Noise response of detection filters: relation between detection space and completion space

An analysis is presented of the noise response of detection filters with zero-mean gaussian white noise. The major focus is placed on the relation between eigenvalues and error covariance. By similarity transformation it is shown that assigning the eigenvalues associated with a detection space does not affect the noise response of a completion space or other detection spaces. Based on this relationship, a brief optimisation procedure is given for an optimal problem in terms of the error covariance.