Bounds on essential support and entropy of Weyl-Heisenberg frames at critical density

The incompatibility between numerical stability and high time-frequency localization for Weyl-Heisenberg systems at critical density is formulated in the context of joint time-frequency analysis. By using the essential support and the entropy of time-frequency transforms (Windowed Fourier, Wigner and Radar-Ambiguity transform) for the description of time-frequency localization it is shown that for the class of Weyl-Heisenberg frames with given frame bounds the lower bounds on essential support and entropy both exceed the corresponding lower bounds for the class of square integrable functions. The stability-localization antagonism is expressed as a relation between the bounds on the time-frequency localization and the frame bounds describing the class of Weyl-Heisenberg frames.