In adaptive filtering theory, the mean-square error of an LMS-type adaptive filter operating with estimated (not optimal) weights decomposes into the minimum mean-square error plus an "excess mean-square error" component arising because the weight vector fluctuates around the optimum. This excess error is governed by the recursive evolution of the diagonal covariance matrix of the transformed weight-error vector, whose steady-state boundedness depends on the eigenvalues of a derived symmetric "F" matrix (built from the step size, the diagonalized recursion matrix P, and the eigenvalue vector of the input autocorrelation matrix) remaining within unit magnitude. The ratio of the steady-state excess mean-square error to the minimum mean-square error, called the misadjustment, is a central figure of merit in adaptive filter design, analyzed using linear algebra results on matrix powers/convergence of Hermitian matrices and the matrix inversion lemma (Woodbury-type identity).
In adaptive filtering theory, the mean-square error of an LMS-type adaptive filter operating with estimated (not optimal) weights decomposes into the minimum mean-square error plus an "excess mean-sq…