Blind source separation and algorithm analysis
Derong Liu, Sanqing Hu · 2006
Sequential blind source extraction (SBSE) and simultaneous blind source separation (SBSS) of linear instantaneous mixtures with arbitrarily mixed source signals were discussed where the mixing matrix has full column rank as well as one of four ill-conditioned cases. Global output convergence of a class of continuous-tune recurrent, neural networks (RNNs) were presented. For SBSE, two general extractability conditions for source extraction of arbitrarily mixed nonzero source signals were given. A sufficient condition was provided to guarantee that sequential extraction can be continued. A necessary and sufficient condition for inseparability of an extracted mixture was presented. For inseparable mixture, all important property was shown: that is, any two extracted signals involving the same sources are proportional to each other. For sub (sup)-Gaussian source signals with only mutual independence, cost functions based on fourth-order cumulants were introduced to sequentially extract all separable single sources and all inseparable mixtures. Gradient-based methods are developed. Simulation results showed the operation characteristic and the effectiveness of our methods. For SBSS, two necessary and sufficient conditions for the identifiability of SBSS were presented. A sufficient condition was also derived for the existence of an optimal partition of the mixing matrix which leads to a unique maximum set of separated signals. Three sufficient conditions are given to describe an important property, of maximum partition(s). According to these conditions one maximum partition corresponds to a unique class of outputs and as a result we can determine the number of maximum partitions of the mixing matrix front the classes of outputs under different separation matrices. For sub (sup)-Gaussian source signals with only mutual independence, a cost function and gradient-based method were developed. Simulation results showed the effectiveness of the present method. For a class of RNNs which have wide applications to BSS and other optimization problems, global output convergence results were discussed. Three sufficient conditions for global output convergence of the RNNs were established. Our results extend the existing results to more general cases of connection weight matrices. As a result, our results expand the application domain of the RNNs.