The cross-bicepstrum: definition, properties, and application for simultaneous reconstruction of three nonminimum phase signals

The recovery of three signals from their cross-bispectrum (or the identification of the impulse responses of three parallel linear time-invariant (LTI) systems from the cross-bispectrum of their system outputs) by computing the complex cepstrum of the cross-bispectrum, as long as the signals (or impulse responses) have no zeros on the unit circle, is discussed. It is shown that the three signals can be separated completely and (approximately) recovered in the cross-bicepstrum domain, except for their magnitude and linear phase factors. The computation of the cross-bicepstrum can be seen as a method for the simultaneous computation of the ordinary complex cepstra of three nonminimum-phase signals without the need for phase unwrapping. Both least-squares and fast Fourier-transform (FFT)-based methods for computing the bicepstral coefficients are presented. Simulation examples of signal reconstruction in Gaussian white and nonGaussian colored noise and of system identification are included. The results are extended to nth-order cross-spectra, and the factorization problem for these spectra is discussed