{"id":"41bafc88-9c23-4ba8-8d99-edc87810b68c","arxiv_id":"1908.08109","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Closed-form thermal noise variance expressions for switched-capacitor integrators and first-order filters are derived using the extended Bode theorem and validated by transient noise simulation.","lead":"This paper gives analog circuit designers simple formulas and a visual method to estimate thermal noise in switched-capacitor filters without solving complex equations. The approach is validated against circuit simulations for an integrator and a first-order low-pass filter.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (5) neglects input-voltage terms in the virtual-ground equation; the paper gives no error bound and validates only single-stage circuits, so the method's applicability to general SC filters is unsupported.","rationale":"The reader's verdict of CONDITIONAL is supported by the same load-bearing assumption: Eq. (5) and the small-ratio conditions of Eqs. (4a)–(4c). My stress-test sharpens the concern by noting that the neglected β1·V1 term is not merely a small capacitance-ratio issue but carries the noise of a preceding stage in a real filter. The paper's own simulations only exercise single-OTA circuits where V1 is ground or V2 is the OTA's own output, so the truncation error in Eq. (3) is never actually tested. The paper gives the extended Bode theorem from Part I as a black box, and the reproduction of simplified formulas in the appendices is careful, but the missing quantitative validation of the central approximation in a multi-stage setting is the weakest point. The proposed concrete test would settle whether the method generalizes. Since the reader already flagged the small-ratio condition as the weak assumption and assigned CONDITIONAL, my analysis does not change the verdict.","tokens_in":15965,"tokens_out":10691,"duration_ms":108731,"concrete_test":"Simulate a second-order SC low-pass filter (two cascaded stray-insensitive integrators with feedback) with inter-stage coupling capacitance αC and α = 0.5, using the same ELDO ideal-component setup as the paper. Compute the output noise variance two ways: (i) the proposed inspection method of Eq. (6), applying V ≈ h_fb·Vout for each OTA and neglecting the cross-coupling capacitor's noise contribution from the first stage; (ii) full transient noise simulation. If the RMS difference exceeds 10% of the simulated value, the approximation in Eq. (5) is not valid for general SC filters and the paper's claim is overbroad.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that the extended Bode theorem (Eq. 6) applies to OTA-based SC filters because the virtual-ground voltage can be approximated as V ≈ h_fb·Vout (Eq. 5). This requires dropping the β1·V1 + β2·V2 terms in Eq. (3). The paper justifies this by C1,C2 ≪ C, but provides no error bound. In a multi-stage SC filter, V1 is the noisy output of a preceding OTA, of the same order as Vout; with α = C1/C = 0.1 the dropped term is 10% of Vout, and for α = 0.5 it is 33%. Such an error directly biases the noise-charge variances computed from Eq. (6) because the OTA conductance becomes h_fb·Gm only if the controlling voltage is strictly proportional to Vout. The examples validate only single-OTA stages where V1 is ground or V2 = Vout (absorbed in h_fb), so the multi-stage, moderate-α regime is untested. No error analysis or sweep over α is reported.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper (Part II) extends the authors' extended Bode theorem, introduced in Part I, to OTA-based switched-capacitor filters. After reviewing the noise-charge mechanism in an SC integrator, the paper approximates the OTA virtual-ground voltage as V approximately h_fb * Vout (Eq. 5), so that the OTA is replaced by a conductance h_fb * Gm and the noise variance between any two nodes is expressed by Eq. (6) in terms of three capacitor-only equivalent circuits. The method is applied to three examples: a passive first-order low-pass filter, a stray-insensitive integrator, and an active OTA-based first-order low-pass filter. Closed-form expressions are derived for the sampled and direct output noise, including recursive convergence to steady state, and these are compared with ELDO transient noise simulations using ideal VCCS OTAs and ideal switches; the agreement is reported as excellent.","tokens_in":16155,"tokens_out":10214,"duration_ms":105621,"significance":"The contribution is potentially useful: it provides closed-form, parameter-free expressions for thermal noise variances in several common SC filter stages, avoiding transfer-function and PSD integrations. The paper makes no fitted-parameter claim, and the validation against transient noise simulations is an independent check of the formulas, not a re-derivation of them. If supported by a quantitative statement of the approximation domain, the method would be a convenient hand-calculation tool for early design. The main strengths are the explicit recursive treatment of non-reset integrating capacitors and the compact Eq. (6) that reduces the calculation to three capacitor-only inspections. The central weakness is that the one approximation enabling the application to OTA filters, Eq. (5), is not accompanied by an error bound, and the demonstrations cover only single-OTA stages at alpha = 0.1.","major_comments":[{"comment":"The approximation V approximately h_fb*Vout is load-bearing: it converts the OTA into a conductance h_fb*Gm and thereby justifies Eq. (6) for OTA-based filters. The paper discards beta1*V1 + beta2*V2 based on C1,C2 being much smaller than C and on the observation that V1, V2, and Vout are bounded by the supply, but for a cascaded filter V1 is the noisy output of another OTA, not a quiet signal source, so bounding by VDD says nothing about the noise contribution. The error in the controlling voltage is of order C1/C and the resulting error in the computed variances is not quantified. The validation in Section III uses only single-OTA examples with alpha = 0.1 and does not sweep alpha or test two cascaded stages. Please add an explicit error analysis (for example, an O((C1/C)^2 + (C2/C)^2) bound on the relative variance error under reasonable independence assumptions) or restrict the claimed domain, and validate the approximation with an alpha sweep or a two-stage example.","section":"Section II, Eq. (5)"},{"comment":"The derivations jump from the three equivalent capacitances to the final beta expressions. For instance, Eq. (53) is introduced as 'applying the extended Bode theorem (6)' and Eqs. (54)-(55) appear without the intermediate substitution of C_infinity, C'_infinity, C0, and h_fb into Eq. (6). Since the central selling point is calculation 'by inspection,' and since these formulas are the ones validated in Figs. 15-16, the reader cannot verify them without repeating the algebra. Please include the exact expressions before the alpha, alpha_in much-less-than 1 reduction and at least one worked substitution as a template.","section":"Appendices V-A and V-C, Eqs. (48)-(55)"}],"minor_comments":[{"comment":"The phrase 'very accurate tanks to the excellent matching' should be 'thanks to the excellent matching'.","section":"Introduction"},{"comment":"The sentence 'confirms than the noise increases through the periods' should read 'confirms that the noise increases through the periods'.","section":"Section III-A"},{"comment":"The text reports Cin = 10 fF for the simulation in Fig. 15a, while the figure caption reports Cin = 20 fF; please harmonize these values.","section":"Section III-C"},{"comment":"There is a typographical double equal sign in the displayed equation; clean up the typesetting.","section":"Eq. (43a)"},{"comment":"The inline typesetting of Eq. (6) is ambiguous; please use explicit fractions with parentheses around (gamma/h_fb - 1) and gamma/h_fb so that the reader can see the intended grouping.","section":"Eq. (6)"},{"comment":"The figure would be more informative if the numerical RMS deviations between calculated and simulated values were listed or shown with error bars, since the plotted curves alone make it difficult to assess the quality of the match at each gamma.","section":"Fig. 16"}],"recommendation":"major_revision","confidential_remarks":"Given that Part II depends centrally on Eq. (6) from Part I and the conference paper [5], the editor may wish to confirm that Part I is available in final form and that the notation is compatible with this manuscript. The paper's claims about general SC filters are stronger than the validation supports; the suggested error analysis and a cascaded example would address this. The manuscript is otherwise within the scope of the journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The useful part of this paper is concrete: closed-form thermal noise variance expressions for a passive SC low-pass filter, a stray-insensitive integrator, and an active OTA-based first-order SC filter. The method itself (extended Bode theorem, from Part I) avoids transfer functions and integrals, which is genuinely helpful for hand design in the SC subfield. The final formulas are compact and the transient noise simulations confirm them for the tested cases, including convergence to steady-state noise over switching periods. That is a real contribution, and the passive LP result reducing to kT/C is a sensible sanity check.\n\nThe main soft spot is the load-bearing approximation in Eq. (5): V ≈ h_fb·Vout, which drops the β1V1 and β2V2 terms in Eq. (3). The paper states that C1, C2 ≪ C and that V1, V2, and Vout are all of the same order, so the dropped terms are comparatively small. That is true for the tested values (α = 0.1, α = 1/4), but no error bound is given and no sweep over α is shown. At α = 0.5 the dropped term is around 30% or more of Vout, and for a multi-stage SC filter where V1 is the noisy output of a preceding OTA, the approximation becomes even more questionable. The paper validates only single-OTA stages, so the method's claimed generality is not actually demonstrated. This is an addressable weakness, not a fatal one: the central derivation is not circular, the approximations are stated, and the validation is independent transient noise simulation rather than fitting.\n\nA second, lesser concern is that the appendix derivations jump from \"applying the extended Bode theorem\" to the final algebraic expressions without showing intermediate steps. A reader cannot verify the application without redoing the algebra. The heavy reliance on Part I and reference [5] is acceptable because the extension here is genuinely new, but a self-contained derivation would be stronger.\n\nThe paper deserves a serious referee. It is a reasonable incremental contribution with practical value for analog designers. My recommendation to the editor would be major revision: ask the authors to include the omitted derivation details, add a sensitivity analysis over α, and ideally demonstrate the method on a two-stage filter where the dropped V1 term is not trivially small. If those are addressed, it would be a solid publication.","headline":"Extends the authors' own Bode-based noise method to SC filters with compact formulas that match transient simulations, but the key small-ratio approximation is unquantified and multi-stage or moderate-α cases are untested.","tokens_in":16691,"tokens_out":1916,"would_cite":true,"duration_ms":20522,"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":"Thermal noise in OTA-based switched-capacitor filters can be read off directly from three capacitor-only circuits, without transfer functions or integrals.","keywords":["switched-capacitor circuits","thermal noise","kTC noise","extended Bode theorem","SC filters","OTA","noise variance estimation","transient noise simulation"],"falsifier":"For an SC integrator with $\\alpha = C_1/C$ not much smaller than 1, compute the output noise variance exactly by numerical integration of the phase PSDs, or by transient noise simulation, and compare with Eq. (26): if the prediction deviates increasingly as $\\alpha$ grows, the small-ratio simplification is the breaking point. A second concrete check is to measure the noise of a fabricated SC filter at a ratio where $C_1 \\approx C$ and see whether the variance still follows the capacitor-only formula.","tokens_in":15775,"feed_emoji":"🎛️","tokens_out":5982,"duration_ms":54844,"temperature":0.7,"pith_summary":"This paper claims that the thermal noise voltage variance in switched-capacitor (SC) filters built around operational transconductance amplifiers (OTAs) can be obtained from the extended Bode theorem, a formula that needs only three capacitances read from capacitor-only schematics. The method replaces the OTA by a conductance $h_{fb}G_m$ during each clock phase and treats each phase as a passive RC network, avoiding transfer functions and frequency integrals. For the basic stray-insensitive integrator and an OTA-based first-order low-pass filter, the paper derives closed-form output noise variances, including the growth of integrator noise with the number of clock periods, and reports close agreement with transient noise simulations. If correct, it gives designers a fast hand-calculation route to optimize noise versus power in SC filters.","feed_headline":"SC-filter thermal noise read off by inspection","feed_subtitle":"Extended Bode theorem turns OTA-based filter noise into three capacitor values, no integrals.","key_machinery":"The extended Bode theorem (Eq. 6) is the central object: it expresses the thermal noise voltage variance across any port as $k_B T$ times a combination of the reciprocals of three capacitances $C_\\infty$, $C'_\\infty$, and $C_0$. Those capacitances are obtained by inspection from three equivalent circuits: with all switches and OTAs removed; with closed switches shorted and OTAs removed; and with closed switches shorted and OTA outputs grounded. The theorem works because the noise sources become equivalent conductances (switch on-conductance and $\\gamma G_m$) in a passive RC network. The second load-bearing mechanism is the small-ratio approximation $C_1,C_2\\ll C$, which justifies replacing the OTA by the conductance $h_{fb}G_m$ and lets the sampled noise charge on each switching capacitor be computed phase by phase; a noise-charge injector and recursive relations then propagate the variances through successive clock periods.","core_discovery":"The paper's central claim is that Eq. (6), the extended Bode theorem, applies to OTA-based SC filters: the thermal noise voltage variance between any two nodes during a clock phase is $V_n^2(kl)=k_B T[1/C_\\infty(kl)+(\\gamma/h_{fb}-1)/C'_\\infty(kl)-(\\gamma/h_{fb})/C_0(kl)]$, where $C_\\infty$, $C'_\\infty$, and $C_0$ are read from three capacitor-only circuits by inspection. This works because in SC filters the integrating capacitor $C$ is much larger than the switched and non-switched capacitors $C_1$ and $C_2$, so the virtual-ground voltage is dominated by the OTA output term, $V\\approx h_{fb}V_{out}$, and the OTA behaves like a conductance $h_{fb}G_m$. The paper applies the theorem to a passive first-order low-pass filter, the stray-insensitive integrator, and an active first-order low-pass filter, deriving recursive relations for the accumulation of noise on the non-reset integrating capacitor and closed-form steady-state variances. It validates these expressions against transient noise simulations and reports close agreement.","pith_inferences":["The same capacitor-inspection route should extend to higher-order SC filters built from cascaded integrators, since each phase still reduces to a passive RC network with the OTA replaced by $h_{fb}G_m$; a second-order biquad transient-noise test would check this.","When $\\alpha$ is not small, the neglected $\\beta_1$ and $\\beta_2$ terms in the virtual-ground voltage should become measurable deviations, and a quantitative error bound could be derived by keeping those terms in Eq. (3).","Finite OTA DC gain or incomplete settling would add noise paths not captured by the infinite-gain VCCS model, so modifying $h_{fb}$ to account for finite gain is a natural extension to test."],"forward_implications":["For a designer, computing the output noise of an SC filter reduces to drawing three capacitor-only schematics per phase and applying a one-line formula, with no PSD transfer functions or integrals.","The basic stray-insensitive integrator's sampled output noise variance grows linearly with the number of switching periods $n$, with slope set by $k_B T\\alpha/C$ and the capacitance ratios, a Wiener-process behavior.","In the passive SC low-pass filter, the output noise converges to $k_B T/C$ after many periods, recovering the continuous-time RC result.","In the OTA-based first-order low-pass filter, the steady-state output noise variance is given by the paper's Eq. (45), and for large load capacitance it simplifies to $\\gamma k_B T/C_L + 2k_B T/C$, separating OTA direct noise from switch and OTA charge-transfer noise."],"supporting_citations":[{"why":"Supplies the stray-insensitive SC integrator topology and its z-domain transfer functions used as the paper's main test case.","marker":"[1]"},{"why":"Provides the transient noise simulation technique used to validate the calculated noise variances.","marker":"[4]"},{"why":"Introduces the extended Bode theorem and the equivalent-circuit capacitance extraction for OTA-based SC circuits.","marker":"[5]"},{"why":"Provides the technique for propagating noise charge through z-transfer functions that motivates the recursive variance approach.","marker":"[6]"},{"why":"Gives the passive SC noise result that the passive LP filter calculations are checked against.","marker":"[7]"}],"fun_headline_variants":["No-integral thermal noise for SC filters","SC filter noise: three caps, no calculus","Thermal noise in SC filters by inspection","Extended Bode theorem cuts SC filter noise math","SC filter noise variance from three capacitors"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything rests on the virtual ground voltage being dominated by the OTA output term, $V\\approx h_{fb}V_{out}$, which holds only when the switched and non-switched capacitors $C_1$ and $C_2$ are much smaller than the integrating capacitor $C$; if that ratio condition fails, the OTA cannot be replaced by a conductance $h_{fb}G_m$ and Eq. (6) no longer applies.","fun_headline_variants_meta":{"raw":{"variants":["No-integral thermal noise for SC filters","SC filter noise: three caps, no calculus","Thermal noise in SC filters by inspection","Extended Bode theorem cuts SC filter noise math","SC filter noise variance from three capacitors"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000218,"raw_usage":{"total_tokens":1418,"prompt_tokens":901,"completion_tokens":517,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":517,"completion_tokens_details":{"reasoning_tokens":450}},"tokens_in":517,"tokens_out":517,"duration_ms":4961,"temperature":1.0,"reasoning_tokens":450,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:48:49.833530+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For an SC integrator with $\\alpha = C_1/C$ not much smaller than 1, compute the output noise variance exactly by numerical integration of the phase PSDs, or by transient noise simulation, and compare with Eq. (26): if the prediction deviates increasingly as $\\alpha$ grows, the small-ratio simplification is the breaking point. A second concrete check is to measure the noise of a fabricated SC filter at a ratio where $C_1 \\approx C$ and see whether the variance still follows the capacitor-only formula.","supporting_citations":[{"cited_title":"Gregorian and G","cited_arxiv_id":null,"evidence_quote":"Supplies the stray-insensitive SC integrator topology and its z-domain transfer functions used as the paper's main test case."},{"cited_title":"A New Approach for Noise Simulation in Transient Analysis,","cited_arxiv_id":null,"evidence_quote":"Provides the transient noise simulation technique used to validate the calculated noise variances."},{"cited_title":"Simple Thermal Noise Estimation of OTA-based Switched-capacitor Filters,","cited_arxiv_id":null,"evidence_quote":"Introduces the extended Bode theorem and the equivalent-circuit capacitance extraction for OTA-based SC circuits."},{"cited_title":"Rauschen von Filtern mit geschalteten Kapazit ¨aten,","cited_arxiv_id":null,"evidence_quote":"Provides the technique for propagating noise charge through z-transfer functions that motivates the recursive variance approach."},{"cited_title":"A New Method for kTC Noise Analysis in Periodic Passive Switched-capacitor Networks,","cited_arxiv_id":null,"evidence_quote":"Gives the passive SC noise result that the passive LP filter calculations are checked against."}],"review_version":1}