Product of second moments in time and frequency for discrete-time signals and the uncertainty limit

Abstract

The product of ordinary second moments in time and frequency for discrete-time signals has a unique global minimum for the unit sample sequence. When the ordinary-second-moment product is used as the measure of localization in the continuous-time case, Gaussian signals are the best localized simultaneously in time and frequency. But sampling of the Gaussian signals provides discrete-time signals which are far from being the best localized. In general, it turns out that the product of ordinary second moments in time and frequency is not a most reasonable measure of simultaneous time–frequency localization for discrete-time signals. Motivated by this discrepancy, in this paper an uncertainty relation for the product of generalized second signal moments in time and frequency is derived, and optimal signals that reach the uncertainty limit are found. The requirement for convolution invariance between the optimal signals offers a new uncertainty relation for the discrete-time case. The set of optimal discrete-time signals, beside the unit sample sequence, binomial sequences, the sampled band-limited minimum-effective-duration signal, contains also, as a limit case, the Gaussian signals. The second moments involved in the uncertainty relation are described in more details.