Postfiltering versus prefiltering for signal sampling and recovery under noise

We consider the extension of the Whittaker-Shannon reconstruction formula to the case of signals sampled in the presence of noise and which are not necessarily band-limited. Observing that in this situation the classical sampling expansion yields inconsistent reconstruction, we introduce a class of signal recovery methods with a smooth correction of the interpolation series. Two alternative data smoothing methods are examined based either on a global post-filtering or a local data presmoothing. We assess the accuracy of the methods by the global L₂ error. Both band-limited and non-band-limited signals are considered. A general class of correlated noise processes is taken into account. The weak and strong rates of convergence of the algorithms are established and their relative efficiency is discussed. The influence of the noise memory and its moment structure on the accuracy is thoroughly examined.