No-Go theorem for sampling-based signal processing

The approximation of linear time-invariant (LTI) systems by sampling series is an important topic in signal processing. However, the convergence of the approximation process is not guaranteed. In this paper we prove that for every sampling pattern that is a complete interpolating sequence there exists a universal stable LTI system such that for every oversampling factor there exists a bandlimited input signal such that the approximation process, which is used to approximate the output signal of the LTI system, diverges. This result shows a fundamental limit for the digital sampling-based implementation of systems.