Comments on "Performance Analysis of a Deterministic Channel Estimator for Block Transmission Systems With Null Guard Intervals"

In the above-mentioned paper a Cramer-Rao bound was derived for the performance of a blind channel estimation algorithm. In this paper an error in the bound is pointed out and corrected. It is observed here that the performance of the said algorithm does not achieve the Cramer-Rao bound. In the above paper [1], important work has been done to analyze the algorithm in [2] which solves a blind channel estimation problem. The performance of the algorithm in [2] in high SNR region was shown to be as in (33) of [1]. The Cramer-Rao bound (CRB) of the above mentioned blind estimation problem was shown to be as in (49) of [1]. The coincidence of (33) and (49) led the authors of [1] to claim that the algorithm in [2] is statistically efficient (i.e., achieves the CRB) at high SNR values. However, we have found an error in the derivation of (49), which invalidates this claim. Eq. (49) of [1] was derived from (80) in Appendix B of [1]. The second equality of (80) is not valid in general since it is conditioned on the validity of the matrix identity (ABAH)−1 = AH†B−1A† (1) where A is a full rank matrix with more columns than rows and B is a square positive definite matrix. But a simple example shows that this identity is not true in general: set A = [ 1 0 0 0 1 0 ] , and B =   1 0 0 0 1 1 0 1 2   , then the left hand side of (1) is I2 whereas the right hand side