Unbiased Efficient Estimator of the Fourth-Order Cumulant for Random Zero-Mean Non-i.i.d. Signals: Particular Case of MA Stochastic Process

Non-Gaussian processes may require not only the information provided by first two moments, but also that given by the higher-order statistics, in particular, by the third- and fourth-order moments or cumulants. This paper addresses a fourth-order cumulant estimation problem for real discrete-time random non-i.i.d. signal, that can be approximated as an MA stochastic process. An unbiased estimator is proposed, studied and compared to two other frequently used estimators of the fourth-order cumulant (natural estimator and fourth k -statistics). Statistical comparative studies are undertaken from both bias and MSE points of view, for different distribution laws and MA filters. Algorithms, aiming to reduce computational complexity of the proposed estimator, as well as that of the fourth k-statistics bias, are also provided.

[1]  Juan José González de la Rosa,et al.  Higher Order Statistics and Independent Component Analysis for Spectral Characterization of Acoustic Emission Signals in Steel Pipes , 2007, IEEE Transactions on Instrumentation and Measurement.

[2]  Pierre-Olivier Amblard,et al.  Adaptive estimation of the fourth-order cumulant of a white stochastic process , 1995, Signal Process..

[3]  Aarnout Brombacher,et al.  Probability... , 2009, Qual. Reliab. Eng. Int..

[4]  ANANTHRAM SWAMI,et al.  Closed-form recursive estimation of MA coefficients using autocorrelations and third-order cumulants , 1989, IEEE Trans. Acoust. Speech Signal Process..

[5]  Antonio Napolitano,et al.  Cyclic higher-order statistics: Input/output relations for discrete- and continuous-time MIMO linear almost-periodically time-variant systems , 1995, Signal Process..

[6]  R. C. Bradley.  Basic properties of strong mixing conditions. A survey and some open questions , 2005, math/0511078.

[7]  Stanislaw Osowski,et al.  Higher order statistics and neural network for tremor recognition , 2002, IEEE Transactions on Biomedical Engineering.

[8]  Gérard Favier,et al.  Recursive estimation of fourth-order cumulants with application to identification , 1998, Signal Process..

[9]  A. Kolmogorov,et al.  On Strong Mixing Conditions for Stationary Gaussian Processes , 1960 .

[10]  Eric Moreau,et al.  Unbiased Adaptive Estimations of the Fourth-Order Cumulant for Real Random Zero-Mean Signal , 2009, IEEE Transactions on Signal Processing.

[11]  J.W.A. Fackrell,et al.  The higher-order statistics of speech signals , 1994 .

[12]  R. C. Bradley.  Basic Properties of Strong Mixing Conditions , 1985 .

[13]  C. Jutten,et al.  Blind source separation of convolutive mixtures by maximization of fourth-order cumulants: the non i.i.d. case , 1998, Conference Record of Thirty-Second Asilomar Conference on Signals, Systems and Computers (Cat. No.98CH36284).

[14]  David R. Brillinger,et al.  Time Series: Data Analysis and Theory. , 1982 .

[15]  Jerry M. Mendel,et al.  Tutorial on higher-order statistics (spectra) in signal processing and system theory: theoretical results and some applications , 1991, Proc. IEEE.

[16]  Antonio Napolitano,et al.  The higher order theory of generalized almost-cyclostationary time series , 1998, IEEE Trans. Signal Process..

[17]  Jean-Christophe Pesquet,et al.  Quadratic Higher Order Criteria for Iterative Blind Separation of a MIMO Convolutive Mixture of Sources , 2007, IEEE Transactions on Signal Processing.

[18]  T.W.S. Chow,et al.  Blind and total identification of ARMA mode in higher order cumulants domain , 1999, IEEE Trans. Ind. Electron..

[19]  Pierre Comon,et al.  Separation Of Sources Using Higher-Order Cumulants , 1989, Optics & Photonics.

[20]  Zhi Ding,et al.  FIR channel estimation through generalized cumulant slice weighting , 2004, IEEE Transactions on Signal Processing.

[21]  Richard A. Davis,et al.  A Central Limit Theorem and a Strong Mixing Condition , 2011 .

[22]  Sophocles J. Orfanidis,et al.  Optimum Signal Processing: An Introduction , 1988 .

[23]  D. Brillinger.  Time series - data analysis and theory , 1981, Classics in applied mathematics.

[24]  Elias Masry.  The Estimation of the Fourth-Order Cumulant for Dependent Data: Consistency and Asymptotic Normality , 2010, IEEE Transactions on Signal Processing.

[25]  Jitendra K. Tugnait,et al.  Identification and deconvolution of multichannel linear non-Gaussian processes using higher order statistics and inverse filter criteria , 1997, IEEE Trans. Signal Process..

[26]  Pierre Comon,et al.  MA identification using fourth order cumulants , 1992, Signal Process..

[27]  Wei Wei,et al.  Speech stream detection based on higher-order statistics , 2003, Proceedings of the 2003 International Conference on Machine Learning and Cybernetics (IEEE Cat. No.03EX693).

[28]  M. Tsatsanis,et al.  On the use of higher-order statistics for robust endpoint detection of speech , 1993, [1993 Proceedings] IEEE Signal Processing Workshop on Higher-Order Statistics.

[29]  Tommy W. S. Chow,et al.  Semiblind identification of nonminimum-phase ARMA models via order recursion with higher order cumulants , 1998, IEEE Trans. Ind. Electron..

[30]  Antonio Napolitano,et al.  Cyclostationarity: Half a century of research , 2006, Signal Process..

[31]  P. Milenkovic,et al.  Statistical analysis of word-initial voiceless obstruents: preliminary data. , 1988, The Journal of the Acoustical Society of America.

[32]  P. Moran,et al.  Mathematics of Statistics , 1948, Nature.

[33]  Maurice G. Kendall,et al.  The advanced theory of statistics , 1945 .

[34]  Antonio Napolitano,et al.  Higher-order statistics for Rice's representation of cyclostationary signals , 1997, Signal Process..

[35]  Philippe Loubaton,et al.  Separation of a class of convolutive mixtures: a contrast function approach , 2001, Signal Process..

[36]  Rory A. Fisher,et al.  Moments and Product Moments of Sampling Distributions , 1930 .

[37]  R. Fisher,et al.  148: Moments and Cumulants in the Specification of Distributions. , 1938 .

[38]  Murray Rosenblatt,et al.  A CLASS OF STATIONARY PROCESSES AND A CENTRAL LIMIT THEOREM. , 1956 .

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

[40]  P. Comon,et al.  ICAR: a tool for blind source separation using fourth-order statistics only , 2005, IEEE Transactions on Signal Processing.

[41]  Antonio Napolitano,et al.  Higher-order cyclostationarity properties of sampled time-series , 1996, Signal Process..

[42]  David R. Brillinger,et al.  Time Series , 2018, Randomization, Bootstrap and Monte Carlo Methods in Biology.

[43]  L. M. M.-T..  Theory of Probability , 1929, Nature.

[44]  V. V. Gorodetskii,et al.  On the Strong Mixing Property for Linear Sequences , 1978 .