Wilson bases for general time-frequency lattices
Gitta Kutyniok, Thomas Strohmer
TL;DR
This work develops a general construction of orthonormal Wilson bases for time-frequency lattices of density $\tfrac{1}{2}$ across continuous, discrete, and finite settings. Central to the method is a metaplectic transform that maps general lattices to canonical forms and reduces the problem to known Wilson/Gabor frame results (notably a Balian–Low-type framework) in each setting. The authors provide explicit canonical generator forms, Zak-transform based equivalences, and bijections to handle non-diagonal cases, ensuring that tight Gabor frames with bound 2 yield Wilson bases. The results have practical implications for OFDM-like systems and cosine-modulated filter banks on general lattices, and suggest extensions to higher dimensions and general LCA groups.
Abstract
Motivated by a recent generalization of the Balian-Low theorem and by new research in wireless communications we analyze the construction of Wilson bases for general time-frequency lattices. We show that orthonormal Wilson bases for $\LtR$ can be constructed for any time-frequency lattice whose volume is $\tfrac12$. We then focus on the spaces $\ell^2(\ZZ)$ and $\CC^L$ which are the preferred settings for numerical and practical purposes. We demonstrate that with a properly adapted definition of Wilson bases the construction of orthonormal Wilson bases for general time-frequency lattices also holds true in these discrete settings. In our analysis we make use of certain metaplectic transforms. Finally we discuss some practical consequences of our theoretical findings.