脉冲噪声环境下快速变化子空间的一种新跟踪方法资源-CSDN下载

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标题中的关键知识点：脉冲噪声环境下快速变化子空间的一种新跟踪方法。描述中的关键知识点：该研究提出了一个新方法来追踪在脉冲噪声环境下的快速变化子空间问题。研究使用了基于最大相关熵准则（MCC）的成本函数来推导出改进的子空间跟踪算法（MCC-PAST）。为了解决快速变化的情况，作者还开发了一种自适应遗忘因子（VFF）技术，并将其融合进算法中。通过仿真实验表明，该方法在广义信噪比较低或噪声极度脉冲时依然具有鲁棒性。标签的知识点：这是一篇关于研究论文的标签，意味着文档是某项科学研究的成果。从【部分内容】中提取的关键知识点如下： 1. 最大相关熵准则（MCC）：这是一种用于信号处理中的统计方法，它通过估计概率密度函数来最大化输入和输出信号之间的互信息。MCC能够有效抵抗非高斯噪声，尤其在处理具有非对称分布的脉冲噪声时表现出优势。 2. 投影近似子空间跟踪（PAST）算法：这是一种子空间跟踪算法，通过递归最小二乘（RLS）技术来估计信号子空间。它通过最小化先前估计的子空间的投影近似与观测值之间的最小二乘误差来进行递归估计。PAST算法被广泛应用于阵列信号处理中，以降低计算复杂性和存储要求。 3. 脉冲噪声环境：脉冲噪声是突发性的、非周期性的噪声，可以表现为短暂的高强度尖峰或脉冲。这种噪声通常会对通信系统和信号处理算法造成严重影响，因为它们可能具有极端的振幅和随机的持续时间。 4. 高斯变换跟踪技术：该技术利用高斯函数来变换原始信号，从而减少脉冲噪声对信号估计的影响。高斯变换通常可以平滑信号，使得估计过程更加稳定，尤其是在噪声功率变化很大时。 5. 自适应遗忘因子（VFF）技术：VFF是一种算法自适应策略，用于调整算法中先前数据的权重。它可以根据信号变化的快慢来动态调整，使得算法能够更好地适应快速变化的环境。在子空间跟踪算法中引入VFF，可以让算法在快速变化的信号环境中仍能保持较好的跟踪性能。 6. 仿真实验和鲁棒性：通过仿真实验来验证算法在不同噪声环境下的性能是必要的步骤。鲁棒性测试尤其重要，它能表明算法在极端或异常条件下仍能维持预定的性能。在脉冲噪声环境中，鲁棒性意味着算法能在广义信噪比较低的情况下依然准确地跟踪信号子空间。综合以上知识点，该论文的主要贡献在于提出了一种基于MCC准则的改进PAST算法，并结合VFF技术，有效提升在脉冲噪声环境下的子空间跟踪能力。在实际应用中，该方法可以用于无线通信、雷达信号处理、生物医学信号分析等需要在噪声环境下准确提取信号特征的领域。通过对仿真实验数据的分析，证明了该方法在信号变化快速、噪声异常脉冲时仍能保持较好的跟踪性能，展现出了极高的实用价值和潜在的广泛应用前景。

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A Novel Tracking Method for Fast Varying

Subspaces in Impulsive Noise Environments

Jinfeng Zhang

Shenzhen Key Laboratory of Antennas and Propagation

Shenzhen University

Shenzhen, China

[email protected]u.cn

Tianshuang Qiu

Faculty of Electronic Information and Electrical

Engineering

Dalian University of Technology

Dalian, China

qiutsh@dlut.edu.cn

Abstract

—

By employing the MCC (maximum correntropy

criterion) based cost function in projection approximation

subspace tracking (PAST) algorithm, the MCC-PAST algorithm

is deduced which can be utilized for the subspace tracking under

impulsive noise environments. The Gaussian transformation

technique is combined to further enhance the tracking

performance. To handle the fast varying subspaces

circumstances, the variable forgetting factor (VFF) technique is

developed and incorporated into the algorithm. Simulation

results show the robustness of the proposed nonlinear MCC-

PAST with VFF algorithm, especially when the GSNR

(generalized signal to noise ratio) is fairly low or the underlying

noise is extremely impulsive

I. I

NTRODUCTION

Subspace tracking, which aims to derive a selected subset

of eigenvectors of a Hermitian matrix, has been developed as a

valuable tool in array signal processing in the few past decades

[1-2]. Different from the traditional subspace-based algorithms

which exploit the whole eigenstructure of the data

autocorrelation matrix, a number of fast subspace tracking

algorithms were developed by only updating the signal or noise

subspace so as to lower the computational complexity and

reduce the storage requirements. One very efficient subspace

tracking algorithm is the project approximation subspace

tracking (PAST) approach [2]. By employing the recursive

least-squares (RLS) technique, the PAST algorithm recursively

estimates the signal subspace by minimizing the least square

errors between the observation and a “projection

approximation” obtained from previously estimated subspace.

Unfortunately, the conventional PAST method has two

distinct shortcomings. First, the RLS technique is vulnerable to

impulsive noise in nature. As a matter of fact, in some

scenarios, sudden bursts or spikes, are exhibited at the array

outputs which can be characterized as impulsive. In such cases,

the PAST algorithm degrades its performance substantially.

The second defect of the conventional PAST is also traced

back to the RLS technique. That is, the RLS with a constant

forgetting factor (FF) is not suitable for tracking fast varying

subspaces. In fact, both of the drawbacks of the conventional

PAST have been studied [3-4]. In [3], Chan derived a robust

subspace tracking method by incorporating the M-estimate

technique into PAST to suppress the impulsive noise. In [4], an

extension of the Kalman filter-based algorithm with variable

number of measurements (KFVNM) is developed to handle the

tracking of time varying subspaces. That is, when the subspace

substantially varies, a small number of past measurements are

employed in the recursion and vice versa. In addition, to deal

with the impulsive noise, the M-estimate technique is also

incorporated into the KFVNM algorithm. To distinguish, it is

called the robust KFVNM algorithm.

However, as we noticed, the robust KFVNM algorithm still

suffers from two limitations. First, the computation complexity

may increase considerably when the signal subspace varies

slowly since a relative large number of past measurements

need to be utilized for computation. Second, due to the

employed M-estimate technique, the approach against the

impulsive noise is to withdraw the impulsive noise

contaminated measurements from computation. It is easily to

deduce that, for the scenario the impulses significantly

increase, the estimation error of the algorithm will increase

considerably because of the excessive withdrawn of the

measurements in the computation.

To overcome the defect of the M-estimate technique in the

robust KFVNM, in this paper, the correntropy, which has been

proposed as a new statistics that can quantify both the time

structure and the statistical distribution of two stochastic

random processes,

is employed to deal with the impulsive

noise. Motivated by the “mix norm” property of CIM

(correntropy induced metric), the maximum correntropy

criterion (MCC) is applied as a substitute for the MSE (mean

square error) criterion in the cost formulation in the PAST

algorithm. Based on the RLS-technique, the MCC based PAST

algorithm is developed.

To extend the tracking capability of the MCC-PAST,

guided by the measure of subspace variations [5], a variable

forgetting factors (VFF) technique is developed and employed

in the recursion. More precisely, when the subspace varies

rapidly, a relative small FF should be used to gain a fast

tracking behavior. On the other hand, when the subspace varies

slowly, a relative large FF should be utilized to achieve a good

convergence property.

Moreover, to further enhance the tracking performance, the

technique of nonlinear principal component analysis (PCA) is

combined in the MCC-PAST algorithm. Before the iteration,

the Gaussian function based nonlinear transformation is

performed on the projection approximation so as to restrain the

outliers more effectively.

Compared with the robust KFVNM algorithm, the new

proposed algorithm, as we call the nonlinear MCC-PAST with

VFF, presents the more robust tracking performance especially

when the SNR (signal to noise ratio) is fairly low or the

underlying noise is extremely impulsive.

This paper is organized as follows, in Section II, we address

the problem of interest and the conventional PAST. In Section

III, we deduce the nonlinear MCC-PAST with VFF.

Simulation results and comparison are demonstrated in Section

IV. Conclusions are drawn in Section V.

The following notations are utilized throughout this paper.

The superscripts “

T ”, “

” denote the transpose, the

conjugate transpose, respectively.

()E ⋅

and

Fro

|| ||⋅

stand for

the mathematical expectation and the Frobenius norm,

respectively.

is the

× identity matrix.

II.

ROBLEM

ORMULATION AND THE

PAST

ALGORITHM

Consider

narrow-band incoherent signals impinging on

a uniform linear antenna array with

elements. The array

output received at the

tht

snapshot consists of the

sensor

outputs, which can be written as

() ( ) () () () ()

t stt tt

=+=+



xa nAsn (1)

where ( )

a is the steering vector toward the thr DOA for the

given array geometry.

[

]

(),(),,( )

ωω ω

=Α aa a

and

[]

() (), (), , ()

tstst st=s  denote the steering matrix and the

random source vector, respectively. ( )tn is an independent and

identically distributed (i.i.d) noise vector with covariance

matrix

It was shown in [2] that subspace tracking can be

considered as an unconstrained optimization with the cost

function of ( )tW described as

{

}

Fro

()

JE=−WxWWx

(2)

where

()tW

is a matrix variable with rank

. It was proved

in [2] that, when the column span of

()tW

is equal to the

signal subspace, (2) is minimized. For online implementation,

utilizing RLS technique, the PAST algorithm can recursively

estimate the subspaces by minimizing the following cost

function of

()tW

Fro

min ( ( )) ( ) ( ) ( ) ( )

ti H

Jt i t ti

−

=−



WxWWx

(3)

where

() () ()

iti=

is the projection approximation, and

<≤

is the FF.

Due to the sensitivity of the MSE criterion to outliers, it is

not surprising to discover that the PAST degrades its

performance in impulsive noise environments.

III.

ONLINEAR

MCC-PAST

WITH

VFF

A. Correntropy and the MCC-PAST algorithm

Inspired by kernel-based methods and information theoretic

learning (ITL) methods [6], based on the information potential

(IP), the correntropy for two arbitrary random variables

and

Y is defined as follows [7]

(,) [ ( )]VXY E XY

σσ

=−

(4)

where ( )

⋅ is the kernel function that satisfies Mercer’s

theorem [8] and

is the kernel size [8]. In this paper, the

widely used Gaussian kernel

() exp( () 2 )

2π

κσ

⋅= −⋅ , is

utilized.

Correntropy induced metric (CIM) can be defined in the

sample space as

[

]

CIM(,) sqrt (0) (,)V

σσ

=−pq pq

(5)

where p and q are given two vectors with

(, , , )

pp p=p 

and

(, , , )

qq q=q 

, and ( , )V

can be computed by

(,) ( )

Vpq

σσ

=−



(6)

Apparently, for Gaussian kernel,

(0)

2π

The metric

CIM possesses a “mix norm” property [7]. 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

an L0 norm when the two points get further apart. Meanwhile,

the level of the outlier suppression of CIM is totally controlled

by the kernel size

Inspired by the “mix norm” property of CIM, the MCC, is

developed as a new cost function criterion for adaptive system

training [9]. i.e.,

MCC( ) max ( )



(7)

where

e (1 iN≤≤ ) represent the errors produced by the

system during supervised learning and N is the number of the

data points.

The MCC demonstrates the inherent robustness to outliers

in adaptive filtering, that is, when the magnitude of the

estimation error is small, just like the MSE criterion, the MCC

measures the L2 norm distance between the estimation results

and the observations, otherwise, when the magnitude of the

estimation error gets larger, the MCC measure the L1 norm

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