Limits of signal processing performance under thresholding

In this paper the reconstruction of bandlimited signals by sampling series is analyzed for the case where the samples of the signal are disturbed by the non-linear threshold operator. This operator sets all samples whose absolute value is smaller than some threshold to zero. It is shown that the reconstruction error can grow arbitrarily large, regardless of how small the threshold is chosen, when no oversampling is used. However, with oversampling, it is possible to upper bound the reconstruction error. Additionally, we consider the approximation of outputs of stable linear time-invariant systems, by sampling series that use only the samples that are larger than the threshold, and show that there exist stable linear time-invariant systems for which the approximation error is unbounded, even if oversampling is applied. Finally, we consider the case of non-equidistant sampling. It is possible to draw conclusions about the approximation behavior of the sampling series with threshold operator by analyzing the convergence behavior of the sampling series with permuted sampling points and without threshold operator. Since the latter does not have a good approximation behavior in general, we conjecture that the sampling series with non-equidistant samples and threshold operator has no good approximation behavior either.