Higher-Order Spectra Analysis
This work has been supported by NSF
under grant MIP-9553227,
It is a well-known fact that second-order
statistics (second-order correlation, or power spectrum) are phase-blind, that
is, they are able to describe minimum-phase systems only. Moreover, only
Gaussian random processes can be completely specified by knowledge of their
first- and second-order statistics. There exist numerous practical situations,
where one has to look at higher-order correlations, or cumulants, (order >
2) of a signal in order to extract information on phase, deviations from
Gaussianity, or presence of nonlinearilies.
Higher-order spectra, or polyspectra, are defined as the multidimensional Fourier transforms of higher-order cumulants. Cumulants, or polyspectra, of Gaussian processes, of order > 2, are identically zero. Thus, in theory, HOS are high signal-to-noise-ratio domains, where system identification or signal reconstruction can be performed.
Low-Variance Estimators of HOS
Identification of a non-minimum phase system driven by a non-Gaussian input is only possible by using HOS of the system output. However, HOS have received a lot of criticism lately, because of the amount of data required to produce low variance estimates, and the amount of computational complexity involved. We are currently developing low-rank estimators of HOS which, for a fixed data length, result in a reduction of variance at the expense of increase in bias. We are developing methods for selection of the optimal value of the rank, in the sense that it will result in the minimization of the mean-square-error in the higher-order cumulants estimates.
System Reconstruction Based on HOS
We considered the problem of system reconstruction form HOS of the system output. We focused on reconstruction of bandlimited systems, whose HOS vanish in certain regions. In the same regions, HOS estimates of bandlimited systems will be nonzero, and will be dominated by noise and finite data length effects. Avoiding such regions can yield better system estimates. To this end, we are developing methods that use selected HOS regions for reconstruction. We are also considering methods for selection of ``good'' slices, that lead to superior system estimates. Since only polyspectra slices are used, significant computational savings can arise by avoiding the computation of the entire polyspectrum.
The importance of selecting ``good'' slices to use for system reconstruction is demonstrated in the following figure. The system considered is bandpass, and the bispectrum of its output is computed in a 64-by-64 grid. Slices 13-14 are in a high amplitude bispectrum region, while 23-24 lie in a low amplitude bispectrum region. The performance is significantly better when ``good'' slices are used.
A collection of Matlab functions for estimation of Bispectrum and Trispectrum.