Uncertainty Principles for the Ambiguity Function

Abstract

We extend uncertainty principles which are valid for the Fourier transform to the setting of the ambiguity function. A general result is established for annihilating sets: strongly/weakly annihilating sets for the Fourier transform yield such sets for the ambiguity function, extending a result known for sets of finite measure. We apply this to sublevel sets of nondegenerate quadratic forms. Our main result is a sharp version of Beurling's uncertainty principle for the ambiguity function.

A Refinement of the Robertson–Schrödinger Uncertainty Principle and a Hirschman–Shannon Inequality for Wigner Distributions

“…Ghobber and Jaming [20,21] derived uncertainty principles for arbitrary integral operators (Fourier, Dunkl, Clifford transforms, etc) which have bounded kernels and satisfy a Plancherel theorem. A sharp version of the Beurling uncertainty principle was proven by B. Demange for the ambiguity function [12].…”

“…Other results for the support of joint position-momentum (or time-frequency) representations can be found in [12] for the ambiguity function and in [45] for the short-time Fourier transform. The continuous wavelet transform, which is a time-scale representation, was also shown to have non-compact support in [45].…”

Dunkl-Gabor transform and time-frequency concentration

“…In quantum mechanics and in signal analysis, uncertainty principles for the windowed Fourier transform are often discussed for simultaneous time-frequency representations on R d × R d (the so-called phase space or time-frequency plane), see for example [1], [3], [9], [19] and the references therein. The most famous of them is the following sharp Heisenberg type uncertainty inequality (see [1], Theorem 5.1):…”

Supports of Representations in the Cohen Class

“…Recently the study of uncertainty principles for time-frequency representations has received the attention of several authors (see for instance [3,7,8,15,16]). It is usually assumed that any time-frequency representation should satisfy some appropriate version of the uncertainty principle.…”