Gabor systems and the Balian-Low Theorem

Abstract

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 {su2πimbt g(t — na)} m,n∈ℤ with ab = 1 forms an orthonormal basis for L 2(ℝ) then 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′)⋀(γ) = 2πiγĝ(γ), 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 {e 2πibmt 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 2πimbt g(t — na)} that form exact frames.