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On the adaptive linear estimators, using biased Cramer-Rao bound

机译：在自适应线性估计量上，使用有偏差的Cramer-Rao界

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The idea of minimizing the variance in biased estimation along with controlling the gradient of bias is well established for the case of singular Fisher information matrix (FIM) in order to find the biased estimators. In this paper, the biased Cramer-Rao lower bound (BCRLB) is used to derive and study the estimate of unknown parameters in a linear model with a known twice differentiable additive noise probability density function (PDF). Even if the additive noise is not Gaussian, we show that the derived linear estimators (not unique) are linear functions of the observations (where a constant number is inserted into observation vector) in a particular form. Examples are included to illustrate the estimators 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 when using this approach. For the case where the PDF of the additive noise is not differentiable, such as uniformly distributed or time invariant magnitude noises, an asymptotical approach is given to find the estimators. As an example, we evaluate the performance of the derived adaptive filter for a first-order Markov time varying system. If the FIM is singular, we use the method of singular value decomposition (SVD) to extract the parameter estimate of the linear models. 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 the bound of the bias gradient matrix.
机译：对于奇异费舍尔信息矩阵（FIM）的情况，为了找到有偏估计量，已经很好地确立了将有偏估计的方差最小化以及控制偏斜率的想法。在本文中，使用偏置的Cramer-Rao下界（BCRLB）来推导和研究带有已知两次可微加性噪声概率密度函数（PDF）的线性模型中未知参数的估计。即使加性噪声不是高斯噪声，我们也证明，导出的线性估计量（不是唯一的）是特定形式的观测值（在观测向量中插入常数）的线性函数。包括示例以说明估计器的性能。我们表明，在一般情况下，通过优化BCRLB获得的有偏估计量并不一定令人满意。因此，使用此方法时必须考虑其他注意事项。对于加性噪声的PDF不可微的情况（例如均匀分布的或时不变的幅度噪声），采用渐近方法找到估计量。例如，我们评估一阶马尔可夫时变系统的派生自适应滤波器的性能。如果FIM为奇异值，则使用奇异值分解（SVD）方法提取线性模型的参数估计。例如，我们表明在线性模型中，基于单个观测值的参数估计会导致归一化最小均方（NLMS）算法。在使用BCRLB优化的示例中，我们找到了NLMS算法的步长与偏置梯度矩阵的界限之间的关系。

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