IEEE SIGNAL PROCESSING LETTERS, VOL. 6, NO. 12, DECEMBER 1999 321

What Should We Say About the Kurtosis?A. Mansour,Member, IEEE,and C. Jutten,Member, IEEE

Abstract—In this work, we point out some important propertiesof the normalized fourth-order cumulant (i.e., the kurtosis). In ad-dition, we emphasize the relation between the signal distributionand the sign of the kurtosis. One should mention that in manysituations, authors claim that the sign of the kurtosis depends onthe nature of the signal (i.e., over- or sub-Gaussian). For unimodalprobability density function, that claim is true and is clearlyproved in the letter. But for more complex distributions, it hasbeen shown that the kurtosis sign may change with parametersand does not depend only on the asymptotic behavior of thedistributions.

Finally, these results give theoretical explanation to techniques,like nonpermanent adaptation, used in nonstationary situations.

Index Terms—Blind identification, blind separation, high orderstatistics, kurtosis, over- and sub-Gaussian, probability densityfunction.

I. INTRODUCTION

I N VARIOUS works [1]–[6] concerning the problem ofblind separation of sources, authors propose algorithms

whose efficacy demands conditions on the source kurtosis, andsometimes that all the sources have the same sign of kurtosis.In fact, this assumption seems very strong and in this work westudied the relationship between the signal distribution and thesign of its kurtosis.

II. DEFINITION AND PROPERTIES

Definition 1: Let us denote the probability densityfunction (pdf) of a random process and the average.By definition [7], [8] the kurtosis is as in (1), shown atthe bottom of the next page. Clearly, the kurtosis signis equal to the fourth-order cumulant sign. Some propertiescan be easily derived.

1) , so is invariant byany linear transformation .

2) Let , where is even andis odd. It is easy to prove that only depends on

and that can be considered as a pdf.

Therefore, in the following, the study may be restricted to azero-mean process whose the pdf is even and hasa variance .

It is well known that the kurtosis of a Gaussian distri-bution is equal to zero. Intuitively, the sign of the kurtosisseems related to the comparison between and Gaussiandistribution, by considering the asymptotic properties of thedistributions and the following definition:

Manuscript received March 1, 1999. The associate editor coordinating thereview of this manuscript and approving it for publication was Prof. H.Messer-Yaron.

A. Mansour is with the BMC Research Center (RIKEN), Nagoya 463, Japan(e-mail: mansour@nagoya.riken.go.jp).

C. Jutten is with INPG-LIS, 38031 Grenoble, France.Publisher Item Identifier S 1070-9908(99)09290-1.

Definition 2: A pdf is said over-Gaussian (respectivelysub-Gaussian) [9]–[11], if

(respectively,), where is the normalized Gaussian pdf. In many

examples, it seems that is positive1 for over-Gaussiansignals and negative for sub-Gaussian signals.

III. T HEORETICAL RESULTS

Let us consider an (even) pdf and a zero-mean nor-malized Gaussian pdf .

Theorem 1: If have only two solutions, then

is over-Gaussian

is sub-Gaussian.

A demonstration is given in the Appendix. This theorem showsthat the intuitive claim, given in the previous section, is trueunder the specific condition of Theorem 1.

When has more than two solutions, then thereis no longer simple rule to predict . More precisely, over-Gaussian as well as sub-Gaussian pdf’s can lead to positiveas well as negative sign of kurtosis. As an example, let usconsider the following over-Gaussian pdf:

(2)

It is easy to compute the kurtosis of (2):

Clearly, is not always negative, but may change accord-ing to the values of the parametersand : if

and if .Finally, we may consider that artificial signals (for instance

telecommunication signals) are bounded, and consequentlytheir pdf are sub-Gaussian. It is often claimed that the kurtosisof such signals is negative. The following example shows thatthis claim is wrong. Let us consider the distribution of Fig. 1.It is easy to evaluate the kurtosis of this signal:

(3)

with the normalization condition .For scale reasons, we do not draw directly but

. Thus, if

1When the signal has a positive kurtosis sign, it is said that it is a positivekurtotic signal.

1070–9908/99$10.00 1999 IEEE

322 IEEE SIGNAL PROCESSING LETTERS, VOL. 6, NO. 12, DECEMBER 1999

Fig. 1. Example ofx-limited pdf.

Fig. 2. Representation ofK?(p(x)) according to parametersc and�, witha = 0:9, b = 1:1, and d = 9.

, , otherwise, if ,. Now, we remark that the sign of the kurtosis

may be easily controlled with adequate values of the pdfparameters (see Fig. 2).

IV. CONCLUSION

In this letter, we point out some relations between pdfand kurtosis sign. First, we show that the kurtosis sign isnot modified by any scale or translation factors, and it onlydepends on the even part of the pdf.

Usually, one associates the kurtosis sign of a distributionto its over-Gaussian or sub-Gaussian nature. Here, we

prove that this claim is only relevant for unimodal pdf .Generally, even for bounded pdf, we show by a few exam-ples that the kurtosis sign can change according to the pdfparameters.

From a practical point of view, kurtosis sign of non sta-tionary signals, which must be estimated on short movingwindows, can change. A previous experimental study provesthat the kurtosis sign of speech signal can be affected by thesilent period [12]. Additionally, this work gives a theoreticalexplanation to the necessity and the efficacy of intermit-tent adaptation which is used for separation of nonstationarysources [13].

APPENDIX

PROOF OF THEOREM 1

Let us consider that for , the equationonly has one solution . It is known that the fourth-ordercumulant of a Gaussian distribution is zero. As a consequence,we can write: . Inaddition, we just may study the sign of .

Using (1) and the unit variance signal, one can prove that

(4)

Let us consider that the pdf is an over-Gaussian signal( , when ). Then, the sign ofremains constant on each interval , and [ ]. Usingthe second mean value theorem,can be rewritten as

(5)

where . Using the fact that andare both pdf, we can deduce that

. Taking intoaccount that is over-Gaussian, we deduce

(6)

Using (5) and (6), we remark that

(7)

Finally, if is an over-Gaussian pdf (with the assumptionof a unique solution to the equation for ,then its kurtosis is positive. Using the same reasoning andunder the same condition, we can claim that a sub-Gaussianpdf has a negative kurtosis.

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