Uniqueness of MSOE estimates in IIR adaptive filtering; a search for necessary conditions

In adaptive IIR (infinite impulse response) filtering, gradient search techniques minimize the mean-square output error (MSOE). The uniqueness of the minimum MSOE estimate for exactly matching adaptive filters is a necessary condition for global convergence of these algorithms. Although the existence of stable degenerated solutions is sufficient for the existence of local minima, it is shown by an example not to be a necessary condition, this uniqueness is also guaranteed if T. Soderstrom´s (1985) condition, n_b-n_e+1⩾0, is met when the input is white. It is proved that n_b-n_c+2⩾0 is a weaker sufficient condition for exactly matching models. In fact, this serves as the weakest sufficient condition