{"id":"ffce9fea-b969-4984-9a7e-3779a87965ea","arxiv_id":"1908.08099","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"An extended Bode theorem lets designers estimate the thermal noise variance of OTA-based switched-capacitor amplifiers by inspecting three equivalent capacitor circuits, validated here for an SC amplifier and a track-and-hold circuit.","lead":"This paper presents a shortcut for estimating thermal noise in switched-capacitor amplifier circuits by reading three capacitor-only diagrams instead of solving integrals. It works through two example circuits, an autozero amplifier and a track-and-hold, and checks the results against transient noise simulations.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (9) relies on the GmRon << 1 regime without a quantitative error bound; the paper's own Fig. 10 shows visible deviation once GmRon reaches about 0.27.","rationale":"I read the paper in good faith and did not find an internal algebraic error: Eq. (9) is coherent, the worked SC-amplifier and track-and-hold examples match the classical transfer-function integrations in the appendices, and the simulation figures are consistent with the stated assumptions. The reader's weakest-assumption analysis correctly identifies the central soft spot: the whole simplification rests on the VCCS being replaceable by a constant conductance Gm·h_fb, which requires GmRon << 1. The paper itself flags this in Fig. 10, where the switch-only curve deviates once GmRon reaches about 0.27. What is missing is a quantitative characterization of the error outside the asymptotic regime and a validation against a non-ideal OTA model. These are exactly the kinds of limitations that justify a CONDITIONAL verdict rather than a full ACCEPT, but they do not change the reader's conclusion. I therefore recommend the verdict remain UNCHANGED and the authors be asked to state the GmRon validity range and provide an independent non-ideal-OTA check.","tokens_in":17353,"tokens_out":20815,"duration_ms":215368,"concrete_test":"Reproduce Fig. 10 for a fixed gain (e.g., |Av|=8, C2=CL=0.5 pF, Cin=20 fF, γ=0) and sweep Ron from 0.1 kΩ to 50 kΩ so that GmRon spans roughly 0.005 to 2.7, keeping Gm scaled to maintain constant settling time. Compare Eq. (9) against the full analytical expressions of Appendix VI-A and against ELDO transient-noise simulations, and extract the GmRon threshold at which the relative error of Eq. (9) exceeds 5%. Additionally, run one comparison using an OTA model with finite output conductance and finite DC gain to test whether the ideal-VCCS assumption is the limiting factor.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central formula Eq. (9) is obtained by replacing the OTA with a constant conductance Gm·h_fb (Section III, Figs. 5-6), which is valid only while the input-node relation V = h_fb·Vout (Eqs. 4-5) holds across the noise bandwidth. The paper states this as GmRon << 1, and the assumption is genuinely load-bearing: in Fig. 10 the γ=0 switch-only curve deviates increasingly from the full calculation as GmRon approaches 0.27, and the deviation grows with amplifier gain. No error bound or design rule is given that tells a user how small GmRon must be for Eq. (9) to be accurate to, say, 5%. The paper also validates the method only against the same ideal VCCS model used in the derivation (Section IV.A.2), so the excellent agreement demonstrates consistency with a classical calculation of that idealized model rather than independent confirmation for real OTAs with finite output conductance and finite DC gain. The abstract and conclusion state the method applies to OTA-based SC circuits without prominently carrying the ideal-OTA and GmRon-qualifications, so the main risk is application outside the validated regime.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper proposes a design-oriented method for estimating the thermal noise voltage variance at any port of an OTA-based switched-capacitor circuit during a given clock phase. The method extends Bode's theorem to active circuits by modelling the ideal OTA as a VCCS, replacing it by a conductance Gm·hfb (where hfb is the capacitive feedback gain), splitting the OTA noise into a part consistent with the ambient temperature and an excess-noise part, and applying Bode's theorem to two passive capacitor-only equivalent circuits. The resulting Eq. (9) expresses the variance in terms of three capacitances, C∞, C'∞, and C0, which can be read by inspection. The paper derives closed-form expressions for a SC autozero amplifier and a SC track-and-hold circuit, and compares them with ELDO transient-noise simulations and with the classical transfer-function integration of Appendix VI-A.","tokens_in":17494,"tokens_out":11452,"duration_ms":113205,"significance":"The proposed formula is parameter-free and falsifiable: Eq. (9) is derived algebraically with no fitted parameters, and Appendix VI-A independently reproduces the variances by classical transfer-function integration, which is a definite strength. If the validity conditions are respected, this gives circuit designers a simple, inspection-based alternative to noise PSD integration for an important class of SC circuits. The main caveat is that the derivation rests on ideal-OTA and GmRon<<1 assumptions, and the paper's own simulations show the approximation degrading at GmRon≈0.27; the method is therefore best viewed as an accurate design estimate within a regime that should be quantified.","major_comments":[{"comment":"The central formula Eq. (9) is derived under the condition Gon >> hfbGm (equivalently GmRon << 1), which is used both to replace the switch conductances by short circuits in Fig. 6(d) and, more fundamentally, to justify the input-node relation V = hfb·Vout in Eq. (4). The paper states this condition but gives no quantitative bound on GmRon for a target accuracy. Fig. 10 shows that for γ=0 the switch-only curve deviates visibly from the full calculation once GmRon reaches about 0.27, and the deviation increases with amplifier gain, so a user of Eq. (9) cannot tell from the stated condition whether the formula is accurate to, say, 5%. The authors should provide a design rule or an approximate correction term for finite GmRon, and should present Eq. (9) as an approximation valid in this regime rather than as an exact equality.","section":"Section III, Eqs. (4)-(9) and Fig. 10"},{"comment":"The validation in Figs. 10-12 is performed with the same ideal VCCS model of the OTA and the same switch model (ideal switch plus series resistor Ron) that is used in the derivation, so the agreement demonstrates internal consistency between Eq. (9) and the classical transfer-function integration of Appendix VI-A, but it does not independently confirm the method for real OTAs with finite output conductance and finite DC gain. Since the abstract and conclusion claim applicability to OTA-based SC circuits without these qualifications, the authors should either add a transistor-level verification or explicitly scope the claim to ideal transconductance amplifiers with GmRon << 1.","section":"Section IV.A.2, Figs. 10-12"}],"minor_comments":[{"comment":"The abstract and conclusion state that the method allows 'precise estimation' without carrying the ideal-OTA and GmRon<<1 qualifications; please add these qualifications to avoid overstating the validated regime.","section":"Abstract and Conclusion"},{"comment":"There is a duplicated '∼=' in Eq. (24b): the text shows '∼= ∼=αin + ...', which should be a single '∼=' symbol.","section":"Eq. (24b)"},{"comment":"In the sentence 'The later circuits can now be considered as passive', 'later' should be 'latter'.","section":"Section III, before Fig. 5"},{"comment":"The caption and text refer to 'Full calculation with Ron > 0' and 'Simulations Ron > 0'; please clarify that the dashed lines are the classical analytical expressions including nonzero GmRon rather than a separate simulator result.","section":"Fig. 10 and Section IV.A.2"}],"recommendation":"major_revision","confidential_remarks":"This is a solid circuit-theory contribution with a transparent derivation and a useful inspection-based formula. The main risk is not the internal logic but the scope of the claims: the method is validated only for an ideal VCCS model, and the central approximation lacks a quantitative validity bound. Both issues are fixable within the paper's scope, so I recommend major revision rather than rejection. The split into Part I/II is reasonable, but Part I should stand alone more clearly by including at least the leading-order correction for finite GmRon or a precise design rule."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a solid, useful methods paper for analog IC designers. The extended Bode theorem in Eq. (9) is a real convenience: you read three capacitances off simple schematics and get the output noise variance without transfer functions or integrals. The most valuable new material is the closed-form beta expressions for the SC amplifier and track-and-hold, especially the phase-Phi2 switch contribution that Murmann (ref. 41) omits. The derivation is coherent: replace the VCCS with conductance Gm·h_fb, split the OTA noise into a thermal part and an excess part, apply Bode to each, and the algebra checks out against the classical integration in Appendix VI-A. The ELDO transient-noise simulations agree well in the regime where GmRon is small, and the authors are upfront that Fig. 10 shows the approximation starting to bend once GmRon reaches about 0.27.\n\nThe soft spots are real but not disqualifying. First, the core formula is an extension of the authors' own conference paper [33]; the new contribution is in the detailed application and closed-form results rather than the method itself. Second, the validation uses the same ideal VCCS OTA model that the derivation assumes, so it is a consistency check, not independent confirmation with a transistor-level or finite-output-conductance OTA. Third, there is no quantitative error bound on the GmRon << 1 condition; a designer reading the abstract might apply it outside the validated range. The paper would be stronger with a sentence like 'for our examples, GmRon must stay below 0.1 for 5% accuracy' — or with one example using a more realistic OTA. No code or data are shipped, but that is normal for this venue and not a flaw. The citation pattern is honest; they explicitly compare to [40] and [41].\n\nOverall, the central claim holds up under the stated assumptions. It is a good paper for people who design or teach switched-capacitor circuits and want an intuitive noise shortcut. It deserves a serious referee; I would send it to peer review with a request to quantify the approximation range and ideally add one non-ideal-OTA validation. I would not cite it myself because I do not work in this area, but I would bring it to a reading group if analog designers are in the room.","headline":"A genuinely useful design-oriented noise estimation method for OTA-based switched-capacitor circuits, sound under its stated assumptions, but the GmRon << 1 regime needs a quantitative bound and one real-OTA validation.","tokens_in":18104,"tokens_out":3072,"would_cite":false,"duration_ms":29166,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims that thermal noise in OTA-based switched-capacitor circuits can be estimated from three capacitances read off capacitor-only equivalent circuits, avoiding transfer functions and integrals.","keywords":["thermal noise","switched-capacitor circuits","Bode theorem","kTC noise","operational transconductance amplifier","track and hold","capacitive feedback","noise estimation"],"falsifier":"Run a transient-noise simulation of the SC amplifier of Fig. 3 with $C_2=C_L=0.5$ pF, $C_{in}=20$ fF, gain $|A_v|=8$, and a switch resistance raised so that $G_m R_{on}$ is near 1; the $\\gamma=0$ switch-only output noise predicted by Eq. (23) will fall increasingly below the simulated value as $R_{on}$ grows, while the full finite-$R_{on}$ calculation tracks the simulation.","tokens_in":17100,"feed_emoji":"🔌","tokens_out":9205,"duration_ms":83039,"temperature":0.7,"pith_summary":"The paper claims that for switched-capacitor circuits built around operational transconductance amplifiers with capacitive feedback, the thermal noise voltage variance between any two nodes can be estimated directly from three capacitances read off capacitor-only equivalent circuits, with no transfer functions and no integrations. If this method is correct, the dominant sampled thermal noise of an SC amplifier or track-and-hold circuit becomes a design-oriented quantity: a few capacitance inspections and a closed-form algebraic expression. The paper demonstrates the recipe on an autozero SC amplifier and an SC track-and-hold circuit and validates the resulting formulas with transient noise simulations.","feed_headline":"Thermal noise of SC amplifiers from three capacitor readings","feed_subtitle":"A three-circuit inspection replaces transfer functions and integrals for switched-capacitor amplifiers and track-and-hold circuits.","key_machinery":"The load-bearing object is the extended Bode theorem expressed by Eq. (9), together with the three capacitor-only inspection circuits that define $C_\\infty$, $C'_\\infty$, and $C_0$. The mechanism is a decomposition of the OTA noise current source into a part at ambient temperature $T$ that joins the switch noises in a passive network, plus an excess-noise part at temperature $(\\gamma/h_{fb}-1)T$ evaluated after the switches are shorted. This reduction converts a cyclostationary sampled-noise problem into three capacitance readings and one algebraic summation.","core_discovery":"On its own terms, the paper's discovery is an extension of Bode's theorem from passive RLC networks to active OTA-based switched-capacitor circuits. The variance of the thermal noise voltage between any two nodes $k$ and $l$ is claimed to be $V_{n(kl)}^2 = k_B T [1/C_\\infty(kl) + (\\gamma/h_{fb}-1)/C'_\\infty(kl) - (\\gamma/h_{fb})/C_0(kl)]$, where $C_\\infty$ is the capacitance seen when all switches and OTAs are removed, $C'_\\infty$ is the capacitance seen when closed switches are shorted and OTAs removed, $C_0$ is the capacitance seen when switches are shorted and OTA outputs are grounded, and $h_{fb}$ is the feedback voltage gain from the OTA output to its input. The OTA's excess thermal noise beyond that of a passive conductance is accounted for by assigning it a noise temperature $(\\gamma/h_{fb}-1)T$. The paper derives closed-form noise variances for the SC amplifier and the track-and-hold circuit from this identity and checks them against transient noise simulations.","pith_inferences":["The paper leaves implicit that the same phase-by-phase three-capacitance recipe should extend to multistage switched-capacitor filters and to each step of a successive-approximation ADC front end, as long as charge transfer between phases is handled the way Part I handles the SC amplifier.","The paper's stated condition $G_m R_{on} \\ll 1$ doubles as a practical error budget: when high-speed designs push the product toward 1, Eq. (9) will under-predict switch noise, and the size of the deviation can be probed by comparing the formula with a nonzero-$R_{on}$ transient simulation.","A natural but unstated stress test is finite OTA DC gain: the derivation assumes an infinite-gain ideal OTA, so quantifying the correction for low-gain inverter-based amplifiers would be a testable next step."],"forward_implications":["For the autozero SC amplifier, the total output noise at the end of the amplification phase is $V_{nout}^2 = (k_B T/C_2)(\\gamma\\beta_{ota}+\\beta_{sw})$, with both beta factors explicit ratios of $C_1$, $C_2$, $C_{in}$, and $C_L$, matching the classical transfer-function integration.","For the SC track-and-hold circuit, the same formula applies with the OTA term coming only from phase 2 and the switch term from both phases; the phase-2 switch contribution is small but is included, refining earlier track-and-hold noise results.","Because the method needs only capacitance ratios, it directly exposes which capacitors set the sampled noise floor, which is the quantity a low-power designer must trade against capacitance area and settling time."],"supporting_citations":[{"why":"Supplies the original Bode theorem for passive RLC networks that the paper extends to OTA-based switched-capacitor circuits.","marker":"[34]"},{"why":"The authors' earlier conference paper introducing the extended Bode theorem for OTA-based switched-capacitor filters, which this paper develops and validates in detail.","marker":"[33]"},{"why":"Provides the total-integrated-noise calculation technique used in the appendices to derive the classical expressions that the new formula is checked against.","marker":"[32]"},{"why":"Describes transient noise simulation, the verification method used throughout the paper to validate the closed-form noise estimates.","marker":"[21]"},{"why":"Gives an earlier kTC noise analysis of correlated-double-sampling circuits whose result the paper's Eq. (11) reduces to when the input capacitance is zero.","marker":"[40]"},{"why":"Provides the track-and-hold thermal noise analysis that the paper's Eq. (33) matches except for the added phase-2 switch contribution.","marker":"[41]"},{"why":"Justifies the thermal-noise-only model by explaining that autozeroing suppresses OTA offset and flicker noise, leaving sampled thermal noise as the dominant contributor.","marker":"[23]"}],"fun_headline_variants":["Bode theorem extended to OTA-based switched-capacitor noise","No transfer functions or integrals for SC amplifier noise","Thermal noise from three capacitor readings in SC circuits","Simple OTA noise variance from three capacitance readings","Bode's theorem for OTA-based switched-capacitor thermal noise"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The derivation assumes the switch on-resistance is much smaller than the OTA's effective output resistance, $G_m R_{on} \\ll 1$, so the OTA input node behaves as a pure capacitive divider and the switches can be shorted in the excess-noise circuit.","fun_headline_variants_meta":{"raw":{"variants":["Bode theorem extended to OTA-based switched-capacitor noise","No transfer functions or integrals for SC amplifier noise","Thermal noise from three capacitor readings in SC circuits","Simple OTA noise variance from three capacitance readings","Bode's theorem for OTA-based switched-capacitor thermal noise"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000419,"raw_usage":{"total_tokens":2137,"prompt_tokens":902,"completion_tokens":1235,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":518,"completion_tokens_details":{"reasoning_tokens":1154}},"tokens_in":518,"tokens_out":1235,"duration_ms":9138,"temperature":1.0,"reasoning_tokens":1154,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:50:50.045863+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run a transient-noise simulation of the SC amplifier of Fig. 3 with $C_2=C_L=0.5$ pF, $C_{in}=20$ fF, gain $|A_v|=8$, and a switch resistance raised so that $G_m R_{on}$ is near 1; the $\\gamma=0$ switch-only output noise predicted by Eq. (23) will fall increasingly below the simulated value as $R_{on}$ grows, while the full finite-$R_{on}$ calculation tracks the simulation.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the original Bode theorem for passive RLC networks that the paper extends to OTA-based switched-capacitor circuits."},{"cited_title":"Calculation of Total Integrated Noise in Analog Circuits,","cited_arxiv_id":null,"evidence_quote":"Provides the total-integrated-noise calculation technique used in the appendices to derive the classical expressions that the new formula is checked against."},{"cited_title":"An Accurate kTC Noise Analysis of CDS Circuits,","cited_arxiv_id":null,"evidence_quote":"Gives an earlier kTC noise analysis of correlated-double-sampling circuits whose result the paper's Eq. (11) reduces to when the input capacitance is zero."},{"cited_title":"Thermal Noise in Track-and-Hold Circuits: Analysis and Simulation Techniques,","cited_arxiv_id":null,"evidence_quote":"Provides the track-and-hold thermal noise analysis that the paper's Eq. (33) matches except for the added phase-2 switch contribution."},{"cited_title":"Circuit Techniques for Reducing the Effects of Op-amp Imperfections: Autozeroing, Correlated Double Sampling, and Chopper Stabilization,","cited_arxiv_id":null,"evidence_quote":"Justifies the thermal-noise-only model by explaining that autozeroing suppresses OTA offset and flicker noise, leaving sampled thermal noise as the dominant contributor."}],"review_version":1}