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

AbstractThe 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