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

9.23 PROPERTIES OF BLU ESTIMATORS

Although the BLU estimator is derived using a technique that differs from the WLS minimization, the result is in fact a special case.

• Equivalence of BLU and WLS estimators. When W = R−1VV, the BLU and WLS estimators are identical:

  (9.388)

  When V is uncorrelated such that , the LS and BLU estimators are equivalent. This is a restatement of the LS property given earlier.
• Unbiased. Since the derivation of the BLU estimator includes an unbiasedness constraint, is obviously unbiased for zero-mean noise:

  (9.389)

• Covariance matrix. The covariance matrix of the BLU estimator is

  (9.390)

  Substituting (9.368) gives

  (9.391)

• Minimum . The minimum value of the cost function is the trace of the covariance matrix in (9.391):

  (9.392)

  It is interesting to note ...