Time-frequency shift invariance and the Amalgan Balian Low Theorem

TL;DR

This paper investigates the limitations on the decay properties of generators in Gabor spaces with certain invariance properties, revealing fundamental constraints related to time-frequency localization.

Contribution

It establishes a new theoretical result linking invariance under translation-modulation pairs to decay limitations of Gabor generators, extending the Balian-Low theorem.

Findings

Generators cannot decay well in both time and frequency under invariance.

Riesz basis Gabor systems with rational density have fundamental decay restrictions.

Additional invariance imposes constraints on the generator's smoothness.

Abstract

We consider smoothness properties of the generator of a principal Gabor space on the real line which is invariant under some additional translation-modulation pair. We prove that if a Gabor system on a lattice with rational density is a Riesz basis for its closed linear span, and if the closed linear span, a Gabor space, has any additional translation-modulation invariance, then its generator cannot decay well in time and in frequency simultaneously.