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arxiv: 2602.17471 · v2 · pith:JDJ4HRFSnew · submitted 2026-02-19 · 📡 eess.SP

Analytical Derivation of Quantization Error in Threshold Level Quantizers Using Bipolar PFM

Ricardo Carrero , Ruben Garvi , Luis Hernandez This is my paper

Pith reviewed 2026-05-21 13:04 UTC · model grok-4.3

classification 📡 eess.SP
keywords quantization errorpulse frequency modulationspectral analysisuniform quantizerlevel crossing ADCFourier transformanalog-to-digital conversion
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The pith

A uniform threshold quantizer is equivalent to a bipolar pulse frequency modulation system, so its quantization error spectrum follows from PFM Fourier properties.

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

The paper shows that modeling a uniform quantizer as a bipolar pulse-frequency-modulation system makes the spectrum of the quantized signal available in closed form. This replaces the usual statistical treatment of quantization noise with a deterministic description based on modulation sidebands. A reader would care because the same model directly supplies performance predictions for level-crossing ADCs that use zero-order-hold reconstruction. The equivalence therefore turns an abstract noise source into a concrete spectral object whose shape can be calculated rather than simulated.

Core claim

By establishing the exact equivalence of a uniform threshold quantizer to a bipolar pulse-frequency-modulation system, the Fourier transform of the quantized output is obtained by direct application of known PFM transform properties, yielding an analytical expression for the spectrum of the quantization error.

What carries the argument

The exact mapping of each threshold crossing to a bipolar PFM pulse train, which transfers the closed-form Fourier analysis of PFM directly onto the quantizer output.

If this is right

  • The quantization error spectrum can be written in closed form without invoking statistical assumptions.
  • Performance of level-crossing ADCs with zero-order-hold interpolators can be estimated analytically.
  • The frequency distribution of quantization noise becomes visible as modulation sidebands rather than a flat floor.

Where Pith is reading between the lines

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

  • The same equivalence could be tested for non-uniform or adaptive quantizers by constructing analogous modulation mappings.
  • The derived spectra might be used to predict how quantization noise interacts with subsequent digital filters in a signal chain.
  • Hardware measurements on asynchronous ADCs could directly validate the sideband structure predicted by the model.

Load-bearing premise

A uniform threshold quantizer produces exactly the same pulse sequence as a bipolar pulse-frequency modulator for any input waveform.

What would settle it

Compute the measured Fourier transform of a quantized sine wave and compare it with the spectrum predicted by the PFM model; a mismatch in the locations or amplitudes of the noise sidebands would show the equivalence does not hold.

Figures

Figures reproduced from arXiv: 2602.17471 by Luis Hernandez, Ricardo Carrero, Ruben Garvi.

Figure 1
Figure 1. Figure 1: a shows an integrator that integrates signal [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: a) Uniform quantizer. b) Differential-Integral model. [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: a) Single polarity PFM b) Bipolar PFM v(t) = vm + B · cos(2πfxt) (4) Then, signal d(t) can be expanded into a trigonometric series as follows: d(t) = f0 + B · cos(2πfxt) + m(t), m(t) = 2f0 · ∆ · ∑∞ q=1 ∑∞ r=−∞ Jr ( q · B fx · ∆ ) · ( 1 + rfx qf0 ) cos(2π(qf0 + rfx)t) (5) where f0 corresponds to the rest frequency of the PFM modulator, which is the frequency produced when the DC value vm is applied, and Jr(… view at source ↗
Figure 3
Figure 3. Figure 3: a) Input sinusoid. b) Input derivative. c) bipolar PFM. d) [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: b shows the equivalent result computed analytically [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: a) Simulated spectrum. b) Analytical spectrum. c) Sideband [PITH_FULL_IMAGE:figures/full_fig_p005_6.png] view at source ↗
read the original abstract

Uniform quantization is a topic that has been extensively studied. However and although an analytical description of quantization noise has been proposed, most descriptions of the spectral properties of quantization error resort to statistical descriptions. In this paper, we show how the spectrum of a quantized signal can be expressed using pulse frequency modulation. We first establish the equivalence of a uniform quantizer with a system based on the bipolar pulse frequency modulation and we define afterwards the Fourier transform of the quantized signal using pulse frequency modulation properties. This model brings a more intuitive understanding of the spectral structure of quantization noise and complements prior research in the topic. The results of the paper can be directly applied to level crossing ADCs with zero-order-hold interpolators, giving an accurate estimation of their performance.

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 manuscript claims to derive an analytical expression for the spectrum of quantization error in threshold-level (uniform) quantizers by first establishing an exact equivalence between a uniform quantizer and a bipolar pulse-frequency-modulation (PFM) system, then using PFM properties to define the Fourier transform of the quantized signal. The resulting model is presented as providing an intuitive view of quantization-noise spectral structure that complements statistical treatments and is directly applicable to level-crossing ADCs employing zero-order-hold reconstruction.

Significance. If the asserted equivalence is shown to hold rigorously for the relevant input class and reconstruction, the work would supply a deterministic, closed-form route to the quantization-noise spectrum that avoids reliance on statistical assumptions. This could be useful for performance analysis of level-crossing ADCs and might offer new design insights by linking quantization to PFM spectral properties.

major comments (2)
  1. [Section establishing the quantizer-PFM equivalence] The central modeling step asserts an exact equivalence between a standard memoryless uniform threshold quantizer and a bipolar PFM system. Because this identity is load-bearing for the subsequent Fourier-transform derivation, the manuscript must supply an explicit proof (or counter-example check) that the two systems produce identical outputs for the same continuous-time input, together with any necessary restrictions on input bandwidth, amplitude range, or reconstruction method. The abstract provides no such conditions or supporting equations.
  2. [Section deriving the Fourier transform] Once the equivalence is accepted, the derivation of the Fourier transform of the quantized signal via PFM properties must be shown to be free of additional modeling assumptions (e.g., specific pulse shapes or hold intervals) that are not part of the original uniform quantizer definition. Any such hidden assumptions should be stated explicitly and their effect on the claimed spectrum quantified.
minor comments (2)
  1. [Application section] The abstract states that the results 'can be directly applied' to level-crossing ADCs with zero-order-hold interpolators; the manuscript should include a brief numerical or analytical example confirming that the derived spectrum matches observed behavior for at least one such architecture.
  2. [Notation and definitions] Notation for the bipolar PFM parameters (pulse rate, amplitude, etc.) should be introduced consistently and cross-referenced to the quantizer step size and threshold levels.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review. The comments highlight the need for greater explicitness in the central equivalence and derivation steps, which we address below by committing to targeted revisions that strengthen the rigor without altering the core contribution.

read point-by-point responses

  1. Referee: [Section establishing the quantizer-PFM equivalence] The central modeling step asserts an exact equivalence between a standard memoryless uniform threshold quantizer and a bipolar PFM system. Because this identity is load-bearing for the subsequent Fourier-transform derivation, the manuscript must supply an explicit proof (or counter-example check) that the two systems produce identical outputs for the same continuous-time input, together with any necessary restrictions on input bandwidth, amplitude range, or reconstruction method. The abstract provides no such conditions or supporting equations.

    Authors: We agree that the equivalence requires an explicit, self-contained proof to be fully rigorous. The manuscript derives the equivalence in Section II by showing that the output pulse train of the bipolar PFM matches the quantized levels under zero-order-hold reconstruction for bandlimited inputs. In the revision we will insert a dedicated lemma with a step-by-step proof, including the precise restrictions (input bandwidth less than half the nominal pulse rate and amplitude within the quantizer range) and a brief counter-example verification for out-of-range signals. The abstract will be updated to mention these conditions concisely. revision: yes

  2. Referee: [Section deriving the Fourier transform] Once the equivalence is accepted, the derivation of the Fourier transform of the quantized signal via PFM properties must be shown to be free of additional modeling assumptions (e.g., specific pulse shapes or hold intervals) that are not part of the original uniform quantizer definition. Any such hidden assumptions should be stated explicitly and their effect on the claimed spectrum quantified.

    Authors: We will revise the Fourier-transform section to list every modeling choice explicitly: the representation uses ideal rectangular pulses of fixed width corresponding to the zero-order-hold interval, and the spectrum is obtained via the known PFM Fourier series. We will add a short paragraph quantifying the effect of the finite pulse width on the high-frequency roll-off and confirm that the in-band quantization-noise spectrum remains identical to the memoryless uniform quantizer for the stated input class. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation rests on asserted modeling equivalence

full rationale

The paper states it first establishes the equivalence of a uniform quantizer with a bipolar PFM system and then defines the Fourier transform using PFM properties. This sequence is presented as a modeling choice that complements prior research rather than a self-referential loop. No equations, fitted parameters, self-citations, or uniqueness theorems are quoted that reduce the central claim to its own inputs by construction. The abstract and reader's summary indicate an independent modeling step, not a renaming or ansatz smuggled via citation. Per hard rules, absent specific quotes exhibiting reduction (e.g., Eq. X defined as Eq. Y), circularity is not flagged. This is the expected honest non-finding for a paper whose core step is an external modeling assumption.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The derivation rests on the unproven equivalence between uniform quantization and bipolar PFM; no free parameters, additional axioms, or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption A uniform threshold quantizer is exactly equivalent to a bipolar pulse-frequency-modulation system.
    This equivalence is stated as the first modeling step before the Fourier-transform definition.

pith-pipeline@v0.9.0 · 5653 in / 1202 out tokens · 25725 ms · 2026-05-21T13:04:37.133336+00:00 · methodology

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Reference graph

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