Second-order blind source separation in the Fourier space of data

Abstract

A variant of the second-order blind identification (SOBI) algorithm is described. It achieves blind source separation of data whose inverse Fourier transform presents the required correlation properties. In a former approach, mixtures were separated using an inverse FT–SOBI–FT processing sequence. The same result is obtained using a new algorithm, named f-SOBI, based on an indirect evaluation of signal correlation functions. The relationship between both approaches is discussed. Their equivalence is established when an appropriate definition of correlation functions in SOBI is used, instead of the currently implemented one.

Introduction

Blind source separation (BSS) is a signal processing technique that consists of retrieving n unobservable sources from m linear combinations (m⩾n) provided by sensors [6], [3], [5]. This topic is related to the more general framework of independent component analysis (ICA) that also encompasses the resolution of blind deconvolution and identification problems. In an environment where no adaptation is needed, batch algorithms are preferred, as they act on whole pre-recorded data sets. Most of these algorithms tend to optimize contrast functions [3] computed from higher order cumulants, like in FastIca [5] or JADE [2]. This approach naturally applies to the separation of statistically independent sources whose probability density function is non-Gaussian. Second-order algorithms reduce the strong hypothesis of source statistical independence to source orthogonality. In turn, each source must be self-correlated. Under this assumption, an efficient separation algorithm, named second-order blind identification (SOBI) [1] was designed. It retained our attention for its robustness when applied to real-world applications, involving nuclear magnetic resonance (NMR) spectroscopy and pyroelectric signal analysis [9], [10], [4].

Formally, the raw data is represented by an m×F matrix X, where m is the number of sensors (and of mixtures) and F the number of recorded signal samples. Each column in X is associated with a particular value of the physical variable that evolves during the recording process: it can be time, frequency, wavelength, wavenumber, etc. It will be assumed here without loss of generality that the acquisition variable is frequency (hence the name F for the number of samples). The “natural” representation of the data is thus referred to as the Fourier space. During the course of our spectroscopic studies, it appeared that the spectral source signals, when represented in the conjugated direct space (the time domain signals), present correlation properties compatible with their extraction from mixtures by means of the SOBI algorithm [8]. This is particularly true in spectroscopic data [9] in which spectral peaks are extremely narrow and sparsely distributed. Under these conditions the correlation function of the spectra is not informative. A variant of the SOBI algorithm was developed so that it is not necessary to resort to the inverse FT, separation, FT sequence to display the separated spectra. This novel approach is of general interest because it allows separation in the Fourier space of data, whatever its physical origin, providing that the correlation constraint on time-domain sources is respected. After a brief description of BSS using second-order only statistics in Section 2, the direct computation of the required correlation matrices from spectra will be detailed in Section 3.