# Time-frequency shift invariance of Gabor spaces generated by integer lattices

## TL;DR

This paper investigates the conditions under which Gabor spaces generated by integer lattices exhibit invariance to additional time-frequency shifts, revealing parallels with shift-invariant spaces.

## Contribution

It provides new characterizations of time-frequency shift invariance in Gabor spaces generated by integer lattices, including extreme cases of full invariance.

## Key findings

- Characterizations of shift invariance for Gabor spaces
- Conditions for full translation and modulation invariance
- Analogy with shift-invariant spaces

## Abstract

We study extra time-frequency shift invariance properties of Gabor spaces. For a Gabor space generated by an integer lattice, we state and prove several characterizations for its time-frequency shift invariance with respect to a finer integer lattice. The extreme cases of full translation invariance, full modulation invariance, and full time-frequency shift invariance are also considered. The results show a close analogy with the extra translation invariance of shift-invariant spaces.