Matrix-Based Higher Order Spectral Analysis for Three-Wave Coupling Processes

David Lee Sherman, M. D. Zoltowski

Research output: Contribution to journalArticlepeer-review


Composite linear and quadratic systems produce three-wave coupling when stimulated by random phase input sinusoids. Due to the nonlinearity of the system, the output frequencies are arithmetically related to the input. Using third-order cumulant statistics and their associated bispectrum, techniques are devised based on phase-insensitive matrix structures for detection and frequency estimation of coupling frequencies. The separation of the third-order cumulant series into symmetric and skew-symmetric portions allows us to exploit their characteristic eigendecompositions for frequency estimation. After symmetrization, biphases can be easily extracted as coefficients of the cumulant sequence. Using a generalized eigenvector representation, we can relate symmetric and skew-symmetric bases by a subspace rotation algorithm. Biphases can be estimated directly from generalized eigenvalues of the matrix pencil formed by symmetric and skew-symmetric matrices. The dimensionality of our matrices can be reduced through the use of cumulant projections that yield a slice of the bispectrum. The Radon transform procedure is related to bispectral processing through an isotropic radial-slice Volterra filter. The compact third-order Kronecker product matrix formulation and algorithms for coupling frequency estimation can also be converted for use in biphase estimation. Simulations showing the performance of the above procedures are also presented for both synthetic and biomedical time series. These include the detection and estimation of specific frequencies exhibiting nonlinearities in electroencephalographic (EEG) data.

Original languageEnglish (US)
Pages (from-to)332-348
Number of pages17
JournalIEEE Transactions on Signal Processing
Issue number2
StatePublished - Feb 1994

ASJC Scopus subject areas

  • Signal Processing
  • Electrical and Electronic Engineering

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