Functional learning in signal processing via least squares

Abstract

AbstractThis 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 variables. 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 sequences 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.