Statistical analysis of some second-order methods for blind channel identification/equalization with respect to channel undermodeling

Many second-order approaches have been proposed recently for blind FIR chan nel identification in a single-input/multi output contest. In practical con ditions, the measured impulse responses usually possess "small" leading and trailing terms, the second-order statistics are estimated from finite samp le size, and there is additive white noise. This paper, based on a function al methodology, develops a statistical performance analysis of any second-o rder approach under these practical conditions. We study two channel models . In the first model, the channel tails are considered to be deterministic We derive expressions for the asymptotic bias and covariance matrix. (when the sample size tends to infinity) of the mth-order estimated significant p art of the impulse response. In the second model, the tails are treated as zero mean Gaussian random variables. Expressions for the asymptotic covaria nce matrix of the estimated significant part of the impulse response are th en derived when the sample size tends to infinity, and the variance of the tails tends to 0. Furthermore, some asymptotic statistics are given for the estimated zero-forcing equalizer, the combined channel-equalizer impulse r esponse, and some byproducts, such as the open eye measure. This allows one to assess the influence of the limited sample size and the size of the tai ls, respectively, on the performance of identification and equalization of the algorithms under study. Closed-form expressions of these statistics are given for the least-squares, the subspace, the linear prediction, and the outer-product decomposition (OPD) methods, as examples. Finally the accurac y of the asymptotic analysis is checked by numerical simulations; the resul ts are found to be valid in a very large domain of the sample size and the size of the tails.