A Novel Method for Complex-Valued Signals in Independent Component Analysis Framework

Abstract

This paper deals with the separation problem of complex-valued signals in the independent component analysis (ICA) framework, where sources are linearly and instantaneously mixed. Inspired by the recently proposed reference-based contrast criteria based on kurtosis, a new contrast function is put forward by introducing the reference-based scheme to negentropy, based on which a novel fast fixed-point (FastICA) algorithm is proposed. This method is similar in spirit to the classical negentropy-based FastICA algorithm, but differs in the fact that it is much more efficient than the latter in terms of computational speed, which is significantly striking with large number of samples. Furthermore, compared with the kurtosis-based

Efficient Optimization of Reference-Based Negentropy for Noncircular Sources in Complex ICA

“…As described in [37], we proposed the introduction of reference signals into the negentropy in [25] and [3], which resulted in the reference-based negentropy used in this paper.…”

“…where J is the negentropy in [25] and I is our reference-based negentropy proposed in [37]. In addition, w and v are the column vectors of W and V, respectively, under the constraint E w H x 2 = w 2 = 1.…”

“…In other words, we search for the conditions under which our contrast function (9) has local maxima or minima. Differing from the work in [3] and [37], we assume noncircular sources and operate in the complex domain by utilizing the definitions of the complex derivative of the nonanalytic functions found in [31] and [4] and the Taylor series expansion in [29]. As proved in "Appendix I," using the information in the pseudo-covariance matrix, we can conclude that a local minimum (respectively, maximum) of our contrast function (9) exists when estimating source one given sources (i = 2, 3, .…”

“…More precisely, our new algorithm was derived through the introduction of reference signals into Novey's nc-FastICA algorithm. In other words, we extended our previous work for circular sources [37] to the case of noncircular sources. A rigorous local stability analysis of our contrast function is provided, and a derivation of our algorithm is presented in detail.…”

“…In [36], an efficient and fast fixed-point algorithm was developed for a linear ICA based on reference-based kurtosis. However, the approaches in [34] and [36] are not robust against outliers, and thus an efficient and robust algorithm was proposed for complex circular sources by introducing reference signals into the negentropy [37], and a similar algorithm was developed for real-valued signals based on nonlinear functions [35] and the introduction of reference signals. These reference-based contrast criteria have a common appealing feature: The corresponding optimization algorithms are quadratic with respect to the searched parameters.…”