On the Computability of System Approximations Under Causality Constraints

Abstract

Approximating the transfer function of stable causal linear systems by a basis expansion is a common task in signal- and system theory. This paper characterizes a scale of signal spaces, containing stable causal transfer functions, with a very simple basis (the Fourier basis) but which is not computable. Thus it is not possible to determine the coefficients of this basis expansion on any digital computer such that the approximation converges to the desired function. Since the Fourier basis is not computable, the second part of the paper investigates whether there exist better bases. To this end, the notion of a computational basis is introduced and it is shown that there exists no computational basis in these spaces. The paper characterizes also subspaces on which computational bases do exist.