Impact of thresholding on signal processing performance with applications

In this paper the approximation behavior of the Shannon sampling series is analyzed for the Paley-Wiener space PW_pi ¹, if the samples 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 peak approximation error can grow arbitrarily large, independently of how small the threshold is. However, if oversampling is applied and an appropriate kernel is chosen, then the re-constructed signal converges to the original signal, uniformly on the whole real axis, as the threshold goes to zero. Furthermore, we analyze the approximation behavior if not the signal itself is to be reconstructed but the output of some stable linear time invariant system. In particular, we show for the Hilbert transform that the peak approximation error is unbounded, even if over-sampling is applied.