[Paper Review] Zeros of the Wigner Distribution and the Short-Time Fourier Transform

Karlheinz Gröchenig, Philippe Jaming+1 more|arXiv (Cornell University)|Nov 9, 2018

Mathematical Analysis and Transform Methods41 references22 citations

TL;DR

This paper investigates the zero sets of the Wigner distribution and short-time Fourier transform, proving that certain non-Gaussian functions—specifically, generalized Gaussians, totally positive functions (e.g., convolutions of exponentials), and carefully constructed bounded step functions—can yield zero-free Wigner distributions. The key contribution is constructing explicit examples where the Wigner distribution never vanishes, even when the functions are not Gaussians, revealing deep connections to Hurwitz polynomials, Bessel functions, and Hudson’s theorem.

ABSTRACT

We study the question under which conditions the zero set of a (cross-) Wigner distribution W (f, g) or a short-time Fourier transform is empty. This is the case when both f and g are generalized Gaussians, but we will construct less obvious examples consisting of exponential functions and their convolutions. The results require elements from the theory of totally positive functions, Bessel functions, and Hurwitz polynomials. The question of zero-free Wigner distributions is also related to Hudson's theorem for the positivity of the Wigner distribution and to Hardy's uncertainty principle. We then construct a class of step functions S so that the Wigner distribution W (f, 1 (0,1)) always possesses a zero f $\in$ S $\cap$ L p for p < $\infty$, but may be zero-free for f $\in$ S $\cap$ L $\infty$. The examples show that the question of zeros of the Wigner distribution may be quite subtle and relate to several branches of analysis.

Motivation & Objective

• To determine conditions under which the Wigner distribution W(f, g) or short-time Fourier transform V_g f has an empty zero set.
• To challenge the belief that only generalized Gaussians yield zero-free Wigner distributions by constructing non-Gaussian examples.
• To explore connections between zero-free Wigner distributions and established theorems in harmonic analysis, such as Hudson’s theorem and Hardy’s uncertainty principle.
• To investigate the role of integrability and smoothness in the existence of zeros, particularly in the case where g is the characteristic function of an interval.
• To analyze the interplay between convexity, almost periodicity, and zero sets in the context of step functions and their Fourier transforms.

Proposed method

• Utilizes the theory of totally positive functions to construct zero-free Wigner distributions from one-sided exponentials and their convolutions.
• Applies properties of Bessel functions and Hurwitz polynomials to verify that certain Wigner distributions (e.g., for f(t) = t^n e^{-t} 1_{(0,∞)}) are zero-free.
• Employs metaplectic operators and symplectic invariance to relate the zero sets of Wigner distributions, ambiguity functions, and short-time Fourier transforms.
• Uses convex combinations and almost periodicity arguments to show that for L^p step functions with discontinuities on Z ∪ αZ (α irrational), the STFT must have zeros.
• Constructs a specific bounded L^∞ step function f with jumps on Z ∪ αZ (α irrational) such that V_{1_{(0,1)}} f is zero-free, using a delicate argument based on irrational rotations and trigonometric identities.
• Applies complex analysis and convexity in the unit disc to analyze the Fourier transform of step functions and derive conditions for vanishing Fourier coefficients.

Experimental results

Research questions

• RQ1 Can non-Gaussian functions f and g yield a zero-free Wigner distribution W(f, g)?
• RQ2 What role do total positivity, Bessel functions, and Hurwitz polynomials play in ensuring the absence of zeros in Wigner distributions?
• RQ3 How does the integrability class (L^p vs. L^∞) affect the existence of zeros in the short-time Fourier transform when g is a characteristic function?
• RQ4 To what extent is the zero set of the Wigner distribution related to Hudson’s theorem on non-negative Wigner distributions?
• RQ5 Can a bounded step function f ∈ L^∞(R) with discontinuities on a set like Z ∪ αZ (α irrational) yield a zero-free short-time Fourier transform with g = 1_{(0,1)}?

Key findings

• There exist non-Gaussian functions, such as convolutions of one-sided exponentials (e.g., f(t) = t^n e^{-t} 1_{(0,∞)}), for which the Wigner distribution is zero-free.
• The Wigner distribution of f(t) = t^n e^{-t} 1_{(0,∞)} contains Bessel functions as a factor, and its zero-freeness is guaranteed by the Hurwitz property of associated polynomials.
• For f ∈ L^p(R) with p < ∞ and discontinuities on Z ∪ αZ (α irrational), the short-time Fourier transform V_{1_{(0,1)}} f always has a zero.
• A bounded L^∞ step function f with jumps on Z ∪ αZ (α irrational) can be constructed such that V_{1_{(0,1)}} f is zero-free, demonstrating the sensitivity of zero sets to integrability.
• The zero set of the Fourier transform of a step function f with three intervals is non-empty if the interval endpoints are rationally related; however, for irrational relations, zeros can be avoided under specific coefficient conditions.
• When the coefficient sequence (c_k) in a step function is monotonic, the STFT V_{1_{(0,1)}} f is zero-free; but if (c_k) is not monotonic, zeros exist due to the failure of the convexity argument in the Fourier transform.