Pole-zero identification based on simultaneous realization of normalized covariances and Markov parameters

Given the pole-zero configuration of a stable and rational transfer function it is trivial to determine the normalized covariances and Markov parameters. It is nontrivial and was recently shown that for finite windows of these parameters there corresponds a unique minimal and stable rational transfer function. Furthermore, small changes in the parameters corresponds to small changes of the transfer function, which makes the method robust. However, the proof was non-constructive and no algorithm for determining the inverse map was known. An efficient algorithm for determining the pole-zero configuration of the interpolating transfer function is the main contribution of this paper. As a corollary a novel and simplified approach to the minimal stochastic realization problem is obtained. Using an example from speech processing it is shown how this realization theory result can be used for identification of time series.