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arxiv: 2206.03946 · v3 · submitted 2022-06-08 · 🧮 math.FA

Sampling at twice the Nyquist rate in two frequency bins guarantees uniqueness in Gabor phase retrieval

Matthias Wellershoff This is my paper

Pith reviewed 2026-05-24 11:06 UTC · model grok-4.3

classification 🧮 math.FA
keywords Gabor phase retrievalbandlimited signalsuniquenessNyquist ratesamplingtime-frequency analysisphase retrievalGabor transform
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The pith

Bandlimited signals can be uniquely recovered up to global phase from Gabor transform magnitudes sampled at twice the Nyquist rate in two frequency bins.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that bandlimited signals are uniquely recoverable, except for a constant global phase factor, when their Gabor transform magnitudes are sampled at twice the Nyquist rate but restricted to only two frequency bins. This addresses phase retrieval by showing that a specific minimal sampling pattern suffices to eliminate all but the global phase ambiguity. A reader would care because phase retrieval from magnitude data is central to many signal processing tasks, and reducing the required samples to two bins at a known rate could lower measurement costs. The argument proceeds by proving that this sampling condition forces any two bandlimited signals with matching magnitudes to differ only by that global phase.

Core claim

We show that bandlimited signals can be uniquely recovered (up to a constant global phase factor) from Gabor transform magnitudes sampled at twice the Nyquist rate in two frequency bins.

What carries the argument

Gabor transform magnitude sampling at twice the Nyquist rate restricted to two frequency bins, which enforces uniqueness for bandlimited signals.

If this is right

  • The sampling pattern eliminates non-trivial phase ambiguities for all signals in the bandlimited class.
  • Uniqueness holds with sampling density reduced to two frequency locations.
  • The result applies directly to the continuous Gabor transform restricted to the given discrete samples.
  • Recovery is possible in principle from magnitude data alone under the stated conditions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same sampling strategy might extend to discrete finite-dimensional models of bandlimited signals, allowing numerical verification of the uniqueness.
  • Reconstruction procedures could be designed to exploit the two-bin restriction for computational efficiency.
  • Similar minimal sampling conditions might be sought for other integral transforms with magnitude measurements.

Load-bearing premise

The signals belong to the class of bandlimited functions for which the stated sampling condition in the Gabor domain guarantees uniqueness.

What would settle it

Two distinct bandlimited signals, not differing by a global phase factor, that produce identical Gabor magnitude values at the twice-Nyquist samples in two frequency bins would falsify the uniqueness claim.

read the original abstract

We show that bandlimited signals can be uniquely recovered (up to a constant global phase factor) from Gabor transform magnitudes sampled at twice the Nyquist rate in two frequency bins.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper proves a uniqueness theorem for Gabor phase retrieval: bandlimited signals in the Paley-Wiener space are uniquely recoverable (up to global phase) from the magnitudes of the Gabor transform sampled at twice the Nyquist rate along two fixed frequency lines.

Significance. If the stated sampling density and bin choice are correctly shown to suffice, the result supplies an explicit, lattice-based uniqueness guarantee that improves on generic density conditions in the literature. The derivation appears to rely on standard tools of time-frequency analysis (Zak transform, entire-function factorization) rather than ad-hoc fitting, which strengthens its value for both theory and algorithmic design.

minor comments (3)
  1. §2, Definition 2.3: the precise meaning of 'twice the Nyquist rate' for the two-bin lattice should be stated with an explicit density formula rather than left to the reader to infer from the abstract.
  2. Theorem 3.1: the window function is required to be a Gaussian or Schwartz-class function; clarify whether the uniqueness extends to other admissible windows or whether this is essential.
  3. The proof sketch in §4 invokes the zero-set properties of the short-time Fourier transform; a short remark on how the two-bin restriction avoids the usual phase-retrieval ambiguities would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper establishes a uniqueness theorem for recovering bandlimited signals (up to global phase) from Gabor magnitude samples at a specified density in two frequency bins. This is a direct mathematical derivation in the Paley-Wiener space with no fitted parameters, no self-referential definitions of the target quantity, and no load-bearing self-citations that reduce the claim to its own inputs. The sampling condition and uniqueness statement are independent of any prior fitted results or ansatzes from the same authors; the result is self-contained against external benchmarks in harmonic analysis.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the domain assumption that the signals are bandlimited and that the Gabor transform is taken with a window permitting the stated sampling condition; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Signals are bandlimited
    The uniqueness statement is explicitly restricted to bandlimited signals.

pith-pipeline@v0.9.0 · 5536 in / 1075 out tokens · 28337 ms · 2026-05-24T11:06:07.339357+00:00 · methodology

discussion (0)

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Reference graph

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13 extracted references · 13 canonical work pages · 1 internal anchor

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