A zero-cumulant random variable and its applications

High Order Statistics (HOS) are widely used in many algorithms ranging from blind identification to signal separation. A well-known identifiability result is that at most one Gaussian source should be present in static mixtures [J. Eriksson, V. Koivunen, Identifiability, separability and uniqueness of linear ICA models, IEEE Signal Process. Lett. (2004) 601–604]. The reason for this is that these algorithms utilize cumulants of order higher than two, and that they are all null for circular Gaussian random variables [M. Kendall, A. Stuart, The Advanced Theory of Statistics, Distribution Theory, vol. 1, C. Griffin, 1977]. Simple examples of non-Gaussian complex random variables having zero cumulants of order three to seven are given, which can be encountered in the real world. r 2006 Elsevier B.V. All rights reserved.

[1]  Antoine Souloumiac,et al.  Jacobi Angles for Simultaneous Diagonalization , 1996, SIAM J. Matrix Anal. Appl..

[2]  J. Cardoso,et al.  Blind beamforming for non-gaussian signals , 1993 .

[3]  Jean-Louis Lacoume,et al.  Statistics for complex variables and signals - Part I: Variables , 1996, Signal Process..

[4]  P. Comon,et al.  ICA-based technique for radiating sources estimation: application to airport surveillance , 1993 .

[5]  Pierre Comon,et al.  Independent component analysis, A new concept? , 1994, Signal Process..

[6]  Pierre Comon,et al.  Separation de signaux ZCM: application en radar SSR , 2003 .

[7]  R. Fisher.  The Advanced Theory of Statistics , 1943, Nature.

[8]  Pierre Comon.  Independent component analysis - a new concept? signal processing , 1994 .

[9]  Nicolas Petrochilos,et al.  Algorithms for Separation of Secondary Surveillance Radar Replies , 2002 .

[10]  Eric Moreau,et al.  A generalization of joint-diagonalization criteria for source separation , 2001, IEEE Trans. Signal Process..

[11]  Jean-Christophe Pesquet,et al.  Cumulant-based independence measures for linear mixtures , 2001, IEEE Trans. Inf. Theory.

[12]  C. L. Nikias,et al.  Higher-order spectra analysis : a nonlinear signal processing framework , 1993 .

[13]  Michael C. Stevens,et al.  Secondary Surveillance Radar , 1988 .

[14]  Visa Koivunen,et al.  Identifiability, separability, and uniqueness of linear ICA models , 2004, IEEE Signal Processing Letters.

[15]  S. S. Wilks,et al.  The Advanced Theory of Statistics. I. Distribution Theory , 1959 .