Higher-Order Statistics in Signal Processing

Abstract

First some definitions and properties with respect to higher-order statistics are highlighted to throw some light on the reasons for strong interests of the signal processing community over the last ten years in this research field. Later blind source separation , blind system identification, time-delay estimation, and blind equalization are introduced as examples of some of the current signal processing applications. 4.1 Introduction Until the mid-1980's, signal processing-signal analysis, system identification, signal estimation problems, etc.-was primarily based on second-order statistical information. Autocorrelations and cross-correlations are examples of second-order statistics (SOS). The power spectrum which is widely used and contains useful information is again based on the second-order statistics in that the power spectrum is the one-dimensional Fourier transform of the autocorrelation function. As Gaussian processes exist and a Gaussian probability density function (pdf) is completely characterized by its first two moments, the analysis of linear systems and signals has so far been quite effective in many circumstances. It has nevertheless been limited by the assumptions of Gaussianity, minimum phase systems, linear systems, etc. When signals are non-Gaussian the first two moments do not define their pdf and consequently higher-order statistics (HOS), namely of order greater than two, can reveal other information about them than SOS alone can. Ideally the entire pdf is needed to characterize a non-Gaussian signal. In practice this is not available but the pdf may be characterized by its moments. It should however be noted that some distributions do not possess finite moments of all orders. As an example, Cauchy distribution, defined as 1 1 p(x) = 7r{3 1 + (x~")2-oo