Gabor Systems and the Balian-Low Theorem

J.J. Benedetto

Department of Mathematics, University of Maryland,College Park, Maryland 20742 andThe MITRE Corporation, McLean, Virginia 22102
C. Heil
School of Mathematics, Georgia Institute of Technology,Atlanta, Georgia 30332-0160 andThe MITRE Corporation, Bedford, Massachusetts 01730
D.F. Walnut
Department of Mathematical Sciences, George Mason University,Fairfax, Virginia 22030
email: jjb@math.umd.edu

The Balian--Low theorem (BLT) is a key result in time-frequency analysis, originally stated by Balian and, independently, by Low, as: If a Gabor system $\{e^{2\pi imbt g(t-na)\}_{m,n \in \Z}$ with $ab=1$ forms an orthonormal basis for $L2$ then $$(\int_{-\infty}^\infty |t g(t)|^2 dt) (\int_{-\infty}^\infty |\gamma \hat g(\gamma)|^2 d\gamma = +\infty$$ The BLT was later extended from orthonormal bases to exact frames. This paper presents a tutorial on Gabor systems, the BLT, and related topics, such as the Zak transform and Wilson bases. Because of the fact that $(g')^\wedge(\gamma) = 2 \pi i \gamma \hat g(\gamma)$, the role of differentiation in the proof of the BLT is examined carefully. We include the construction of a complete Gabor system of the form $\{see^{2\pi ib_mt} g(t-a_n)\}$ such that $\{(a_n,b_m)\}$ has density strictly less than 1, and an Amalgam BLT that provides distinct restrictions on Gabor systems $\{e^{2\pi imbt} g(t-na)\}$ that form exact frames.