(non-)existence of 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(R) can be constructed for any time-frequency lattice whose volume is 1 2 . While this may be expected, our second result may come as a surprise. Namely we prove that there cannot exist Wilson bases (orthogonal or non-orthogonal) for general non-rectangular time-frequency lattices for l(Z) nor for the finite case C. These results should be compared to the fact that tight Gabor frames for general time-frequency lattices exist in abundance for L(R), l(Z), and C. We discuss some practical consequences of our theoretical findings.