{"id":"34f683a2-c3e4-41bc-bd35-e96db5a3bbed","arxiv_id":"2506.07519","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Ternary pseudo-random test signals with a DFT-eigenvector property let battery impedance be measured during charging, with drift and transient artifacts removed by combining oppositely signed spectral components.","lead":"The paper shows that two special three-level test signals, called quadratic-residue and direct-synthesis ternary sequences, can measure a battery's impedance while it is charging, and a simple averaging trick removes drift and transient distortion. The value is in making operando impedance checks cheap enough to embed in battery management systems.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The single-experiment reconstruction (Eq. 40) hinges on interpolating Z± across adjacent harmonics; the contamination term (V0+L)/(I0±Iexc) in Eq. (42) is not shown to be smooth, and low-frequency interpolation error is unquantified.","rationale":"The reader identified the interpolation/smoothness assumption as the weakest point, and I agree. The derivation of Eq. (40) is algebraically sound under the assumption that the two virtual experiments differ only by the sign of the excitation, but in the single-experiment implementation the missing Z± values are obtained by interpolation across frequencies. The quantity being interpolated includes the drift/transient contamination, whose spectral smoothness is not guaranteed and is not analyzed. The paper's own acknowledgment that low frequencies are worst, and its discarding of the first two harmonics, indicate the authors are aware of the limitation but offer no quantitative bound. The simulation is too smooth to test the assumption, and the experiment lacks a baseline. A targeted simulation with a non-smooth drift would directly reveal whether the method's central promise—accurate single-period operando impedance—holds beyond the benign test cases shown. Therefore the reader's CONDITIONAL verdict remains appropriate; no change is needed.","tokens_in":33696,"tokens_out":6818,"duration_ms":87748,"concrete_test":"Repeat the Section VI simulation with the same DST sequence and noise, but modify the drift term to include a component with fast spectral variation, e.g., v0(t) = OCV(SOC(t)) + A·exp(−t/τ) with τ=1 s (so L and V0 have comparable low-frequency content), and also a case with a quadratic ramp i0(t)=2.5−(t/Tp)^2. Compute the maximum relative error of |\\hat Z(ωk)| over 1.05 Hz–1 kHz against the true Z. If the error exceeds a few percent in either case, the interpolation assumption is violated in realistic conditions and the paper must quantify a bound or restrict the method's operating range.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that Eq. (40) recovers Z from one operando period using QRT/DST sign-alternating spectra. Algebraically this is Eq. (37) with the missing Z± values filled by interpolation (41). But from Eq. (42), each interpolated quantity is Z±(ωk) = Z(ωk) + [V0(k)+L(ωk)]/[I0(k)±Iexc(k)], so the interpolation must be accurate for the contamination term, not just for Z. The paper asserts smoothness of Z, V0, L, I0, Iexc (Section V.B) but provides no bound or sensitivity analysis. The simulation (Section VI) uses a linear ramp i0(t) and smooth OCV(SOC), so V0(k) is spectrally smooth; the experiment (Section VII) has no reference impedance, so an interpolation-induced bias would go undetected. At low frequencies, where drift and transient dominate and Iexc is relatively small, the contamination term can vary fast; the method already discards the first two harmonics because extrapolation is unreliable (43), and the text concedes 'interpolation error will typically be worst at low frequencies.' This is not a fatal flaw, but it makes the headline claim of accurate single-period suppression of drifts/transients contingent on an unverified regularity condition.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":33978,"tokens_out":5157,"duration_ms":66972,"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":[{"comment":"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.","section":"Section V.B, Eq. (42)"},{"comment":"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.","section":"Section VII"},{"comment":"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.","section":"Section V.A, Eqs. (36)-(39) and Section III, Eq. (25)"},{"comment":"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.","section":"Section VI and V.B"}],"minor_comments":[{"comment":"The phrase 'It's low-cost hardware requirements' should be 'Its low-cost hardware requirements'.","section":"Abstract and Section I"},{"comment":"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.","section":"Section V.B"},{"comment":"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.","section":"Section VII"},{"comment":"Reference [19] is listed as 'Early Access March 2025' without volume or article number; please provide the final citation if available.","section":"References"},{"comment":"The entry 'N 1.0002e6' is hard to read; format it as '1.0002×10^6'.","section":"Table I"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is within scope for a control-systems or instrumentation journal and the core idea is potentially valuable. The main concern is that the experimental section does not validate the method against a reference, and the interpolation error is unquantified. These are fixable with additional analyses rather than a fundamental flaw. I would also encourage the editor to ask for code or detailed pseudo-code, as the methodology is algorithmic in nature."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read Sihvo et al. on QRT/DST sequences for operando battery impedance. Bottom line: the K+/K- pairing trick is real, the algebra in Eqs. (36)-(41) checks out, and the DST eigenvector proof in the appendix is sound. But the paper's central claim - that a single short period suppresses drift and transients - depends on an interpolation smoothness assumption that is asserted, not quantified. I'd send it to review, but I'd ask for an error bound and an experimental baseline first.\n\nWhat's new is narrower than the abstract suggests. QRT and DST sequences were known, and the DST operando demo appears in the authors' own [19]. The genuine contribution is the explicit single-experiment reconstruction: splitting Z+ and Z- across K+ and K-, interpolating the gaps, and combining via Eq. (40). That formulation is clean and the simulation validates it against a known model. The appendix proof of (16) is correct as far as I can tell, and it doesn't depend on the self-citation - that's just motivation.\n\nThe soft spot is exactly where the stress-test note lands. From Eq. (42), each Z+/- is the true impedance plus a contamination term (V0+L)/(I0+/-Iexc). The interpolation has to be smooth for that whole ratio, not just for Z. At low frequencies Iexc is relatively small and drift/transient spectra can be steep, so the error can be large there. The paper concedes this and drops the first two harmonics, but it gives no bound, no sensitivity analysis, and no experiment with a reference. Section VII's spectra have no error bars and no comparison to a steady-state EIS or known model, so a systematic bias would go unnoticed. No code or data is shipped. These are addressable gaps, not fatal flaws.\n\nThe method is simple, computationally cheap, and plausibly useful for BMS-embedded impedance. The authors are honest about limitations, and the derivation stands on its own. It's a solid practical contribution for battery diagnostics, not a paradigm shift.\n\nWho benefits: researchers working on operando EIS, BMS state estimation, and PRS-based identification. I'd take it to a reading group, but I'd want to discuss the interpolation risk.\n\nRecommendation: send it to peer review. Require the authors to quantify the interpolation error - either a worst-case bound on the contamination term or a Monte Carlo sweep - and to add a reference-baseline comparison in the experiments. Also ask for code/data.","headline":"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.","tokens_in":34550,"tokens_out":3157,"would_cite":true,"duration_ms":37199,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["operando impedance spectroscopy","lithium-ion battery","pseudo-random sequences","quadratic-residue ternary sequence","direct-synthesis ternary sequence","DFT eigenvector","drift and transient suppression","battery management system"],"falsifier":"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.","tokens_in":33487,"feed_emoji":"🔋","tokens_out":9603,"duration_ms":98586,"temperature":0.7,"pith_summary":"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.","feed_headline":"One pseudo-random burst measures battery impedance mid-charge","feed_subtitle":"DFT-eigenvector sequences cancel drift and transients, so cheap BMS hardware can run operando EIS mid-charge.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the DFT eigenvalue/eigenvector decomposition theory that underlies the claim that QRT and DST sequences are DFT eigenvectors at excited harmonics.","marker":"[22]"},{"why":"Source of the quadratic-residue ternary sequence definition and its comparison framework.","marker":"[23]"},{"why":"Defines the direct-synthesis ternary sequence with second- and third-order harmonics suppressed, the excitation used in the experiments.","marker":"[16]"},{"why":"Prior demonstration of DST-based battery impedance monitoring under dynamic current, which this paper extends with the full theory and drift-transient suppression derivation.","marker":"[19]"},{"why":"The frequency-domain system identification reference that justifies the measurement model, the transient term $L(\\omega_k)$, leakage avoidance, and smoothness of transients.","marker":"[7]"},{"why":"Provides the nonlinear, time-varying battery model (operando EIS review) on which the operando derivations in Section V rest.","marker":"[8]"},{"why":"The operando EIS frequency-domain modelling method for commercial Li-ion batteries that the proposed lightweight cancellation replaces.","marker":"[9]"},{"why":"Gives the quadratic-residue connection to DFT eigenvalues, fixing the allowed eigenvalue set $\\{1, -1, j, -j\\}$ used in the derivation.","marker":"[24]"}],"fun_headline_variants":["Pseudo-random burst cancels drift in low-cost battery impedance","DFT-eigenvector sequences make cheap BMS track battery health live","Two ternary sequences cut drift, cheap operando EIS for battery BMS","Random-sequence trick enables fast, low-cost battery impedance monitoring","Ternary sequences beat drift in low-cost battery impedance sensing"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Pseudo-random burst cancels drift in low-cost battery impedance","DFT-eigenvector sequences make cheap BMS track battery health live","Two ternary sequences cut drift, cheap operando EIS for battery BMS","Random-sequence trick enables fast, low-cost battery impedance monitoring","Ternary sequences beat drift in low-cost battery impedance sensing"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000392,"raw_usage":{"total_tokens":2046,"prompt_tokens":918,"completion_tokens":1128,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":534,"completion_tokens_details":{"reasoning_tokens":1040}},"tokens_in":534,"tokens_out":1128,"duration_ms":9291,"temperature":1.0,"reasoning_tokens":1040,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T05:33:40.329448+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Eigenvalue and eigenvector decomposition of the discrete Fourier transform,","cited_arxiv_id":null,"evidence_quote":"Supplies the DFT eigenvalue/eigenvector decomposition theory that underlies the claim that QRT and DST sequences are DFT eigenvectors at excited harmonics."},{"cited_title":"Comparison of perturbation signals for linear system identification in the frequency domain,","cited_arxiv_id":null,"evidence_quote":"Source of the quadratic-residue ternary sequence definition and its comparison framework."},{"cited_title":"Direct synthesis of pseudo-random ternary perturbation sig- nals with harmonic multiples of two and three suppressed,","cited_arxiv_id":null,"evidence_quote":"Defines the direct-synthesis ternary sequence with second- and third-order harmonics suppressed, the excitation used in the experiments."},{"cited_title":"Real-time impedance monitoring of Li-ion batteries under dynamic operating conditions: The discrete fourier trans- form eigenvector approach,","cited_arxiv_id":null,"evidence_quote":"Prior demonstration of DST-based battery impedance monitoring under dynamic current, which this paper extends with the full theory and drift-transient suppression derivation."},{"cited_title":"Pintelon and J","cited_arxiv_id":null,"evidence_quote":"The frequency-domain system identification reference that justifies the measurement model, the transient term $L(\\omega_k)$, leakage avoidance, and smoothness of transients."},{"cited_title":"Electrochemical impedance spectroscopy beyond linearity and station- arity—A critical review,","cited_arxiv_id":null,"evidence_quote":"Provides the nonlinear, time-varying battery model (operando EIS review) on which the operando derivations in Section V rest."},{"cited_title":"Operando electrochemical impedance spec- troscopy and its application to commercial Li-ion batteries,","cited_arxiv_id":null,"evidence_quote":"The operando EIS frequency-domain modelling method for commercial Li-ion batteries that the proposed lightweight cancellation replaces."},{"cited_title":"Quadratic residues: Ap- plication to chirp filters and discrete fourier transforms,","cited_arxiv_id":null,"evidence_quote":"Gives the quadratic-residue connection to DFT eigenvalues, fixing the allowed eigenvalue set $\\{1, -1, j, -j\\}$ used in the derivation."}],"review_version":1}