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The alpha-discretization map is stable only for alpha in [0.5,1], and the paper argues no single alpha removes both amplitude and phase distortion.

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The alpha-approximation has stable range [0.5, 1], admits a 'hexagonal' integration interpretation, and the paper conjectures no single alpha minimizes both amplitude and phase distortion across all frequencies.

T0 review reviewed 2026-08-05 challenge →

load-bearing objection The stability analysis and hexagonal framing are solid, but the paper's headline exclusion hypothesis is ill-posed and fails on the trivial case H(s)=1. the 3 major comments →

arxiv 2509.02054 v2 pith:24THDJN6 submitted 2025-09-02 eess.SY cs.SY

Comprehensive Analysis and Exclusion Hypothesis of $\alpha$-Approximation Method for Discretizing Analog Systems

Shen Chen , Jisong Wang , Dejun Liu , Jiaxi Ying , Shuai Wang This is my paper
classification eess.SY cs.SY
keywords discretizationalpha-approximationhexagonal approximationbilinear transformationstability analysisdistortion analysisexclusion hypothesis
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The reading

The paper aims to establish three things about the alpha-approximation family of s-to-z substitutions. First, the map can be interpreted as numerical integration of the error function over one sampling interval, giving a hexagonal approximation in which alpha is the fraction of the interval weighted by the current error sample. Second, the map is stable exactly when alpha lies in [0.5,1], which is proved by requiring that the left half of the s-plane maps inside the unit circle. Third, a distortion study across several controllers suggests that no single alpha can minimize amplitude distortion and phase distortion simultaneously at all frequencies below Nyquist; this is stated as an exclusion hypothesis, supported by examples but not proved. If the hypothesis is right, choosing alpha is inherently a trade-off, and discretization quality cannot be improved by tuning this parameter alone.

Core claim

The authors prove that the alpha-approximation substitution s=(1/T)(z-1)/(alpha*z+1-alpha) preserves stability of a continuous-time system only when alpha>=0.5. The proof follows from setting the real part of s to be non-positive; the set of z values mapping to the left half-plane is a disk whose boundary crosses the real axis at z=1 and at a second point, and requiring that disk to stay inside the unit circle forces alpha>=0.5. The paper also re-derives the approximation as the hexagonal approximation: over one sampling interval, the error function is piecewise constant, equal to e(n-1) for the first (1-alpha)T seconds and e(n) for the remaining alpha*T seconds, so u(n)=u(n-1)+(1-alpha)*T*e

What carries the argument

The central object is the rational map s=(1/T)(z-1)/(alpha*z+1-alpha), a first-order s-to-z substitution whose endpoints are the backward-difference map and the bilinear map. The load-bearing piece of the stability argument is the circle in the z-plane: the preimage of the closed left half-plane is {z : (Re z - (1 - 1/(2 alpha)))^2 + (Im z)^2 <= (1/(2 alpha))^2}, and requiring this disk to lie inside the unit circle yields alpha>=0.5. The complementary interpretive machinery is the hexagonal approximation, which gives alpha a physical meaning as the weight on the current error sample in a one-step numerical integration of the error function.

Load-bearing premise

The exclusion hypothesis rests on an unformalized meaning of 'minimize amplitude and phase distortion simultaneously'; with that notion left undefined, the statement is not yet a precise mathematical claim, and the paper does not prove it.

What would settle it

Take a fixed stable plant and a fixed frequency, compute the amplitude error and phase error of the alpha-approximation as functions of alpha in [0,1], and find their minimizers. If the two minimizers coincide at any frequency in (0, f_nyquist), the exclusion hypothesis is false; a numerical sweep over alpha for the low-pass filter used in the paper would already test the strongest reading of the claim.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • If the stability proof is correct, any internally stable analog system discretized with alpha in [0.5,1] retains continuous-time stability, so the safe design region is known exactly.
  • The hexagonal interpretation turns alpha from a curve-fitting parameter into a physical weight on the current versus previous error sample, which can guide its choice.
  • For the cases studied, amplitude error is smallest around alpha=0.7 and phase error at alpha=0.5, so a designer must decide which distortion to prioritize.
  • If the exclusion hypothesis holds universally, searching over alpha alone cannot produce a discretization that is simultaneously best in amplitude and phase, pushing design toward frequency-selective or multi-objective methods.

Where Pith is reading between the lines

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

  • The exclusion hypothesis becomes a sharp theorem only after 'minimize simultaneously' is given a concrete meaning; under a weighted integrated-error objective, a single alpha could plausibly be optimal for a particular plant, so the unrestricted statement likely needs qualification.
  • Plotting the per-frequency minimizers alpha_A(f) and alpha_phi(f) would show where the two curves cross; those crossing frequencies are the natural targets for adaptive or scheduled discretization.
  • Re-running the circle proof for alpha outside [0,1], as the generalized bilinear transformation allows, suggests the stable range remains alpha>=0.5, so extending the parameter range does not add stable maps.
  • A direct frequency-domain test, checking whether the unit circle maps into the stable half-plane for each alpha, would verify the stability threshold more cheaply than Bode-plot examples.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper studies the alpha-approximation discretization mapping s = (1/T)(z-1)/(alpha*z + 1 - alpha) for continuous-to-discrete conversion. It derives the mapping from a first-order Taylor truncation of z = e^{sT}, identifies it as a special case of several known transformations (Euler, Tustin, Al-Alaoui, and the generalized bilinear transformation), and gives a geometric interpretation as numerical integration of a piecewise-constant error function, which the authors call the hexagonal approximation. Section 4 presents a stability analysis concluding that the mapping is stable for alpha in [0.5, 1]. Section 5 gives qualitative Bode-plot distortion comparisons for a low-pass filter, and Section 6 proposes an 'exclusion hypothesis' claiming that no single alpha can minimize both amplitude and phase distortion simultaneously for every frequency in (0, f_Nyquist). The conclusion lists the hypothesis as a main contribution and states that machine-learning verification is pending.

Significance. If the stability proof and geometric interpretation are correct, they provide a self-contained and useful tutorial treatment of a known result, and the physical interpretation of alpha as the percentage of backward-rectangular area is a nice pedagogical contribution. The table of relationships among existing discretization methods is also helpful. However, the paper's headline new contribution, the exclusion hypothesis, is neither formally stated nor proven, and is contradicted by a trivial internally stable system. The distortion analysis is anecdotal rather than systematic. The authors are honest that the hypothesis is unverified, but this does not make the claim acceptable as a main result. The strongest elements are the parameter-free stability derivation and the explicit relationships in Table 1, which deserve credit.

major comments (3)
  1. [Section 6] The Exclusion Hypothesis is not a well-formed mathematical statement and is false under the natural literal reading. 'Minimize amplitude distortion and phase distortion simultaneously' is never defined as an objective (pointwise Pareto optimality, weighted sum, minimax norm, or integrated error), and the quantifier structure of 'for every frequency f ... no single parameter alpha' is ambiguous. More concretely, for the internally stable system H(s)=K with K constant (e.g., K=1), the alpha-approximation gives H_d(z)=K for every alpha, so amplitude and phase distortion are identically zero at all frequencies. Hence alpha=0.5 (indeed every alpha in [0,1]) simultaneously minimizes both, contradicting the hypothesis. The statement needs a non-triviality condition (e.g., H not constant) and a formal definition of the distortion measures and of 'minimize'.
  2. [Sections 5 and 6] The distortion analysis is purely qualitative. No equations define amplitude distortion or phase distortion as functions of alpha and frequency, and no systematic search over alpha or over a class of systems is reported. The exclusion hypothesis is asserted on the basis of a small set of examples (low-pass filter, PI, PR, notch) and the authors explicitly postpone verification to future machine-learning work. For a journal contribution, the hypothesis should either be proved for a well-defined class of systems or be supported by a reproducible exhaustive computational search with a stated objective. As written, this is an anecdotal conjecture rather than a demonstrated result.
  3. [Section 4, Eq. (30)] The real-axis crossing points of the circle in Eq. (29) are miscomputed. Completing the square gives center 1 - 1/(2alpha) and radius 1/(2alpha), so the intersections with the real axis are sigma_{z1}=1 and sigma_{z2}=1 - 1/alpha, not -1/(2alpha) as stated. The subsequent condition sigma_{z2} >= -1 still yields alpha >= 0.5, so the final conclusion is correct, but the displayed equation is wrong and should be corrected.
minor comments (5)
  1. [Eq. (9)] The series in the denominator uses index k with m in the expression; it should be (-1)^m (s alpha T)^m / m!. The fraction is also typeset ambiguously and should be parenthesized.
  2. [Abstract and Introduction] Several grammatical and typographical issues: 'the error function We defined' should be a sentence; '0alpha in [0,1]' is malformed; and the subscript notation for alpha_p in Eq. (5) is inconsistent.
  3. [References] Reference [2] is a paper on digital twins and does not appear to support the claim that neural ODEs use bilinear discretization to preserve gradient flow. Please check the citation or replace it with an appropriate reference.
  4. [Section 2.2] The stability statement in Figure 1 is asserted graphically; the rigorous proof appears in Section 4. A cross-reference to Section 4 would help the reader.
  5. [Section 6] The notation 'f(0, fnyquist)' should be 'f in (0, f_Nyquist)'.

Circularity Check

0 steps flagged

No significant circularity: the stability analysis is self-contained and the exclusion hypothesis is an open conjecture, not a derived result.

full rationale

The paper's positive derivation chain is self-contained. The α-approximation is introduced in Eq. (4)/(10) and derived from z=e^{sT} via first-order Taylor truncation in Eqs. (8)-(9). The stability range α∈[0.5,1] follows from a direct algebraic inequality: substituting s=σ_s+jω_s into the transform gives Eq. (26), and the condition σ_s≤0 is rearranged into the disk equation (29), whose leftmost real intercept yields α≥0.5 in Eq. (32). No fitted parameter is involved; α is swept, not estimated from data. The mathematical-interpretation section is also non-circular: the piecewise error function in Eq. (21) is integrated to produce exactly the same transform, which is an interpretation rather than a circular prediction. The distortion analysis in §5 is empirical and the §6 exclusion hypothesis is explicitly presented as a conjecture, not as a proven consequence; the paper even states work on verification is ongoing. There are no author self-citations that carry a load-bearing argument. The main weakness is that the exclusion hypothesis is not formally defined ('minimize' lacks a concrete objective) and is supported only by four qualitative examples, but this is a completeness/validity concern, not circularity. One can even construct trivial counterexamples (e.g., H(s)=1 gives zero distortion for all α), but refutation is not circularity. Accordingly, the circularity score is low.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

The paper introduces no new physical entities or mediators. The 'hexagonal approximation' is a geometric label for the same first-order transformation. The main unstated reliance is on the first-order Taylor truncation defining the method and on the informal notion of simultaneous distortion minimization in the exclusion hypothesis.

axioms (4)
  • domain assumption z = e^{sT} is the reference mapping between s-domain and z-domain (Eq. 7, Section 2.1).
    All stability and distortion conclusions are relative to this mapping; alternative discretization criteria would change the analysis.
  • domain assumption Truncation to first order in sT for both numerator and denominator (Eq. 9) defines the alpha-approximation; the analysis concerns this first-order method, not the exact exponential mapping.
    The truncation error is not quantified, but it is the definition of the method under study.
  • standard math For discrete systems, stability requires poles inside the unit circle, so the left half-plane of s must map inside the unit circle.
    Used in Section 4 to derive the alpha >= 0.5 condition.
  • ad hoc to paper The error function e(t) = du/dt and the piecewise-constant reconstruction in Eq. (21) are assumed for the geometric interpretation.
    This reconstruction is chosen to reproduce the alpha-approximation formula; it is an interpretation, not a measured physical law.

reviewed 2026-08-05 · how reviews work

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Cite this review

Pith. "Pith review of Comprehensive Analysis and Exclusion Hypothesis of $\alpha$-Approximation Method for Discretizing Analog Systems." pith.science (2026). https://pith.science/paper/24THDJN6

@misc{pith2026250902054,
  author       = {Pith},
  title        = {Pith review of: Comprehensive Analysis and Exclusion Hypothesis of $\alpha$-Approximation Method for Discretizing Analog Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/24THDJN6}},
  note         = {Machine review of arXiv:2509.02054}
}
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read the original abstract

A popular method for designing digital models is transforming the transfer function of the corresponding analog models from continuous domain (s-domain) into discrete domain (z-domain) using the s-to-z transformation. The alpha-approximation is a generalized form of these transformations. When alpha is set to 0.5, the result is the well-known Tustin transformation or bi-linear transformation. In this paper, we provided a comprehensive analysis of the alpha-approximation method, including mathematical interpretation, stability analysis and distortion analysis. Through mathematical interpretation, we revealed that it can be derived by numerically integrating the error function We defined this as the hexagonal approximation. We demonstrated that the stable range of alpha was [0.5, 1] by doing stability analysis. Through distortion analysis, we found that minimizing amplitude and phase distortion simultaneously seemed impossible by regulating alpha alone. Finally, We proposed an exclusion hypothesis hypothesizing that there is no single parameter alpha to minimize the amplitude distortion and phase distortion simultaneously across all frequency points within the Nyquist frequency range. This paper demonstrates that designing parameter alpha involves balancing amplitude and phase distortion.

Figures

Figures reproduced from arXiv: 2509.02054 by Dejun Liu, Jiaxi Ying, Jisong Wang, Shen Chen, Shuai Wang.

Figure 1
Figure 1. Figure 1: shows the mapping of the s-plane to the z-plane. Ac￾cording to the theory of stability of discrete systems, the map￾ping is stable unless the left half of the s-plane is mapped into the unit circle of the z-plane [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (a)-(b): Rectangular approximation (Euler), (c)-(d): Trapezoidal approximation (Tustin), (e)-(f): Hexagonal approx￾imation (α-approximation) 6. Exclusion Hypothesis Based on our research into several cases, including low-pass filter, the proportional-integral controller, the proportional￾resonant controller, and the notch filter, we found that it was impossible to minimize the amplitude distortion and phas… view at source ↗
Figure 3
Figure 3. Figure 3: Amplitude and phase distortion with different α [12] G. Zhang, X. Chen, and T. Chen, “Digital redesign via the gener￾alised bilinear transformation,” International Journal of Control, vol. 82, no. 4, pp. 741–754, 2009. [13] G. Raayoni, S. Gottlieb, Y. Karni, Y. Fest, Y. Manor, D. Goldin, T. Michaeli, and I. Lior, “Generating conjectures on fundamen￾tal constants with the ramanujan machine,” Nature, vol. 59… view at source ↗

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This paper was first reviewed by deepseek-v4-flash on August 5, 2026.