利用自适应积分算子增强信号分离的稳健性

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"这篇研究论文探讨了自适应积分算子在信号分离中的应用，通过利用微分算子及其初值和边界值来构建相应的积分算子。与微分算子相比，积分算子对噪声信号更具鲁棒性。此外，论文提出了一种将积分算子与基于匹配滤波的方法相结合的策略，用于处理调频（FM）信号，并将其应用于Null Space Pursuit（NSP）算法中，以估计核函数并提取信号的子成分。通过与其他先进方法的比较，展示了该提议算法的稳健性和有效性。" 本文的核心知识点包括： 1. **自适应积分算子**：这是一种用于信号分离的技术，它允许将信号分解为一组加性子组件。与传统的微分算子不同，自适应积分算子在处理噪声信号时表现出更高的鲁棒性。 2. **微分算子与积分算子的关系**：尽管两者具有相同的零空间，但积分算子在处理噪声环境下的信号时更稳定。这表明积分算子在某些情况下可能比微分算子更适合于信号处理任务。 3. **FM信号的特征展开**：通过特征分解，可以将调频信号的内核展开，这使得可以将基于操作器的方法与匹配滤波法相结合。在这里，特征函数被用作匹配滤波器，增强了信号处理的精确度。 4. **匹配滤波器**：在本文中，积分算子与匹配滤波器的概念相融合，利用特征函数作为匹配滤波器，进一步优化了信号分离过程。 5. **Null Space Pursuit (NSP) 算法**：NSP是一种稀疏恢复算法，常用于寻找信号的稀疏表示。论文将积分算子集成到NSP算法中，以估计信号的内核并有效地提取子信号。 6. **性能评估与比较**：为了验证所提方法的有效性，论文对比了该算法与当前最先进的信号分离方法。这种比较展示了该算法在处理噪声和复杂信号环境时的优越性能。这篇研究论文对信号处理领域有重要的贡献，特别是对于噪声环境中信号的分离和恢复，提供了新的思考和工具。通过使用自适应积分算子和改进的NSP算法，它为未来的研究提供了一个强大且鲁棒的框架。

IEEE SIGNAL PROCESSING LETTERS, VOL. 22, NO. 9, SEPTEMBER 2015 1383

Adaptive Integral Operators for Signal Separation

Xiyuan Hu, Silong Peng, and Wen-Liang Hwang, Senior Member, IEEE

Abstract—The operator-based signal separation approach uses

an adaptive operator to separate a signal into a set of additive sub-

components. In this paper, we show that differential operators and

their initial and boundary values can be exploited to derive cor-

responding integral operators. Although the differential operators

and the integral operators have the same null space, the latter are

more robust to noisy signals. Moreover, after expanding the ker-

nels of Frequency Modulated (FM) signals via eigen-decomposi-

tion, the operator-based approach with the integral operator can

be regarded as the matched ﬁlter approach that uses eigen-func-

tions as the matched ﬁlters. We then incorporate the integral op-

erator into the Null Space Pursuit (NSP) algorithm to estimate the

kernel and extract the subcomponent of a signal. To demonstrate

the robustness and efﬁcacy of the proposed algorithm, we com-

pare it with several state-of-the-art approaches in separating mul-

tiple-component synthesized signals and real-life signals.

Index Terms—Integral equation, narrow band signal, null space

pursuit (NSP), operator-based.

I. INTRODUCTION

N RECENT years, several approaches [1]–[10] have

been proposed to separate a single-channel signal into

a mixture of several additive coherent subcomponents. The

method used to separate signals depends on the deﬁnition of the

subcomponents. For example, in the empirical mode decom-

position (EMD) approach [2], [10]–[12], the subcomponents

are Intrinsic Mode Functions (IMFs); in the Synchrosqueezed

Wavelet Transform (SWT) approach, the subcomponents are

IntrinsicModeTypeFunctions(IMT)[7],[8],[13];andinthe

operator-based approach [5], [6], a subcomponent is deﬁned as

being in the null space of an operator, which is characterized

by some parameters that are estimated from the input (residual)

signal.

To improve the robustness and efﬁcacy of the operator-based

approach, the Null Space Pursuit (NSP) algorithm was proposed

[6]. It separates a signal

into and such that is

Manuscript received April 07, 2014; revised August 10, 2014; accepted Au-

gust 22, 2014. Date of publication August 26, 2014; date of current version

February 27, 2015. This work was supported by the Natural Science Foundation

of China under Grants 61032007 and 61201375, and by the China Scholarship

Council under Grant 201304910129. The associate editor coordinating the re-

view of this manuscript and approving it for publication was Prof. Alexander

M. Powell.

X. Hu and S. Peng are with the High Technology Innovation Center (HITIC),

Institute of Automation, Chinese Academy of Sciences, Beijing, 100190,

China(e-mail: xiyuan.hu@ia.ac.cn,silong.peng@ia.ac.cn).

W.-L. Hwang is with the Institute of Information Science, Academia Sinica,

Taipei 11529, Taiwan (e-mail: whwang@iis.sinica.edu.tw)

Color versions of one or more of the ﬁgures in this paper are available online

at http://ieeexplore.ieee.org.

Digital Object Identiﬁer 10.1109/LSP.2014.2352340

in the null space of an operator by minimizing the following

problem:

(1)

The ﬁrst and second terms of Eq. (1) are the same as in the

operator-based approach. The parameter

in the third term of

Eq. (1) determines the amount of

to be retained in the null

space of

. The last term is the Lagrange term for the param-

eters of the operator

. Based on some assumptions, the NSP

algorithm can adaptively estimate the parameters

and and

derive the optimal solution of Eq. (1) [6].

An attractive feature of the operator-based approach is that

the operator design can be customized based on the characteris-

tics of the signal’s subcomponents. Let

be a subcomponent.

Then, any operator

with (i.e., is in the null

space of the operator) can be used in the proposed approach to

“annihilate” the subcomponent signal. For instance, to annihi-

late a frequency modulated (FM) signal

,where is

a local linear function, we can use the operator

(asdeﬁnedin[6]).Here, is the instanta-

neous frequency (IF) of the signal. In addition, the operator

described in [14] can be used to

annihilate an amplitude modulated and frequency modulated

(AM-FM) signal

, with the parameters

( is the instantaneous bandwidth (IB))

and

( istheIF).In[5],the

general form of a differential operator is deﬁned as

(2)

where

is a square summable sequence. For a mixture

of narrow band signals, FM or AM-FM, the proposed differ-

ential operators can separate each subcomponent successfully;

however, in some instances, particularly low SNR scenarios, the

differential operators tend to amplify the noise component when

estimating the parameters of the operator.

Also, a kind of integral operator using a simple local meanas

has been proposed in [5]. However, this

kind of integral operator can only annihilate the type of narrow

band signals that has only one frequency or a narrow range of

frequencies varying as a function of time, as deﬁned in [15].

Therefore, we propose the following general form for an integral

operator:

(3)

where

is the parameterized integral kernel and is the

integral interval at time

.Asignal is in the null space of the

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

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