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arxiv: 2505.05388 · v3 · submitted 2025-05-08 · 📡 eess.SP

On Multiangle Discrete Fractional Periodic Transforms

Christian Oswald , Franz Pernkopf This is my paper

Pith reviewed 2026-05-22 15:47 UTC · model grok-4.3

classification 📡 eess.SP
keywords multiangle discrete fractional Fourier transformM-periodic transformsdiscrete Fourier transformcomputational efficiencysymmetry exploitationtime-frequency analysissignal processingfractional transforms
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The pith

The multiangle centered discrete fractional Fourier transform generalizes to arbitrary M-periodic cases, covering standard discrete transforms while halving the FFT count through symmetry exploitation.

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

This paper extends the multiangle centered discrete fractional Fourier transform to general M-periodic transforms. The generalization includes the discrete Fourier, sine, cosine, Hadamard, and Hartley transforms among others. It also introduces a multiangle standard discrete fractional Fourier transform and uses symmetries present in both versions to reduce the number of required FFT operations by half. The resulting efficiency supports applications where computational resources are limited.

Core claim

The MA-CDFRFT generalizes to M-periodic transforms that encompass standard discrete Fourier, sine, cosine, Hadamard, and Hartley transforms, and the symmetries inherent to the MA-CDFRFT and the new MA-DFRFT halve the number of FFTs needed to compute them.

What carries the argument

Generalization of the multiangle centered discrete fractional Fourier transform to M-periodic transforms, together with symmetry-based reduction of FFT operations in both the centered and standard variants.

If this is right

  • Standard discrete transforms including DFT, DST, DCT, Hadamard, and Hartley become computable within the same efficient framework.
  • The halved FFT requirement applies equally to the newly defined multiangle standard discrete fractional Fourier transform.
  • Resource-constrained devices gain practical access to these time-frequency tools without increased computational overhead.

Where Pith is reading between the lines

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

  • Real-time signal processing pipelines could adopt the reduced-FFT versions for faster execution on embedded hardware.
  • The symmetry reduction technique may extend to other fractional or periodic operators beyond those treated here.
  • Parameter choices for the generalized transforms could be tuned to optimize specific applications like filtering or feature extraction.

Load-bearing premise

The symmetries that halve the FFT count for the original MA-CDFRFT continue to apply without change or loss of correctness when the method is extended to arbitrary M-periodic transforms.

What would settle it

A direct implementation count for the generalized transform on a non-Fourier M-periodic case such as the Hadamard transform that shows the number of FFT calls remains unchanged from a baseline method.

read the original abstract

The efficient multiangle centered discrete fractional Fourier transform (MA-CDFRFT) [1] has proven to be a useful tool for time-frequency analysis; in this paper, we generalize the MA-CDFRFT to general M -periodic transforms, which, among others, include the standard discrete Fourier, discrete sine, discrete cosine, Hadamard and discrete Hartley transform. Furthermore, we exploit the symmetries inherent to the MA-CDFRFT and our novel multiangle standard discrete fractional Fourier transform (MA-DFRFT) to halve the number of FFTs needed to compute these transforms, which paves the way for applications in resource-constrained environments.

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

2 major / 2 minor

Summary. The paper generalizes the multiangle centered discrete fractional Fourier transform (MA-CDFRFT) to general M-periodic transforms, which include the standard discrete Fourier, discrete sine, discrete cosine, Hadamard, and discrete Hartley transforms. It introduces the multiangle standard discrete fractional Fourier transform (MA-DFRFT) and exploits symmetries from both the MA-CDFRFT and MA-DFRFT to halve the number of FFTs needed for computation.

Significance. If the generalization is rigorously derived and the symmetry-based FFT reduction holds uniformly, the work offers a unified efficient framework for fractional versions of multiple classical orthogonal transforms. This could be valuable for time-frequency analysis in resource-constrained signal processing applications, building directly on prior MA-CDFRFT results with potential for reproducible implementations.

major comments (2)
  1. [Generalization to M-periodic transforms and symmetry exploitation sections] The central claim that symmetries enabling FFT halving in the MA-CDFRFT transfer without modification to arbitrary M-periodic cases (including DCT, DST, Hadamard, and Hartley) is load-bearing for the efficiency result. The eigenstructures of these transforms differ in sign patterns and periodicity from the centered fractional Fourier kernel, so explicit verification or re-derivation of the conjugation/sign-flip identities is needed for each family rather than assuming uniform applicability.
  2. [MA-DFRFT definition and complexity analysis] The manuscript states that the MA-DFRFT symmetries also contribute to halving the FFT count across the listed transforms, but without a concrete breakdown (e.g., via equations showing how the fractional angle parameters map under each basis), it is unclear whether the reduction is parameter-free or requires case-specific adjustments.
minor comments (2)
  1. [Abstract] The abstract lists specific transforms after 'among others'; clarify whether the M-periodic generalization applies to all M-periodic orthogonal bases or is demonstrated only for the enumerated ones.
  2. Ensure consistent notation for the fractional angle parameters across the MA-CDFRFT, MA-DFRFT, and generalized cases to avoid ambiguity in the symmetry relations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and constructive feedback on our generalization of the MA-CDFRFT to M-periodic transforms and the associated efficiency improvements. We address each major comment below and will incorporate clarifications in the revised manuscript.

read point-by-point responses

  1. Referee: [Generalization to M-periodic transforms and symmetry exploitation sections] The central claim that symmetries enabling FFT halving in the MA-CDFRFT transfer without modification to arbitrary M-periodic cases (including DCT, DST, Hadamard, and Hartley) is load-bearing for the efficiency result. The eigenstructures of these transforms differ in sign patterns and periodicity from the centered fractional Fourier kernel, so explicit verification or re-derivation of the conjugation/sign-flip identities is needed for each family rather than assuming uniform applicability.

    Authors: We agree that eigenstructures vary across M-periodic families. Our generalization defines the multiangle transforms at the level of the general M-periodic kernel, deriving the conjugation and sign-flip identities directly from the M-periodicity condition. This framework ensures the FFT-halving symmetries hold uniformly for the listed transforms without separate re-derivations. To improve clarity and address the referee's concern, we will add a new subsection with explicit specialization of the general identities to DCT, DST, Hadamard, and Hartley cases. revision: yes

  2. Referee: [MA-DFRFT definition and complexity analysis] The manuscript states that the MA-DFRFT symmetries also contribute to halving the FFT count across the listed transforms, but without a concrete breakdown (e.g., via equations showing how the fractional angle parameters map under each basis), it is unclear whether the reduction is parameter-free or requires case-specific adjustments.

    Authors: The MA-DFRFT symmetries are derived analogously to the centered case within the same M-periodic framework, with fractional angle parameters mapping uniformly under the basis for all listed transforms; the FFT reduction is therefore parameter-free once the general form is fixed. We will revise the complexity analysis to include explicit equations illustrating the angle mappings for representative bases (e.g., DFT and DCT) and confirm the uniform applicability. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation builds on external prior result

full rationale

The paper cites the MA-CDFRFT from reference [1] as the foundation and states that it generalizes this to M-periodic transforms while exploiting symmetries from both the original and the new MA-DFRFT. No equations or claims in the provided text reduce a derived quantity to a fitted parameter or self-defined input by construction. The central claims (generalization and FFT halving) rest on explicit extension of prior work rather than re-deriving the base result from the paper's own outputs. This is a standard, non-circular citation of independent prior results.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review performed from abstract only; no explicit free parameters, axioms or invented entities are identifiable. Full manuscript would be required to audit any implicit assumptions about periodicity or symmetry preservation.

pith-pipeline@v0.9.0 · 5626 in / 1120 out tokens · 46803 ms · 2026-05-22T15:47:04.811156+00:00 · methodology

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Reference graph

Works this paper leans on

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