A Balian–Low type theorem for Gabor Riesz sequences of arbitrary density

Abstract

Gabor systems are used in fields ranging from audio processing to digital communication. Such a Gabor system (g,varLambda ) consists of all time-frequency shifts pi (lambda ) g of a window function g in L^2({mathbb {R}}) along a lattice varLambda subset {mathbb {R}}^2. We focus on Gabor systems that are also Riesz sequences, meaning that one can stably reconstruct the coefficients c = (c_lambda )_{lambda in varLambda } from the function sum _{lambda in varLambda } c_lambda , pi (lambda ) g. In digital communication, a function of this form is used to transmit the digital sequence c. It is desirable for g to be well localized in time and frequency, since the transmitted signal will then be almost compactly supported in time and frequency if the sequence c has finite support. In this paper, we study what additional structural properties the signal space mathcal {G}(g,varLambda ), i.e., the span of the Gabor system, satisfies in addition to being a closed subspace of L^2({mathbb {R}}). The most well-known result in this direction—the Balian–Low theorem—states that if g is well localized in time and frequency and if (g,varLambda ) is a Riesz sequence, then mathcal {G}(g,varLambda ) is necessarily a proper subspace of L^2({mathbb {R}}). We prove a generalization of this result related to the invariance of mathcal {G}(g,varLambda ) under time-frequency shifts. Precisely, we show that if (g,varLambda ) is a Riesz sequence with g being well localized in time and frequency (precisely, g should belong to the so-called Feichtinger algebra), then pi (mu ) mathcal {G}(g,varLambda ) subset mathcal {G}(g,varLambda ) holds if and only if mu in varLambda . For lattices of rational density, this was already known, with the proof based on Zak transform techniques. These methods do not generalize to arbitrary lattices, however. Instead, our proof for lattices of irrational density relies on combining methods from time-frequency analysis with properties of a special C^*-algebra, the so-called irrational rotation algebra.