edu.cn

Tianshuang Qiu

Faculty of Electronic Information and Electrical

Engineering

Dalian University of Technology

Dalian, China

e-mail: qiutsh@dlut.edu.cn

Abstract—This paper derives the stochastic Cramér-Rao

bound (CRB) for the direction of arrival (DOA) estimation

accuracy for noncircular sources in the noise modelled by

Cauchy process, and expands it to the general alpha-stable

noise situations. The performance of the EXC-NC-MUSIC,

which is proposed by the authors, in a wide range of impulsive

noise environments is examined via simulation experiments

and evaluated with the CRB.

Keywords-stochastic cramér -rao bound; stable noise;

direction of arrival; noncircular

I. INTRODUCTION

In recent years, developing direction finding methods for

noncircular signals, e.g., binary phased-shift keying (BPSK),

offset-quadrature-phase-shift keying (OQPSK) modulated

signals, has aroused plenty of interest. Various high-

resolution parameter estimation methods such as NC-

MUSIC [1], NC root-MUSIC [2], NC-ESPRIT [3], NC-

unitary-ESPRIT [4], NC-

-MUSIC [5], and RNC-ESPRIT

[6] have been developed to exploit the structure of signals

from noncircular sources. And then, to evaluate the

performance of these methods, several stochastic CRBs

(Cramér-Rao bound) have been derived. In [7-8], the CRB is

derived under the noncircular Gaussian sources assumption.

In [9], the deterministic CRB for strictly noncircular sources

has been derived. However, as we described in [10], all the

above methods and the CRBs are derived under the same

assumption on the underlying noise, i.e. Gaussian distributed.

In fact, the impulsive characteristic of noise has been

found to be commonly encountered in many problems,

including atmospheric noise and underwater problems such

as sonar and submarine communication. Taking these

scenarios into account, in [10], by modelling the underlying

noise as symmetric alpha-stable (



) processes, we

address the direction finding problem in the presence of

impulsive noise by utilizing the extended covariation based

matrix of the sensor outputs for noncircular sources. The

robust performance of the proposed method, as we call it the

EXC-NC-MUSIC in [10], for noncircular sources’ DOA

estimation in impulsive noise has been demonstrated by

simulation experiments.

To assess the performance of our proposed method more

precisely, in this paper, we derive the closed-form expression

of the stochastic CRB for DOA parameters for noncircular

sources in



noise. The paper is organized as follows: in

Section II, the problem of DOA estimation for noncircular

sources is formulated and the



distribution is introduced,

and In Section III, the stochastic CRB for DOA parameters is

derived. Finally, illustrative examples are presented in

Section IV, and conclusions are drawn in Section V.

The following notations are utilized throughout this paper.

The superscripts “

”, “

” and “



” denote the transpose,

the conjugate transpose, and the conjugate, respectively.

()E 

()

and

()

stand for the expectation, the real and

imaginary part operators, respectively.

is the

MM

identity matrix.

II. PROBLEM FORMULATION AND



DISTRIBUTION

A. Problem formulation

Consider an array of

sensors that receives the signals

emitted by

far-field narrowband sources with locations

 

, , ,

  

. The array output can be expressed as

( ) ( ) ( )t t tx As n

(1)

where

[ ( ), ( ), , ( )]

  

A a a a

is the steering matrix, in

which

( ) ( )

( ) [1, , , ]

k M k

   







is the steering

vector toward direction



, with



and

()



denoting the

center frequency and the propagation delay between the first

and the

thi

sensor for a waveform impinging from direction



, respectively.

 

( ) ( ), ( ), , ( )

t s t s t s ts

is the signal

vector with

()

denoting the signal emitted by the

thk

source.

 

( ) ( ), ( ), , ( )

t n t n t n t

is the noise vector.

Considering that

()ts

is noncircular with the noncircularity

rate





, the second order statistics information is

embedded not only in the conventional covariance function

[ ( ) ( )]

E t tss

but also in the unconjugated spatial covariance

function

[ ( ) ( )]

E t tss

. Correspondingly, the NC-MUSIC [1]

is to retrieve the estimation of the DOA parameters from the

2017 9th IEEE International Conference on Communication Software and Networks

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