This paper addresses certain functional learning tasks in signal processing using familiar algorithms and analytical tools of least squares for autoregressive moving average exogenous input (ARMAX) models. the models can be viewed as conventional ARMAX models but with parameters dependent on variables such as inputs or states, termed function input variables. The functional dependence of the parameters on these variables is represented in terms of basis function expansions or, more generally, interpolation function representations. The interpolation functions in a least-squares identification of coefficients also turn out to be in essence spread functions that spread learning throughout the space of function input Lariables. Thus for a set of training sequences or trajectories in function input space, system parameters and thereby system functionals can be updated. The idea is that these will have relevance for similar sequence5 or neighbouring trajectories.
The concept of persistence of excitation to achieve complete function learning or, equivalently, signal model learning is studied using least-squares convergence results. Application of the proposed algorithms and theory within the signal-processing context is addressed by means of simple illustrative examples. KEY u'OKDS Functional learning Persistence of excitation Recursive approximation Identification Parameter estimation This paper was recommended for publication by editor M . J. Grimble