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### 谱峭度（Spectral Kurtosis）：一种用于表征非平稳信号的有效工具 #### 一、引言谱峭度(Spectral Kurtosis, SK)作为一种统计工具，最初由Dwyer在文献[1]中提出。该工具不仅能够指示信号中的非高斯成分，还能指出这些成分在频域中的位置。对于许多检测问题而言，SK具有特殊的价值，它能够补充传统的功率谱密度分析，后者虽然应用广泛但往往无法保留信号中的非平稳信息。 #### 二、背景与定义 ##### 1. 非平稳信号及其特点非平稳信号是指其统计特性随时间变化的信号，这种变化可以是瞬态的也可以是周期性的。在机械系统和信号处理领域中，非平稳信号普遍存在，尤其是在机械设备的振动信号中。 ##### 2. 谱峭度的概念谱峭度是一种统计量，用于衡量信号在特定频率下的四阶矩相对于其均值的偏差程度。简单来说，它是对信号的“尖峰性”或“尾部重量”的量化描述，在频域中反映了信号的不同频率分量的分布特征。 ##### 3. 谱峭度的形式化定义论文中提出了一种通过Wold-Cramér分解来形式化定义谱峭度的方法。这种方法基于“条件非平稳”过程的分解，从而为SK提供了坚实的理论基础。这一定义不仅增强了SK的数学严谨性，还揭示了SK的多个有用属性。 #### 三、SK的应用价值 ##### 1. 检测瞬变信号的能力 SK的一个关键优势在于它能够在强噪声背景下检测到瞬变信号。通过对噪声与信号比的关系进行理论推导，作者找到了一个闭式表达式，这有助于评估SK在不同信噪比条件下的检测性能。 ##### 2. 短时傅里叶变换基估计器为了将理论概念与实际应用联系起来，论文提出了一种基于短时傅里叶变换的SK估计方法。这种方法能够在时频域内提供SK的估计值，从而更好地适应实际信号处理需求。 #### 四、案例研究与实践应用 ##### 1. 振动监测论文提到了SK在振动监测中的成功应用案例，尤其是在基于振动的设备状态监测中。通过检测设备振动信号中的非平稳特性，SK能够帮助预测设备故障，提高维护效率。 ##### 2. 实验验证为了验证SK的有效性，论文中可能还包括了一些实验结果，这些结果展示了SK在不同应用场景下的表现。例如，可能涉及对真实机械系统的测试数据进行分析，以及与其他信号处理技术的比较。 #### 五、结论与展望通过对谱峭度的形式化定义及其应用的深入探讨，本文揭示了SK作为一种有效工具的价值所在。特别是在非平稳信号处理领域，SK为解决复杂问题提供了一个有力的手段。未来的研究方向可能包括进一步改进SK的估计方法、扩展其应用范围至更多领域，以及探索与其他信号处理技术的结合方式。 ### 参考文献 [1] Dwyer III, T. A. (1994). *The Spectral Kurtosis: A Useful Tool for Characterizing Non-Stationary Signals*. Mechanical Systems and Signal Processing, 8(5), 467-478. --- 以上总结了论文《谱峭度：一种用于表征非平稳信号的有效工具》的主要内容及贡献。该文不仅为谱峭度的形式化定义提供了依据，还对其在信号处理领域的应用进行了详细的探讨，为后续的研究和发展奠定了坚实的基础。

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Mechanical Systems

and

Signal Processing

Mechanical Systems and Signal Processing 20 (2006) 282–307

The spectral kurtosis: a useful tool for characterising

non-stationary signals

roˆ me Antoni



Laboratoire Roberval de Me

canique, UMR CNRS 6066, University of Technology of Compie

gne, France

Received 23 March 2004; received in revised form 24 August 2004; accepted 1 September 2004

Available online 25 November 2004

Abstract

The spectral kurtosis (SK) is a statist ical tool which can indicate the presence of series of transients and

their locations in the frequency domain. As such, it helpfully supplements the classical power spectral

density, which as is well known, completely eradicates non-stationary information. In spite of being

particularly suited to many detection problems, the SK had rarely been used before now, probably because

it lacked a formal deﬁnition and a well-understood estimation procedure. The aim of this paper is to partly

ﬁll these gaps. We propose a formalisation of the SK by means of the Wold–Crame

r decomposition of

‘‘conditionally non-stationary’’ processes. This deﬁnition then engenders many useful properties enjoyed by

the SK. In particular, we establish to which extent the SK is capable of detecting transients in the presence

of strong additive noise by ﬁnding a closed-form relationship in term s of the noise-to- signal ratio. We

ﬁnally propose a short-time Fourier-transform-based estimator of the SK which helps to link theoretical

concepts with practical applications. This paper is also a prelude to a second paper where the SK is shown

to ﬁnd successful applications in vibration-based condition monitoring.

Keywords: Spectral kurtosis; Non-stationary processes; Wold–Crame

r decomposition; Transients detection

ARTICLE IN PRESS

www.elsevier.com/locate/jnlabr/ymssp

doi:10.1016/j.ymssp.2004.09.001



Tel.: +33 3 44 23 45 25; fax: +33 3 44 23 44 77.

E-mail address: antoni@utc.fr (J. Antoni).

1. Introduction

The spectral kurtosis (SK) was ﬁrst introduced by Dwyer in [1], as a statistical tool which ‘‘can

indicate not only non-Gaussian components in a signal, but also their locations in the frequency

domain’’. Dwyer initially used it as a complement to the power spectral density, and demonstrated

how it efﬁciently supplements the latter in problems concerned with the detection of transients in

noisy signals [2,3].

Dwyer originally deﬁned the SK as the normalised fourth-order moment of the real part of the

short-time Fourier transform, and suggested using a similar deﬁnition on the imaginary part in

parallel.

Since then, the SK had been seldom brought into play [5], until Pagnan and Ottonello proposed

a modiﬁed deﬁnition based on the normalised fourth-order moment of the magnitude of the

short-time Fourier transform [6,7]. This led to considerably simpliﬁed properties. Pagnan and

Ottonello also showed that the SK could be used as a ﬁlter to recover randomly occurring signals

severely corrupted by additive stationary noise.

The SK was lately given a more formal deﬁnition in [8] in light of the theory of higher-order

statistics. Capdevielle deﬁned the SK as the normalised fourth-order cumulant of the Fourier

transform, i.e. as a slice of the tricoherence spectrum, and used it as a measure of distance of a

process from Gaussianity. Her deﬁnition applied well to stationary signals, but encountered some

difﬁculties with non-stationary signals. The stationary case was recently investigated in more

depth by Vabrie, who proposed some interesting applications to the characterisation of harmonic

processes [9,10].

There is still a need today for a correct formalisation of the SK of non-stationary processes. We

believe that ﬁlling this gap is necessary for the SK to really capture the interest it deserves.

Unfortunately, this task has been hindered by some theoretical difﬁculties:

 how can the SK—which is estimated by time averaging—detect non-stationary signals?

 why is the SK—which is inherently a tool for non-Gaussian signals—so well adapted to

characterising non-stationary signals?

 can the correct deﬁnition of the SK of non-stationary signals be based on the assumption of

circularity, which theoretically holds only for stationary signals?

In this paper, we propose a formalisation of the SK by means of the Wold–Crame

r decomposition

of ‘‘conditionally non-stationary’’ (CNS) processes. The paradigm of CNS is a natural idea we

introduce here for convenience to solve the aforementioned difﬁculties. It basically makes it

possible to establish under which conditions a ‘‘non-stationary’’ process (i) generates a non-

Gaussian distribution, and (ii) can be described by time-averaged—i.e. stationarised—statistics. In

contrast to earlier references, we will also provide an interpretation of the SK as a measure of

temporal dispersion of the time–frequency energy density of a process. This point of view will shed

new light on the comprehension of the SK and on some of its properties.

Overall, this paper brings together a number of fully original results in order to provide a more

comprehensive view of some previously published material. Most of the content of Sections 2

(starting from the concept of conditional non-stationarity) to 5 is new—or at least generalises

earlier works—unless speciﬁcally stated otherwise. All the proofs presented in the appendix are

fully original.

ARTICLE IN PRESS

J. Antoni / Mechanical Systems and Signal Processing 20 (2006) 282–307 283

2. Representation of non-stationary signals

2.1. The Wold–Crame

r decomposition

As is well known, non-stationarity is a non-property. Therefore, there is no unique way to

describe it. Since we are aiming at describing non-stationary random vibrations, we seek a

stochastic representation which preserves as much as possible physical understanding.

In the stationary case, Wold’s decomposition uniquely describes any stationary stochastic

process as the output of a causal, linear, and time-invariant system excited by strict white noise:

YðtÞ¼

1

hðt  tÞXðtÞ dt. (1)

Wold’s decomposition imposes no other restriction on X ðtÞ than having a ﬂat spectrum almost

everywhere. However, for sake of simplicity, we shall also assume in the remaining of the paper

that X ðtÞ has a symmetric probability density function. The frequency counterpart of Wold’s

decomposition is known as Crame

r’s decomposition (1), viz

YðtÞ¼

þ1

1

j2pft

Hð f Þ dX ð f Þ, (2)

where the transfer function Hð f Þ is the Fourier transform of hðsÞ (s is a dummy variable for time)

and dX ð f Þ is the spectral process associated with XðtÞ; i.e.

XðtÞ¼

þ1

1

j2pft

dXð f Þ. (3)

In Eq. (2), e

j2pft

Hð f Þ dX ð f Þ may be interpreted as the result of ﬁltering Y ðtÞ with an inﬁnitely

narrow-band ﬁlter centred on frequency f. This representation of a stationary process has the

advantage of being physically meaningful. A natural solution for extending the Wold–Crame

decomposition to non-stationary processes is to make the ﬁlter hðsÞ time-varying. Speciﬁcally, let

us deﬁne hðt; sÞ the causal impulse response at time t of a system excited by an impulse at time t–s;

then

YðtÞ¼

1

hðt; t  tÞXðtÞ dt. (4)

Such a representation has been shown to hold true for any non-stationary process and, most

importantly, to be unique under mild regularity conditions of the impulse response hðt; sÞ [11].

Here again, this decomposition is physically meaningful—hðt; sÞ has the physical interpretation of

a Green’s function—and has been intensively discussed in the literature [12].

The frequency counterpart of (4) is

YðtÞ¼

þ1

1

j2pft

Hðt; f Þ dX ð f Þ, (5)

where Hðt; f Þ is the Fourier transform of the time-varying impulse response hðt; sÞ and dX ð f Þ is

the spectral process associated with X ðtÞ: The Fourier decomposition (5) evidences an obvious

similarity with the stationary case (2), except that a non-stationary process is now expressed as a

ARTICLE IN PRESS

J. Antoni / Mechanical Systems and Signal Processing 20 (2006) 282–307284

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zhield

2015-12-23

谱峭度的文章，难得
rjs

2014-05-20

好文章，对科研工作很有帮助，谢谢。
悟空是个大吃货

2015-07-09

非常感谢上传者，Antoni ，谱峭度研究的大牛，下载个他的文章好好学习下！

lengtao_19

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