脉冲噪声下基于肾上腺素相关性的新DOA估计算法

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在脉冲噪声环境下，到达方向（DOA）估计是一个重要的信号处理问题，特别是在无线通信和雷达系统中。传统方法在面对这种高瞬变、非高斯噪声时往往表现不佳。本文提出了一个创新的算法，名为"基于熵的DOA估计算法"，其灵感来源于"稳健统计"的特性，尤其是由肾上腺素优势所代表的当前相关性（correntropy）。当前相关性，作为非线性统计工具，能够更好地处理非高斯噪声，如在脉冲噪声环境中常见的对称α稳定（SαS）随机变量。SαS分布广泛存在于自然界中的许多信号中，因此对它的分析对于实际应用至关重要。作者设计了一种新的算子——基于肾上腺素的相关性（CRCO），这是一种针对iid SαS随机变量的处理手段。 CRCO的主要贡献在于将这种新算子应用于阵列传感器输出，构建出适应于脉冲噪声环境的矩阵形式。相比于传统的基于分数低阶统计量（FLOS）的MUSIC算法，CRCO-MUSIC在处理具有对称α稳定噪声的场景下表现出更高的稳健性和准确性。蒙特卡洛仿真结果详尽地验证了这一点，尤其是在高脉冲噪声的极端条件下，CRCO-MUSIC算法显示出明显的优势。这篇论文的创新之处在于它提出了一种利用非线性统计方法（如correntropy）改进DOA估计的技术，尤其适用于那些难以用传统方法有效处理的复杂噪声环境。这对于提升信号处理系统的鲁棒性，尤其是在通信和雷达等领域的应用有着显著的意义。通过CRCO，研究者能够在保持估计精度的同时，增强抗干扰能力，为实际工程应用提供了新的理论支持。

Both covariation and FLOM are proved to be bounded

for jointly SαS random variables. The operator ðU Þ

〈β〉

mainly used to suppress the outliers in a SαS distribution.

Based on the well defined covariation and FLOM, the ði; jÞth

entries of two M  M matrices for the array output xðtÞ are

formulated by ½x

ðtÞ; x

ðtÞ

in ROC-MUSIC and E fx

ðtÞj

p 2

ðtÞg in FLOM-MUSIC respectively. However, we

can see from both Eqs. (3) and (4) that ðU Þ

〈β〉

only acts on

the random variable Y but not on X at all. So the

performance of either covariation or FLOM based MUSIC

algorithms will degenerate significantly when the impul-

siveness of both x

ðtÞ and x

ðtÞ is remarkable [12].

Belkacemi [12] defined the conjugate of the FLOM

operator X

〈β〉

¼ðX

〈β〉

¼ðX

〈β〉

and proposed the PFLOM

based operator for X and Y by

PFLOM

¼EfX

〈

〉

〈 

〉

g 0o bo α ð5Þ

Thereby, the corresponding matrix for the array output

xðtÞ based on PFLOM can be derived with the ði; jÞth entry

given by Efx

〈

〉

ðtÞx

〈 

〉

ðtÞg. Since the impulsiveness in x

ðtÞ

and x

ðtÞ has been restrained simultaneously by the opera-

tor ðU Þ

〈

〉

and ðU Þ

〈 

〉

, it is easy to understand why the

PFLOM based MUSIC algorithm performs superior to the

covariation and FLOM based ones especially in highly

impulsive noise environments. However, compared with

the performance in less impulsive noise situations (e.g.

α4 1:8), all the FLOS based MUSIC algorithms (including

the PFLOM based MUSIC) exhibit a distinct degradation in

the strong impulsive noise environments (e.g. α o 1:5).

In addition, for the scenario that noncircular signals (e.g.

BPSK, AM signals) are contaminated by SαS noise, these

FLOS based MUSIC algorithms perform poorer due to the

model mismatch than they do in the scenario of circular

signals [11].

3. Correntropy based correlation

Here, we propose a new operator based on correntropy,

namely the correntropy based correlation, and then apply

it with MUSIC to gain a more robust DOA estimation

results in highly impulsive noise envrionments.

3.1. Correntropy

Inspired by kernel-based methods [15] and information

theoretic learning (ITL) methods [16], based on the infor-

mation potential (IP), the correntropy for two arbitrary

random variables X and Y is defined as follows [14]:

ðX; YÞ¼E½κ

ðX YÞ ð6Þ

where κ

ðU Þ is the kernel function that satisfies Mercer's

theory [17] and E½U  denotes the mathematical expecta-

tion. A brief interpretation for Mercer' theory and kernel

function can be found in Appendix A.

Using a Taylor series expansion for the widely used

Gaussian kernel, the correntropy can be rewritten as

ðX; YÞ¼

ﬃﬃﬃﬃﬃﬃ

2π

∑

n ¼ 0

ð1Þ

E½ðX YÞ

ð7Þ

which involves all the even-order moments of the random

variable ðX YÞ. It is notable that the term corresponding

to n ¼1in(7) is proportional to EðX

ÞþEðY

Þ2EðXY Þ,

which indicates that the conventional covariance function

(the autocorrelation for zero-mean processes) is also

included within correntropy. Liu [13] has compared cor-

rentropy with the constrained covariance [18] and con-

cluded that the former is a much simpler, while possibly

weaker measure of independence than the latter but a

much stronger measure than the traditional covariance.

Correntropy can also induce a metric which can be

described as

CIMðX; YÞ¼sqrtfE½κð0Þκ

ðX YÞg ð8Þ

Apparently, for Gaussian kernel, κð0Þ¼1=

ﬃﬃﬃﬃﬃﬃ

2π

s. The metric

correntropy induced metric (CIM) possesses a “mix norm”

property, that is, it behaves like an L2 norm when two

points are close, and like an L1 norm when the two points

are getting apart and eventually like a L0 norm when the

two points get further apart. This property demonstrates

the inherent robustness to outliers of correntropy. Liu [13]

analyzed the influence to CIM of the kernel size and

concluded that a small kernel size leads to a tight linear

region and to a large L0 region, while a larger kernel size

will enlarge the linear region and shrink the L0 region,

that is, the kernel size controls the level of the outlier

suppression.

In practice, the joint probability density function is

often unknown, and only a finite number of data

fðx

; y

Þg

i ¼ 1

for X and Y are available, where the sample

estimator of correntropy can be obtained through

ðX; YÞ¼

∑

i ¼ 1

ðx

y

Þð9Þ

The merit of correntropy is that it can convey informa-

tion not only about the correlation but also the statistical

distribution for stochastic processes as the correntropy

possesses a lot of properties quantifying the PDF of the

data directly. Meanwhile, correntropy is robust against

outliers due to the inner product in the feature space

computed via the Gaussian kernel. All these features would

inspire us to develop new array signal processing methods

using conventional correlation matrices, such as, signal and

noise subspace decompositions, projections, etc.

3.2. Correntropy based correlation

In this paper, we define a new operator based on

correntropy which can be applied in SαS processes for a

wide range of 1o α r 2, and that would make it as an

effective substitute for the conventional correlation

functions.

Theorem 1. Let X and Y be i.i.d SαS random variables with

the characteristic exponent 1o α r 2, the correntropy-based

correlation (CRCO) by applying Gaussian kernel of X and Y

given by

CRCO

¼E exp 

jX Yj





ð10Þ

is bounded. Where s is the kernel size. See Appendix B for the

proof of the boundedness of CRCO.

J. Zhang et al. / Signal Processing 104 (2014) 346–357348

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