International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Article ID 48185, 22 pages
doi:10.1155/IJMMS/2006/48185
On the uncertainty inequality as applied to discrete signals
1Computer Vision and Artificial Intelligence Laboratory, Department of Electrical Engineering, Indian Institute of Science, Bangalore 560012, India
2Department of Electrical and Computer Engineering, Faculty of Engineering, National University of Singapore, 117576, Singapore
Received 11 June 2005; Revised 24 October 2005; Accepted 28 November 2005
Copyright © 2006 Y. V. Venkatesh et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Given a continuous-time bandlimited signal, the Shannon sampling theorem provides an interpolation scheme for exactly reconstructing it from its discrete samples. We analyze the relationship between concentration (or compactness) in the temporal/spectral domains of the (i) continuous-time and (ii) discrete-time signals. The former is governed by the Heisenberg uncertainty inequality which prescribes a lower bound on the product of effective temporal and spectral spreads of the signal. On the other hand, the discrete-time counterpart seems to exhibit some strange properties, and this provides motivation for the present paper. We consider the following problem: for a bandlimited signal, can the uncertainty inequality be expressed in terms of the samples, using thestandard definitions of the temporal and spectral spreads of the signal? In contrast with the results of the literature, we present a new approach to solve this problem. We also present a comparison of the results obtained using the proposed definitions with those available in the literature.