Decoding of Linear Instantaneous Mixtures

Coding and decoding methods are founded on three notions: orthogonality, uncorrelatedness, and independence. In this chapter we review the mathematics and basic principles associated with these properties, for both vectors and functions, especially as they relate to the coding and decoding methods presented in this book. We also provide some basic background on the uncorrelatedness and independence of statistical variables. These two notions lie at the heart of the statistical decoding solutions described in this and subsequent chapters.

[1]  Peter J. Smith A Recursive Formulation of the Old Problem of Obtaining Moments from Cumulants and Vice Versa , 1995 .

[2]  Solomon Kullback,et al.  Information Theory and Statistics , 1960 .

[3]  J. R. Cruz,et al.  Simultaneous Vibroseis Encoding Techniques , 1988 .

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

[5]  Dimitri P. Bertsekas,et al.  Constrained Optimization and Lagrange Multiplier Methods , 1982 .

[6]  H. Hotelling Analysis of a complex of statistical variables into principal components. , 1933 .

[7]  Karl Pearson,et al.  Mathematical Contributions to the Theory of Evolution. XIX. Second Supplement to a Memoir on Skew Variation , 1901 .

[8]  Andrzej Cichocki,et al.  A New Learning Algorithm for Blind Signal Separation , 1995, NIPS.

[9]  Terrence J. Sejnowski,et al.  An Information-Maximization Approach to Blind Separation and Blind Deconvolution , 1995, Neural Computation.

[10]  Philippe Garat,et al.  Blind separation of mixture of independent sources through a quasi-maximum likelihood approach , 1997, IEEE Trans. Signal Process..

[11]  P. McCullagh Tensor Methods in Statistics , 1987 .

[12]  Roger M. Ward,et al.  Phase Encoding of Vibroseis Signals For Simultaneous Multisource Acquisition , 1990 .

[13]  Eric Moulines,et al.  A blind source separation technique using second-order statistics , 1997, IEEE Trans. Signal Process..

[14]  Pierre Comon,et al.  Improved contrast dedicated to blind separation in communications , 1997, 1997 IEEE International Conference on Acoustics, Speech, and Signal Processing.

[15]  D. Luenberger Optimization by Vector Space Methods , 1968 .

[16]  A. Tarantola Inverse problem theory : methods for data fitting and model parameter estimation , 1987 .

[17]  L. Carin,et al.  A new algorithm for independent component analysis with or without constraints , 2002, Sensor Array and Multichannel Signal Processing Workshop Proceedings, 2002.

[18]  K. Horadam Hadamard Matrices and Their Applications , 2006 .

[19]  R. A. Leibler,et al.  On Information and Sufficiency , 1951 .

[20]  Laurenz Wiskott,et al.  CuBICA: independent component analysis by simultaneous third- and fourth-order cumulant diagonalization , 2004, IEEE Transactions on Signal Processing.

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

[22]  R. Yarlagadda,et al.  Hadamard matrix analysis and synthesis: with applications to communications and signal/image processing , 1996 .

[23]  C. E. SHANNON,et al.  A mathematical theory of communication , 1948, MOCO.

[24]  Lawrence C. Wood Seismic data compression methods , 1974 .

[25]  B. Noble Applied Linear Algebra , 1969 .

[26]  A. Bunse-Gerstner,et al.  Numerical Methods for Simultaneous Diagonalization , 1993, SIAM J. Matrix Anal. Appl..

[27]  Gene H. Golub,et al.  Matrix computations , 1983 .

[28]  Simon Haykin,et al.  Image Denoising by Sparse Code Shrinkage , 2001 .

[29]  E. Oja,et al.  Independent Component Analysis , 2013 .

[30]  Aapo Hyvärinen,et al.  Fast and robust fixed-point algorithms for independent component analysis , 1999, IEEE Trans. Neural Networks.

[31]  F. Murnaghan The unitary and rotation groups , 1962 .

[32]  Jean-Franois Cardoso High-Order Contrasts for Independent Component Analysis , 1999, Neural Computation.