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REVIEW 4 major objections 5 minor 30 references

Pseudo-random sequences for low-cost operando impedance measurements of Li-ion batteries

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Ternary pseudo-random sequences that are eigenvectors of the discrete Fourier transform recover battery impedance from a single operando measurement during charging, cancelling drift and transient contamination with a simple averaging…

desk verdict The K+/K- pairing trick for drift/transient suppression is real and the algebra checks out, but the unquantified low-frequency interpolation error makes the experimental claim thinner than the paper suggests. read the letter →

arxiv 2506.07519 v1 pith:OLMMOQBJ submitted 2025-06-09 eess.SY cs.SY

Jussi Sihvo , Noël Hallemans , Ai Hui Tan , David A. Howey , Stephen. R. Duncan , Tomi Roinila This is my paper · ORCID
classification eess.SYcs.SY
keywords operandoimpedancespectroscopylithium-ionbatterypseudo-randomsequencesquadratic-residueternarysequencedirect-synthesisDFTeigenvectordriftandtransientsuppressionmanagementsystem
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Operando impedance spectroscopy could monitor batteries while they charge or drive, but conventional measurements require steady state and heavy processing to remove voltage drift and transients. This paper claims that two three-level pseudo-random sequences — the quadratic-residue ternary (QRT) sequence and the direct-synthesis ternary (DST) sequence — are eigenvectors of the discrete Fourier transform at the harmonics they excite, so their spectra satisfy $U(k_+) = -U(k_-)$. The paper exploits that sign complement to split one measured spectrum into two virtual sign-opposite experiments whose average cancels the drift-plus-transient term, yielding the true impedance $Z(\omega)$ from a single 6.7-second excitation via a lightweight formula (Eq. (40)). The claim is backed by a derivation, a simulation with an equivalent-circuit battery model, and operando measurements on a commercial NMC 18650 cell during 1C fast charging. If correct, this makes operando EIS cheap and fast enough to embed in battery management systems.

What carries the argument

The DFT-eigenvector property of ternary sequences. For the QRT sequence (a Legendre-symbol ternary sequence of prime length) and the DST sequence (the product of a fixed six-sample special sequence with a repeated QRT basic sequence), the normalized DFT maps the sequence to a scalar multiple of itself at the excited harmonics, so the spectral values at the interleaved sets $K_+$ and $K_-$ are exact opposites, $U(k_+) = -U(k_-)$. Under zero-order-hold reconstruction this becomes the approximation $I(\omega_{k_+}) \approx -I(\omega_{k_-})$ for close harmonics, and that is what lets Eq. (40) reconstruct the true impedance from a single experiment: it reproduces the two-experiment cancellation of drift $V_0(k)$ and transient $L(\omega_k)$ using only a DFT, a division, and linear interpolation.

What would settle it

Add a narrow resonance to the true impedance of the paper's own simulation model, placed between two adjacent excited harmonics near 10 Hz: if Eq. (40) misses the feature while the two-experiment formula (37) captures it, the interpolation assumption is the failure point. Alternatively, on the hardware setup, apply both formulas during a deliberate charging-current step at fixed SOC and look for spectral differences near the step frequency, which would be a direct measurement of the interpolation error.

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Extended reading notes

Core claim

The paper's central claim is that QRT and DST sequences are eigenvectors of the DFT matrix at every harmonic they excite, giving the exact sign-complement property $U(k_+) = -U(k_-)$ on the interleaved index sets $K_+$ and $K_-$. Under zero-order-hold reconstruction the continuous excitation satisfies the approximation $I(\omega_{k_+}) \approx -I(\omega_{k_-})$ for close harmonics. Starting from the two-experiment identity (37), the paper treats the measured ratio $V(k)/I(k)$ at the $K_+$ indices as one component $Z_+(\omega_k)$ and at the $K_-$ indices as the other $Z_-(\omega_k)$, interpolates each component across the opposite index set, and forms the average in Eq. (40). Because both components carry the same drift and transient contamination $V_0(k) + L(\omega_k)$, that average cancels the contamination and leaves the period-average impedance $Z(\omega)$. The contribution is the full chain: sequence definitions, the DFT-eigenvector proof, the measurement setup, the estimator, simulation validation, and operando fast-charging experiments.

Load-bearing premise

The reconstruction assumes the impedance, drift, transient, and slow operating current are smooth across neighbouring excited harmonics, so linear interpolation of the $Z_+$ and $Z_-$ components in Eq. (41) is accurate; if the true spectrum changes sharply between two adjacent excited harmonics, Eq. (40) carries an interpolation error the paper does not quantify.

Editorial extensions

If this is right

  • Operando impedance becomes a one-shot measurement: a single 6.668 s DST burst superimposed on a 1C charging current yields the spectrum from about 1.05 Hz to 1 kHz, with no steady-state preconditioning or separate drift model.
  • The processing load — a DFT, a division, and linear interpolation — plus the three-level current excitation fits the hardware and memory budget of typical BMS processors, which is the cost barrier the paper targets.
  • Because DST suppresses second- and third-order harmonics, the same burst exposes nonlinear distortion levels at $V(2k)$ and $V(3k)$ while still giving the best linear approximation of the impedance.
  • The estimator corrects for an arbitrary slow operating current $I_0(k)$ (Eq. (40)), so it applies to time-varying profiles such as EV driving, not only constant-current charging.
  • Measured operando spectra differ systematically from steady-state spectra (the 80% SOC semicircle is smaller mid-charge), so the method opens the dynamic operating regime to state-of-health monitoring and fast-charging model parameterization.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A self-check the paper does not include: run two sign-opposite bursts at one operating point and compare the two-experiment estimator (37) against the one-experiment estimator (40); the gap directly measures the interpolation error on that cell, which the paper leaves unbounded.
  • The cancellation mechanism is not tied to the specific QRT/DST construction — any perturbation whose spectrum flips sign between two interleaved index sets could feed the same Eq. (40), so the estimator could transfer to the fuel-cell and power-converter diagnostic settings already cited in the paper.
  • The interpolation bounds (43)-(44) are what cost the method its lowest frequencies; chaining bursts of different periods could push the floor below 0.15 Hz, but that would relax the single-period time-invariance assumption the whole derivation rests on.
  • Because a burst lasts 6.7 s and rides on current the battery is already drawing, a BMS could schedule one at every charging step at near-zero cost and use the resulting impedance stream as a continuous internal-state trend — an operational payoff the authors only gesture at.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper presents a method for operando battery impedance measurements using quadratic-residue ternary (QRT) and direct-synthesis ternary (DST) sequences. The key idea is that these sequences are eigenvectors of the DFT matrix at the excited harmonics, so that the excitation spectrum satisfies U(k+) = -U(k-). The authors show that this property can be used to separate the measured voltage-to-current ratio into two components Z+ and Z-, taken from interleaved harmonic sets, and then to reconstruct the true impedance Z from a single excitation period through an interpolation-based formula, Eq. (40). The derivation of the algebraic reconstruction, Eq. (37), for two experiments with opposite excitations is correct. The method is validated in simulation on an equivalent-circuit battery model with a smooth charging ramp and demonstrated experimentally during 1C fast charging of a commercial NMC cell, yielding spectra at several SOC levels.

Significance. If the method works as claimed, it is practically attractive: it needs only one short ternary perturbation burst, low-cost hardware, and simple DFT-based processing, which is feasible for BMS embedded implementation. The paper contains a genuine theoretical contribution in the Appendix, proving the DFT-eigenvector property of the DST sequence, and the algebraic derivation of Eq. (37) is sound. However, the central claim of drift and transient suppression in a single experiment rests on an unquantified interpolation regularity assumption, and the experimental section provides no reference measurement or error analysis. These gaps currently limit the strength of the paper's conclusions, but they are addressable within the manuscript's scope.

major comments (4)
  1. [Section V.B, Eq. (42)] The interpolation step is the load-bearing element of the single-experiment reconstruction, but its error is never quantified. Equation (42) shows that each interpolated quantity Z±(ωk) equals Z(ωk) plus the contamination term [V0(k)+L(ωk)]/[I0(k)±Iexc(k)], so the interpolation must be accurate for the contamination term, not just for Z itself. The paper states that Z, V0, L, I0, and Iexc are smooth over the excited harmonics and concedes that 'the interpolation error will typically be worst at low frequencies,' but it provides no bound, no sensitivity analysis, and no numerical experiment that varies the regularity of the drift or transient. Without such an analysis, the claim that the method 'suppresses' drifts and transients is contingent on an unverified regularity condition.
  2. [Section VII] The experimental validation has no reference baseline. Figure 7 reports 20 operando impedance spectra during fast charging, but there is no comparison to a conventional steady-state EIS measurement at the same SOC and temperature, and no repeated measurements or error bars are shown. Because the simulation uses a linear ramp and a smooth OCV curve, the interpolation-induced bias that the authors themselves say is worst at low frequencies would not be detected. The paper should add a reference-based validation, for example by interleaving steady-state EIS at matching operating points or by comparing against a known load, and should report the repeatability of the operando spectra.
  3. [Section V.A, Eqs. (36)-(39) and Section III, Eq. (25)] The derivation of the two-experiment formula (37) assumes the transient term L(ωk) is identical for opposite excitations; the authors note that end conditions may differ but do not bound this difference. For the single-experiment method, the analogous assumption is that the excitation spectra satisfy Iexc(k+) ≈ -Iexc(k-), but Eq. (25) shows this is only approximate because of the zero-order-hold factor. The text says the ZOH effect can be 'taken into account for increased accuracy' but does not show how. The magnitude of the error introduced by these approximations is not assessed, and this is central to the claimed accuracy of the single-period reconstruction.
  4. [Section VI and V.B] No code or data are provided, and the reconstruction algorithm is not fully specified. The interpolation method is described only as 'e.g. linear interpolation' in Section V.B, and the simulation section does not state how many Monte-Carlo realizations were used for the noise or how the interpolation was configured. For a methods paper, the absence of an explicit algorithm or pseudo-code makes it difficult to reproduce the results, and the single simulation with one noise realization is insufficient to establish statistical behavior. Please provide the code or a detailed pseudocode for the reconstruction, and add a repeatability analysis of the simulation.
minor comments (5)
  1. [Abstract and Section I] The phrase 'It's low-cost hardware requirements' should be 'Its low-cost hardware requirements'.
  2. [Section V.B] The sentence beginning 'Interpolation is reasonable when...' lists the smoothness assumptions but does not define a quantitative measure of smoothness; consider adding a metric such as the maximum frequency gap or a Lipschitz constant on the contamination term.
  3. [Section VII] The sentence 'The current of 1C is twice the standard charging current of the cell which was considered appropriate...' is awkward and would be clearer as two sentences; also, the paper does not report cell temperature during the fast-charging experiment, which is a relevant variable for impedance interpretation.
  4. [References] Reference [19] is listed as 'Early Access March 2025' without volume or article number; please provide the final citation if available.
  5. [Table I] The entry 'N 1.0002e6' is hard to read; format it as '1.0002×10^6'.

Circularity Check

0 steps flagged · score 1.0 of 10

Core reconstruction is algebraically self-contained; only minor non-load-bearing self-citations appear in motivating and concluding text.

full rationale

No load-bearing circular step was found. Equation (37) is an exact algebraic inversion of Eq. (36): substituting Z± = Z + (V0+L)/(I0±Iexc) into (37) cancels the V0+L term identically. The single-experiment version, Eq. (40), uses the sign-alternating property (20) to supply the missing Z± components by interpolation, as stated in Eqs. (41)-(42). This relies on an explicit smoothness assumption, which the paper discloses in Section V.B ('Interpolation is reasonable when the impedance Z(ω), the drift signal V0(k), the transient L(ω), slow excitation I0(k), and Iexc(k) are smooth over the excited frequencies'), and it is a correctness risk rather than a hidden fit. The DST eigenvector property (16) is proved in the Appendix from the sequence construction (12)-(15), and the QRT property (11) follows directly from the eigenvalue equation (8); neither is imported solely by citation. The simulation uses an independently parametrized equivalent-circuit model, so the validation is not a re-statement of the method's inputs. The only self-citations, [18] and [19], are used for background motivation and for the non-central assertion of applicability to time-varying currents; they are not needed to establish Eq. (40). The acknowledged low-frequency interpolation error is an assumption limitation, not a circular step.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The method relies on standard DFT eigenvector properties and on linearity and smoothness assumptions for interpolation. The simulation parameters are validation inputs, not fitted degrees of freedom, and no new physical entities are introduced.

free parameters (4)
  • Equivalent circuit parameters R0, R1, C1, R2, C2 = 5 mΩ, 8 mΩ, 0.1 F, 20 mΩ, 1 F
    Chosen for the simulation validation in Section VI; not fitted to an experiment and not part of the method itself.
  • Voltage and current noise standard deviations = σv = 0.5 mV, σi = 0.5 mA
    Chosen for the simulation noise level in Section VI; not fitted to the target system.
  • Slow charging current profile i0(t) = 2.5 - t/(2 Tp) A
    Chosen for the simulation to emulate a ramped charging current; in practice i0 is the arbitrary user-defined operating signal.
  • Measurement design parameters (Np, fzoh, fs, excitation amplitude) = Np = 10002, fzoh = 1.5 kHz, fs = 150 kHz, C = 1 A (2 A for Fig. 4)
    Design choices in Table I for the experimental setup; they set the frequency band and SNR but are not fitted to the measured outcome.
assumptions (6)
  • standard math The QRT sequence is an eigenvector of the normalized DFT matrix (Eq. 4).
    Known property from [24] and used to derive the K+/K- sign-flip relation (Eq. 11).
  • standard math The DST sequence satisfies UDST(k) = λDST uDST(k) for k in K and zero elsewhere (Eq. 16).
    Proved in the appendix using the DFT multiplication-convolution property and the QRT eigenvector property.
  • domain assumption The battery is approximately linear and time-invariant within one excitation period when the perturbation amplitude is small (Eqs. 31-32).
    Standard small-signal linearization for impedance measurements; the paper relies on it to write the output as a convolution with a time-invariant impedance.
  • domain assumption The transient term L(ωk) is the same for the plus and minus excitations, with initial-condition effects dominating end-condition effects (after Eq. 36).
    Needed for the two-experiment cancellation in Eq. (37); the paper states that the effect of the excitation sign on end conditions is negligible.
  • domain assumption The impedance, drift, transient, slow current, and excitation spectrum are smooth enough over the excited harmonics that linear interpolation of Z± and tilde-I is accurate.
    This is the load-bearing interpolation assumption in Eq. (41), discussed qualitatively in Section V.B but not quantified with error bounds.
  • domain assumption Zero-order hold effects are negligible or correctable so that Iexc(k+) is approximately -Iexc(k-) for nearby harmonics (Eqs. 25 and 39).
    The paper notes the exact continuous-spectrum relation requires the ZOH ratio, but uses the approximation for nearby harmonic pairs.

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Cite this review

Pith. "Pith review of Pseudo-random sequences for low-cost operando impedance measurements of Li-ion batteries." pith.science (2026). https://pith.science/paper/OLMMOQBJ

@misc{pith2026250607519,
  author       = {Pith},
  title        = {Pith review of: Pseudo-random sequences for low-cost operando impedance measurements of Li-ion batteries},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OLMMOQBJ}},
  note         = {Machine review of arXiv:2506.07519}
}
read the original abstract

Operando impedance measurements are promising for monitoring batteries in the field. In this work, we present pseudo-random sequences for low-cost operando battery impedance measurements. The quadratic-residue ternary sequence and direct-synthesis ternary sequence exhibit specific properties related to eigenvectors of the discrete Fourier transform matrix that allow computationally efficient compensation for drifts and transients in operando impedance measurements. We describe the application of pseudo-random sequences and provide the data processing required to suppress drift and transients, validated on simulations. Finally, we perform experimental operando impedance measurements on a Li-ion battery cell during fast-charging, demonstrating the applicability of the proposed method. It's low-cost hardware requirements, fast measurements, and simple data-processing make the method practical for embedding in battery management systems.

Figures

Figures reproduced from arXiv: 2506.07519 by the authors.

Figure 2
Figure 2. Time- and frequency-domain characteristics of the DST se￾quence of length 42. Top: time-domain signal. Middle: DFT magnitude response. Bottom: Phase response. The values at the sets K+ = {1, 5, 17, 25, 37, 41} and K− = {11, 13, 19, 23, 29, 31} are, re￾spectively, indicated in blue and red. with λDST = j √ 2rλQRT, r ∈ {−1, 1} depending on the QRT sequence, and excited harmonics K ={1 + 6p, 5 + 6p | p = 0, 1, . . . , … view at source ↗
Figure 4
Figure 4. DFT magnitude spectra of the steady-state current and voltage measurements with highlighted 2nd- and 3rd-order nonlinearities (2 A excitation amplitude). ties at V (2k) (purple) and odd nonlinearities at V (3k) (green) for k ∈ Kexc. Odd nonlinearities dominate in this case. An improvement in SNR therefore comes at the cost of increased total harmonic distortion. The best linear approximation can still be obtained by… view at source ↗
Figure 3
Figure 3. Measured current, voltage, and impedance data during one steady-state period (after discarding preceding period). From top to bottom: time-domain current- (1) and -voltage profiles (2), their DFT magnitudes (3), and the impedance spectrum (4). where upper case variables refer to the DFT of the correspond￾ing time domain data, ωk = 2πk/T, and L(ωk) is a transient term originating from different initial and end condit… view at source ↗
Figures from the paper (3 more)
Figure 5
Figure 5. Figure 5: Measured current, voltage and impedance data during one operando period. From top to bottom: time-domain current- (1) and - voltage profiles (2), their DFT magnitude spectra (3), and the impedance spectra (4). Input-output data over this period was measured with a sam￾…
Figure 7
Figure 7. Figure 7: Results of the operando impedance measurements during fast￾charging across 20% to 80% of SOC. Top figure: current profile. Middle figure: voltage profile. Bottom figure: measured impedances. to the cell from 20 to 80% SOC. The current of 1C is twice the standard chargi…
Figure 6
Figure 6. Figure 6: Results and configuration of the simulations. From top to bottom: Simulation model (1), DFT magnitude spectra of the voltage and current (2), DFT magnitude- (3) and phase spectra (4) of the impedances, and absolute error of the estimated impedance (5). VII. OPERANDO IM…

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