TY - JOUR

T1 - Unboundedness of thresholding and quantization for bandlimited signals

AU - Boche, Holger

AU - Mönich, Ullrich J.

PY - 2012/12

Y1 - 2012/12

N2 - It is well-known that the Shannon sampling series is locally uniformly convergent for all signals in the Paley-Wiener space PWπ1. An interesting question is how this good local approximation behavior is affected if the samples are disturbed by the non-linear threshold operator. This operator, which sets to zero all samples with absolute value smaller than some threshold, arises in the modeling of many applications, e.g., in wireless sensor networks. Moreover, it constitutes an essential part of a large class of quantizers, and consequently is important for all digital signal processing applications that involve conversion between analog and digital domains. In this paper, the approximation behavior of the Shannon sampling series that only uses the samples with absolute value larger than or equal to some threshold is analyzed. It is shown that there exists a signal in PWπ1 such that the local approximation error increases unboundedly as the threshold tends to zero. Moreover, for a fixed threshold, the local approximation error can grow arbitrarily large on the set of signals whose norm is bounded by one. With this, we generalize results of Butzer et al. that were given in the paper On quantization, truncation and jitter errors in the sampling theorem and its generalizations, Signal Processing (2) 1980 [1]. We conclude the paper with a discussion about the differences in the reconstruction behavior between the sampling series which is truncated in the domain of the sampled signal, i.e., time-domain truncation, and the sampling series which is truncated in the range of the sampled signal.

AB - It is well-known that the Shannon sampling series is locally uniformly convergent for all signals in the Paley-Wiener space PWπ1. An interesting question is how this good local approximation behavior is affected if the samples are disturbed by the non-linear threshold operator. This operator, which sets to zero all samples with absolute value smaller than some threshold, arises in the modeling of many applications, e.g., in wireless sensor networks. Moreover, it constitutes an essential part of a large class of quantizers, and consequently is important for all digital signal processing applications that involve conversion between analog and digital domains. In this paper, the approximation behavior of the Shannon sampling series that only uses the samples with absolute value larger than or equal to some threshold is analyzed. It is shown that there exists a signal in PWπ1 such that the local approximation error increases unboundedly as the threshold tends to zero. Moreover, for a fixed threshold, the local approximation error can grow arbitrarily large on the set of signals whose norm is bounded by one. With this, we generalize results of Butzer et al. that were given in the paper On quantization, truncation and jitter errors in the sampling theorem and its generalizations, Signal Processing (2) 1980 [1]. We conclude the paper with a discussion about the differences in the reconstruction behavior between the sampling series which is truncated in the domain of the sampled signal, i.e., time-domain truncation, and the sampling series which is truncated in the range of the sampled signal.

KW - Paley-Wiener space

KW - Quantization

KW - Shannon sampling series

KW - Threshold operator

UR - http://www.scopus.com/inward/record.url?scp=84864281006&partnerID=8YFLogxK

U2 - 10.1016/j.sigpro.2012.04.003

DO - 10.1016/j.sigpro.2012.04.003

M3 - Article

AN - SCOPUS:84864281006

SN - 0165-1684

VL - 92

SP - 2821

EP - 2829

JO - Signal Processing

JF - Signal Processing

IS - 12

ER -