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

Many second-order approaches have been proposed for blind FIR channel identification in single-input/multi-output context. In practical conditions, the measured impulse responses usually possess “small” leading and trailing terms, the second-order statistics are estimated from finite sample size, and there is additive white noise. This paper, based on a functional methodology, develops a statistical performance analysis of any second-order 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 ∞) of the mth-order estimated significant part of the impulse response. In the second model, the tails are treated as zero mean Gaussian random variables. Expressions for the asymptotic covariance matrix of the estimated significant part of the impulse response are then derived when the sample size tends to ∞, 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 response, 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 tails, 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 accuracy of the asymptotic analysis is checked by numerical simulations; the results are found to be valid in a very large domain of the sample size and the size of the tails