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

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.