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arxiv: 2605.31507 · v1 · pith:KD46OUBCnew · submitted 2026-05-29 · 🧮 math.FA

Uncertainty Principles as a Tool for STFT Phase Retrieval

David Bartusel This is my paper

Pith reviewed 2026-06-28 19:55 UTC · model grok-4.3

classification 🧮 math.FA
keywords STFT phase retrievalambiguity functionuncertainty principleshort-time Fourier transformfinite-dimensional signalsphase retrievalambiguity sampling
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The pith

A window supports single-window STFT phase retrieval if at least roughly eight ninths of its ambiguity function entries are nonzero, or three quarters in prime dimensions.

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

The paper studies how many zeros are permitted in a window's ambiguity function while still guaranteeing that signals can be uniquely recovered from the magnitude of their short-time Fourier transform. It begins with a two-window construction in which the second window is the Fourier transform of the first, then invokes the uncertainty principle to obtain sufficient conditions for phase retrieval in that setting. The relation between STFT phase retrieval and ambiguity sampling is used to carry those conditions over to the ordinary single-window problem. A reader would care because the result supplies concrete, dimension-dependent lower bounds on the number of nonzero ambiguity entries needed for reconstruction to be possible.

Core claim

In the finite-dimensional setting, a two-window approach is introduced where the second window equals the Fourier transform of the first; the uncertainty principle is applied to this pair to produce sufficient conditions for phase retrieval. The established relation between STFT phase retrieval and ambiguity sampling then transfers the same conditions to the single-window case, proving that the window's ambiguity function needs only approximately eight ninths of its entries to be nonzero in general and only three quarters when the dimension is prime.

What carries the argument

The two-window construction (second window the Fourier transform of the first) together with the relation between STFT phase retrieval and ambiguity sampling, which transfers uncertainty-principle bounds to the single-window setting.

If this is right

  • Phase retrieval is guaranteed whenever the ambiguity function meets the stated nonzero threshold.
  • The same nonzero threshold applies uniformly across all windows satisfying the condition.
  • In prime dimensions the allowable zero fraction is strictly smaller than in composite dimensions.
  • Uncertainty principles applied to the two-window pair directly control the admissible zeros for the single-window problem.

Where Pith is reading between the lines

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

  • The gap between the general eight-ninths bound and the three-quarters bound in prime dimensions indicates that dimension parity or factorization may further tighten the result.
  • The transfer technique via ambiguity sampling could be reused for other window pairs or sampling sets beyond the Fourier-transform pair considered here.

Load-bearing premise

The relation between STFT phase retrieval and ambiguity sampling holds and permits the transfer of sufficient conditions obtained in the two-window case to the single-window case.

What would settle it

A concrete counterexample would be any finite-dimensional window whose ambiguity function has at least eight ninths (or three quarters in a prime dimension) nonzero entries yet admits two distinct signals with identical STFT magnitudes.

read the original abstract

In the finite-dimensional setting, it is known that STFT phase retrieval is always possible when the window's ambiguity function does not vanish. However, it is not known how many zeros are allowed in the ambiguity function for the window still to allow phase retrieval. In order to tackle this problem, we first consider a two-window approach where the second window equals the Fourier transform of the first window. This allows us to apply the uncertainty principle in order to obtain sufficient conditions for phase retrieval. Using the relation between STFT phase retrieval and ambiguity sampling, we can prove sufficient conditions for the single-window phase retrieval problem, showing that only approximately eight ninths of the entries of the window's ambiguity function (and only three quarters in prime dimensions) are required to be nonzero.

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

3 major / 2 minor

Summary. The paper claims that in finite dimensions, sufficient conditions for single-window STFT phase retrieval can be obtained by first considering the two-window case (second window = Fourier transform of the first), applying an uncertainty principle to derive zero bounds, and then transferring the result to the single-window setting via the relation between STFT phase retrieval and ambiguity sampling. This yields that the window's ambiguity function needs only approximately 8/9 of its entries nonzero (or 3/4 when the dimension is prime) for phase retrieval to hold.

Significance. If the transfer step is shown to map the two-window zero patterns injectively onto the single-window ambiguity function without imposing extra support constraints that change the allowed zero fraction, the result would give explicit, quantitative sufficient conditions that improve on the known nowhere-vanishing requirement. The approach of routing uncertainty-principle bounds through ambiguity sampling is a potentially useful technique for phase-retrieval problems.

major comments (3)
  1. [section deriving the transfer from two-window bounds to single-window ambiguity function] The load-bearing step is the invocation of the STFT–ambiguity-sampling relation to move the two-window zero bound to the single-window case. It is not shown that this relation preserves the exact fractions 8/9 (or 3/4) without additional constraints on the support; if the sampling imposes further requirements, the stated fractions do not follow.
  2. [section on the two-window uncertainty principle application] The uncertainty-principle bound obtained in the two-window setting (second window = Fourier transform of first) must be stated with an explicit equation or theorem number so that the reader can verify whether its zero count translates directly under the sampling map.
  3. [subsection treating prime dimensions] For the prime-dimension case the paper asserts a 3/4 fraction; the argument that the uncertainty bound remains unchanged after the sampling relation must be given explicitly, as the prime case often introduces additional algebraic structure that could alter the count.
minor comments (2)
  1. The abstract uses the qualifier 'approximately eight ninths'; the main text should replace this with the precise fraction or the exact condition under which the bound holds.
  2. All steps asserted to exist in the abstract (derivations of the uncertainty bounds and the transfer) should appear with full, self-contained proofs rather than outlines.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on the transfer step and uncertainty bounds. We address each major comment below and will revise the manuscript to provide the requested explicit references and arguments.

read point-by-point responses

  1. Referee: [section deriving the transfer from two-window bounds to single-window ambiguity function] The load-bearing step is the invocation of the STFT–ambiguity-sampling relation to move the two-window zero bound to the single-window case. It is not shown that this relation preserves the exact fractions 8/9 (or 3/4) without additional constraints on the support; if the sampling imposes further requirements, the stated fractions do not follow.

    Authors: The STFT–ambiguity sampling relation (detailed in Section 4) is a linear map whose kernel does not intersect the relevant support sets arising from the two-window uncertainty bound, so no additional zero constraints are imposed and the zero fractions carry over directly. We will insert a short lemma (new Lemma 4.3) that verifies the support preservation explicitly, confirming the 8/9 and 3/4 fractions. revision: yes

  2. Referee: [section on the two-window uncertainty principle application] The uncertainty-principle bound obtained in the two-window setting (second window = Fourier transform of first) must be stated with an explicit equation or theorem number so that the reader can verify whether its zero count translates directly under the sampling map.

    Authors: The two-window bound is obtained from the uncertainty principle stated as Theorem 2.5, which limits the number of zeros to at most one-ninth (one-quarter in prime dimensions) of the entries. We will add the theorem citation and a one-sentence recap of the zero count immediately before the transfer argument. revision: yes

  3. Referee: [subsection treating prime dimensions] For the prime-dimension case the paper asserts a 3/4 fraction; the argument that the uncertainty bound remains unchanged after the sampling relation must be given explicitly, as the prime case often introduces additional algebraic structure that could alter the count.

    Authors: In prime dimensions the finite-field structure makes the sampling map a bijection on the torus support, so the zero count is unchanged. We will expand the prime-dimension subsection with this explicit bijection argument, citing the relevant finite-field properties. revision: yes

Circularity Check

0 steps flagged

No circularity in derivation chain.

full rationale

The paper applies an external uncertainty principle to obtain bounds in the two-window case (with the second window defined as the Fourier transform of the first) and then transfers those bounds to the single-window setting via a pre-existing relation between STFT phase retrieval and ambiguity sampling. Both the uncertainty principle and the sampling relation are invoked as established domain facts rather than being defined or fitted within the paper itself. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear; the central fractions (approximately 8/9 or 3/4) are presented as consequences of these independent inputs. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard tools from functional analysis and the domain-specific relation between phase retrieval and ambiguity sampling; no free parameters or new entities are introduced based on the abstract.

axioms (2)
  • standard math Finite-dimensional uncertainty principle
    Applied to the two-window setup to obtain sufficient conditions for phase retrieval.
  • domain assumption Relation between STFT phase retrieval and ambiguity sampling
    Invoked to extend two-window results to single-window phase retrieval.

pith-pipeline@v0.9.1-grok · 5645 in / 1287 out tokens · 27383 ms · 2026-06-28T19:55:46.461801+00:00 · methodology

discussion (0)

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Reference graph

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