Discussion

An EM iteration is particularily simple, consisting of the calculation of the sufficient satistics ${\bf g}_k$ and ${\bf G}_k$ in the E-step and the maximization with respect to ${\bf\theta}$ in the M-step.

Concerning an E-step, for temporally indepedent signal amplitudes, the ${\bf g}_k$ and ${\bf G}_k$ are computed independently over k using only simple matrix/vector operations and a D-dimensional matrix inverse for each k (e.g. for Gaussian signal amplitudes, (20, 21) ). While more computationally complex for the Gauss-Markov case, since the ${\bf g}_k$ and ${\bf G}_k$ are forward/backward recursive functions of all the data (i.e. (25-27) ), computation of these quantities is still based just on simple matrix/vector operations and D-dimensional matrix inverses.

Concerning an M-step, maximization of (12) is comparable in computation to the solution of the CML problem, where initialization is not an issue assuming the EM algorithm has been properly initialized. Computation per M-step iteration can be reduced using Generalized EM (GEM).

Initialization and convergence are important issues for any EM algorithm, or for any ML algorithm for that matter. For each case addressed above we suggest an approach to EM algorithm initialization. Concerning convergence, the required number of E-step/M-step iterations will depend on the application scenario and on the required accuracy. We do not address this important issue in this paper, except to rationalize our selection of the number-of-iterations used for each simulation.