Wilson Bases for General Time-Frequency Lattices

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 $L^2(\mathbb{R})$ can be constructed for any time-frequency lattice whose volume is $\frac12$. We then focus on the spaces $\ell^2(\mathbb{Z})$ and $\mathbb{C}^L$ 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.

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