The Linear Model Of The Minimum Variance Estimation Problem

Linear minimum variance estimate and optimally weighted LS estimate are often used in many fields such as signal processing,control and communications. Kalman filtering is the recursive version of ihe first estimate. In this paper we mainly discuss the problem about minimum variance estimation for linear model and bring out the relevance and difference of performance between the two methods in order to provide theoretic foundation for choosing appropriate estimation method.For a general linear model(input matrix is deterministic),under a certain conditions on variance matrix invertibility,the two estimates can be identical provided that they have the same priori information on the parameter under estimation. Even if the above information is unknown only for the optimally weighted LS estimate,the sufficient condition and necessary condition,under which the two estimates are identical,is derived. More significantly,we know how to design input of the linear system to make the performance of the optimally weighted LS estimation identical to that of the linear minimum variance estimation in case of being lack of prior information. Finally,using the above results,we can understand and explain well the connections between some of spueifie estimates.Moreover,we compare the performances between these two estimations for a linear model with random input. In this case optimally weighted LS estimate is not a linear estimate of a parameter given input and observation anymore and can not be compared with linear minimum variance estimate. Under a certain conditions on variance matrix invertibility,we show that the optimally weighted LS estimate outperforms the linear minimum variance estimate provided that they have the same priori information. Then we give the necessary and sufficient condition under which the optimally weighted LS estimate is identical to thu conditionalmean of the parameter given input and observation,i.e.,the optimally weighted LS estimate could be optimal nonlinear estimate in the minimum variance sense.Finally,we discuss application of Kalman filtering. The optimality of multi-sensor Kalman filtering fusion with feedback is presented and a filter bank based on wavelets and equipped with a miltiscale Kalman filter is proposed for estimating fractal signal in additive Gaussian white noise.