International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 10, Pages 1589-1599
On bandlimited signals with minimal product of effective spatial and spectral widths
Department of Electrical Engineering, Indian Institute of Science, Bangalore 560012, Karnataka, India
Received 31 July 2004; Revised 5 January 2005
Copyright © 2005 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.
It is known that signals (which could be functions of space or time) belonging to -space cannot be localized simultaneously in space/time and frequency domains. Alternatively, signals have a positive lower bound on the product of their effective spatial andeffective spectral widths, for simplicity, hereafter called the effective space-bandwidthproduct (ESBP). This is the classical uncertainty inequality (UI), attributed to many, but, from a signal processing perspective, to Gabor who, in his seminal paper, established the uncertainty relation and proposed a joint time-frequency representation in which the basis functions have minimal ESBP. It is found that the Gaussian function is the only signal that has the lowest ESBP. Since the Gaussian function is not bandlimited, no bandlimited signal can have the lowest ESBP. We deal with the
problem of determining finite-energy, bandlimited signals which have the lowest ESBP. The main result is as follows. By choosing the convolution product of a Gaussian signal (with as the variance parameter) and a bandlimited filter with a continuous spectrum, we demonstrate that there exists a finite-energy, bandlimited signal whose ESBP can be made to be arbitrarily close (dependent on the choice of ) to the optimal value specified by the UI.