Ill-conditioned inverse problems: a solution using the complex cepstrum of higher order spectra

The application of higher-order spectral methods to certain multichannel inverse problems where the observed time signals are linear combinations of unknown sources is proposed. The transfer matrix is static and known beforehand but is highly ill-conditioned, leading to the failure of standard least squares regularization techniques or low rank approximation when there is additive noise. The noise suppression properties of third order cumulants are used, along with the ability to obtain multiple estimates by extracting individual signals from their third order cross-correlations via cross-bicepstrum operations. Variations of this approach are presented, and normalization issues are discussed