The use of bispectrum and other higher order statistics in the analysis of one dimensional signals

摘要

A huge body of literature has been published on the use of second order statistics in signal processing, mainly through the study of the power spectrum. This is a sensible way to analyse signals since all non-trivial signals possess variance and the second order statistics are simple to calculate. A relatively small amount has been published on the use of higher order statistics and most of this has been produced in the last decade. There are two main reasons for the current surge in interest in the higher order statistics. The first is that the higher order statistics contain information not present in the second order statistics and as a relatively new field there is much to be discovered. The second is that with the availability of powerful computers the effort involved in calculating higher order statistics has been reduced to the level where the rewards justify it. The family of higher order spectra is presented and various problems are examined with reference to the properties of these spectra (more specifically, the second and third-members of the polyspectra family, the bispectrum and trispectrum). The use of the bispectrum is examined to assist the detection of continuous unknown signals in noise. Methods for making full use of the third order statistics of the signal are examined and their potential assessed. In the case of ship noise hidden in ambient sea noise, it was found that although the ship noise possesses significant levels of skewness it is not present at a high enough level to appreciably improve detection. udVarious aspects of the bispectrum are investigated and complete expressions for the variance and covariance of the bispectral estimate are derived. New tests for stationarity of the sampled and continuous signal are presented. Fault is found with the use of the linear model to simulate samples from non-Gaussian continuous stationary signals. The effects of bandlimiting on the third and fourth order statistics of signals is examined.