Novel Techniques for the Detection and Estimation of Three-wave Coupling with Application to Human Brain Waves

Composite linear and quadratic systems produce three-wave coupling when stimulated by random-phase input sinusoids. Due to the non-linearity 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 phaseinsensitive 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 which 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.