Reconstruction Behavior of Shannon Sampling Series with Oversampling - Fundamental Limits

For practical applications it is important to have a stable signal reconstruction in the sense of an sampling series that is uniformly convergent on the whole real axis. Therefore, it is interesting to know the largest signal space, for which a stable reconstruction is possible. Recently it was shown that signals in the Paley-Wiener space PW1pi cannot be stably reconstructed from its samples taken equidistantly at Nyquist rate. However, if the sampling rate is greater than the Nyquist rate, i.e., when oversampling is applied, a stable reconstruction is possible and even the Shannon sampling series is a stable reconstruction process. This demonstrates that no elaborate kernel design is necessary as far as only convergence is concerned. In the proof an upper bound, which depends on the oversampling factor, is derived for the peak value of the Shannon sampling series. Furthermore, it is shown that the redundancy in the set of samples is not sufficient for a stable reconstruction, because a projection of the reconstruction process onto the range of signal frequencies is not possible without loosing stability. Additionally, a sufficient condition, in the form of an integrability criterion in the vicinity of the bandlimit, is given for the stability of the Shannon sampling series without oversampling.