A Survey of Kurtosis Optimization Schemes for MISO Source Separation and Equalization

Blind source separation and equalization aim at recovering a set of unknown source signals from their linearly distorted mixtures observed at a sensor array output, with little or no prior knowledge about the sources or the distorting channel. This fundamental signal processing problem arises in a broad range of applications such as multiuser digital communications, biomedical data analysis, and seismic exploration. Put forward over three decades ago, the normalized fourth-order cumulant, also known as kurtosis, has arguably become one of the most popular blind source separation and equalization criteria. Using multiple-input single-output (MISO) filter structures for single source extraction combined with suitable deflation procedures, the kurtosis contrast yields separation algorithms free of spurious extrema in ideal system conditions. The lack of closed-form solutions, however, calls for numerical optimization schemes. The present chapter reviews some of the iterative algorithms most widely used for MISO source separation and equalization based on kurtosis. These include gradient and Newton search methods, algorithms with optimal step-size selection, as well as techniques based on reference signals. Their main features are briefly summarized and their performance is illustrated by some numerical experiments in digital communications and biomedical signal processing.