[Paper Review] Balayage and Short time Fourier transform frames

Enrico Au-Yeung, John J. Benedetto|arXiv (Cornell University)|Sep 2, 2013

Mathematical Analysis and Transform Methods20 references3 citations

TL;DR

This paper establishes non-uniform sampling theorems for the short-time Fourier transform (STFT) using Beurling's theory of balayage, spectral synthesis, and strict multiplicity. It proves that if balayage is possible for a separated set $E = \{(s_m, t_n)\} \subset \mathbb{R}^{2d}$ and a compact, convex, symmetric S-set $\Lambda \subset \widehat{\mathbb{R}}^{2d}$, then the STFT samples at $E$ form a frame for $L^2(\mathbb{R}^d)$, ensuring stable signal reconstruction from non-uniform samples.

ABSTRACT

Using his formulation of the potential theoretic notion of balayage and his deep results about this idea, Beurling gave sufficient conditions for Fourier frames in terms of balayage. The analysis makes use of spectral synthesis, due to Wiener and Beurling, as well as properties of strict multiplicity, whose origins go back to Riemann. In this setting and with this technology, we formulate and prove non-uniform sampling formulas in the context of the short time Fourier transform (STFT).

Motivation & Objective

• To generalize non-uniform sampling theory to the short-time Fourier transform (STFT) setting using advanced harmonic analysis tools.
• To establish sufficient conditions for the existence of STFT frames using balayage, spectral synthesis, and strict multiplicity.
• To provide a general framework for STFT sampling that extends classical Fourier frame results to irregular, non-uniform sampling sets.
• To connect the STFT sampling problem to the theory of bounded Radon measures and the Wiener-Pitt type theorems via balayage.

Proposed method

• Utilizes Beurling's formulation of balayage to transfer measures from $\mathbb{R}^{2d}$ to a separated set $E \subset \mathbb{R}^{2d}$, ensuring that the Fourier transform of a measure on $E$ agrees with that on $\Lambda$.
• Applies spectral synthesis (S-set property) of $\Lambda$ to ensure that vanishing Fourier transform on $\Lambda$ implies vanishing integral against functions in $\mathcal{C}(\Lambda)$.
• Employs the concept of strict multiplicity to control the decay of the inverse Fourier transform of measures on $\Lambda$, ensuring non-triviality and stability.
• Uses the Feichtinger algebra $\mathcal{S}_0(\mathbb{R}^d)$ as the function space where $V_g f \in L^1(\mathbb{R}^{2d})$, enabling the use of $L^1$-based frame bounds.
• Constructs a reconstruction formula via the STFT sampling at $E = \{(s_m, t_n)\}$, expressing $V_g f(y, \omega)$ as a convergent sum over $E$ using coefficients $a_{mn}(y, \omega)$ with absolutely summable $L^1$-norm.
• Derives frame bounds $A, B > 0$ such that $A \|f\|_{L^2}^2 \leq \sum_{m,n} |V_g f(s_m, t_n)|^2 \leq B \|f\|_{L^2}^2$ under the assumption that $\mathcal{F}(V_g f)$ is supported in $\Lambda$.

Experimental results

Research questions

• RQ1 Under what conditions on a separated set $E \subset \mathbb{R}^{2d}$ and a compact, convex, symmetric S-set $\Lambda \subset \widehat{\mathbb{R}}^{2d}$ does balayage exist?
• RQ2 How can balayage and spectral synthesis be used to derive frame bounds for STFT samples at non-uniform points?
• RQ3 What function space conditions ensure that the STFT of a function is integrable and that sampling at $E$ yields a stable reconstruction?
• RQ4 How does the theory of strict multiplicity contribute to the existence of non-trivial measures with decaying Fourier transforms on $\Lambda$?

Key findings

• Balayage is possible for $(E, \Lambda)$ if and only if for every $\mu \in M_b(\mathbb{R}^{2d})$, there exists $\nu \in M_b(E)$ such that $\widehat{\mu} = \widehat{\nu}$ on $\Lambda$.
• If $\Lambda$ is a compact, convex, symmetric S-set and balayage is possible for $(E, \Lambda)$ with $E$ separated, then the STFT samples at $E$ form a frame for $L^2(\mathbb{R}^d)$.
• For $f \in \mathcal{S}_0(\mathbb{R}^d)$ such that $\mathcal{F}(V_g f)$ is supported in $\Lambda$, the frame inequality $A \|f\|_{L^2}^2 \leq \sum_{m,n} |V_g f(s_m, t_n)|^2 \leq B \|f\|_{L^2}^2$ holds with positive constants $A$ and $B$.
• The upper frame bound $B$ can be taken as $C \|V_{G_0} G\|_1$, where $G_0$ is a Gaussian window and $C$ is a supremum over a convolution-type expression involving $V_{G_0} G_0$.
• The reconstruction of $f$ from its STFT samples at $E$ is stable and unconditional in $\mathcal{S}_0(\mathbb{R}^d)$, provided $E$ is separated and the covering condition of Gröchenig is satisfied.