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Linear Estimation

1. Kalman Filter Tutorial (Cont'd...)

1.4 Properties of Kalman filter

Unbiased estimator

An estimator is said to be unbiased, if the statistical expectation of an error is equal to zero. i.e. $$\label{4k} E[e_{k|k}]=E[e_{k|k-1}]=0.$$ The expression of expectation of prior error could be written as: $$\label{abc} E[e_{k|k-1}]=E[\hat{X}_{k|k-1}-X_{k}]=E[\hat{X}_{k|k-1}]-E[X_{k}].$$ We know that the predicted value depends on current state and previous measurement, i.e. \begin{equation*} E[\hat{X}_{k|k-1}]= E[E[X_{k}|Y_{k-1}]], \end{equation*} and also $X_{k}$ and $Y_{k-1}$ are independent to each other. So, we write \begin{equation*} E[E[X_{k}|Y_{k-1}]]=E[X_{k}]. \end{equation*} Now, substitute this value in the above equation, $$E[e_{k|k-1}]=0.$$