Consistent analogs of the Fourier uncertainty relation

To resolve an issue that was raised in our earlier paper in this journal [69, 340-347 (2001)], a direct analog of the standard uncertainty relation is derived for the discrete Fourier transform (DFT). This inequality gives a simple lower bound for the degree of localization of the DFT of any sequence in terms of just the localization of the original sequence. It is also shown that the earlier uncertainty relations for the Fourier transform and the Fourier series can be derived from this new relation by taking appropriate limits. Because the same cannot be said of the others, this new relation is arguably the most fundamental of the three. [DOI: 10.1119/1.1383599]