Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

9.22 BEST LINEAR UNBIASED ESTIMATION

In Section 9.21, we assumed so that the LS estimator is the BLU estimator. When the noise covariance matrix does not have this diagonal form, it is still possible to achieve BLU estimation by using WLS with weighting matrix W = R−1VV. The resulting estimator is

(9.368)

This result is derived below using a constrained optimization under the following conditions.

• The observation model for Y is given by (9.310) where is nonrandom but unknown, is known (measurable), and is a random vector with zero mean and covariance matrix .
• The form of the BLU estimator is a linear combination of the measurements:

  (9.369)

  where .
• It is unbiased:

  (9.370)

  which requires ...