Adaptive linear estimators, using biased cramer-RAO bound

In this paper, the biased Cramer-Rao lower bound (BCRLB) is used to derive the estimate of unknown parameters in a linear model with an arbitrary known additive noise probability density function (PDF).We show that the derived linear estimators (not unique) are linear functions of the observations. Examples are included to illustrate their performances. We show that a biased estimator obtained by optimization of BCRLB is not necessary satisfactory in a general case; therefore, additional considerations must be taken into account. If the Fisher information matrix (FIM) is singular, we use the method of singular value decomposition (SVD) to extract the parameter estimate of linear model. For example we show that in a linear model, parameter estimation based on single observation leads to the normalized least mean square (NLMS) algorithm. In this example using BCRLB optimization, we find the relation between the step size of the NLMS algorithm and bound of bias gradient matrix