Cost functions used in blind source separation are often defined in terms of expectations, i.e., an infinite number of samples is assumed. An open question is whether the local minima of finite sample approximations to such cost functions are close to the minima in the infinite sample case. To answer this question, we develop a new methodology of analyzing the finite sample behavior of general blind source separation cost functions. In particular, we derive a new probabilistic analysis of the rate of convergence as a function of the number of samples and the conditioning of the mixing matrix. The method gives a connection between the number of available samples and the probability of obtaining a local minimum of the finite sample approximation within a given sphere around the local minimum of the infinite sample cost function. This shows the convergence in probability of the nearest local minima of the finite sample approximation to the local minima of the infinite sample cost function. We also answer a long-standing problem of the mean-squared error (MSE) behavior of the (finite sample) least squares constant modulus algorithm (LS-CMA), namely whether there exist LS-CMA receivers with good MSE performance. We demonstrate how the proposed techniques can be used to determine the required number of samples for LS-CMA to exceed a specified performance. The paper concludes with simulations that validate the results.
Bibliographical noteFunding Information:
Manuscript received March 8, 2007; revised January 24, 2008. Published August 13, 2008 (projected). The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Philippe Loubaton. This research has been partially supported by the EU-FP6 under contract 506790 and by NWO-STW under the VICI programme (DTC.5893). Parts of this paper were presented at the IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Istanbul, Turkey, June 2000.
- Blind source separation
- Constant modulus algorithm
- Finite sample analysis