Simple Thermal Noise Estimation of Switched Capacitor Circuits Based on OTAs -- Part II: SC Filters

Assim Boukhayma; Christian Enz; Fran\c{c}ois Krummenacher; Sammy Cerida Rengifo

REVIEW 2 major objections 6 minor 8 references

Simple Thermal Noise Estimation of Switched Capacitor Circuits Based on OTAs -- Part II: SC Filters

T0 review · 2 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Thermal noise in OTA-based switched-capacitor filters can be read off directly from three capacitor-only circuits, without transfer functions or integrals.

desk verdict 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. read the letter →

arxiv 1908.08109 v1 pith:TYMHND2N submitted 2019-08-21 eess.SP

Christian Enz , Sammy Cerida Rengifo , Assim Boukhayma , François Krummenacher This is my paper · ORCID

classification eess.SP

keywords switched-capacitorcircuitsthermalnoisekTCextendedBodetheoremSCfiltersOTAvarianceestimationtransientsimulation

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

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the 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.

The 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.

A 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.

The 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.

Referee Report

2 major / 6 minor

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.

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 (2)

[Section II, Eq. (5)] 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.
[Appendices V-A and V-C, Eqs. (48)-(55)] 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.

minor comments (6)

[Introduction] The phrase 'very accurate tanks to the excellent matching' should be 'thanks to the excellent matching'.
[Section III-A] The sentence 'confirms than the noise increases through the periods' should read 'confirms that the noise increases through the periods'.
[Section III-C] The text reports Cin = 10 fF for the simulation in Fig. 15a, while the figure caption reports Cin = 20 fF; please harmonize these values.
[Eq. (43a)] There is a typographical double equal sign in the displayed equation; clean up the typesetting.
[Eq. (6)] 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.
[Fig. 16] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: Eq. (6) is self-cited but independently validated; downstream formulas are algebraic consequences and match transient noise simulations without parameter fitting.

full rationale

The paper's central tool, the extended Bode theorem (Eq. 6), is imported from the authors' Part I and Ref. [5]. This is a self-citation, but it is not a circular reduction: Eq. (6) is a parameter-free theorem with stated assumptions (ideal OTA, full settling, capacitive feedback), and it is not fitted to the target results of Part II. Every subsequent variance expression (Eqs. (19)-(28), (31)-(46), and the appendix derivations) is obtained by substituting capacitance values C_inf, C'_inf, C0 and the feedback factor h_fb into Eq. (6) and manipulating algebraically; no free parameter is adjusted to make the formulas match the simulations. The validation uses ELDO transient noise simulations with ideal VCCS OTAs and the same thermal-noise PSD 4kBT*gamma*Gm used in the analytical model; that is an independent numerical integration, not a fit to the analytical predictions. The approximation V ~ h_fb*Vout (Eq. 5), based on C1,C2 << C, is a stated modeling assumption and a limitation for larger capacitor ratios; it is an accuracy/correctness concern (no error bound, validated only for alpha=0.1), not a circularity. No self-definitional use, fitted-input-as-prediction, or ansatz-smuggling-by-citation is present. The passive LPF result kT/C is acknowledged as a known limit, not repackaged as new. Hence no specific circular step can be exhibited under the paper's own equations.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper leans on the extended Bode theorem from the authors' own Part I and on ideal-component assumptions. The noise excess factor gamma is an input parameter, not fitted. No new physical entities are introduced.

assumptions (5)

domain assumption The extended Bode theorem (Eq. 6) gives the thermal noise voltage variance for OTA-based SC circuits with capacitive feedback.
This is the central formula from the authors' Part I and [5]; its derivation is not reproduced in this Part II paper, so the reader must accept it as a starting point.
domain assumption The virtual ground voltage can be approximated as V ≈ h_fb * Vout, neglecting the V1 and V2 terms (Eqs. 3-5).
Requires C1, C2 << C; the paper relies on this to replace the VCCS with a conductance h_fb * Gm.
domain assumption OTA is ideal: infinite gain, zero offset, linear VCCS, no slew-rate, and the circuit fully settles in each phase.
Stated in Section II and III; real OTAs deviate from this.
domain assumption Noise sources in different phases are uncorrelated, so variances add (Eq. 1).
Standard for thermal noise in SC circuits, but not proven here.
domain assumption Switch on-conductance is large enough that transfer functions are frequency-independent.
Assumption stated before Eq. (3).

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Cite this review

Pith. "Pith review of Simple Thermal Noise Estimation of Switched Capacitor Circuits Based on OTAs -- Part II: SC Filters." pith.science (2026). https://pith.science/paper/TYMHND2N

@misc{pith2026190808109,
  author       = {Pith},
  title        = {Pith review of: Simple Thermal Noise Estimation of Switched Capacitor Circuits Based on OTAs -- Part II: SC Filters},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TYMHND2N}},
  note         = {Machine review of arXiv:1908.08109}
}

read the original abstract

In Part I of this paper, we have shown how to calculate the thermal noise voltage variances in switched-capacitor (SC) circuits using operational transconductance amplifiers (OTAs) with capacitive feedback by using the extended Bode theorem. The method allows a precise estimation of the thermal noise voltage variances by simple circuit inspection without the calculation of any transfer functions nor integrals. While Part I focuses on SC amplifiers and track&hold circuits, Part II shows how to use the extended Bode theorem for SC filters. It validates the method on the basic integrator and then on a first-order low-pass filter by comparing the analytical results to transient noise simulations showing an excellent match.

Figures

Figures reproduced from arXiv: 1908.08109 by the authors.

**Figure 3.** Capacitances calculation for the extended Bode theorem [5]. [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

**Figure 4.** Passive SC first-order LP filter [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

Figures from the paper (11 more)

**Figure 5.** Figure 5: Equivalent circuit schematics for noise variance calculation in phase [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

**Figure 6.** Figure 6: Equivalent circuit schematics for noise variance calculation in phase [PITH_FULL_IMAGE:figures/full_fig_p005_6.png]

**Figure 8.** Figure 8: Stray-insensitive non-inverting SC integrator [1]. [PITH_FULL_IMAGE:figures/full_fig_p005_8.png]

**Figure 7.** Figure 7: Simulated transient noise output RMS voltage [PITH_FULL_IMAGE:figures/full_fig_p005_7.png]

**Figure 9.** Figure 9: Equivalent circuit schematics for noise variance calculation in phase [PITH_FULL_IMAGE:figures/full_fig_p006_9.png]

**Figure 10.** Figure 10: Equivalent circuit schematics for noise variance calculation in phase [PITH_FULL_IMAGE:figures/full_fig_p006_10.png]

**Figure 12.** Figure 12: OTA-based SC first-order low-pass filter. [PITH_FULL_IMAGE:figures/full_fig_p007_12.png]

**Figure 13.** Figure 13: Equivalent circuit schematics for the calculation of the noise voltage variances of the first-order LP filter in phase [PITH_FULL_IMAGE:figures/full_fig_p008_13.png]

**Figure 14.** Figure 14: Equivalent circuit schematics for the calculation of the noise voltage variances of the first-order LP filter in phase [PITH_FULL_IMAGE:figures/full_fig_p008_14.png]

**Figure 15.** Figure 15: OTA-based SC first-order LP filter output RMS [PITH_FULL_IMAGE:figures/full_fig_p009_15.png]

**Figure 16.** Figure 16: Simulated and calculated output noise RMS voltage [PITH_FULL_IMAGE:figures/full_fig_p009_16.png]

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Reference graph

Works this paper leans on

8 extracted references · 8 canonical work pages

[5]

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C. Enz, F. Krummenacher, and A. Boukhayma, “Simple Thermal Noise Estimation of OTA-based Switched-capacitor Filters,” in 2015 Interna- tional Conference on Noise and Fluctuations (ICNF) , June 2015, pp. 1–4

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[3]

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J. Goette and C. A. Gobet, “Exact Noise Analysis of SC Circuits and an Approximate Computer Implementation,” IEEE Trans. on Circuits and Systems, vol. 36, no. 4, pp. 508–521, April 1989

work page 1989
[4]

A New Approach for Noise Simulation in Transient Analysis,

P. Bolcato and R. Poujois, “A New Approach for Noise Simulation in Transient Analysis,” in Proc. of the IEEE International Symposium on Circuits and Systems (ISCAS) , vol. 2, May 1992, pp. 887–890

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B. Furrer, “Rauschen von Filtern mit geschalteten Kapazit ¨aten,” PhD, ETHZ, Date 1983, no. 7284

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A. Boukhayma and C. Enz, “A New Method for kTC Noise Analysis in Periodic Passive Switched-capacitor Networks,” in 2015 IEEE 13th International New Circuits and Systems Conference (NEWCAS) , June 2015, pp. 1–4

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A. Papoulis, Probability, Random Variables and Stochastic Processes , 1st ed. New York: Mc Graw-Hill, 1981. Christian Enz (M84, S’12) received the M.S. and Ph.D. degrees in Electrical Engineering from the EPFL in 1984 and 1989 respectively. He is currently Professor at EPFL, Director of the Institute of Micro- engineering and head of the IC Lab. Until Apr...

work page 1981

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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Simple Thermal Noise Estimation of OTA-based Switched-capacitor Filters,

C. Enz, F. Krummenacher, and A. Boukhayma, “Simple Thermal Noise Estimation of OTA-based Switched-capacitor Filters,” in 2015 Interna- tional Conference on Noise and Fluctuations (ICNF) , June 2015, pp. 1–4

work page 2015

[2] [1]

Gregorian and G

R. Gregorian and G. C. Temes, Analog MOS Integrated Circuits for Signal Processing. Wiley, 1986

work page 1986

[3] [2]

Design-oriented Estimation of Thermal Noise in Switched-capacitor Circuits,

R. Schreier, J. Silva, J. Steensgaard, and G. C. Temes, “Design-oriented Estimation of Thermal Noise in Switched-capacitor Circuits,” IEEE Transactions on Circuits and Systems I: Regular Papers , vol. 52, no. 11, pp. 2358–2368, Nov. 2005

work page 2005

[4] [3]

Exact Noise Analysis of SC Circuits and an Approximate Computer Implementation,

J. Goette and C. A. Gobet, “Exact Noise Analysis of SC Circuits and an Approximate Computer Implementation,” IEEE Trans. on Circuits and Systems, vol. 36, no. 4, pp. 508–521, April 1989

work page 1989

[5] [4]

A New Approach for Noise Simulation in Transient Analysis,

P. Bolcato and R. Poujois, “A New Approach for Noise Simulation in Transient Analysis,” in Proc. of the IEEE International Symposium on Circuits and Systems (ISCAS) , vol. 2, May 1992, pp. 887–890

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B. Furrer, “Rauschen von Filtern mit geschalteten Kapazit ¨aten,” PhD, ETHZ, Date 1983, no. 7284

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A New Method for kTC Noise Analysis in Periodic Passive Switched-capacitor Networks,

A. Boukhayma and C. Enz, “A New Method for kTC Noise Analysis in Periodic Passive Switched-capacitor Networks,” in 2015 IEEE 13th International New Circuits and Systems Conference (NEWCAS) , June 2015, pp. 1–4

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Papoulis, Probability, Random Variables and Stochastic Processes , 1st ed

A. Papoulis, Probability, Random Variables and Stochastic Processes , 1st ed. New York: Mc Graw-Hill, 1981. Christian Enz (M84, S’12) received the M.S. and Ph.D. degrees in Electrical Engineering from the EPFL in 1984 and 1989 respectively. He is currently Professor at EPFL, Director of the Institute of Micro- engineering and head of the IC Lab. Until Apr...

work page 1981