llseresult

# llseresult - Boston University Department of Electrical and...

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Boston University Department of Electrical and Computer Engineering EC505 STOCHASTIC PROCESSES Notes on a LLSE Result Consider the following problem. We want to find the Bayes linear least squares estimate of the random vector Z based on observation of the random vector Y , with mean m Y and covariance Λ Y . Further, suppose the random vector Z is related to the random vectors X and W as follows: Z = FX + HW + b (1) where F and H are deterministic matrices, b is a deterministic vector, E [ X ] = m X , Cov( X , X ) = λ X , Cov( X , Y ) = λ XY , E [ W ] = 0, Cov( W , W ) = λ w , and W is uncorrelated with both X and Y . Show that b z L ( y ), the LLSE of Z based on Y , can be written as a linear function of b x L and b w L ( y ), the LLSE estimates of X and W based on Y , respectively. Further show that the corresponding error covariance matrix Λ L z , can be written as a Λ L x and Λ L w . This result will be important later, as it shows that we may use the estimate of one random variable to easily obtain the estimate of another if the two are linearly related! Solution: First note that b
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