Identifiability of unknown noise covariance matrices for some special cases of a linear, time-invariant, discrete-time dynamic system

This paper investigates the identification of unknown noise covariance matrices
Q
and
R
of an LTI discrete-time dynamic system. Two algorithms based on maximum a posteriori (MAP) and maximum likelihood (ML) cost functions are evaluated. It is demonstrated that the cost functions exhibit local minima versus those elements of
Q
and
R
which dominate the steady-state output covariance matrix. The following three special cases are considered: 1) single-input single-output (SISO) system; 2) multiinput single-output (MISO) systems; and 3) single-input multioutput (SIMO) systems with a diagonal
R
. For these special cases, specific identifiability criteria are presented and verified by examples. The improvement of the MAP algorithm over the ML algorithm is also demonstrated.