Non-uniform sampling — signal and system representation

In this paper we analyze the approximation of the outputs of stable linear time-invariant (LTI) systems T by sampling series that use only the samples of the input signal f. For our analysis we use the Paley-Wiener space of signals with absolutely integrable Fourier transform PW_pi ¹ and consider non-uniform sampling patterns. We completely characterize all stable LTI systems T and sampling patterns for which the sampling series converges to Tf for all f isin PW_pi ¹. In addition, we show that there are stable LTI systems and signals in PW_pi ¹ such that the approximation error grows arbitrarily large as the number of samples that are used for the approximation is increased. Furthermore, we analyze the approximation behavior of the sampling series for bandlimited wide-sense stationary processes that have a power spectral density and give a necessary and sufficient condition for the convergence in the mean square sense. Surprisingly, there is a close connection between the convergence behavior of the sampling series for deterministic signals in PW_pi ¹ and for bandlimited wide-sense stochastic processes.