Causal discrete‐time system approximation of non‐bandlimited continuous‐time systems by means of discrete prolate spheroidal wave functions

Abstract

Abstract In general, linear time‐invariant (LTI) continuous‐time (CT) systems can be implemented by means of LTI discrete‐time (DT) systems, at least for a certain frequency band. If a causal CT system is not bandlimited, the equivalent DT system may has be to non‐causal for perfectly implementing the CT system within a certain frequency band. This paper studies the question to which degree a causal DT system can approximate the CT system. By reducing the approximation frequency band, the approximation accuracy can be increased‐‐‐at the expense of a higher energy of the impulse response of the DT system. It turns out, that there exists a strict trade‐off between approximation accuracy, measured in the squared integral error, and energy of the impulse response of the DT system. The theoretically optimal trade‐off can be achieved by approximations based on a weighted linear combination of the discrete prolate spheroidal wave functions (DPSWFs). The results are not limited to the case of approximating a CT system by means of a causal DT system, but they generally hold for the approximation of an arbitrary spectrum by means of a spectrum of an indexlimited time sequence. Copyright © 2008 John Wiley & Sons, Ltd.