Asymptotic properties of subband identification

The purpose of this paper is to study the asymptotic properties (i.e., strong convergence and asymptotic convergence rate) of the subband identification method in every subband and in the overall method. The study of strong convergence aims to answer the question whether the “best possible” model is retrieved, on the limit, with probability one. The study of the asymptotic convergence rate aims to give an expression that quantifies how fast the model approaches the “best possible” value as the number of samples goes to infinity. To do this, we need to generalize existing results for fullband identification. In the process of doing so, we come up with a new notion of ergodicity, which we call strong ergodicity. Strongly ergodic signals not only satisfy the assumptions required for our analysis but also enjoy an interesting property, which is that strong ergodicity is invariant under a number of transformations. In particular, the subband components of a strongly ergodic signal are guaranteed to be strongly ergodic, therefore, ergodic, which is not true for an ergodic signal in general.