About the cumulants of periodic signals

Mechanical Systems and Signal Processing 99 (2018) 684–690

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Short communication

About the cumulants of periodic signals Axel Barrau a,⇑, Mohammed El Badaoui b a b

SAFRAN TECH, Groupe Safran, Rue des Jeunes Bois, Chteaufort, 78772 Magny Les Hameaux Cedex, France Univ Lyon, UJM-St-Etienne, LASPI, EA3059, F-42023 Saint-Etienne, France

a r t i c l e

i n f o

Article history: Received 18 November 2016 Received in revised form 13 June 2017 Accepted 17 June 2017 Available online 14 July 2017 Keywords: Kurtosis Health monitoring Statistical independence Source separation JADE

a b s t r a c t This note studies cumulants of time series. These functions originating from the probability theory being commonly used as features of deterministic signals, their classical properties are examined in this modified framework. We show additivity of cumulants, ensured in the case of independent random variables, requires here a different hypothesis. Practical applications are proposed, in particular an analysis of the failure of the JADE algorithm to separate some specific periodic signals. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction Cumulants of a random variable are a widespread tool of signal processing due to their mathematical properties, especially regarding their connections to the concept of statistical independence [11]. They are at the core of some successful approaches to blind channel identification [18,23,10,9] and blind source separation [9,8,21,19,15,20,22]. In mechanical vibration analysis, fourth-order normalized cumulant (a.k.a. kurtosis) is certainly one of the most popular indicators for bearing and gear spalling fault detection [17,12,1,5,13,4,16,3]. Although cumulants have been initially developed as features of random variables, the time series they are computed on are frequently dominated by their deterministic components. Rotating machine signals for instance, are cyclostationary [14] due to the inherent periodicities of the mechanisms producing them [2,7,6]. The issue raised by this note is the relevance of the notion of independence when signals at play are basically periodic. The shared dependence on time prevents direct transposition of usual probabilistic computations. Our focus in the present work is on fourth-order cumulant. We study in which cases it can be considered an additive operator, as occurs with independent random variables. A novel condition is proposed, involving the Lower Common Multiple (LCM) of the frequencies of the signals added. As a concrete application, we show satisfaction of the proposed hypothesis determines the success of the JADE algorithm [8] for blind source separation. This paper is organized as follows. In Section 2 we recall the definition and main properties of cumulants of a random variable. In Section 3 we write the natural extension of this notion to (periodic) deterministic signals and explain under what conditions they behave additively. Section 4 displays two applied examples of the property derived in Section 3. In particular, simulations show the importance of the proposed hypothesis in the success of JADE algorithm [8] for blind source separation. Finally, Section 5 gives some concluding remarks.

⇑ Corresponding author. E-mail address: [email protected] (A. Barrau). http://dx.doi.org/10.1016/j.ymssp.2017.06.019 0888-3270/Ó 2017 Elsevier Ltd. All rights reserved.

A. Barrau, M. El Badaoui / Mechanical Systems and Signal Processing 99 (2018) 684–690

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2. Cumulants of random variables 2.1. Definition The cumulants ðjn Þn2N of a real random variable X 2 R are a sequence of scalars defined through their generating function g : R ! R:

gðuÞ ¼ ln E euX ;

ð1Þ

where EðÞ denotes the expected value of a random variable. The series ðjn Þn2N is given by the Taylor expansion:

gðuÞ ¼

1 X

un : n!

jn ðX Þ

n¼1

ð2Þ

For each n, the obtained scalar jn ðX Þ is called n th cumulant. The following explicit formulas for the first cumulants are useful in practice, and will be referred to in the present note.

j1 ðX Þ ¼ EðX Þ;

j2 ðX Þ ¼ E X 2 EðX Þ2 ;

j3 ðX Þ ¼ E X 3 3EðX ÞE X 2 þ 2EðX Þ3 ;

ð3Þ

j4 ðX Þ ¼ E X 4 4E X 3 EðX Þ þ 12EðX Þ2 E X 2 3E X 2

2

6EðX Þ4 :

j2 ðX Þ is also known as the variance of X, and rather denoted by rðX Þ2 . In many fields, cumulants are used as a shape indicator of the probability density function (p.d.f.) of X. But due to the properties recalled below they are also used to characterize statistical independence. 2.2. Properties Defining the cumulants through the Taylor expansion (2) allows immediate checking of a classical property: Proposition 1 (Additivity of cumulants). Let X 2 R; Y 2 R be two independent random variables. Then we have:

8n 2 N; jn ðX þ Y Þ ¼ jn ðX Þ þ jn ðY Þ: Proof. As X and Y are independent we have, for any t 2 R; E euðXþY Þ ¼ E euX euY ¼ E euX E euY , thus log E euðXþY Þ ¼ uX uY þ log E e . Introducing (1) into the latter equality gives g XþY ðuÞ ¼ g X ðuÞ þ g Y ðuÞ, which implies in turn equality log E e of the coefficients of the Taylor expansions of the two terms of the equality. Hence the result (see Eq. (2)).h

3. Cumulants of deterministic signals 3.1. Statistical definition For time series, probabilistic expectation is replaced with time average EðÞ:

1 T!1 2T

EðxÞ :¼ lim

Z

T

ð4Þ

xðtÞdt: T

The definitions of cumulants are obtained transposing formulas (3):

j1 ðxÞ ¼ EðxÞ; j2 ðxÞ ¼ Eðx2 Þ EðxÞ2 ; j3 ðxÞ ¼ E x3 3EðxÞE x2 þ 2EðxÞ3 ;

2

j4 ðxÞ ¼ E x4 4E x3 EðxÞ þ 12EðxÞ2 E x2 3E x2 6EðxÞ4 :

ð5Þ

The signal xðtÞ can be random or deterministic, usually a combination of both. More precisely, most situations involved in the literature mentioned in the introduction involve periodic components. The point of this note is to show that even if the random parts of xðtÞ and yðtÞ are independent, nothing ensures Proposition 1 holds for statistical cumulants of time series. In particular, the specific case where xðtÞ and yðtÞ are fully deterministic and periodic is pivotal to gain insight into more complicated cases. The following examples show the issue is not trivial.

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Example 1 (Additivity of 4th cumulants). We choose here xðtÞ ¼ sinðtÞ and yðtÞ ¼ sinð4tÞ. Analytic integration gives: j4 ðxÞ ¼ 3=8; j4 ðyÞ ¼ 3=8 and j4 ðx þ yÞ ¼ 3=4. Here, the cumulants of two periodic signals with different periods seem to behave additively.

Example 2 (Couter-example to additivity of 4th cumulants). We choose here xðtÞ ¼ sinðtÞ and yðtÞ ¼ sinð3tÞ. Analytic integration gives: j4 ðxÞ ¼ 3=8; j4 ðyÞ ¼ 3=8 and j4 ðx þ yÞ ¼ 5=4. Here, additivity is not verified. We propose in the next section to derive a sufficient condition providing additivity of statistical cumulants for deterministic periodic signals. 3.2. Main result We give in this section a sufficient condition ensuring fourth cumulants of deterministic periodic signals can be added. It requires first to introduce the definition of the Lowest Common Multiple (LCM) of two real numbers: Definition 1 (Lowest Common Multiple). Let a; b denote two positive real numbers. Their Lowest Common Multiple (LCM) is defined as:

LCMða; bÞ ¼ minfaN \ bN g: Note that a and b having an irrational ratio implies LCMða; bÞ ¼ þ1. Now we can formulate the condition: Proposition 2 (Additivity of fourth cumulant). Let xðtÞ and yðtÞ denote two signals having periods T x and T y respectively, and frequencies f x and f y , and assume there exist nx ; ny 2 N such that: The support of the Fourier series of xðtÞ is in [nx ; nx ]. The support of the Fourier series of yðtÞ is in [ny ; ny ]. LCMðf x ;f y Þ LCMðf x ;f y Þ We have nx < or ny < . 3f x 3f y Then we have:

j4 ðx þ yÞ ¼ j4 ðxÞ þ j4 ðyÞ: The proof has been moved to the appendix. The proposed condition is accurate, in the sense that building deterministic signals violating it although verifying additivity of fourth cumulant is tricky. Note that, a similar result can be derived for any order although we focused here on j4 . The notion of LCM arising here is especially relevant in gear fault diagnosis, where it already appears in the computation of the joint period of different cogwheels. As cumulant-based indicators such as Kurtosis are very popular, we will show in the next section how Proposition 2 can give useful insight into the behavior of these tools. 4. Examples The two examples presented in this section display practical situations where Proposition 2 is of interest. 4.1. Kurtosis-based diagnosis We consider here monitoring a system using excess Kurtosis, defined for a signal xðtÞ as:

cðxÞ ¼

j4 ðxÞ : rðxÞ4

Usually, an alarm is triggered if a given threshold is exceeded. But a very natural issue to be raised during the design phase is the sensitivity of the method to the presence of additional sources. Thus, we assume the presence of a second periodic signal yðtÞ perturbs the obtained Kurtosis. If xðtÞ; yðtÞ verify Proposition 2 we can write:

cðx þ yÞ ¼

j4 ðx þ yÞ j4 ðxÞ þ j4 ðyÞ rðxÞ2 ¼ 2 ¼ 4 rðx þ yÞ rðxÞ2 þ rðyÞ2 rðxÞ2 þ rðyÞ2

!2

cðxÞ þ

rðyÞ2 rðxÞ2 þ rðyÞ2

!2

cðyÞ:

ð6Þ


687

Remark 1. Additivity of the second cumulant (i.e. variance) has also been used in the above computation. Actually, under the hypothesis of Proposition 2, all four first cumulants behave additively. The proof is the same as for the fourth cumulant so we did not include it. The obtained relation allows to easily anticipate the impact of some classes of perturbations, and to identify the situations requiring further thought. Assume for instance a motor is monitored using the kurtosis of the sound signal it produces. To investigate the impact of a second similar motor running close-by, we can be guided by some immediate considerations: both motors can be expected to produce similar signals, but with a frequency related to their rotation speed. Kurtosis being unaffected by time scaling, both signals have the same kurtosis. If frequencies are the same, total Kurtosis is not changed by the presence of yðtÞ (Proposition 2 does not apply in this case) but if their frequencies becomes slightly different their LCM will be high, the hypotheses of Proposition 2 will hold and relation (6) shows the measured Kurtosis will drop by half. If the two frequencies continue to spread they can reach values where their LCM is low and the total kurtosis changes again (because Proposition 2 doesn’t hold any more). The important point here is that none of these variations is due to a defect in one of the motors. 4.2. JADE algorithm Another usual application of cumulants is source separation. Algorithm JADE, which has proved extremely efficient in many configurations, creates combinations of a set of signals such that their joint cumulants vanish. It relies on the observation that this property should be verified by independent signals. In practice, JADE has been applied very often to separation of deterministic periodic signals. The idea that having different periods is a form of independence is widespread and reasonable, but should be used with care as will be illustrated by this example. Let xðtÞ and yðtÞ denote two sine waves:

xðtÞ ¼ sinðtÞ; yðtÞ ¼ sinðrtÞ; with r 2 R a parameter we will set to different values. These functions are mixed into a new (2-dimensional) signal SðtÞ through a random square matrix:

SðtÞ ¼

1

w1

w2

1

xðtÞ yðtÞ

;

where w1 and w2 are random real numbers. The diagonal coefficients of the mixing matrix have been set to one to avoid ambiguity. JADE algorithm is then applied to SðtÞ, and the results are normalized to ensure the diagonal coefficients of the estimated mixing matrix are ones. Figs. 1 and 2 illustrate the results for two values of r: r ¼ 4: initial signals are correctly recovered. r ¼ 3: initial signals are not recovered. The failure of JADE for r ¼ 3 is not surprising considering Proposition 2. Let us recall the summation formula of cumulants in the general case:

Fig. 1. Blind separation of a random mixture of signals xðtÞ ¼ sinðtÞ and yðtÞ ¼ sinð3tÞ using JADE algorithm. Although having different periods, we see the signals are not correctly recovered.

688


Fig. 2. Blind separation of a random mixture of signals xðtÞ ¼ sinðtÞ and yðtÞ ¼ sinð4tÞ using JADE algorithm. The signals are correctly recovered due to sufficient spectral separation.

jn ðx þ yÞ ¼ jn ðxÞ þ jn ðyÞ þ

n X n k¼1

k

jðx; . . . ; x; y; . . . ; yÞ; |fflfflfflffl{zfflfflfflffl} |fflfflfflffl{zfflfflfflffl} k

nk

where jð. . .Þ denotes the joint cumulant. Its immediate consequence is that two signals having vanishing joint cumulants also have additive cumulants (the sum over k above becomes zero). But we know (Example 2) that cumulants cannot be added for r ¼ 3. This means the joint cumulants of sinðtÞ and sinð3tÞ do not vanish and JADE cannot separate these signals. Remark 2. This example immediately raises a new question: under what conditions do periodic signals have vanishing joint cumulants? Of course, the proof given in Appendix A could be extended but this is left for future work.

5. Conclusion This note investigated the behavior of fourth cumulant when applied to deterministic and periodic signals. In this context, having different periods is sometimes considered the twin of being statistically independent in a probabilistic context. This idea has been studied with care and a more elaborated hypothesis has been proposed, ensuring additivity of the fourth cumulant. The consequences of not taking it into account have been illustrated through failure of the JADE algorithm applied to a very simple problem. Obviously, the result presented here should be extended to all cumulants (i.e. higher than 4), and to joint cumulants. This is left for future work. Appendix A. Proof of Proposition 2 The proof is the combination of Lemmas 1 and 2 below. Lemma 1. Let z1 ðtÞ and z2 ðtÞ denote two periodic signals. If their spectral supports meet only at zero then we have Eðz1 z2 Þ ¼ Eðz1 ÞEðz2 Þ; EðÞ being defined by Eq. (4). Proof. Let T 1 ; T 2 denote the periods of z1 ðtÞ and z2 ðtÞ respectively, and x1 ; x2 the corresponding pulses. We introduce ða1 ½kÞk2Z and ða2 ½kÞk2Z , the Fourier coefficients of z1 and z2 . We have: 0 ! !1 ! þ1 þ1 þ1 X þ1 þ1 X þ1 X X X X Eðz1 z2 Þ ¼ E@ a1 ½keikx1 t a2 ½keikx2 t A ¼ E a1 ½ka2 ½peiðkx1 px2 Þt ¼ a1 ½ka2 ½pE eiðkx1 px2 Þt : k¼1

k¼1

k¼1p¼1

k¼1p¼1

As we have E eixt ¼ 0 for any x – 0 we can write:

Eðz1 z2 Þ ¼

X

a1 ½ka2 ½p ¼ a1 ½0a2 ½0 ¼ Eðz1 ÞEðz2 Þ;

kx1 ¼px2

where the removal of the sum is allowed by the hypothesis that the spectral supports of z1 ðtÞ and z2 ðtÞ meet only at zero.h In particular, the condition on spectral supports of z1 ðtÞ and z2 ðtÞ cannot be violated if one of them does not reach LCMðx1 ; x2 Þ, i.e. the lowest positive real where multiples of x1 and x2 meet. Lemma 2. If the spectral support of a function zðtÞ is in L; L½ then the spectral support of zn is in nL; nL½.


689

Proof. This is an immediate consequence of the spectrum of zn being the one of z convolved n times.h Proposition 2 is proved now through a direct computation. Explicit formula (5) for the fourth cumulant gives:

j4 ðx þ yÞ ¼E ðx þ yÞ4 4E ðx þ yÞ3 Eðx þ yÞ þ 12Eðx þ yÞ2 E ðx þ yÞ2 3E ðx þ yÞ2

2

6Eðx þ yÞ4 :

Expanding the powers using the binomial formula then re-arranging the terms we obtain:

2

j4 ðx þ yÞ ¼E x4 4E x3 EðxÞ þ 12EðxÞ2 E x2 3E x2 6EðxÞ4

2 þ E y4 4E y3 EðyÞ þ 12EðyÞ2 E y2 3E y2 6EðyÞ4 3 3

2 2 2 2

þ 4 E x y E x EðyÞ þ 6 E x y E x E y h i

þ 4 E xy3 EðxÞE y3 12 E x2 y EðyÞ E x2 EðyÞ2 h i 12 E xy2 EðxÞ EðxÞ2 E y2 h i þ 12 4EðxÞEðyÞEðxyÞ EðxyÞ2 3EðxÞ2 EðyÞ2 h i h i þ 12 2EðxÞ2 EðxyÞ 2EðxÞ3 EðyÞ þ 12 2EðyÞ2 EðxyÞ 2EðyÞ3 EðxÞ

12 E x2 y EðxÞ 2EðxÞEðyÞE x2 þ E x2 EðxyÞ

12 E y2 x EðyÞ 2EðyÞEðxÞE y2 þ E y2 EðxyÞ :

Using Lemma 2 and the hypotheses of Proposition 2 we know the spectral supports of the following pairs of signals meet only at zero: x and y; x2 and y; x3 and y; x2 and y2 ; x and y2 ; x and y3 . Thus Lemma 1 can be applied in the expression above and all brackets vanish. Then the expansion boils down to:

2

j4 ðx þ yÞ ¼E x4 4E x3 EðxÞ þ 12EðxÞ2 E x2 3E x2 6EðxÞ4

2 þ E y4 4E y3 EðyÞ þ 12EðyÞ2 E y2 3E y2 6EðyÞ4 :

And finally re-injecting explicit formula (5) of the fourth cumulant into the right-hand side of the equality we obtain:

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