KNOW-HOW

Mixed-Mode S-parameter

12 May 2021 Introduction to Mixed-Mode S-parameters [https://blog.teledynelecroy.com/2021/05/introduction-to-mixed-mode-s-parameters.html]

Troy Beukema (IBM Research, Yorktown Heights, NY). 03-Sep-2009. Topics in Design and Analysis of High Data Rate SERDES Systems [https://ewh.ieee.org/r5/denver/sscs/Presentations/2009_09_Beukema.pdf]

Bert Simonovich. A Guide for Single-Ended to Mixed-Mode S-parameter Conversions [https://www.signalintegrityjournal.com/articles/1832-a-guide-for-singleended-to-mixedmode-s-parameter-conversions]

single-ended S-parameters

Mixed-mode S-parameters

Missing Term in KVL

Prof. Kolb/Whites. EE 382 Applied Electromagnetics Lecture 8: Maxwell's Equations and Electrical CIrcuits [http://montoya.sdsmt.edu/ee382/lectures/382Lecture8.pdf]

Transmission-line

Telegrapher's equations

EECS 723- Microwave Engineering Spring 2.1 -The Lumped Element Circuit Model for Transmission Lines

1/20/2005 [https://www.ittc.ku.edu/~jstiles/723/handouts/2_1_Lumped_Element_Circuit_Model_package.pdf]

note [https://www.ittc.ku.edu/~jstiles/723/handouts/section_2_1_The_Lumped_Element_Circuit_Model_package.pdf]

present [https://www.ittc.ku.edu/~jstiles/723/handouts/section_2_1_The_Lumped_Element_Circuit_Model_present.pdf]

Transmission Line Wave Equation

Characteristic Impedance (\(Z_0\))

Remember, S-parameters don't mean much unless you know the value of the reference impedance (it's frequently called Z0).

simulator will read sp file's Z0 parameter

The default Z0 exported by EMX is 50

Complex Propagation Constant \(\gamma\)

TODO 📅

Input impedance (Line Impedance)

Reflection Coefficient

TODO 📅

Steady-State Solution (DC voltage division)

Sam Palermo. [https://people.engr.tamu.edu/spalermo/ecen689/lecture3_ee689_tlines.pdf]

Kyoung-Jae Chung. Special Topics in Radiation Engineering (High-voltage pulsed power engineering) [https://ocw.snu.ac.kr/sites/default/files/NOTE/Lecture_03_Transmission%20line%20theory.pdf]

David R. Jackson. [https://courses.egr.uh.edu/ECE/ECE3317/SectionJackson/Class%20Notes/Notes%208%203317%20Transmission%20Lines%20(Bounce%20Diagram).pdf]

Shouri Chatterjee [https://web.iitd.ac.in/~shouri/ell112/material/txline.pdf]

How can I go from transmission line model to lumped elements model? [https://physics.stackexchange.com/a/386603]

E157 Introduction to Radio Frequency Circuit Design [https://pages.hmc.edu/mspencer/e157/fa24/]

Shen Lin. On-Chip Inductance and Coupling Effects [http://eda.ee.ucla.edu/pub/asic.pdf]

A. Deutsch et al., "When are transmission-line effects important for on-chip interconnections?," in IEEE Transactions on Microwave Theory and Techniques, vol. 45, no. 10, pp. 1836-1846, Oct. 1997

Ho, Ron. “Chip Wires: Scaling and Efficiency.” (2003). [https://www-vlsi.stanford.edu/people/alum/pdf/0303_Ho_Wires.pdf]

—. ISSCC 2007 T3: Dealing with Issues in VLSI Interconnect Scaling, by Ron Ho

Tony Chan Carusone. ISSCC 2017 T6: Signal Integrity Analysis for Gb/s Links

Byungsub Kim ISSCC 2022 T11: "Basics of Equalization Techniques: Channels, Equalization, and Circuits"

Power Wave Equations

Peter J. Pupalaikis (Ciena). DesignCon 2026: Port Referencing in S-Parameters – Critical Insights You Need to Know

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Transmission Line Theory
        │
        ▼
v = v⁺ + v⁻,  i = (v⁺ - v⁻)/Z0      ← Physical decomposition
        │
        ▼
v⁺ = (v + iZ0)/2,  v⁻ = (v - iZ0)/2   ← Solve for forward/backward waves
        │
        ▼
a = v⁺/√Z0,  b = v⁻/√Z0               ← Normalize so |a|² = power
        │
        ▼
v = √Z0·(a+b),  i = (a-b)/√Z0          ← Invert to recover v and i

Z0 is a chosen reference impedance (typically 50Ω), which is an arbitrary normalization choice. It does not have to equal the characteristic impedance of the transmission line

Voltage scattering

transmitted voltage \[ V= \frac{2Z_l}{Z_l+R_0}\frac{V_s}{2}= \frac{Z_l}{Z_l+R_0}\cdot V_s \]

CHAPTER 6 Transmission-Line Essentials for Digital Electronics [https://ws.engr.illinois.edu/sitemanager/getfile.asp?id=178]

CHAPTER 7 Transmission-Line Analysis [https://ws.engr.illinois.edu/sitemanager/getfile.asp?id=199]

Voltage Transfer Function

Troy Beukema (IBM Research, Yorktown Heights, NY). 03-Sep-2009. Topics in Design and Analysis of High Data Rate SERDES Systems [https://ewh.ieee.org/r5/denver/sscs/Presentations/2009_09_Beukema.pdf]

Pupalaikis, Peter. (2012). The Relationship Between Discrete-Frequency S-parameters and Continuous-Frequency Responses. [pdf]

Impulse Response from S-Parameters (channel)

David Banas. A comparison of different techniques (i.e. - windowing, vector fitting, etc.) for extracting the impulse response from S-parameters. [https://github.com/capn-freako/ImpulseResponseFromSparameters/tree/main]

—, Impulse Response from Insertion Loss [https://www.signalintegrityjournal.com/articles/1847-impulse-response-from-insertion-loss]

Sam Palermo. ECEN720: High-Speed Links Circuits and Systems Spring 2025 - Lecture 3: Time-Domain Reflectometry & S-Parameter Channel Models [https://people.engr.tamu.edu/spalermo/ecen689/lecture3_ee720_tdr_spar.pdf]

Troy Beukema (IBM Research, Yorktown Heights, NY). 03-Sep-2009. Topics in Design and Analysis of High Data Rate SERDES Systems [https://ewh.ieee.org/r5/denver/sscs/Presentations/2009_09_Beukema.pdf]

Cadence application note: 7 Habits of Highly Successful S-Parameters, Spectre 21.1

Derek Rowell, 2.161 Signal Processing: Continuous and Discrete Fall 2008: Determining a System’s Causality from its Frequency Response [https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/142cd928b3b3959721198872ab97b647_causality.pdf]

Causality

Kramers–Kronig / Hilbert-transform causality relation — Magnitude/real part and phase/imaginary part cannot be chosen independently. \[ \boxed{H_I(\omega) = -\mathcal{H}\{H_R(\omega)\}} \qquad \boxed{H_R(\omega) = \mathcal{H}\{H_I(\omega)\}} \] where \(H_R(\omega) = \operatorname{Re}\{H(\omega)\}\) and \(H_I(\omega) = \operatorname{Im}\{H(\omega)\}\)

Conjugate symmetry only ensures the time-domain signal is real

Hz-domain formula

angular frequency formula

single pole low pass filter

Passivity

causality-passivity correction

P. Triverio, S. Grivet-Talocia, M. S. Nakhla, F. G. Canavero and R. Achar, "Stability, Causality, and Passivity in Electrical Interconnect Models," in IEEE Transactions on Advanced Packaging, vol. 30, no. 4, pp. 795-808, Nov. 2007 [https://sci-hub.ru/10.1109/TADVP.2007.901567]

S. Sercu, C. Kocuba, J. Nadolny, "Causality Demystified", in DesignCon 2015, Jan. 2015 [pdf]

Vinod Arjun Huddar. Causality Problems in Power Delivery Networks [https://www.signalintegrityjournal.com/articles/1217-causality-in-power-delivery-network-in-package-board]

Tyler Huddleston, Signal Edge Solutions. Causality in Practice: How Frequency Sampling and Bandwidth Shape Time-Domain Fidelity [https://www.signalintegrityjournal.com/articles/4061-causality-in-practice-how-frequency-sampling-and-bandwidth-shape-time-domain-fidelity]

Interpolation Methods

jinghua Huang SYNOPSYS. Optimum Frequency Sampling in S-Parameter Extraction and Simulation [https://ibis.org/summits/nov08a/huang.pdf]

• convolution-based method (ifft): linear, spline

• rational approximation method: bbspice, rational

Broadband SPICE (bbspice) – a Rational Interpolation Method

Real/Imaginary(RI) Magnitude/Angle(MA) interpolation

Rational Fit

Use the rational function to fit data defined in the frequency domain with an equivalent Laplace transfer function. Using rational function fitting you can create simple models for a required accuracy, model order reduction, zero phase on extrapolation to DC, and causal modeling system among other advantages

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filename  = 'touchstone/ISI.S4P';
s4p = read(rfdata.data, filename);
sdd_params = s2sdd(s2p.S_Parameters, 2);
sdd21 = squeeze(sdd_params(2, 1, :));   % s21
freq = s4p.Freq;

% rational fitting
weight = ones(size(sdd21));
weight(floor(end*3/4):end) = 0.2;
weight(2:10) = 0;

[hfit, errb] = rationalfit(freq, sdd21, 'IterationLimit', [4, 16], 'Delayfactor', 0.98, ...
    'Weight', weight, 'Tolerance', -38, 'NPoles', 32);
[sdd21_fit, ff] = freqresp(hfit, freq);

figure(1)
plot(freq/1e9, db(sdd21), 'b-'); hold on;
plot(ff/1e9, db(sdd21_fit), 'r-'); hold off; grid on;
legend('sdd21', 'sdd21\_fit');
xlabel('Freq (GHz)');
ylabel('magnitude (dB)');


ts = 1e-12;
n = 2^18;
trise = 4e-14;
[yout, tout] = stepresp(hfit, ts, n, trise);
figure(2)
plot(tout*1e12, yout, 'b-'); grid on;
xlabel('Time (ps)');
ylabel('V');
title('Step Response');


% write verilog-A
writeva(hfit, 'channel_32poles.va');

Cascading S-Parameters

Sam Palermo, Lecture 6: S-Parameter Channel Examples [https://people.engr.tamu.edu/spalermo/ecen689/lecture6_ee689_sparam_channels.pdf]

Cascading S-Parameters in Plain English: Part 3: T-Parameters in Plain English [http://thinkinitthrough.com/blogs/details/6]

S-parameters by definition require very specific control over the ports. But this breaks down when we chain multiple devices together

Cascading with ABCD Matrix

aka. transmission matrix

With load impedance \(v_3=i_3 Z_L\), cascade transfer function can be derived

Cascading with T-Matrix

aka. scattering transfer parameters, T-Parameters, transmission parameters

When you cascade two networks by multiplying their T-matrices, you're implicitly assuming that port 2 of network A and port 1 of network B share the same wave definitions

S-parameter Renormalization

[https://swb.skku.edu/emc/infromation.do?mode=download&articleNo=21913&attachNo=19783]

Gustavo Blando, S-parameter Renormalization, The Art of Cheating [https://www.signalintegrityjournal.com/articles/270-s-parameter-renormalization-the-art-of-cheating]

Spar in Tran simulation

Spar in AC simulation

reference

Bogatin, E. (2018). Signal and power integrity, simplified. Prentice Hall. [pdf]

Oh, Kyung, and Xing Yuan. High-Speed Signaling: Jitter Modeling, Analysis, and Budgeting. 1st edition. Prentice Hall, 2011. [pdf]

Pupalaikis, P. (2020). S-Parameters for Signal Integrity. Cambridge: Cambridge University Press.

microwaves101, S-parameters (https://www.microwaves101.com/encyclopedias/s-parameters)

Coelho, C. P., Phillips, J. R., & Silveira, L. M. (n.d.). Robust rational function approximation algorithm for model generation. Proceedings 1999 Design Automation Conference (Cat. No. 99CH36361). [https://sci-hub.ru/10.1109/dac.1999.781313]

Cadence IEEE IMS 2023, Introducing the Spectre S-Parameter Quality Checker and Rational Fit Model Generator

The Complex Art Of Handling S-Parameters: The importance of extraction and fitting to circuit simulation involving S-parameters [https://semiengineering.com/the-complex-art-of-handling-s-parameters]

Dr. John Choma. EE 541, Fall 2006: Course Notes #2 Scattering Parameters: Concept, Theory, and Applications [https://www.ieee.li/pdf/essay/scattering_parameters_concept_theory_applications.pdf]

Dr. Ray Kwok . Network Techniques: Conversion between Filter Transfer Function and Filter Scattering (SMatrix) Parameters [https://www.sjsu.edu/people/raymond.kwok/docs/project172/FTF%20to%20S-Matrix%20Spring%202011.pdf]

田庆诚教授 台湾中华大学 射频电路基础（公司培训）[https://www.bilibili.com/video/BV1LA41177wr/?p=3&share_source=copy_web&vd_source=5a095c2d604a5d4392ea78fa2bbc7249]

Three fast time-domain system simulation techniques:

• single-bit response method (SBR)
• double-edge response method (DER)
• multiple-edge response method (MER)

Symmetric Rising and Falling Edges

Single-Bit Response (SBR) Method

Overlapping portions of a pulse response from neighboring bits are referred to as intersymbol interference (ISI). A received waveform is formed by superimposing, in time, the pulse responses of each bit in the sequence, as illustrated in Figure 9, assuming symmetric positive and negative pulses are transmitted for 1s and 0s

To avoid spurious glitches between consecutive ones, rising and falling edge responses shall be symmetric. This is the limitation of SBR method.

Let \(p(t)\) be the SBR of the channel, \(t_s\) be the data sampling phase, \(T\) be the bit time, \(N_c\) is the number of UI in stored pulse response and \(b_m\) be the \(m\)th transmitted symbol. The voltage seen by the receiver's data sampler at the \(m\)th data sample is determined by \[ y_m = \sum_{k=m-N_c+1}^{m}b_kp(t_s+(m-k)T) \] where \(b_k \in [0, 1]\) and \(p(t) \ge 0\)

We always prepend \(Nc-1\) 0s in random bit stream for consistency.

For computation convenient, the pulse need to be positive. For differential signal and amplitude \(V_{peak}\), the peak to peak is \(-V_{peak}\) to \(+V_{peak}\). After pulse added by \(V_{peak}\), peak to peak is \(0\) to \(+2V_{peak}\).

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hold on;

yy_sum = zeros(OSR*Ns, Ns);
for idxBit = 1:Ns
    bs_split = zeros(1, Ns+Nc-1);
    bs_split(idxBit) = bs(idxBit);
    yy = zeros(OSR, Ns);
    for ii = Nc:Nc+Ns-1
        bb = bs_split(ii:-1:ii-Nc+1);
        yy(:,ii-Nc+1) = sum(bb.*yrps, 2);
    end
    yy_cont2 = reshape(yy, [], 1);
    h = plot(yy_cont2);
    h.Annotation.LegendInformation.IconDisplayStyle = 'off';
    yy_sum(:, idxBit) = yy_cont2;
end
yy_sum = sum(yy_sum, 2);    % merge
plot(yy_sum, 'k--');
plot(yy_cont, 'm-.');
grid on;
legend('sum', 'syn');
title('merge all single bit');
ylabel('mag');
xlabel('Time (\times Ts)');

The pulse response contain rising and falling edge. The 1 bit first rise from -1 to 1, then fall to -1; The 0 bit just do nothing for synthesized waveform with the help of falling edge of 1 bit.

The DC shift help deal with continuous 0 bits.

another SBR example

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A = zeros(10,21);
n = [1:10];

% post cursor
for m = 1:3
    A(m, 11+n(m)) = 0.5;
    A(m, 11-n(m)) = 0.5;
end

% one
for m = 4:10
    A(m, 11) = 1;
end

% h0 or main cursor
h0 = zeros(1, 21);
h0(1, 1) = 0.5;
h0(1, 21) = 0.5;
out = h0;

for m = 1:10
    out = conv(out, A(m, :), "full");
end


stem(out)

S-Parameter to Single Bit Response (SBR)

Mike Li, "S-Parameter to Single Bit Response (SBR) Transformation and Convergence Study" [https://ieee802.org/3/bj/public/may12/li_01_0512.pdf]

TODO 📅

Double-Edge Response (DER) Method

To handle the more general cases, with asymmetric rising and falling edges, the system response can be constructed in terms of edge transitions instead of bit responses.

The DER method decomposes the input data pattern, in terms of rising and falling edge transitions. The system response can be calculated by superimposing the shifted versions of the rising and falling edge responses : \[ y_m = \sum_{k=m-N_c+1}^{m}(b_k-b_{k-1})s_k(t_s+(m-k)T) + y_{int} \] where

\[\begin{align} s_i(t) &= r(t) -V_{low} \quad \text{if} \: (b_i\gt b_{i-1}) \\ &= V_{high}-f(t) \quad \text{otherwise} \end{align}\]

\(r(t)\) and \(f(t)\) are the rising and falling edge responses,respectively. \(V_{high}\) and \(V_{low}\) are the steady state DC levels, in response to a constant stream of ones and zeros, respectively. \(y_{int}\) is the initial DC state (either \(V_{high}\) or \(V_{low}\) ).

We always prepend \(Nc\) 0s in random bit stream for consistency.

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figure(1)
subplot(3, 1, 1)
plot(yrc);
hold on;
plot(yfc);
hold off;
legend('rising', 'falling')
grid on;
ylabel('mag');
xlabel('Time (\times Ts)');
title('step response');

subplot(3, 1, 2)
stem(bs, 'k'); grid on;
hold on;
stem((idxPreRspStart:idxPreRspEnd), bs(idxPreRspStart:idxPreRspEnd), "filled", 'r');
stem(idxPreRspStart, bs(idxPreRspStart), 'go');
stem(idxCurData, bs(idxCurData), "filled", 'm');
stem((idxPreRspStart:idxPreRspEnd)+0.5, 0.1.*bd(idxPreRspStart:idxPreRspEnd), 'bd-.');
hold off;
legend('', 'Nc bits', 'y_{int}', 'Current bit', 'Edge Transitions');
ylabel('mag');
xlabel('Time (\times UI)');
title('input stream');

subplot(3, 1, 3)
yy_cont = reshape(yy, [], 1);   % continuous version
plot(yy_cont); grid on;
title('continuous yout')
ylabel('mag');
xlabel('Time (\times Ts)');

figure(2)
hold on;
for idx = idxPreRspStart+1-Nc:idxCurData+32-Nc
    ys = yy(:, idx);
    tt = ((idx-1)*OSR+1:idx*OSR);
    h = plot(tt(:), ys(:), 'LineWidth',3);
    h.Annotation.LegendInformation.IconDisplayStyle = 'off';
end
plot(yy_cont, 'm--', 'LineWidth',1);
hold off;
legend('syn')
ylabel('mag');
xlabel('Time (\times Ts)');
title('synthesize with step response');
grid on;

Cross Talk using Convolution

Charles Moore, Computing effect of cross talk using Convolution [https://grouper.ieee.org/groups/802/3/ap/public/channel_adhoc/moore_c1_0305.pdf]

Huang Chunxing, Statistical Eye Simulation Requirements [https://ibis.org/summits/oct06a/huang.pdf]

Reference

T. C. Carusone, "Introduction to Digital I/O: Constraining I/O Power Consumption in High-Performance Systems," in IEEE Solid-State Circuits Magazine, vol. 7, no. 4, pp. 14-22, Fall 2015

Oh, Kyung Suk Dan, and Xing Chao Chuck Yuan. High-Speed Signaling: Jitter Modeling, Analysis, and Budgeting. Prentice Hall, 2011. [pdf]

Ren, Jihong and Kyung Suk Oh. "Multiple Edge Responses for Fast and Accurate System Simulations." IEEE Transactions on Advanced Packaging 31 (2008) [https://sci-hub.jp/10.1109/TADVP.2008.2002201]

Shi, Rui. "Off-chip wire distribution and signal analysis." (2008). [pdf]

X. Chu, W. Guo, J. Wang, F. Wu, Y. Luo and Y. Li, "Fast and Accurate Estimation of Statistical Eye Diagram for Nonlinear High-Speed Links," in IEEE Transactions on Very Large Scale Integration (VLSI) Systems, vol. 29, no. 7, pp. 1370-1378, July 2021, [https://sci-hub.ru/10.1109/TVLSI.2021.3082208]

Tingting Pang, DesignCon 2025: Fast BER Analysis Technique for Next Generation Chiplet Simultaneous Bi-Directional Transceiver

Voltage to Excess Phase Transformations

Excess phase as Random Noise

excess phase around \(n\)-th harmonic

\(\Delta t\) is same for any n-th harmonic

Excess phase as deterministic periodic signal

Nicola Da Dalt, ISSCC 2012: Jitter Basic and Advanced Concepts, Statistics and Applications

C. Samori, "Tutorial: Understanding Phase Noise in LC VCOs," 2016 IEEE International Solid-State Circuits Conference (ISSCC), San Francisco, CA, USA, 2016

FM \(\Delta \omega_0 \cos\omega_m t\) to PM \(\frac{\Delta \omega_0}{\omega_m} \sin\omega_m t\) depend on FM modulation frequency \(\omega_m\)

Jacobi-Anger expansions with Bessel functions

Bessel function

Spurious Tones in Spectrum & Spur-to-Carrier Ratio (SCR)

single tone PM \(A\sin\omega_m t\) to DJ don't depend on PM modulation frequency \(\omega_m\)

P.E. Allen - 2003 ECE 6440 - Frequency Synthesizers: Lecture 150 – Phase Noise-I [https://pallen.ece.gatech.edu/Academic/ECE_6440/Summer_2003/L150-PhaseNoise-I(2UP).pdf]

\[ f = \frac{\mathrm{d}(\omega_0 t + A\sin\omega_mt)}{2\pi \mathrm{d}t}=\frac{\omega_0 + A\cdot 2\pi f_m \cos\omega_m t}{2\pi} \] therefore \[ \Delta f_{pk} = Af_m \] and \[ P_{spur} = 10\log\left(\frac{\Delta f_{pk}}{2f_m}\right)^2= 10\log\left(\frac{A}{2}\right)^2 \]

Phase Noise to Jitter

Note that \(L(f )\) is defined over positive frequencies only \((f \ge 0)\)

\[\begin{align} S_{jACC(N)}(f) &= |1-z^{-N}|^2\cdot S_{jABS}(f) \\ &= |1-\cos\theta +j\sin\theta|^2\cdot S_{jABS}(f) = ((1-\cos\theta)^2 + \sin^2\theta)\cdot S_{jABS}(f) \\ &= 2(1-\cos\theta)\cdot S_{jABS}(f) = 4\sin^2(\theta/2)\cdot S_{jABS}(f) \end{align}\]

where \(\theta = 2\pi f N/f_0\)

As EQ(3.44), EQ(3.45)

the autocorrelation is the inverse Fourier transform of the PSD

\[ R_{\varphi}(t) = \int_{-\infty}^{+\infty} S_{\varphi} (f) e^{j2\pi f t}df \]

Then, \[ R_{\varphi}(0) = \int_{-\infty}^{+\infty} S_{\varphi} (f) df \qquad R_{\varphi}(NT_0) = \int_{-\infty}^{+\infty} S_{\varphi} (f) e^{j2\pi f NT_0} df \] Thus, yield EQ(3.48)

Simplified PLL Phase Noise Profile

Absolute Jitter

TODO 📅

Period Jitter

Given Simple PLL \(\frac{\mathcal{L}_0}{1 + (f / f_{3dB})^2}\) and \(\sigma_{p(N)}^2=\frac{\mathcal{L}_0 f_{3dB}}{2\pi f_0^2} \left(1 - \exp\left(-2\pi f_{3dB} N / f_0\right)\right)\)

where \(\sigma_{\mathbf{p}}^2 = \frac{\mathcal{L}_0 f_{3dB}^2}{f_0^3}\)

a random-walk DCO - \(1/f^2\) Phase Noise Profile

L. Avallone, M. Mercandelli, A. Santiccioli, M. P. Kennedy, S. Levantino and C. Samori, "A Comprehensive Phase Noise Analysis of Bang-Bang Digital PLLs," in IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 68, no. 7, pp. 2775-2786, July 2021 [https://sci-hub.st/10.1109/TCSI.2021.3072344]

Mozhgan Mansuri “Low-Power Low-Jitter On-Chip Clock Generation” thesis UCLA [https://people.engr.tamu.edu/spalermo/ecen689/pll_thesis_mansuri_ucla_2003.pdf]

[https://people.engr.tamu.edu/spalermo/ecen689/PRBS_&_PLL_model.pdf]

\(1/f^2\)$ phase noise corresponds to white frequency/period fluctuations

• \(L(f) \propto \frac{1}{f^2}\) phase noise corresponds to white frequency noise.
• Frequency/period fluctuations are related to the derivative of phase, introducing an \(f^2\) factor.

Therefore:

\[ S_{\phi}(f) \propto \frac{1}{f^2} \quad\Rightarrow\quad S_{\Delta f}(f) \propto f^2 S_{\phi}(f) = \text{constant} \]

Intersymbol interference (ISI)

\[ \color{red}\phi = 2\pi D \cdot f \]

Even-odd Jitter (EOJ)

Even-odd jitter, also known as F/2 jitter, arises from a clock signal's duty cycle not being perfectly 50%

Even-odd jitter has been referred to as duty cycle distortion by other Physical Layer specifications for operation over electrical backplane or twinaxial copper cable assemblies

Comparing DCD and F/2 Jitter Using a BERTScope® Bit Error Rate Testing Application Note [https://download.tek.com/document/65W_26040_0_Letter.pdf]

Pulse Width Jitter (PWJ)

Jeff Morriss Updated 10/25/07. Analysis of 8G PCIe Pulse Width Jitter (UI to UI Jitter_10_25.ppt)

Duty Cycle Distortion – DCD

\[ \boxed{DJ_{DCD,pp} = 2 \times UI \times \left| DCD\% - 50\% \right|} \]

Jitter fundamental & How Isolating Root Causes of Jitter [https://picture.iczhiku.com/resource/eetop/ShKgzTEiUfdFOcvn.pdf]

There are two primary causes of DCD jitter which are usually generated within a transmitter

• If the data input to a transmitter is theoretically perfect, but if the transmitter sampling threshold is offset from its ideal level, then the output of transmitter will have duty cycle distortion as a function of the slew rate of the data signal
• Another cause of duty cycle distortion can be a mismatch/asymmetry in rising and falling edge speeds

Unfortunately, other sources such as ISI almost always exist making it sometimes difficult to isolate the DCD component. One technique to test for DCD is to stimulate your system/components with a repeating 1-0-1-0… data pattern. This technique will eliminate inter-symbol interference (ISI) jitter and make viewing the DCD within the spectrum display much easier

Why clock pattern? That's because all symbols experience same inter-symbol interference, which are canceled out

[https://scdn.rohde-schwarz.com/ur/pws/dl_downloads/dl_application/application_notes/1td03/1TD03_2e_RTO_Jitter_Analysis.pdf]

Correlated vs. Uncorrelated

If the PDF of one jitter source changes when the PDF of another source is changed, then those two sources are dependent or correlated

Inter-Symbol Interference (ISI)

The primary cause of Data Dependent Jitter

Jitter measurements can be classified into three categories: cycle-to-cycle jitter, period jitter, and long-term jitter

Jitter is a key performance parameter. Need to know what matters in each case:

• PJ for digital timing
• LTJ for data converters and serial data
• Phase noise for communications (not all bandwidths matter)

The above Cycle-Cycle Jitter equation is wrong, \(\tau_1\) and \(\tau_2\) are not independent

Short Term Jitter

Period jitter, Jper is the short term variation in clock period compared to the average (mean) clock period.

Cycle-to-Cycle, Jcc is the time difference of two adjacent clock periods

Long Term Jitter (LTJ)

[https://people.engr.tamu.edu/spalermo/ecen689/PRBS_&_PLL_model.pdf]

absolute jitter is also known as long-term jitter

measuring LTJ

Jitter Calculation Examples

Jcc vs Jper

Estimating the RMS cycle-to-cycle jitter if all you have available is the RMS period jitter.

• Cycle-to-cycle jitter - The short-term variation in clock period between adjacent clock cycles. This jitter measure, abbreviated here as \(J_{CC}\), may be specified as either an RMS or peak-to-peak quantity.
• Period jitter - The short-term variation in clock period over all measured clock cycles, compared to the average clock period. This jitter measure, abbreviated here as \(J_{PER}\), may be specified as either an RMS or peak-to-peak quantity.

Let the variable below represent the variance of a single edge's timing jitter, i.e. the difference in time of a jittery edge versus an ideal edge, \(\sigma^2_j\)

If each edge's jitter is independent then the variance of the period jitter can be written as \[\begin{align} \sigma^2_\text{jper} &= (\sigma_\text{j(n+1)}-\sigma_\text{j(n)})^2 \\ &= \sigma_\text{j(n+1)}^2-2\sigma_\text{j(n+1)}\sigma_\text{j(n)})+\sigma_\text{j(n)})^2\\ &= \sigma_\text{j(n+1)}^2+\sigma_\text{j(n)})^2 \\ &=2\sigma^2_j \end{align}\]

In every cycle-to-cycle measurement we use one "interior" clock edge twice and therefore we must account for this

\[\begin{align} \sigma^2_\text{jcc} &= (\sigma_\text{jper(n+1)}-\sigma_\text{jper(n)})^2 \\ &=(\sigma_\text{j(n+2)}-2\sigma_\text{j(n+1)}+\sigma_\text{j(n)})^2 \end{align}\]

Since each edge's jitter is assumed to be independent and have the same statistical properties we can drop the cross correlation terms and write:

\[\begin{align} \sigma^2_\text{jcc} &=(\sigma_\text{j(n+2)}-2\sigma_\text{j(n+1)}+\sigma_\text{j(n)})^2 \\ &=\sigma_\text{j(n+2)}^2+4\sigma_\text{j(n+1)}^2+\sigma_\text{j(n)}^2 \\ &=6\sigma_\text{j}^2 \end{align}\]

The ratio of the variances is therefore \[ \frac{\sigma^2_\text{jcc}}{\sigma^2_\text{jper}} = \frac{6\sigma_\text{j}^2} {2\sigma_\text{j}^2}=3 \] Then \[ \sigma_\text{jcc} = \sqrt{3}\sigma_\text{per} \]

[Timing 101 #8: The Case of the Cycle-to-Cycle Jitter Rule of Thumb, Silicon Labs]

Phase Noise Modeling

w/ noisefile

Tawna, "Modeling Oscillators with Arbitrary Phase Noise Profiles"[https://community.cadence.com/cadence_blogs_8/b/rf/posts/modeling-oscillators-with-arbitrary-phase-noise-profiles]

—, "How to Specify Phase Noise as an Instance Parameter in Spectre Sources (e.g. vsource, isource, Port)" [https://community.cadence.com/cadence_blogs_8/b/rf/posts/how-to-specify-phase-noise-as-an-instance-parameter-in-spectre-sources-e-g-vsource-isource-port]

driving an otherwise ideal oscillator with direct phase modulation makes its noise purely PM — a good model of near-carrier oscillator noise

In verilog-A model oscwphnoise.va, Norton-equivalent circuit \[ \boxed{v(t) = A\cos(\omega_0 t + \phi(t))\approx A[\cos\omega_0 t - \sin\omega_0 t \cdot \varphi(t)]} \]

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`define db10_real(x) pow(10, (x)/10)
`define dbm2pow(x) `db10_real(((x)-30))
`define pow2v(x,r) sqrt(8*(r)*(x))

The first two macros convert the available power from dBm to watts: \[ P_{\mathrm{W}} = 10^{(P_{\mathrm{dBm}}-30)/10}. \] The third macro calculates the open-circuit peak voltage \[ V_{\text{oc,pk}}=\sqrt{8R_{\text{out}}P_{\mathrm{W}}}. \] The factor \(8\) follows from a matched source: \[ V_{\text{load,pk}}=\frac{V_{\text{oc,pk}}}{2}, \] and therefore \[ P_{\mathrm{avail}} = \frac{V_{\text{load,rms}}^2}{R_{\text{out}}} = \frac{\left(V_{\text{oc,pk}}/2\sqrt{2}\right)^2}{R_{\text{out}}} = \frac{V_{\text{oc,pk}}^2}{8R_{\text{out}}}. \] For the defaults, \[ P=10\ \mathrm{dBm}=10\ \mathrm{mW}, \qquad R_{\text{out}}=50\ \Omega, \] so \[ V_{\text{oc,pk}} = \sqrt{8(50)(0.01)} = 2\ \mathrm{V}. \] With a matched \(50\ \Omega\) load, the load voltage is \(1\ \mathrm{V_{pk}}\), corresponding to \(10\ \mathrm{dBm}\).

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`include "constants.vams"
`include "disciplines.vams"

// Behavioral oscillator with an arbitrary phase noise profile.

// fundname: the name of the fundamentail frequency default: ""
// power: available power (dBm) 	default: 10 dBm
// freq: output frequency (Hz)          default: 1 GHz
// rout: output impedance (Ohm)        	default: 50 Ohm

`define db10_real(x) pow(10, (x)/10)
`define dbm2pow(x) `db10_real( ((x)-30) )
`define pow2v(x,r) sqrt(8*(r)*(x))  // open-circuit peak voltage

module oscwphnoise(out, ph);
inout out;
input ph;
electrical out;
electrical ph;
electrical gnd;
ground gnd;
electrical int;

parameter real power = 10 ;
parameter real rout = 50 ;
parameter real freq = 1e+09 ;
  
isource #(.type("sine"), .ampl(`pow2v(`dbm2pow(power),rout)/rout), .freq(freq) ) is1(gnd,out);
vsource #(.type("sine"), .ampl(`pow2v(`dbm2pow(power),rout)/rout), .sinephase(-90), .freq(freq) ) vs1(gnd,int);
 
analog begin
    I(out) <+ -V(int)*V(ph); // - \sin\omega_c t \cdot \phi(t)
    I(out) <+ V(out)/rout;
end

endmodule

Note that int is purely internal — no current flows there, and it never appears at the output; vs1 exists only to be sampled by the multiplier

isource and vsource are not built-in Verilog-A language constructs. They are Cadence-provided behavioral source modules \[ \boxed{ \texttt{isource},\ \texttt{vsource} \text{ are instantiated source models supplied by Cadence} } \]

1. vsource (p, n) forces V(p) − V(n) = w(t), where for type="sine" the waveform is w(t) = ampl·sin(2πf(t − delay) + sinephase·π/180), with sinephase in degrees.

​ So in the model, vs1(gnd, int) means V(gnd) − V(int) = w(t), i.e. V(int) = −w(t) — the polarity flips because ground was listed first.

2. isource (p, n): positive current flows from p, through the source, to n — it is pulled out of node p and injected into node n.

​ So is1(gnd, out) pumps ampl·sin(ωt) into node out, which is exactly why the blog wired it as (gnd, out): to drive the output. Writing is1(out, gnd) with the same amplitude would sink that current from out instead

3. I(a,b) <+ expr adds a current expr flowing a → b through that branch.

​ So I(out) <+ V(out)/rout is a resistor from out to ground,

​ and **I(out) <+ -X** injects X into out (positive contribution = current leaving the node)

For modern spectreRF, PORTs and other sources with noisefiles or instance parameter are easier and correct methods

Starting in MMSIM 13.1, you can specify the phase noise as an instance parameter in Spectre sources, including port, vsource and isource

The Generate noise? button (corresponds to isnoisy parameter on the port) is by default set to "yes"

w/ periodic jitter inject

我想流tsmc28nm, 振荡器噪声建模：从Phase Noise 到Jitter [xiaohongshu]

Claude Fable5 [Github gist]

Matthew Schubert. Colouring Noise - Generating coloured noise to simulate physical processes [https://blog.ioces.com/matt/posts/colouring-noise/] [https://gist.github.com/m-schubert/45c562146c6607b8990f1e8f34ff87b0]

The same jitter standard deviation \(\sigma_a^2\) can correspond to completely different phase noise shapes. A single \(\sigma_a^2\) cannot define the PN shape, because σ is an integral of the spectrum, and integration throws away shape information.

The shape lives in the correlation structure of the jitter sequence — equivalently, in how \(\sigma^2_{p(N)}\) grows with \(N\) — and in your simulation it is set by how you generate and inject the samples, not by the \(\sigma_a^2\) value itself

Flat: \(\mathcal{L}(f)=\mathcal{L}_0\)

statistically independent edges, the generator therefore perturbs each ideal edge directly, with no accumulation \[ t_n \;=\; nT_0 + a_n, \qquad a_n \stackrel{\text{iid}}{\sim} \mathcal{N}\!\left(0,\;\sigma_a^2\right), \qquad \sigma_a^2 = \frac{\mathcal{L}_0}{4\pi^2 f_0} \]

\(1/f^2\): \(\mathcal{L}(f)=\mathcal{L}_1 f_1^2/f^2\) (white FM)

\(\sigma_p^2(N)\;=\;\frac{\mathcal{L}_1 f_1^2}{f_0^3}\,N\) — Linear growth in \(N\) is the signature of independent increments, the generator injects an iid error into every period and accumulates edges: \[ t_n \;=\; t_{n-1} + T_0 + d_n, \qquad d_n \stackrel{\text{iid}}{\sim} \mathcal{N}\!\left(0,\;\sigma_\Delta^2\right), \qquad \sigma_\Delta^2 \;=\; \sigma_p^2(N{=}1) \;=\; \frac{\mathcal{L}_1 f_1^2}{f_0^3} \]

\(1/f^3\): \(\mathcal{L}(f)=\mathcal{L}_1 f_1^3/f^3\) (flicker FM)

shaping noise in the frequency domain

fft and ifft are just tool to transform between time domain and frequency domain — we only care about input PSD (uniformly sampled discrete white noise) and final output PSD

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W = np.fft.rfft(rng.standard_normal(M))

This generates white Gaussian noise and transforms it to the frequency domain, i.e. Unit-variance sampled white noise has one-sided PSD \[ S_W = \frac{2}{f_s} \]

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H[1:] = 1.0 / np.sqrt(f[1:])

Filtering amplitudes by \(1/\sqrt f\) filters power by \[ |H(f)|^2=\frac{1}{f} \]

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return x * np.sqrt(c * fs / 2.0)

Therefore, the resulting sequence has a \(1/f\) PSD. DC is set to zero because ideal \(1/f\) noise diverges at \(f=0\).

Unit-variance sampled white noise has one-sided PSD \(2/f_s\). Thus the output PSD is \[ S_o(f) = \frac{2}{f_s}\frac{1}{f} \left(\frac{c f_s}{2}\right) = \frac{c}{f} \]

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c_fl  = 2 * L3 * f1 ** 3 / f0 ** 4      # 1/f^3: period-error PSD S_d(f)=c/f [s^2/Hz]

\[ c_{fl} = \frac{2L_3f_1^3}{f_0^4} \]

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J["1_f3_only"] = np.cumsum(d3)

Accumulation is discrete-time integration. Its exact transfer function is \[ |H_{\mathrm{acc}}(f)|^2 = \frac{1}{4\sin^2(\pi f/f_0)} \approx \frac{f_0^2}{4\pi^2f^2}, \qquad f\ll f_0 \] Consequently, the accumulated time-error PSD is

\[ S_J(f) \approx \frac{c}{f}\frac{f_0^2}{4\pi^2f^2} = \frac{c f_0^2}{4\pi^2f^3} \]

Timing error is converted to phase using \(\phi=2\pi f_0J\)

Therefore,

\[ S_\phi(f) =(2\pi f_0)^2S_J(f) \approx \frac{c f_0^4}{f^3} \] Since the code uses the usual phase-noise definition \(L(f)=S_\phi(f)/2\)

\[ L(f)=\frac{c f_0^4}{2f^3} \] The coefficient is selected as \(c_{fl} = \frac{2L_3f_1^3}{f_0^4}\). Substitution gives exactly the desired target:

\[ \boxed{L(f)=L_3\left(\frac{f_1}{f}\right)^3} \] In short:

\[ \underbrace{\frac1f}_{\text{flicker period noise}} \times \underbrace{\frac1{f^2}}_{\text{accumulation}} = \underbrace{\frac1{f^3}}_{\text{phase noise}} \]

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EULER_GAMMA = 0.5772156649015329

# ----------------------------------------------------------------- targets ----
f0 = 5.015e9                # DCO frequency [Hz]
T0 = 1.0 / f0
f1 = 1e6                    # reference offset for the sloped lines [Hz]
L0_dB = -155.0              # flat floor            [dBc/Hz]
L2_dB = -128.0              # 1/f^2 line @ 1 MHz    [dBc/Hz]
L3_dB = -130.0              # 1/f^3 line @ 1 MHz    [dBc/Hz]
L0, L2, L3 = (10 ** (x / 10) for x in (L0_dB, L2_dB, L3_dB))

M    = 2 ** 22              # number of simulated periods (~0.84 ms)
fmin = f0 / M               # low-frequency cutoff set by record length
rng  = np.random.default_rng(7)

# ----------------------------- sigma_p^2(N)  ->  generator parameters ---------
var_a = L0 / (4 * np.pi ** 2 * f0)      # flat : edge-jitter variance   [s^2]
var_d = L2 * f1 ** 2 / f0 ** 3          # 1/f^2: per-period variance = sigma_p^2(1)
c_fl  = 2 * L3 * f1 ** 3 / f0 ** 4      # 1/f^3: period-error PSD S_d(f)=c/f [s^2/Hz]

print(f"sigma_a  (flat, per edge)    = {np.sqrt(var_a)*1e15:8.3f} fs")
print(f"sigma_d  (1/f^2, per period) = {np.sqrt(var_d)*1e15:8.3f} fs")
print(f"flicker coefficient c        = {c_fl:.3e}  s^2  (S_d = c/f)")

def flicker(M, fs, c, rng):
    """Zero-mean Gaussian sequence with one-sided PSD c/f, sampled at fs."""
    W = np.fft.rfft(rng.standard_normal(M)) # Unit-variance sampled white noise has one-sided PSD 2/fs
    f = np.fft.rfftfreq(M, d=1.0 / fs)
    H = np.zeros_like(f)
    H[1:] = 1.0 / np.sqrt(f[1:])            # |H|^2 = 1/f, DC removed
    x = np.fft.irfft(W * H, n=M)
    return x * np.sqrt(c * fs / 2.0)        # c/f

# --------------------------------------------- generate the three sequences ---
a  = rng.standard_normal(M) * np.sqrt(var_a)    # flat : direct edge displacement
d2 = rng.standard_normal(M) * np.sqrt(var_d)    # white period errors
d3 = flicker(M, f0, c_fl, rng)                  # flicker period errors

# --------------------- per-period injection  +  edge accumulation -------------
#   t_n = n*T0 + J(n).  Only the deviation J is accumulated -> no precision loss.
J = {
    "flat_only": a,                              # no accumulation (white PM)
    "1_f2_only": np.cumsum(d2),                  # random walk     (white FM)
    "1_f3_only": np.cumsum(d3),                  # flicker walk    (flicker FM)
}
J["all_noise"] = J["1_f2_only"] + J["1_f3_only"] + a

def Lref_dB(name, f):
    if name == "flat_only":
        return np.full_like(f, L0_dB)
    if name == "1_f2_only":
        return L2_dB - 20 * np.log10(f / f1)
    if name == "1_f3_only":
        return L3_dB - 30 * np.log10(f / f1)
    return 10 * np.log10(L0 + L2 * f1 ** 2 / f ** 2 + L3 * f1 ** 3 / f ** 3)

Let the input period error be (d[n]), and let the accumulated timing error be \(J[n]=J[n-1]+d[n]\)

This is exactly what np.cumsum(d) implements, assuming \(J[-1]=0\)

Taking the (z)-transform: \(J(z)=z^{-1}J(z)+D(z)\)

Rearranging \(J(z)(1-z^{-1})=D(z)\)

so the accumulator transfer function is \[ H_{\mathrm{acc}}(z) =\frac{J(z)}{D(z)} =\frac{1}{1-z^{-1}} \]

To obtain its frequency response, evaluate it on the unit circle: \(z=e^{j\omega}\)

Therefore, \[ H_{\mathrm{acc}}(e^{j\omega}) = \frac{1}{1-e^{-j\omega}} \] Its power gain is the squared magnitude: \[ |H_{\mathrm{acc}}(e^{j\omega})|^2 = \frac{1}{|1-e^{-j\omega}|^2} \] The denominator can be simplified as

\[\begin{aligned} |1-e^{-j\omega}|^2 &=(1-e^{-j\omega})(1-e^{j\omega})\\ &=2-e^{j\omega}-e^{-j\omega}\\ &=2-2\cos\omega\\ &=4\sin^2\left(\frac{\omega}{2}\right) \end{aligned}\]

Thus the exact discrete-time power gain is

\[ \boxed{ |H_{\mathrm{acc}}(e^{j\omega})|^2 = \frac{1}{4\sin^2(\omega/2)} } \]

references

AN10007 Clock Jitter Definitions and Measurement Methods, SiTime [pdf]

SERDES Design and Simulation Using the Analog FastSPICE Platform, Silicon Creations [pdf]

Flexible clocking solutions in advanced processes from 180nm to 5nm, Silicon Creations [pdf]

One-size-fits-all PLLs for Advanced Samsung Foundry Processes, Silicon Creations [pdf]

Circuit Design and Verification of 7nm LowPower, Low-Jitter PLLs, Silicon Creations, [pdf]

Lecture 10: Jitter, ECEN720: High-Speed Links Circuits and Systems Spring 2023 [pdf]

Jitter 360° Knowledge Series [pdf, slides]

N. Da Dalt, "Tutorial: Jitter: Basic and Advanced Concepts, Statistics, and Applications," 2012 IEEE International Solid-State Circuits Conference, San Francisco, CA, USA, 2012 [slides, transcript ]

Two-port parameters of MOS

\(Y_{11},Y_{12},Y_{21},Y_{22}\) are same with four AC simulation, but more elegant and concise

Gate (thermal) noise

Two-Side Poly Contact & folding

Both scheme yield a total distributed resistance of \(R_G/4\) for gate noise calculation

folding

finger 0 \[\begin{align} \overline{i_{tot,0}^2} &= \left(\frac{g_m}{2} \right)^2(4kT\frac{R_G/2}{3}) \\ &= g_m^2\left(4kT\frac{R_G}{3}\right)\frac{1}{2^2\cdot 2} \end{align}\]

similarly finger 1 \[ \overline{i_{tot,1}^2} = g_m^2\left(4kT\frac{R_G}{3}\right)\frac{1}{2^2\cdot 2} \] Assuming uncorrelated \[ \overline{i_{tot}^2} = \sum_{N=0}^1\overline{i_{tot,N}^2} =g_m^2\left(4kT\frac{R_G}{3}\right)\frac{1}{2^2\cdot 2} \cdot 2 = g_m^2\left(4kT\frac{R_G}{3}\right)\frac{1}{2^2} \] Generally \[ \overline{i_{tot}^2} = g_m^2\left(4kT\frac{R_G}{3}\right)\frac{1}{N^2} \] where the gate is decomposed into \(N\) parallel fingers

two-side poly contact

We fracture Gate poly at the center point, then we obtain 2 segments, both have same \(\frac{g_m}{2}\) and \(R_G/2\).

The derivation procedure is same with folding structure, i.e. plug \(N=2\) into \(\overline{i_{tot}^2} = g_m^2\left(4kT\frac{R_G}{3}\right)\frac{1}{N^2}\)

That is \[ \overline{i_{tot}^2} = g_m^2\left(4kT\frac{R_G}{12}\right) \] The input referred noise of gate resistance \[ \overline{V_{nRG}^2} = 4kT\frac{R_G}{12} \]

four equal gate fingers

\[ \overline{i_{tot}^2} = g_m^2\left(4kT\frac{R_G}{3}\right)\frac{1}{4^2} \] Then \[ \overline{V_{nRG}^2} = \frac{\overline{i_{tot}^2}}{g_m^2} =4kT\frac{R_G}{48} \]

Gate Res noise contribution

Representaion of resistor thermal noise by a current source

\(R_s\): previous stage output impedance

\(R_G\), \(I_n\): gate resistance and its noise current

\(R_L\): next stage input impedance

with \(R_SI_o + (I_o - I_n)R_G + I_oR_L=0\), we have \(I_o=I_n\frac{R_G}{R_S+R_G+R_L}\), then \[\begin{align} V_S &= -I_n\frac{R_GR_S}{R_S+R_G+R_L} \overset{R_L\to \infty}{\longrightarrow} 0\\ V_G &= I_n\frac{R_GR_L}{R_S+R_G+R_L} \overset{R_L\to \infty}{\longrightarrow} I_nR_G \end{align}\]

The above equation show that gate resistance noise don't contribute to previous stage output noise, however contribute to next stage

resistor thermal noise by a voltage source give same conclusion

Gate resistance in PEX

They fracture the poly line at the intersection with the active (diffusion) layer, breaking it into "gate poly"(poly over active) and "field poly" (poly outside active)

gploy, fpoly

Gate poly is also fractured at the center point. Gate instance pin of the MOSFET (SPICE model) is connected to the center point of the gate poly. Gate poly is described by two parasitic resistors, connecting the fracture points.

MOSFET extrinsic parasitic capacitance between gate poly and source / drain diffusion and contacts is calculated by parasitic extraction tools, and assigned to the nodes of the resistive networks.

Different extraction tools do this differently - some tools connect these parasitic capacitances to the center point of the gate poly, while some other tools connect them to the end points of the gate poly resistors.

\(\Delta\) gate model

This distributed network has a different AC and transient response than a simple lumped one-R and one-C circuit.

It was shown [B. Razavi] that such RC network behaves approximately the same as a network with one R and one C element, where C is the total capacitance, and R=1/3 * W/L rsh for single-side connected poly, and R=1/12 W/L * rsh for double-sided connected poly.

These coefficients - 1/3 and 1/12 - effectively enable an accurate reduced order model for the gate, reducing a large number of R and C elements to two (or three) resistors and one capacitor.

Gate Delta Model: where a gate is described by two positive and one negative resistors

only applicable to contacts on gate overhangs

invalid for self-aligned gate contacts, where gate contact land directly on top of gate, not gate overhang

1. 1-side gate contact \[ R_{eq,1side} =R_1 \parallel (R_2+R_1)= \frac{R_G}{6}\parallel (-\frac{R_G}{2}+\frac{R_G}{6})=\frac{R_G}{3} \]

2. 2-side contact \[ R_{eq,2side}= R_1 \parallel R_1 = \frac{R_G}{6}\parallel \frac{R_G}{6} = \frac{R_G}{12} \]

Some SPICE simulators have problems handling negative resistors, that's possibly why this model did not get a wide adoption. Some foundries and PDKs support delta gate model, while some others don't.

Vertical component of gate resistance

In "old" technologies (pre-16nm), gate resistance was dominated by lateral resistance. However, in advanced technologies, multiple interfaces between gate material layers lead to a large vertical gate resistance.

It's very easy to check this in DSPF file - if gate instance pin is connected directly to the center of the gate poly - vertical resistance is not accounted for. If it is connected by a positive resistor to the center of the gate poly - that resistors represents the vertical gate resistance.

decap

leakage current is determined by \(R_s + R_p\), and \(R_p \gg R_s\)

• low freq: \(Z=R_s + R_p\)
• high freq: \(Z=R_s\)

An example: \(R_s=200\space \Omega\), \(R_p = 8 \space M\Omega\) and \(C_\text{gate}=10\space fF\)

reference

⭐ B. Razavi, Y. Ran, and K. F. Lee, “Impact of Distributed Gate Resistance on the Performance of MOS Devices,” IEEE Trans. Circuits and Systems, Part I, pp. 750–754, Nov. 1994.

⭐ Maxim Ershov, Diakopto. "Gate Resistance in IC design flow", [link, pdf], [https://www.synopsys.com/blogs/chip-design/gate-resistance-ic-design-flow.html]

A.J.Sholten et al., "FinFET compact modelling for analogue and RF applications", IEDM'2010 [https://sci-hub.se/10.1109/IEDM.2010.5703322]

Saha, Samar K.. “FinFET Devices for VLSI Circuits and Systems.” (2020).

Harpe, Pieter J. A., Andrea Baschirotto and Kofi A. A. Makinwa. “Hybrid ADCs, Smart sensors for the IoT, and Sub-1V and Advanced node analog circuit design: Advances in Analog Circuit Design 2017.” (2018).

Chauhan, Yogesh Singh. FinFET Modeling for IC Simulation and Design: Using the BSIM-CMG Standard. London, UK: Academic Press, 2015.

A.J.Sholten et al., "FinFET compact modelling for analogue and RF applications", IEDM'2010, p.190.

W. Wu and M. Chan, “Gate resistance modeling of multifin MOS devices,” IEEE Electron Device Letters, vol. 27, no. 1, pp. 68-70, Jan. 2006.

A. L. S. Loke, C. K. Lee and B. M. Leary, "Nanoscale CMOS Implications on Analog/Mixed-Signal Design," 2019 IEEE Custom Integrated Circuits Conference (CICC), Austin, TX, USA, 2019, pp. 1-57, doi: 10.1109/CICC.2019.8780267.

Nonzero FFT bins

everynanocounts. Memos on FFT With Windowing. [https://a2d2ic.wordpress.com/2018/02/01/memos-on-fft-with-windowing]

The problem with sine-wave scaling is that the noise power is, on average, evenly distributed over all FFT bins, whereas the sine-wave power is concentrated in only a few bins. With sine-wave scaling, the power of individual sine-wave components can be read directly from the spectral plot, but in order to determine the noise power, the powers of all the noise bins must be added together.

We can't use Fourier transform of random signal

[https://raytroop.github.io/2023/11/10/random/#lti-systems-on-wss-processes]

\[ \mathrm{SNR} = \frac{X_\text{sig}^2}{X_\text{n}^2N} = \frac{X_\text{sig}^2\cdot \sum_k W_k^2}{X_\text{n}^2N\cdot \sum_k W_k^2} = \frac{\sum_\text{nb} X_\text{w,sig}^2}{N X_\text{w,n}^2} \] where \(\text{nb}\) is number of non-zero FFT bins

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subplot(3,1,1);
for N = [16 256 1024]
    wrect = rectwin(N);
    Wrect = fftshift(fft(wrect));
    Wrect_mag = abs(Wrect)/sum(wrect);
    nb_rect = sum(Wrect_mag > 0.1);
    fprintf('Number of nonzero FFT bin(rect window N=%d): %d\n', N, nb_rect);
    stem(1:N, Wrect_mag, LineWidth=2)
    hold on
end
grid on
legend('16', '256', '1024')
title('rect window')

subplot(3,1,2);
for N = [16 256 1024]
    whann = hann(N);
    Whann = fftshift(fft(whann));
    Whann_mag = abs(Whann)/sum(whann);
    nb_hann = sum(Whann_mag > 0.1);
    fprintf('Number of nonzero FFT bin(hann window N=%d): %d\n', N, nb_hann);
    stem(1:N, Whann_mag, LineWidth=2)
    hold on
end
grid on
legend('16', '256', '1024')
title('Hanning window')

subplot(3,1,3);
for N = [16 256 1024]
    whann2 = (1-cos(2*pi*(0:N-1)/N)).^2/2^2;
    Whann2 = fftshift(fft(whann2));
    Whann2_mag = abs(Whann2)/sum(whann2);
    nb_hann2 = sum(Whann2_mag > 0.1);
    fprintf('Number of nonzero FFT bin(hann2 window N=%d): %d\n', N, nb_hann2);
    stem(1:N, Whann2_mag, LineWidth=2)
    hold on
end
grid on
legend('16', '256', '1024')
title('Hann2 window')


% Number of nonzero FFT bin(rect window N=16): 1
% Number of nonzero FFT bin(rect window N=256): 1
% Number of nonzero FFT bin(rect window N=1024): 1
% Number of nonzero FFT bin(hann window N=16): 3
% Number of nonzero FFT bin(hann window N=256): 3
% Number of nonzero FFT bin(hann window N=1024): 3
% Number of nonzero FFT bin(hann2 window N=16): 5
% Number of nonzero FFT bin(hann2 window N=256): 5
% Number of nonzero FFT bin(hann2 window N=1024): 5

[https://web.engr.oregonstate.edu/~temes/ece627/Lecture_Notes/FFT_for_delta_sigma_spectrum_estimation.pdf]

where \(\mathrm{NBW} = \frac{\sum_{n}|w[n]|^2}{ \left| \sum_{n} w[n] \right|^2}\)

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% excerpt of A.4 An Example in Pavan, Schreier and Temes, "Understanding Delta-Sigma Data Converters, Second Edition" ISBN 978-1-119-25827-8
%
% Compute modulator output and actual NTF
%
OSR = 32;
ntf0 = synthesizeNTF(5,OSR,1);
N = 64*OSR;
fbin = 11;
u = 1/2*sin(2*pi*fbin/N*[0:N-1]);
[v tmp1 tmp2 y] = simulateDSM(u,ntf0);
k = mean(abs(y)/mean(y.^2))
ntf = ntf0 / (k + (1-k)*ntf0);
%
% Compute windowed FFT and NBW
%
w = hann(N); % or ones(1,N) or hann(N).^2
nb = 3; % 1 for Rect; 5 for Hann^2
w1 = norm(w,1);
w2 = norm(w,2);
NBW = (w2/w1)^2
V = fft(w.*v)/(w1/2);
%
% Compute SNR
%
signal_bins = fbin + [-(nb-1)/2:(nb-1)/2];
inband_bins = 0:N/(2*OSR);
noise_bins = setdiff(inband_bins,signal_bins);
snr = dbp(sum(abs(V(signal_bins+1)).^2)/sum(abs(V(noise_bins+1)).^2))

Noise Floor

[http://individual.utoronto.ca/schreier/lectures/2015/1.pdf]

General Formula

signal tone power \[ P_{\text{sig}} = 2 \frac{X_{w,sig}^2}{S_1^2} \]

noise power \[ P_n = \frac{X_{w,n}^2}{N\cdot S_2}N=\frac{X_{w,n}^2}{S_2} \]

where white noise, \(X_n(i) = X_n(j)\) for any \(i \neq j\)

Therefore, displayed SNR is obtained \[\begin{align} \mathrm{SNR} &= 10\log10\left(\frac{X_{w,sig}^2}{X_{w,n}^2}\right) \\ &= 10\log_{10}\left(\frac{P_{\text{sig}}}{P_n}\right) + 10\log_{10}\left(\frac{S_1^2}{2S_2}\right) \\ &= \mathrm{SNR}'-10\log_{10}\left(\frac{2S_2}{S_1^2}\right) \\ &= \mathrm{SNR}'-10\log_{10}(2\cdot\mathrm{NBW}) \end{align}\]

DFT's output \(\mathrm{SNR}\)

\[ \text{PS(k)} = \text{PSD(k)} \cdot \text{ENBW} \]

where Effective Noise BandWidth \(\text{ENBW} =f_{\text{res}} \cdot \frac{N S_2}{S_1^2}\)

yield \[ \text{PS(k)} = \text{PSD(k)} \cdot f_{\text{res}} \cdot \frac{N S_2}{S_1^2} = \left\{\text{PSD(k)} \cdot f_{\text{s}} \right\} \cdot \frac{S_2}{S_1^2} \] where \(\left\{\text{PSD(k)} \cdot f_{\text{s}} \right\} =\text{const}\), i.e. \(x_\text{n,rms}\)

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ratio_list = [];
for len = 6:12
 wnd = hanning(2^len);
 S2 = sum(wnd.^2);
 S1 = sum(wnd);
 ratio = S2./S1.^2;
 ratio_list(end+1) = ratio;
end

ratio_db = 10*log10(ratio_list);
plot(2.^[6:1:12], ratio_db, 'ro-', LineWidth=4)
grid on; grid minor;

xlabel('N'); ylabel("$10\log(S_2/S_1^2)$",'Interpreter', 'latex')

FFT Noise Floor

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N = 2048;
cycles = 67;
fs = 1000;
fx = fs*cycles/N;
LSB = 2/2^10;
%generate signal, quantize (mid-tread) and take FFT
x = cos(2*pi*fx/fs*[0:N-1]);
x = round(x/LSB)*LSB;
s = abs(fft(x));
s = s(1:end/2)/N*2;
% calculate SNR
sigbin = 1 + cycles;
noise = [s(1:sigbin-1), s(sigbin+1:end)];
snr = 10*log10( s(sigbin)^2/sum(noise.^2) );

sdb = 20*log10(s);

% How to plot a series of numbers which some of them are inf?
% https://www.mathworks.com/matlabcentral/answers/476643-how-to-plot-a-series-of-numbers-which-some-of-them-are-inf
plot([0:N/2-1]/N, max(sdb, -120), LineWidth=4)
hold on;
plot([0 0.5], [-61.9 -61.9], 'r--', LineWidth=2)
plot([0 0.5], [-92 -92], 'm--', LineWidth=2)
grid on; grid minor;
ylim([-120 0]); xlim([0 0.5]);
xlabel('Frequency [f/fs]'), ylabel('DFT Magnitude [dBFS]');
title('2048 point FFT, SNR=61.90dB')

Rectangular Window

DFT bin's output noise standard deviation (rms) value is proportional to \(\sqrt{N}\), and the DFT's output magnitude for the bin containing the signal tone is proportional to \(N\)

signal tone power \[ P_{\text{sig}} = 2 \frac{X_{w,sig}^2}{N^2} \]

noise power \[ P_n = \frac{X_{w,n}^2}{N} \]

The displayed SNR \[ \mathrm{SNR} = \mathrm{SNR}' - 10\log_{10}(2/N) \] If we increase a DFT's size from \(N\) to \(2N\), the DFT's output SNR increased by 3dB. So we say that a DFT's processing gain increases by 3dB whenever \(N\) is doubled.

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for N=[2^6 2^8 2^10 2^12]
  wd = rectwin(N);
  nbw = enbw(wd)/N;
  snr_shift = 10*log10(nbw * 2);
  disp(snr_shift);
end

output:

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-15.0515

-21.0721

-27.0927

-33.1133

How spectrum analyzer work?

We tried to plot a power spectral density together with something that we want to interpret as a power spectrum

• spectrum of a periodic signal
• spectral density of a broadband signal such as noise

Sine-wave components are located in individual FFT bins, but broadband signals like noise have their power spread over all FFT bins!

The noise floor depends on the length of the FFT

PS and PSD

The spectral density format is appropriate for random or noise signals but inappropriate for discrete frequency components because the latter theoretically have zero bandwidth

Amplitude Correction

1. A finite-duration window \(w[n]\)

   DTFT is \(W[e^{j\omega}]\) and the maximum magnitude is is at DC frequency, which \(\sum_n w_n\)

2. Sinusoidal signal \(x[n]\)

   DFS is \(X_k\), and DTFT shall be \(\frac{2\pi}{N}X_k(e^{j\omega})\)

3. the windowed sequence \(v[n] = x[n]w[n]\)

   with multiplication property, DTFT of \(v[n]\) shall be \(\frac{X_k(e^{j\omega})}{N}\sum_n w_n\)

   As we know, DFT of \(v[n]\) is samples of its DTFT, that is \[ \frac{X_k(e^{j\omega})}{N}\sum_n w_n = X_v[k] \] Therefore, \[ \frac{X_k(e^{j\omega})}{N} = \frac{X_v[k]}{\sum_n w_n} \]

Effective Noise BandWidth (ENBW)

General derivation

The relationship between a power spectrum (\(PS, V^2\)) and a power spectral density (\(PSD, V^2/Hz\)) is given by the effective noise bandwidth (ENBW), which can easily be determined at the time when the DFT is computed.

ENBW should always be recorded when a spectrum or spectral density is computed, such that the result can be converted to the other form at a later stage, when the information about the frequency resolution \(f_{res}\) and the window that was used is normally not easily available any more.

The normalized equivalent noise bandwidth (NENBW) of the window is given by

\[ \text{NENBW} = \frac{NS_2}{S_1^2} \] where \(S_1 = \sum _{k=0}^{N-1}w_k\) and \(S_2 = \sum _{k=0}^{N-1}w_k^2\)

The ENBW is given by

\[ \text{ENBW} = \text{NENBW}\cdot f_{res} = \text{NENBW}\cdot \frac{f_s}{N} = f_s\frac{S_2}{S_1^2} \]

Equivalent noise bandwidth (ENBW) compares a window to an ideal, rectangular time-window. It is the bandwidth of the rectangular window's frequency-domain shape that passes the same amount of white noise energy as the frequency-domain shape defined by the other window.

Therefore, the equivalent noise bandwidth \(B_{enbw}\) is given by

\[ B_{enbw} = \frac{\int_{-f}^{f} |W(f)|^2 df}{|W(f_0)|^2} \]

Translating to discrete domain, the equivalent noise bandwidth can be computed using DFT samples as

\[ B_{enbw} =\frac{\sum_{k=0}^{N-1}|W[k]|^2}{|W[k_0]|^2} \]

where \(k_0\) is the index at which the magnitude of FFT output is maximum and \(N\) is the window length, i.e. \(k_0=0\).

Applying Parseval's theorem and \(W[0]=\sum_{n}w[n]\), \(B_{enbw}\) can also be computed using time domain samples as

\[ B_{enbw} = N \frac{\sum_{n}|w[n]|^2}{ \left| \sum_{n} w[n] \right|^2} \]

scale \(w[n]\) don't change \(B_{enbw}\)

Noise power inside window: \(\int_{-f}^{f} |W(f)|^2 df \to N\cdot\sum_{n}|w[n]|^2\)

peak amplitude: \(|W(f_0)|^2 \to \left| \sum_{n} w[n] \right|^2\)

An alternative derivation

Assuming the windowed sequence \(v[n] = x[n]w[n]\)

• \(W[k]\): Fourier Transform of finite sequence window

• \(X_{sig}\): Fourier Transform of signal

• \(X_{n}\): Fourier Transform of noise

• \(X_{v,sig}\): Fourier Transform of windowed signal

• \(X_{v,n}\): Fourier Transform of windowed noise

From Fig. 6,, we observe that the amplitude of the harmonic estimate at a given frequency is biased by the accumulated broad-band noise included in the bandwidth of the window.

In this sense, the window behaves as a filter, gathering contributions for its estimate over its bandwidth

The Fourier Transform of windowed signal can be expressed as

\[\begin{align} X_{v,sig} &= W_{max}\cdot X_{sig} \\ &= W[0]\cdot X_{sig} \end{align}\]

For a typical window, \(W_{max}\) occurs at \(\omega = 0\)

And the Fourier Transform of windowed noise can be expressed as

\[ X_{v,n}^2 = \sum_k (W[k])^2 \cdot X_n^2 \]

divided by \((W[0])^2\) on both sides of the above equation

\[ \frac{X_{v,n}^2}{(W[0])^2} = \frac{\sum_k (W[k])^2}{(W[0])^2} \cdot X_n^2 \]

By Parseval's theorem

\[ \frac{X_{v,n}^2}{\left(\sum_n w[n]\right)^2} = \frac{N\sum_n w^2[n]}{\left(\sum_n w[n]\right)^2} \cdot X_n^2 \]

where \(X^2_n\) is what is deserved and

\[ X_n^2 = \frac{PS_{n}}{B_{enbw}} \]

where \(B_{enbw} = N \frac{\sum_{n}|w[n]|^2}{ \left| \sum_{n} w[n] \right|^2}\) and \(PS_{n}=\left| \frac{X_{v,n}}{\sum_n w[n]}\right|^2\)

example

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lw = 128;
win = hann(lw);
lt = 2048;
windft = fftshift(fft(win,lt));

ad = abs(windft).^2;
mg = max(ad);

fs = 1000;

bw = enbw(win,fs);

bdef = sum((win).^2)/sum(win)^2*fs;
fprintf("bw: %.3f\n", bw);
fprintf("bdef: %.3f\n", bdef);

freq = -fs/2:fs/lt:fs/2-fs/lt;

plot(freq,ad, bw/2*[-1 -1 1 1],mg*[0 1 1 0],'--')
xlim(bw*[-1 1])

Adiff = trapz(freq,ad)-bw*mg;
fprintf("Adiff: %.3e\n", Adiff);

Verify that the area of the rectangle contains the same total power as the window.

1

Adiff = trapz(freq,ad)-bw*mg

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bw: 11.811
bdef: 11.811
Adiff: 7.276e-12

Unpingco, José. Python for Signal Processing: Featuring IPython Notebooks. Cham: Springer, 2013. [pdf]

[https://github.com/unpingco/Python-for-Signal-Processing/blob/master/Windowing_Part2.ipynb]

demo

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%https://aaronscher.com/Course_materials/Communication_Systems/documents/PSD_Autocorrelation_Noise.pdf

%Clear variables. clear command window, close all figures:
clc;
clear all;
close all;
%%%Setup and define variables
f0=10; %frequency of sinusoidal signal (Hz)
fs=100; %sampling frequency (Hz)
Ts=1/fs; %sampling period (seconds)
N0=3000; %number of samples
t=[0:Ts:Ts*(N0-1)]; %Sample times
noise_PSD=.5; %This is the desired noise power spectral density in W/Hz.
variance=noise_PSD*fs;% Variance = sigma^2
sigma=sqrt(variance);
noise=transpose(sigma*randn(N0,1));%create sampled white Gaussian noise.
xsignal=20*sin(2*pi*f0*t); %create sampled sinusoidal signal
x=xsignal+noise; %Add signal to noise
figure(1)
histogram(noise,30) %plot histogram
set(gca,'FontSize',14) %set font size of axis tick labels to 18
xlabel('Noise amplitude','fontsize',14)
ylabel('Frequency of occurance','fontsize',14)
title('Simulated histogram of white Gaussian noise','fontsize',14)
SNR_try1=snr(xsignal,noise); %calculate SNR using built in "snr" function.
SNR_try2=10*log10(sum(xsignal.^2)/sum(noise.^2)); %manually calculate SNR.
%If everything is correct, the two SNR calculations above should agree.
%Plot noise in time-domain
figure(2)
plot(t,x)
set(gca,'FontSize',14) %set font size of axis tick labels to 18
xlabel('Time (s)','fontsize',14)
ylabel('Amplitude','fontsize',14)
title('Noisey sinusoid','fontsize',14)
grid on
%Plot power spectral density (PSD) of noise using three different methods:
%
%Method 1. Calcululate PSD from amplitude spectrum
N=2^16; %Number of discrete points in the FFT
y=fft(x,N)/fs; %fft of noise
z=fftshift(y);%center noise spectrum
f_vec=[0:1:N-1]*fs/N-fs/2; %designate sample frequencies
amplitude_spectrum=abs(z); %compute two-sided amplitude spectrum
ESD1=amplitude_spectrum.^2; %ESD = |F(w)|^2;
PSD1=ESD1/((N0-1)*Ts);% PSD=ESD/T where T = total time of sample
figure(3)
plot(f_vec,10*log10(PSD1));
xlabel('Frequency [Hz]','fontsize',14)
ylabel('dB/Hz','fontsize',14)
title('Power spectral density - method 1','fontsize',14)
grid on
set(gcf,'color','w'); %set background color from grey (default) to white
axis tight
%calculate average power using PSD calclated from method 1:
Average_power_method_1=sum(PSD1)*fs/N; % Pav=sum(PSD)*delta_f where delta_f=fs/N;
%
%Method 2 - Calculate PSD from autocorrelation
time_lag=((-length(x)+1):1:(length(x)-1))*Ts;
auto_cor=xcorr(x,x)/fs; %Use xcorr function to find PSD
y=1/fs*fft(auto_cor,N); %fft of auto correlation function
PSD2=abs(1/(N0-1)*fftshift(fft(auto_cor,N)));
figure(4)
plot(f_vec,10*log10(PSD2));%use convolution
xlabel('Frequency [Hz]','fontsize',14)
ylabel('dB/Hz','fontsize',14)
title('Power spectral density - method 2','fontsize',14)
grid on
set(gcf,'color','w'); %set background color from grey (default) to white
axis tight
%calculate average power using PSD calclated from method 1:
Average_power_method_2=sum(PSD2)*fs/N; %Pav=sum(PSD)*delta_f where delta_f=fs/N;
%
%Method 3 - Calculate PSD using built in pwelch function
figure(5)
PSD3=periodogram(x,[],N,fs,'centered');
plot(10*log10(PSD3))
xlabel('Frequency [Hz]','fontsize',14)
ylabel('dB/Hz','fontsize',14)
title('Power spectral density - method 3','fontsize',14)
grid on
set(gcf,'color','w'); %set background color from grey (default) to white
axis tight
Average_power_method_3=sum(PSD3)*fs/N; %Pav=sum(PSD)*delta_f where delta_f=fs/N;
%
%Calculate mean and average PSD of noise:
PSD_noise=periodogram(noise,[],N,fs,'centered');
Average_noise_PSD=mean(PSD_noise);
Mean_noise=mean(noise);

The power spectral density plots for methods 2 and 3 exactly match that for method 1 (shown above).

reference

enbw Matlab URL:https://www.mathworks.com/help/signal/ref/enbw.html

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N = 256;
wn = hanning(N);
s1 = sum(wn)^2;
s2 = sum(wn.^2);

(N*s2)/s1
enbw(wn)

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ans =

    1.4942


ans =

    1.4942

Schmid, H. (2012). How to use the FFT and Matlab's pwelch function for signal and noise simulations and measurements. URL:http://www.schmid-werren.ch/hanspeter/publications/2012fftnoise.pdf

Bonnie C.Baker. Reading and Using Fast Fourier Transforms (FFT) URL:http://ww1.microchip.com/downloads/en/appnotes/00681a.pdf

FFT analysis: SNR and Noise level vary with FFT Window size URL:https://www.virtins.com/forum/viewtopic.php?f=7&t=1382

Lyons, R. G. (2011). Understanding digital signal processing (3rd ed.). Prentice Hall

Stefan Scholl, "Exact Signal Measurements using FFT Analysis",Microelectronic Systems Design Research Group, TU Kaiserslautern, Germany. [pdf]

Aaron Scher. PSD, Autocorrelation, and Noise in MATLAB [pdf]

Aaron Scher. FFT, total energy, and energy spectral density computations in MATLAB [pdf]

Amplitude Estimation and Zero Padding URL:https://www.mathworks.com/help/signal/ug/amplitude-estimation-and-zero-padding.html

Harris, F. (1978). On the use of windows for harmonic analysis with the discrete Fourier transform. Proceedings of the IEEE, 66, 51-83. [pdf]

Harris, F. J. . (1976). Windows, harmonic analysis and the discrete fourier transform. [pdf]

Wang Hongwei. Virtins. Evaluation of Various Window Functions using Multi-Instrument [pdf]

Properties of FFT Windows Used in Stable32 [pdf]

Solomon, Jr, O M. PSD computations using Welch's method. [Power Spectral Density (PSD)]. United States. https://doi.org/10.2172/5688766 [pdf]

Measure Power of Deterministic Periodic Signals [https://www.mathworks.com/help/signal/ug/measuring-the-power-of-deterministic-periodic-signals.html]

Mathuranathan. Equivalent noise bandwidth (ENBW) of window functions. [https://www.gaussianwaves.com/2020/09/equivalent-noise-bandwidth-enbw-of-window-functions/]

recordingblogs, Equivalent noise bandwidth [https://www.recordingblogs.com/wiki/equivalent-noise-bandwidth]

unpingco, Python for Signal Processing [https://github.com/unpingco/Python-for-Signal-Processing/blob/master/Windowing_Part2.ipynb]

Pavan, Schreier and Temes, "Understanding Delta-Sigma Data Converters, Second Edition" ISBN 978-1-119-25827-8

A finite-length data record = an infinite record multiplied by a rectangular window

Windowing is unavoidable

Applying the Hanning window (or any window) to a periodic signal creates leakage.

leakage:

​ The component at one frequency leaks into the vicinity of another compnent owing to the spectral smearing introdued by window.

Notice side lobes adding out of phase can reduce the heights of the peaks

periodic vs symmetric Hann

why windowing length is N+1 but drop the last one? #10 [https://github.com/wulffern/aic2023/issues/10#issuecomment-3242582671]

• Symmetric Hann: include both ends of the window

• Periodic Hann: sample one period and exclude the duplicated endpoint

Normalizing by \(\sum w\) of the window recovers only the carrier amplitude;

Only periodic window np.hanning(N+1)[:N] cleans the spectrum noise floor, harmonics, THD, SNDR, SFDR

The important point is not simply that one has zero at both array ends. The key difference is that the periodic Hann uses denominator \(N\), so its cosine is exactly commensurate with the \(N\)-point DFT: \[ w_{\text{periodic}}[n] = \frac{1}{2} \left( 1-\cos\left(\frac{2\pi n}{\textcolor{blue}{N}}\right) \right), \qquad n=0,\ldots,N-1. \]

Thus,

\[ w_{\text{periodic}}[n] \sim \cos\left(\frac{2\pi n}{N}\right). \]

\[ w_{\mathrm{per}}[n] = \frac{1}{2} -\frac{1}{4}e^{j2\pi n/N} -\frac{1}{4}e^{-j2\pi n/N} \]

These three terms are exactly three \(N\)-point DFT basis functions: \[ 1, \qquad e^{j2\pi n/N}, \qquad e^{-j2\pi n/N} \]

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import numpy as np N = 16 w = np.hanning(N + 1)[:N] W = np.fft.fft(w) for k, value in enumerate(W): print(f"k={k:2d}: {value.real:+.12f} {value.imag:+.12f}j") # k= 0: +8.000000000000 +0.000000000000j # k= 1: -4.000000000000 +0.000000000000j # k= 2: +0.000000000000 +0.000000000000j # k= 3: +0.000000000000 -0.000000000000j # k= 4: +0.000000000000 +0.000000000000j # k= 5: -0.000000000000 -0.000000000000j # k= 6: +0.000000000000 +0.000000000000j # k= 7: +0.000000000000 +0.000000000000j # k= 8: +0.000000000000 +0.000000000000j # k= 9: +0.000000000000 +0.000000000000j # k=10: +0.000000000000 +0.000000000000j # k=11: -0.000000000000 -0.000000000000j # k=12: +0.000000000000 +0.000000000000j # k=13: -0.000000000000 -0.000000000000j # k=14: +0.000000000000 +0.000000000000j # k=15: -4.000000000000 +0.000000000000j

This cosine is exactly one of the \(N\)-point DFT basis frequencies.

In contrast, the symmetric Hann uses \[ w_{\text{symmetric}}[n] = \frac{1}{2} \left( 1-\cos\left(\frac{2\pi n}{\textcolor{blue}{N-1}}\right) \right), \qquad n=0,\ldots,N-1, \]

so

\[ w_{\text{symmetric}}[n] \sim \cos\left(\frac{2\pi n}{N-1}\right). \]

The \(N\)-point DFT basis is

\[ e^{-j2\pi kn/N}, \]

so the denominator is \(N\), not \(N-1\).

Therefore:

\[ \boxed{ \text{Periodic Hann: denominator }N \rightarrow \text{aligned with the }N\text{-point DFT grid} } \]

while

\[ \boxed{ \text{Symmetric Hann: denominator }N-1 \rightarrow \text{exact sample symmetry} } \]

So the essence of the distinction is

\[ \boxed{ N \quad \text{vs.} \quad N-1 } \]

rather than simply whether the last stored sample is zero.

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# Symmetric Hann
w_sym = np.hanning(N)

# Periodic Hann
w_per = np.hanning(N + 1)[:N]

[https://github.com/wulffern/aic2026/blob/3e36f104be4caef0315160222f54abcd46c948ab/ex/q.py#L59-L82]

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def freqDomain(x,hann=True):
    N = len(x)
    # Use hanning window to prevent FFT bin energy spread.
    # N+1 with the last sample dropped is deliberate. It is exactly the
    # periodic window 0.5 - 0.5*cos(2*pi*n/N), which is a sum of three
    # DFT basis vectors: bins 0 and +-1, with weights 1/2, -1/4, -1/4.
    # Multiplying by it therefore convolves the spectrum with three
    # taps, so a tone sitting on an exact bin lands in exactly three
    # bins and leaks nothing anywhere else. np.hanning(N) is the
    # symmetric window, 0.5 - 0.5*cos(2*pi*n/(N-1)), which is not a
    # combination of DFT basis vectors - it has N-1 nonzero DFT
    # coefficients - so the same tone smears across every bin at about
    # -45 dBc. Measured, not asserted (aic2023 issue #10).
    if(hann):
        w = np.hanning(N+1)
    else:
        w = np.ones(N+1)

    # Convert to frequency domain
    X= np.fft.fftshift(np.fft.fft(np.multiply(w[0:N],x)))

    # Normalize to max output power
    X = X/np.max(np.abs(X))
    return X

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import numpy as np
import matplotlib.pyplot as plt

# ------------------------------------------------------------
# Parameters
# ------------------------------------------------------------
N = 16
n = np.arange(N)

# Symmetric Hann
w_sym = np.hanning(N)

# Periodic Hann
w_per = np.hanning(N + 1)[:N]

# Equivalent explicit expressions:
#
# w_sym = 0.5 * (1 - np.cos(2*np.pi*n/(N-1)))
# w_per = 0.5 * (1 - np.cos(2*np.pi*n/N))


# ============================================================
# 2. Frequency domain
#
# Dense zero-padded FFT ≈ samples of DTFT
# ============================================================
Nfft = 65536

W_sym_dense = np.fft.fft(w_sym, Nfft)
W_per_dense = np.fft.fft(w_per, Nfft)

# Frequency in cycles/sample
f_dense = np.arange(Nfft) / Nfft

# Normalize each response to its DC value
W_sym_dense = np.abs(W_sym_dense) / np.abs(W_sym_dense[0])
W_per_dense = np.abs(W_per_dense) / np.abs(W_per_dense[0])

eps = 1e-14

W_sym_dense_dB = 20*np.log10(np.maximum(W_sym_dense, eps))
W_per_dense_dB = 20*np.log10(np.maximum(W_per_dense, eps))


# ------------------------------------------------------------
# Actual N-point DFT
# ------------------------------------------------------------
W_sym_DFT = np.fft.fft(w_sym, N)
W_per_DFT = np.fft.fft(w_per, N)

f_DFT = np.arange(N) / N

W_sym_DFT = np.abs(W_sym_DFT) / np.abs(W_sym_DFT[0])
W_per_DFT = np.abs(W_per_DFT) / np.abs(W_per_DFT[0])

W_sym_DFT_dB = 20*np.log10(np.maximum(W_sym_DFT, eps))
W_per_DFT_dB = 20*np.log10(np.maximum(W_per_DFT, eps))

Windowed Signal

Short transient signals in the time domain produce high, broadband frequency content.

To reduce leakage, a mathematical function called a window is applied to the data. Windows are designed to reduce the sharp transient in the re-created signal as much as possible.

Because the sharp transients are reduced and smoothed, the broadband frequency of the spectral leakage is also reduced.

Periodic versus Non-Periodic Background

When performing a Fourier Transform on measurement data, a window affects periodic and non-periodic data differently:

• Periodic (No Window needed): A signal captured in a periodic manner does not require a window, and a resulting Fourier Transform has no leakage. Applying a window alters the resulting Fourier transform, and even creates spectral leakage where there would have been no leakage otherwise.

• Non-periodic (Window needed): Windows are used on signals that are captured in a non-periodic manner to reduce spectral leakage and get closer to the periodic results. A window can minimize the leakage present in a non-periodic signal, but cannot eliminate it.

The signal is repeated and appended mathematically because the measured data is assumed to be representative of the entire original signal

Periodic

When a measurement signal is captured in a periodic manner, the Fourier Transform of the captured signal will have no leakage in the frequency domain.

A window is not recommended for a periodic signal as it will distort the signal in an unnecessary manner, and actually creates spectral leakage.

Maloberti, F. Data Converters. Dordrecht, Netherlands: Springer, 2007.

Non-periodic

The same sine wave, with a different measurement time, results in a non-periodic captured signal. Here, when the captured signal is repeated, the original sine wave signal is not re-created.

In fact, several broadband transient events (circled in red) are introduced. These transients create a broadband response, or leakage.

Windows are used to minimize this leakage effect in the frequency domain.

Hanning

When doing operational noise and vibration measurements, the Hanning window is commonly used.

Random data has spectral leakage due to the abrupt cutoff at the beginning and end of the time block. It is non-periodic.

There is no way to ensure that the captured random signal is periodic by varying the measurement time.

Hanning windows are often used with random data because they have moderate impact on the frequency resolution and amplitude accuracy of the resulting frequency spectrum.

• The maximum amplitude error of a Hanning window is 15%

  In the cited article, all spectral data had an amplitude correction factor applied.

• while the frequency leakage is typically confined to 1.5 spectral lines to each side of the original sine wave signal

periodic signal

Applying the Hanning window (or any window) to a periodic signal creates leakage.

The periodically captured sine wave with the Hanning window (blue) is wider in frequency than the original signal (red)

In the figure, the sine wave with the Hanning window (blue) is wider in frequency than the original signal (red).

non-periodic signal

When a Hanning window is applied to a non-periodic signal, the leakage is greatly reduced and the amplitude is higher.

A non-periodically captured sine wave (magenta) has a spectral leakage over the entire bandwidth, applying a Hanning window minimized the leakage (green)

RMS calculation

A RMS calculation sums up the energy within a frequency range.

• both the RMS of the periodic and non-periodic signals with a Hanning window are equal to the RMS of the leakage-free sine wave.

• Only the RMS of the non-periodic sine wave without a window applied is not equal to the others

With the leakage spread over a smaller frequency range, doing analysis calculations like RMS yields more accurate results.

Flattop

• The Flattop window has a better amplitude accuracy in frequency domain compared to the Hanning window,

  The maximum amplitude error of a Flattop window is less than 0.01%. By contrast, the Hanning window maximum amplitude error is 15%.

• A Flattop window confines leakage to 3.43 spectral lines to each side of the original signal.

amplitude errors

These maximum amplitude errors assume that amplitude correction factors are applied to the frequency spectrums. These amplitude correction factors compensate for any reduction caused by applying a window.

leakage

The frequency accuracy of the Flattop window is more coarse compared to a Hanning window. As a result, the Flattop window is typically employed on data where frequency peaks are distinct and well separated from each other.

When the frequency peaks are not guaranteed to be well separated, the Hanning window is preferred because it is less likely to cause individual peaks to be lost in the spectrum

Spectrum of two periodically captured tones that are \(4Hz\) apart with a \(1Hz\) frequency resolution. The spectrum with a Hanning window (green) shows two peaks while the spectrum with a Flattop window (blue) shows one peak.

Note that at the original frequencies of the tones the amplitude is correct and equal to one for both windows.

One common application for a flattop window is performing calibration. For example, a sound pistonphone only produces one single and distinct frequency during microphone calibration.

Uniform

A Uniform window has a value of 1.0 across the entire measurement time. In reality, a Uniform window could be called no window.

Depending on the data acquisition system used, sometimes the term Rectangular window is also used.

• A Uniform window creates no frequency or amplitude distortion when the measured signal is periodic.

• When a measured signal is not periodic, the amplitude is reduced by a maximum of 36% and the frequency content is spread over the entire bandwidth of the measurement.

  This is due to sharp transients that are created by repeating and appending the measured signal.

Whenever a measurement signal is periodic, a Uniform window is preferred.

Applying a Hanning or Flattop window to a periodic signal will actually create amplitude and frequency distortion.

Benefit of Reducing Leakage

The benefit is not that the captured signal is perfectly replicated.

The main benefit is that the leakage is now confined over a smaller frequency range, instead of affecting the entire frequency bandwidth of the measurement.

With the leakage spread over a smaller frequency range, doing analysis calculations like RMS yields more accurate results.

It is impossible to calculate the proper RMS amplitude estimate over a limited frequency range of the un-windowed sine wave, since the leakage is over the full frequency range. Therefore the RMS amplitude is not correct.

Two tones

In the case of two closely spaced sine tones, without a window being applied, two tones frequencies would leak into each other, which make determining the true amplitude of individual peaks very difficult.

The window makes it easier to separate and distinguish each tone so a proper analysis could be performed.

window function in frequency domain

The transfer function \(a(f)\) of a window \(w_j, j \in [0, N-1]\) expresses the response of the window to a sinusoidal signal at an offset of \(f\) frequency bins, i.e. DFT .

real part: \[ a_r(f)=\sum_{j=0}^{N-1}w_j\cos (2\pi f j/N) \]

imaginary part: \[ a_i(f)=\sum_{j=0}^{N-1}w_j\sin (2\pi f j/N) \]

frequency response can be obtained as \[ a(f) = \frac{\sqrt{a_r^2+a_i^2}}{S_1} \] where \(S_1 = \sum _{k=0}^{N-1}w_k\)

Rectangular window example

aka. Uniform window, "Rectangular" window, "no window"

Whenever a measurement signal is periodic, a Uniform window is preferred. Applying a Hanning or Flattop window to a periodic signal will actually create amplitude and frequency distortion.

1. When \(f=0\)

\[ a_r(f) + ja_i(f) = \sum_{k=0}^{N-1}w_k = N \]

3. When \(f \neq 0\)

\[\begin{align} a_r(f) + ja_i(f) &= \sum_{k=0}^{N-1} e^{\frac{j2\pi k f}{N}} \\ &= \sum_{k=0}^{N/2} e^{\frac{j2\pi k f}{N}} + e^{\frac{j2\pi (k+N/2) f}{N}} \\ &= \sum_{k=0}^{N/2} e^{\frac{j2\pi k f}{N}} + e^{j\pi} e^{\frac{j2\pi k f}{N}} \\ &= \sum_{k=0}^{N/2} e^{\frac{j2\pi k f}{N}} - e^{\frac{j2\pi k f}{N}} \\ &= 0 \end{align}\]

A Uniform window creates no frequency or amplitude distortion when the measured signal is periodic.

However, if the signal cannot be guaranteed to be periodic, a Uniform window should be avoided.

Window Properties

There is no possibility of trade-off between main-lobe width and sied-lobe amplitude, since the window length is the only variable parameter.

The rectangular window has the narrowest main lobe for a given length, i.e. \(\Delta _{ml}=4\pi/L\)

Other windows include the Bartlett, Hann, and Hamming windows. The DTFTs of all these windows have main-lobe width \(\Delta _{ml}=8\pi/(L-1)\), which is approximately twice that of the rectangular window, but they have significantly smaller side-lobe amplitudes.

Demo

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clc;
clear all;

N = 512;
fs = 40*1e3; % 40kHz
fres = fs/N; % 78.125
tt = (0:N-1)*1/fs;
ff = (0:N/2)*fres;
fin = 390.625;

whan = hanning(N); % hanning window
wrect = rectwin(N); % rect window


% fin/fs = 5/N, periodically captured sine wave
wv = cos(2*pi*fin*tt);
power = periodogram(wv, whan, N, fs, 'power');
X = (power).^0.5*2^0.5;
psd = periodogram(wv, whan, N, fs, 'psd');
rms = sum(psd*fres)^0.5;
fprintf('RMS@periodic & hanning: %.5f\n', rms);

power_rect = periodogram(wv, wrect, N, fs, 'power');
X_rect = (power_rect).^0.5*2^0.5;
psd_rect = periodogram(wv, wrect, N, fs, 'psd');
rms_rect = sum(psd_rect*fres)^0.5;
fprintf('RMS@periodic & rect: %.5f\n', rms_rect);


figure(1)
plot(ff, X, 'r-o', ff, X_rect, 'b-s');
xlabel('Frequency(Hz)');
ylabel('|X|')
title('Amplitude spectrum of periodically captured sine wave');
legend('w/ hanning', 'w/ rect');
grid on
grid minor
% rectangular window provide higher frequency resolution
% hanning window induce leakage for the periodically captured sine wave



% fin - 0.5fres
fin_lkg0d5 = fin - 0.5*fres;
wv_lkg0d5 = cos(2*pi*fin_lkg0d5*tt);
power_lkg0d5 = periodogram(wv_lkg0d5, whan, N, fs, 'power');
X_lkg0d5 = (power_lkg0d5).^0.5*2^0.5;
psd_lkg0d5 = periodogram(wv_lkg0d5, whan, N, fs, 'psd');
rms_lkg0d5 = sum(psd_lkg0d5*fres)^0.5;
fprintf('RMS@-0.5fres & hanning: %.5f\n', rms_lkg0d5);

power_lkg0d5_rect = periodogram(wv_lkg0d5, wrect, N, fs, 'power');
X_lkg0d5_rect = (power_lkg0d5_rect).^0.5*2^0.5;
psd_lkg0d5_rect = periodogram(wv_lkg0d5, wrect, N, fs, 'psd');
rms_lkg0d5_rect = sum(psd_lkg0d5_rect*fres)^0.5;
fprintf('RMS@-0.5fres & rect: %.5f\n', rms_lkg0d5_rect);

figure(2)
plot(ff, X_lkg0d5, 'r-o', ff, X_lkg0d5_rect, 'b-s');
xlabel('Frequency(Hz)');
ylabel('|X|')
title('Amplitude spectrum of -0.5fres');
legend('w/ hanning', 'w/ rect');
grid on
grid minor
% hanning reduce leakage and max amplitude error 15%



% fin - 0.25fres
fin_lkg0d25 = fin - 0.25*fres;
wv_lkg0d25 = cos(2*pi*fin_lkg0d25*tt);
power_lkg0d25 = periodogram(wv_lkg0d25, whan, N, fs, 'power');
X_lkg0d25 = (power_lkg0d25).^0.5*2^0.5;
psd_lkg0d25 = periodogram(wv_lkg0d25, whan, N, fs, 'psd');
rms_lkg0d25 = sum(psd_lkg0d25*fres)^0.5;
fprintf('RMS@-0.25fres & hanning: %.5f\n', rms_lkg0d25);

power_lkg0d25_rect = periodogram(wv_lkg0d25, wrect, N, fs, 'power');
X_lkg0d25_rect = (power_lkg0d25_rect).^0.5*2^0.5;
psd_lkg0d25_rect = periodogram(wv_lkg0d25, wrect, N, fs, 'psd');
rms_lkg0d25_rect = sum(psd_lkg0d25_rect*fres)^0.5;
fprintf('RMS@-0.25fres & rect: %.5f\n', rms_lkg0d25_rect);

figure(3)
plot(ff, X_lkg0d25, 'r-o', ff, X_lkg0d25_rect, 'b-s');
xlabel('Frequency(Hz)');
ylabel('|X|')
title('Amplitude spectrum of -0.25fres');
legend('w/ hanning', 'w/ rect');
grid on
grid minor
% hanning reduce leakage

output

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RMS@periodic & hanning: 0.70711
RMS@periodic & rect: 0.70711
RMS@-0.5fres & hanning: 0.70711
RMS@-0.5fres & rect: 0.70711
RMS@-0.25fres & hanning: 0.70711
RMS@-0.25fres & rect: 0.70780

rectangular window provide higher frequency resolution

hanning reduce leakage and max amplitude error 15%

hanning reduce leakage and reduce amplitude error

reference

Windows and Spectral Leakage. URL:https://community.sw.siemens.com/s/article/windows-and-spectral-leakage

Article (20416822) Title: How to Utilize a Windowing Technique for Accurate DFT

B.P. Lathi, Roger Green. Linear Systems and Signals (The Oxford Series in Electrical and Computer Engineering) 3rd Edition

Window Types: Hanning, Flattop, Uniform, Tukey, and Exponential URL:https://community.sw.siemens.com/s/article/window-types-hanning-flattop-uniform-tukey-and-exponential

Window Correction Factors URL:https://community.sw.siemens.com/s/article/window-correction-factors

Root Mean Square (RMS) and Overall Level. URL:https://community.sw.siemens.com/s/article/root-mean-square-rms-and-overall-level

Alan V Oppenheim, Ronald W. Schafer. Discrete-Time Signal Processing, 3rd edition

Stefan Scholl, "Exact Signal Measurements using FFT Analysis",Microelectronic Systems Design Research Group, TU Kaiserslautern, Germany. [ pdf ]

Harris, F. (1978). On the use of windows for harmonic analysis with the discrete Fourier transform. Proceedings of the IEEE, 66, 51-83. [pdf]

Equivalent noise bandwidth (ENBW) of window functions URL:https://www.gaussianwaves.com/2020/09/equivalent-noise-bandwidth-enbw-of-window-functions/

Why should I zero-pad a signal before taking the Fourier transform? URL:https://dsp.stackexchange.com/q/741

enbw function in MATLAB URL:https://www.mathworks.com/help/signal/ref/enbw.html

Window function – figure of merits URL:https://www.gaussianwaves.com/2020/09/window-function-figure-of-merits/

Memos on FFT With Windowing URL:https://a2d2ic.wordpress.com/2018/02/01/memos-on-fft-with-windowing/

Jens Ahrens, "Some Notes on Windows in Spectral Analysis," Tech. Report, Chalmers Univeristy of Technology, 2020. URL:https://appliedacousticschalmers.github.io/scaling-of-the-dft/notes_on_windows/

Karnaugh Map

Pinaki Mazumder; Idongesit E. Ebong, "Lectures on Digital Design Principles," in Lectures on Digital Design Principles , River Publishers, 2023

Combinational Logic Minimization using Karnaugh maps (K-maps)

TODO 📅

IJTAG

While JTAG connects chips externally, IJTAG extends it inside the chip — linking embedded instruments (MBIST, sensors, monitors, etc.) through a reconfigurable network.

Pin-Mux Logic Pins are precious in SoC design!

• Pin-Mux Logic lets functional and test signals share the same pins depending on the mode.
• During test mode, JTAG/IJTAG signals are routed internally via multiplexers — saving pins and silicon area.

Digital Delay Model

Akio Kitagawa, Analog layout design https://mixsignal.files.wordpress.com/2013/03/analog-layout.pdf

THE WIRE http://bwrcs.eecs.berkeley.edu/Classes/icdesign/ee141_f01/Notes/chapter4.pdf

Anoop Veliyath, Design Engineer, Cadence Design Systems. Accurately Modeling Transmission Line Behavior with an LC Network-based Approach [pdf]

Mark Horowitz. Lecture 2: Wires and Wire Models [pdf]

Neil Weste and David Harris. 2010. CMOS VLSI Design: A Circuits and Systems Perspective (4th. ed.). Addison-Wesley Publishing Company, USA.

Cheng-Kok Koh. EE695K Modeling and Optimization of High Performance Interconnect [lec3a_pdf]

Vishal Saxena. ECE 445 Intro to VLSI Design: Lectures for Spring 2019 https://www.eecis.udel.edu/~vsaxena/courses/ece445/s19/ECE445.htm

Effective Switching resistance

https://www.eecis.udel.edu/~vsaxena/courses/ece445/s19/Lecture%20Notes/lec15_ece445.pdf

wire delay

Elmore Delay

Basic idea: use of mean of \(v'(t)\) to approximate median of \(v'(t)\)

Elmore delay approximates the median of \(h(t)\) by the mean of \(h(t)\)

Distributed RC-Line

Lumped approximations

\(rc\)-models

If your simulator does not support a distributed \(rc\)-model, or if the computational complexity of these models slows down your simulation too much, you can construct a simple yet accurate model yourself by approximating the distributed \(rc\) by a lumped RC network with a limited number of elements

The accuracy of the model is determined by the number of stages. For instance, the error of the \(\Pi -3\) model is less than 3%, which is generally sufficient.

Why use "\(\Pi\) Model"

examples

Wire Inductive Effect

• RC delay increases quadratically with length
• LC delay (speed of light flight time) increases linearly with length

Inductance will only be important to the delay of low-resistance signals such as wide clock lines

wave

Signal propagates over the wire as a wave (rather than diffusing as in RC only models)

Signal propagates by alternately transferring energy from capacitive to inductive modes

Glitches & Hazards

CPE166/EEE 270 Advanced Logic Design-Digital Design: Time Behavior of Combinational Networks: https://www.csus.edu/indiv/p/pangj/166/f/sram/Handout_Hazard.pdf

John Knight, ELEC3500 Glitches and Hazards in Digital Circuits http://www.doe.carleton.ca/~shams/ELEC3500/hazards.pdf

• A glitch is an unwanted pulse at the output of a combinational logic network – a momentary change in an output that should not have changed
• A circuit with the potential for a glitch is said to have a hazard
• In other words a hazard is something intrinsic about a circuit; a circuit with hazard may or may not have a glitch depending on input patterns and the electric characteristics of the circuit.

When do circuits have hazards ?

Hazards are potential unwanted transients that occur in the output when different paths from input to output have different propagation delays

Types of Hazards (on an output)

static 1-hazard, static 0-hazard, dynamic hazard

Hazard's Concern

• Hazards do not hurt synchronous circuits
• Hazards Kill Asynchronous Circuits
• Glitches Increase Power Consumption

Isolation cells

Isolation cells and Level Shifter cells URL: https://vlsitutorials.com/isolation-cells-level-shifter-cells-low-power-vlsi/

Isolation cells are additional cells inserted by the synthesis tools for isolating the buses/wires crossing from power-gated domain of a circuit to its always-on domain (AON).

To prevent corruption of always-on domain, we clamp the nets crossing the power domains to a value depending upon the design.

A simple circuit having a switchable (or gated) power domain

The circuit shown in Figure 1, after isolation cells are inserted

Always-On Buffer

Clock Gating

The Ultimate Guide to Clock Gating https://anysilicon.com/the-ultimate-guide-to-clock-gating/

Clock Gating is defined as: "Clock gating is a technique/methodology to turn off the clock to certain parts of the digital design when not needed".

AND gate-based clock gating

In simplest form a clock gating can be achieved by using an AND gate as shown in picture below

However, this simplest form of clock gating technique has some problem of generating glitches in the clock provide to the FF, which are not desirable.

Glitches in enable/gated clock

Latch based clock gating

These glitches can be removed by introducing a negative edge triggered FF (assuming downstream FFs are positive edge) or low-level sensitive latch at the output of the clock enable signal.

This will make sure that any glitch in the clock enable signal will not be visible to the gated clock output. The Latch output will only be updated during the negative clock cycle and thus input to AND gate will be stable high.

Glitch Free Gated Clock

OCV Derating With AOCV

Genus Attribute Reference 22.1

Innovus Text Command Reference 22.10

Article (20416394) Title: Analysis with Advanced On-chip Variation (AOCV) derating in EDI system and ETS

timing_aocv_derate_mode

1

timing_aocv_derate_mode{aocv_multiplicative | aocv_additive}

Default: aocv_multiplicative

Controls the AOCV derating mode.

When set to aocv_multiplicative, the derating factor will be calculated as AOCV derating * OCV derating, which is set using the set_timing_derate command.

When set to aocv_additive, the derating factor will be calculated as AOCV derating + OCV derating values.

When you use this global variable, the report_timing command shows the total_derate column in the timing report output, which allows you to view and cross-check the calculated total derate factor.

To set this global variable, use the set_global command.

Sulege Vivek [link]

preserve hand-instantiated cells

To preserve the hand-instantiated cells

1

set_dont_touch [get_cells -hierarchical *dont_touch_*]

The instances whose name contain "dont_touch_" shall be preserved during synthesis

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// no performace concerns, rest sync use sync3 is enough

module CN_resetb_sync_cell
(
    input resetb_in,
    input clkdst,
    output resetb_out
);

`ifdef USE_VERILOG
reg [2:0] resetb_dly;
`else
wire [2:0] resetb_dly;
`endif

`ifdef USE_VERILOG
always @(posedge clkdst or negedge resetb_in)
    if (~resetb_in) resetb_dly <= 3'b000;
    else resetb_dly <= {resetb_dly[1:0], 1'b1};
`else
SDFCNQD4 dont_touch_sync_flop0 (
    .SI(1'b0),
    .SE(1'b0),
    .CP(clkdst),
    .CDN(resetb_in),
    .D(1'b1),
    .Q(resetb_dly[0])
);
SDFCNQD4 dont_touch_sync_flop1 (
    .SI(1'b0),
    .SE(1'b0),
    .CP(clkdst),
    .CDN(resetb_in),
    .D(resetb_dly[0]),
    .Q(resetb_dly[1])
);
SDFCNQD4 dont_touch_sync_flop2 (
    .SI(1'b0),
    .SE(1'b0),
    .CP(clkdst),
    .CDN(resetb_in),
    .D(resetb_dly[1]),
    .Q(resetb_dly[2])
);
`endif

assign resetb_out = resetb_dly[2];

endmodule

Gain-boosted cascode

TODO 📅

Bode's Analysis

Note that ABCD here does not denote the transmission matrix of a two-port network

total sigma and correlation

\[ \sigma_{X+Y} = \sqrt{\sigma_{X}^2 + \sigma_{Y}^2 + 2\rho\sigma_{X}\sigma_{Y}} \]

\(\rho\): Pearson correlation coefficient between \(X\) and \(Y\) (ranges from \(-1\) to \(1\))

\(\rho\sigma_{X}\sigma_{Y}\): Covariance between \(X\) and \(Y\)

• Perfect Positive Correlation \(\rho = 1\): the variables move exactly together, \(\sigma_{total} = \sigma_{X} + \sigma_{Y}\)

• Zero Correlation \(\rho = 0\): the geometric sum of individual variations, \(\sigma_{total} = \sqrt{\sigma_{X}^2 + \sigma_{Y}^2}\)

• Perfect Negative Correlation \(\rho = -1\): variables offset each other's fluctuations, \(\sigma_{total} = \vert{}\sigma_X - \sigma_Y\vert{}\)

Zero-Value Time Constant Analysis

TODO 📅

MOS as resistor

MOSFET can be on and carry no current

Two capacitor paradox

Intuitive examination of the two-capacitors "paradox"l [https://youtu.be/WNpxV5qAUMU]

The two-capacitor paradox is a classic puzzle in circuit theory where energy seems to vanish

1. If there is any resistance $R $ (always true in reality). $R $ dissipates heat

2. If the resistance is genuinely zero, Then you can't ignore inductance — With pure $LC $, the circuit becomes an oscillator

Two capacitors \(C\) in series around the loop give an effective capacitance \(C_{eq} = C/2\). With inductance \(L\) in the loop, the resonant frequency is \[ \omega_0 = \frac{1}{\sqrt{LC_{eq}}} = \frac{1}{\sqrt{L(C/2)}} = \sqrt{\frac{2}{LC}} \]

\[ v_1(t) = \frac{V_0}{2}\left(1 + \cos\omega_0 t\right), \qquad v_2(t) = \frac{V_0}{2}\left(1 - \cos\omega_0 t\right) \] the \(y\)-axis is in units of \(V_0\), so the curves run between 0 and 1 \[ v_1(t) + v_2(t) = V_0 \quad \text{at all times} \]

RC High-Pass Filter (R in parallel with \(C_p\))

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import numpy as np
import matplotlib.pyplot as plt

# 1. Define circuit parameters
R = 100e3            # Resistance in ohms
C = 10e-12           # Main capacitance in Farads
Cp = 500e-15         # Parasitic capacitance in Farads

# 2. Calculate corner (cutoff) frequency and angular frequency
fc = 1 / (2 * np.pi * R * (C + Cp))
wc = 2 * np.pi * fc

# 3. Frequency vector (logarithmic scale from 10 Hz to 1 GHz)
f = np.logspace(1, 9, 2000)
w = 2 * np.pi * f

# 4. Transfer function with parasitic capacitance
# H(s) = (s*C*R) / (1 + s*R*(C + Cp))
numerator = 1j * w * C * R
denominator = 1 + 1j * w * R * (C + Cp)
H_s = numerator / denominator

# Convert to magnitude in decibels (dB)
magnitude_dB = 20 * np.log10(np.abs(H_s))

# 5. Plotting the frequency response
plt.figure(figsize=(10, 6))
plt.semilogx(f, magnitude_dB, 'b', linewidth=2, label='Filter Response')

# 6. Add vertical line for corner frequency
plt.axvline(x=fc, color='r', linestyle='--', label=f'Corner Freq: {fc/1e6:.2f} MHz')

# Formatting the plot
plt.title('Bode Plot: RC High-Pass Filter (R in parallel with $C_p$)')
plt.xlabel('Frequency (Hz)')
plt.ylabel('Gain (dB)')
plt.grid(True, which="both", ls="-", color='0.85')
plt.legend(loc='lower right')
plt.ylim(-40, 5); plt.xlim(1e3, 1e9)

# Show the plot
plt.show()

Transmission Gate

Equivalent Resistance is defined by large signal

[https://www.ece.ucdavis.edu/~ramirtha/EEC116/F11/TGlecture.pdf]

latch borrowed time

Kunal Saraswat [link]

Digital DC Offset Correction

\[ X- Y\cdot \beta z^{-1}\cdot \frac{1}{1-z^{-1}} = Y \] therefore \[ \frac{Y}{X} = \frac{1-z^{-1}}{1-(1-\beta)z^{-1}} \]

Temperature Inversion

Adam Teman. Digital Integrated Circuits (83-313) Lecture 3: MOSFET Modeling [pdf]

Zero Temperature Coefficient (ZTC) Bias Points

M. Coelho et al., "Is There a ZTC Biasing Point in the Leading-Edge FET Intrinsic Gain gmrDS?," 2025 9th International Young Engineers Forum on Electrical and Computer Engineering (YEF-ECE), Caparica / Lisbon, Portugal, 2025

M. Coelho et al., "Analysis of the ZTC Bias Points in the FinFET Gate Capacitance and Transition Frequency," 2025 37th International Conference on Microelectronics (ICM), Cairo, Egypt, 2025, pp. 1-6, doi: 10.1109/ICM66518.2025.11322461

there's a specific bias point where the MOSFET transition frequency (fT) becomes almost temperature‑independent

CMRR (Common-Mode Properties)

MOS Noise

MOS Thermal Noise

Michael H. Perrott. Analysis and Design of Analog Integrated Circuits Lecture 14 Noise Spectral Analysis for Circuit Elements [https://www.cppsim.com/CircuitLectures/Lecture14.pdf]

Legacy Long-Channel Model

\[ \color{red} \overline{i^2_d} \propto \gamma \propto \eta \propto \frac{1}{V_{DS}} \]

[https://www.eecg.toronto.edu/~johns/ece512/lecture_notes/04_noise_2_per_page.pdf]

[https://people.engr.tamu.edu/spalermo/ecen474/lecture12_ee474_noise.pdf]

Short-Channel Effects

Gildenblat, G. S. (2010). Compact modeling : principles, techniques and applications. Springer.

VDS Effect On Channel Noise

\[ \color{red} \overline{i^2_d} \propto V_{DS} \]

K. Ohmori and S. Amakawa, "Direct White Noise Characterization of Short-Channel MOSFETs," in IEEE Transactions on Electron Devices, vol. 68, no. 4, pp. 1478-1482, April 2021 [pdf, slides]

X. Ding, G. Niu, A. Zhang, W. Cai and K. Imura, "Experimental Extraction of Thermal Noise γ Factors in a 14-nm RF FinFET technology," 2021 IEEE 20th Topical Meeting on Silicon Monolithic Integrated Circuits in RF Systems (SiRF), San Diego, CA, USA, 2021[https://sci-hub.se/10.1109/SiRF51851.2021.9383331]

NF50

TODO 📅

\(\gamma\) vs VDS, VGS in simulation

N28

fix VDS, sweep VGS

fix VGS, sweep VDS

Gamma vs current density

MOS Flicker Noise

T. Noulis, "CMOS process transient noise simulation analysis and benchmarking," 2016 26th International Workshop on Power and Timing Modeling, Optimization and Simulation (PATMOS), Bremen, Germany, 2016 [https://sci-hub.ru/10.1109/PATMOS.2016.7833428]

Dr. Thomas Gneiting, AdMOS GmbH. Flicker Noise Extraction for Scalable MOS Simulation Models [https://www.mos-ak.org/premstaetten/papers/MOS-AK_T.Gneiting.pdf]

Noise analysis ON,IN cfg	input noise, noise gain, output noise
		i: Id o: Vg
		i: V5 o: Id
		i: V1 o: Id

Above simulation demonstrate that flicker noise is represented by a drain-source current in BSIM model, however modeled as a voltage source in series with the gate is just for calculating convenience

David Lee, Berkeley DA. Device Noise Analysis of Switched-Cap Circuits [https://semiwiki.com/eda/2900-device-noise-analysis-of-switched-cap-circuits/]

---

Sam Palermo. Lecture 12: Noise. ECEN474: (Analog) VLSI Circuit Design Fall 2012 [https://people.engr.tamu.edu/spalermo/ecen474/lecture12_ee474_noise.pdf]

Notice, input referred noise unit follow output noise's unit

陈铖颖，杨丽琼，王统. CMOS模拟集成电路设计与仿真实例：基于CadenceADE. 电子工业出版社，2013

Ma, Mb are parallel MOS, with width \(W_a\) and \(W_b\), Since flicker noise current \(\overline{I_{D}^2}(\Delta f) = K_f\frac{g_{m}^2}{C_{ox}WL}\frac{\Delta f}{f}\) and \(g_{m,tot}=g_{m,a} + g_{m,b}\), \(W_{tot} = W_a + W_b\), \(g_{m,a} = \beta W_a\), \(g_{m,b}=\beta W_b\) \[ \overline{I_{D,a}^2}(f) =K_f\frac{\beta^2W_a}{C_{ox}L}\frac{\Delta f}{f} \quad \overline{I_{D,b}^2}(f) = K_f\frac{\beta^2W_b}{C_{ox}L}\frac{\Delta f}{f} \quad \overline{I_{D,tot}^2}(f) = K_f\frac{\beta^2(W_a+W_b)}{C_{ox}L}\frac{\Delta f}{f} \\ \]

Then \[ \color{green}\overline{I_{D,tot}^2} \equiv \overline{I_{D,a}^2} + \overline{I_{D,b}^2} \]

MOS Shot Noise

Gate-Referred Noise Lemma

inverter capacitance simulation

inverter input

R-C, series equivalent circuit

inverter output

R-C, parallel equivalent circuit

AC simulation

@vi = 0

sweep vi from 0 to 800mV (vdd)

SP simulation

EEStream. Cadence - How to find device capacitance - DC simulation, SP simulation and Large-signal SP simulation [https://www.youtube.com/watch?v=M3zP6eJnONk]

50-ohm?

Why is 50-ohm characteristic impedance standardized in RF and Microwave systems? [link]

[https://www.microwaves101.com/encyclopedias/why-fifty-ohms]

It was a compromise made and a tradeoff between power handling (red curve) and losses (blue curve)

Gm Rout Lemma

\[ A_V = -G_m R_{out} \]

Relative Sensitivity

Olivier de Weck, Karen Willcox. MIT, Gradient Calculation and Sensitivity Analysis [pdf]

Karti Mayaram, ECE 521 Fall 2016 Analog Circuit Simulation, Sensitivity and noise analyses [https://web.engr.oregonstate.edu/~karti/ece521/lec16_11_09.pdf]

Assuming Target \(T\) ( for example, the total resistance) is function of \(x_1,x_2,...,x_N\), then total variation can be expressed as

\[\begin{align} \mathrm{d}T &= \sum_{n=1}^N\frac{\partial T}{\partial x_n}dx_n = \sum_{n=1}^N\frac{\partial T}{\partial x_n}x_n\cdot \frac{\mathrm{d}x_n}{x_n} \end{align}\]

Then, we obtain relative variation \[\begin{align} \frac{\mathrm{d}T}{T} &= \sum_{n=1}^N\frac{\partial T}{\partial x_n}\frac{x_n}{T}\cdot \frac{\mathrm{d}x_n}{x_n} = \sum_{n=1}^N S_{x_n}^T \cdot \frac{\mathrm{d}x_n}{x_n} \end{align}\]

⭐ where \(S_{x_n}^T=\frac{\partial T}{\partial x_n}\frac{x_n}{T}\) is relative sensitivity

relative sensitivity connect \(\frac{\mathrm{d}x_n}{x_n}\) with total relative variation \(\frac{\mathrm{d}T}{T}\)

And \(\mathrm{d}T\) can be expressed as \[ dT =\sum_{n=1}^N S_{x_n}^T T\cdot \frac{\mathrm{d}x_n}{x_n} = \sum_{n=1}^N x_n'\cdot \frac{\mathrm{d}x_n}{x_n} \] ⭐ where \(x_n'= S_{x_n}^T T\) is the contribution of \(x_n\) in \(T\)

⭐ For parallel or series resistors, it can prove \(\sum_{n=1}^N S_{x_n}^T = 1\) and \(\sum_{n=1}^N x_n'=T\)

Here \(T= R_1 \parallel R_2 = \frac{R_1R_2}{R_1+R_2}\), and \(T|_{R_1=8000, R_2=2000} = 1600\)

We obtain relative sensitivity: \[\begin{align} S_{R_1}^T = \frac{R_2}{R_1+R_2} \qquad S_{R_2}^T = \frac{R_1}{R_1+R_2} \end{align}\]

The contribution of \(R_1\) and \(R_2\) to \(T\) \[\begin{align} R_1' &= S_{R_1}^T T | _{R_1=8000, R_2=2000} = 320 \\ R_2' &= S_{R_2}^T T | _{R_1=8000, R_2=2000} = 1280 \end{align}\]

Normalized sensitivity captures relative sensitivity

change in objective per change in design variable

Device Current Components

common gate amplifiers

[https://www.linkedin.com/posts/chembiyan-t-0b34b910_analog-analogdesign-rfdesign-activity-7126946716938878976-GeW6?utm_source=share&utm_medium=member_desktop]

Level Shifter

TIA

\[ I_{in} = \frac{V_i}{R_S} + \frac{V_i - V_o}{R_F} \qquad \frac{V_i - V_o}{R_F} = g_m V_i \]

Then

\[ V_o = \frac{I_{in}R_F}{\frac{R_S+R_F}{R_S}\frac{1}{1-g_mR_F}- 1} \qquad V_i= \frac{I_{in}R_F}{\frac{R_F}{R_S}+g_mR_F} \] If \(R_S \gg R_F\) \[ V_o = \frac{I_{in}}{g_m}(1-g_mR_F) \qquad V_i = \frac{I_{in}}{g_m} \]

linearity

TIA stage allows for improved gain with better linearity, as mostly signal current passes through \(R_F\) TODO 📅 ??? Quantitative analysis

Switched-Capacitor Resistor

\[ R_{eq} = \frac{1}{f_sC} \]

[https://youtu.be/SL3-9ZMwdJQ&t=1339]

Channel-Length Modulation & Pinched off

• \(\lambda \propto \frac{1}{L_g}\)
• \(\lambda \propto \frac{1}{V_{DS}}\)

• If \(V_{DS}\) is slightly greater than \(V_{GS} - V_{TH}\), then the inversion layer stops at \(x \leq L\), and we say the channel is "pinched off"
• Upon passing the pinchoff point, the electrons simply shoot through the depletion region near the drain junction and arrive at the drain terminal

\(L^{'}\) is the function of \(V_{DS}\)

with \(\frac{1}{L^{'}} = \frac{1}{L-\Delta L}=\frac{L+\Delta L}{L^2-\Delta L^2}\approx \frac{1}{L}\left(1+\frac{\Delta L}{L}\right)\), we have \[ I_D \approx \frac{1}{2}\mu_n C_{ox}\frac{W}{L}\left(1+\frac{\Delta L}{L}\right)(V_{GS}-V_{TH})^2 = \frac{1}{2}\mu_n C_{ox}\frac{W}{L}(V_{GS}-V_{TH})^2 (1+\lambda V_{DS}) \] assuming \(\frac{\Delta L}{L} = \lambda V_{DS}\)

\(\lambda\) represents the relative variation in length for a given increment in \(V_{DS}\). Thus, for longer channels, \(\lambda\) is smaller

In reality, however, \(r_O\) varies with \(V_{DS}\). As \(V_{DS}\) increases and the pinch-off point moves toward the source, the rate at which the depletion region around the source becomes wider decreases, resulting in a higher incremental output impedance.

Early Voltage indicator

\[ g_m r_o = \frac{g_m}{I_D}I_D \cdot \frac{V_A}{I_D} = \frac{g_m}{I_D} \cdot V_A \]

$g_m r_o $ is the indicator of \(V_A\), if \(\frac{g_m}{I_D}\) is same

Cgd of Common-Source Stage

Miller effect of Cgd during layout

Nonlinearity of Differential Circuits

\[ \cos^3\omega t = \frac{3\cos \omega t + \cos(3\omega t)}{4} \]

Zero in differential pair with active current mirror

Noting the circuit consists of a "slow path" (M1, M3, M4) in parallel with a "fast path" (M2)

• "slow path" \[ H_\text{slow}(s) = \frac{A_0}{(1+s/\omega _{pE})(1+s/\omega _{pO})} \]

• "fast path" \[ H_\text{fast}(s) = \frac{A_0}{1+s/\omega _{pO}} \]

Then \[\begin{align} \frac{V_\text{out}}{V_\text{in}} &= H_\text{slow}(s) + H_\text{fast}(s) \\ &= \frac{A_0}{1+s/\omega _{pO}}\left(\frac{1}{1+s/\omega _{pE}} + 1 \right) \\ &= \frac{A_0(1+s/2\omega _{pE})}{(1+s/\omega _{pO})(1+s/\omega _{pE})} \end{align}\]

That is, the system exhibits a zero at \(2\omega_{pE}\)

signals traveling through two paths within an amplifier may cancel each other at one frequency, creating a zero in the transfer function

\[ \omega_z = \frac{(A_1+A_2)\omega_{p1}\omega_{p2}}{A_1\omega_{p1}+A_2\omega_{p2}} \] noting \(\omega_{p1}\lt \omega_z \lt \omega_{p2}\)

"Zero" by Inspection

a method to predict the existence of "zero" by inspection, based on the concept of "Analog Phase Interpolation"

TODO 📅

Debashis Dhar, How to Recognize "Zero" by Inspection (Utilizing Analog Phase Interpolation) [https://www.linkedin.com/posts/debashis-dhar-12487024_how-to-recognize-zero-by-inspection-activity-7163364364329160704-9qOq?utm_source=share&utm_medium=member_desktop]

Random offset

Art Zirger, Random Offset in CMOS IC Design [https://designers-guide.org/forum/Attachments/mismatch_presentation.pdf]

The dependence of offset voltage and current mismatches upon the overdrive voltage is similar to our observations for corresponding noise quantities

current mirror

To minimize current mismatch, the overdrive voltage must be maximized, a trend opposite to that in differential pair.

This is because as \(V_{GS}-V_{TH}\) increases, threshold mismatch has a lesser effect on the device currents

\(\Delta I_D= g_m \Delta V_{TH} = \frac{2I_D}{V_{OV}}\Delta V_{TH}\)

differential pair

In reality, since mismatches are independent statistical variables

Above shows that the input transistors must be designed for high gain (\(g_mr_o = \frac{2}{V_{OV}\lambda}\)), which means they must be designed for small \(V_{GS}-V_{TH}\).

It is desirable to minimize \(V_{GS}-V_{TH}\) by lowering the tail current or increasing the transistor widths

For \(\frac{\Delta K}{K}\)

\[\begin{align} v_{os} g_m &= \Delta K \frac{W}{L}(V_{GS}-V_{TH})^2 \\ v_{os} 2K\frac{W}{L}(V_{GS}-V_{TH}) &= \Delta K \frac{W}{L}(V_{GS}-V_{TH})^2 \\ v_{os} &= \frac{V_{GS}-V_{TH}}{2} \frac{\Delta K}{K} \end{align}\]

The derivation for \(\frac{\Delta W/L}{W/L}\) is same with \(\frac{\Delta K}{K}\)

alternative derivation

\[\begin{align} \Delta V_\beta \cdot g_m &= \frac{\partial I_D}{\partial \beta} \Delta \beta \\ &= I_D \frac{\Delta \beta}{\beta} \end{align}\]

That is \(\Delta V_\beta = \frac{I_D}{g_m}\frac{\Delta \beta}{\beta}\)

\[ \Delta V_R \cdot g_m R = I_D \cdot \Delta R \]

That is \(\Delta V_R = \frac{I_D}{ g_m} \cdot \frac{\Delta R}{R}\)

[https://electronicengineering.phd.upc.edu/en/courses-and-seminars/courses-materials/2008-2009/slides-makinwa-1]

RC charge & discharge

• charge: \[ V_o(t) = V_{X}(1-e^{-\frac{t}{\tau}}) + V_{o,0}\cdot e^{\frac{-t}{\tau}} \]

• discharge: \[ V_o(t) = V_{o,0}\cdot e^{-\frac{t}{\tau}} + V_{o,\infty}\cdot(1-e^{-\frac{t}{\tau}}) \]

1. \(e^{-\frac{t}{\tau}}\) item determine the initial state
2. \((1-e^{-\frac{t}{\tau}})\) item determine the final state

AC coupling

\(V_m=\frac{1}{4},\space \frac{3}{4}\) and its common voltage \(\frac{1}{2}\)

\(V_o=-\frac{1}{4},\space \frac{1}{4}\) and its common voltage \(0\)

\[ \tau = 200 \text{nF} \times (50+50)\text{ohm} = 20 \mu s \]

high level envelope:

Current mirror with source degeneration

Razavi 2nd, problem 14.15

Thevenin and Norton Equivalent Circuits

戴维南定理

等效电阻的计算方法

使用外加电源法时， 全部独立电源需要置零

诺顿定理

Miller's Approximation: right-half-plane zero

A quick inspection of this circuit reveals that a zero lies at a frequency where the current through \(C_{12}\) becomes equal to \(g_2V_1\).

When this occurs, the current through the parallel combination of \(C_2\) and \(R_2\) becomes zero, creating a zero in the transfer function.

In other words, we can write

\[\begin{align} g_2V_1 &= V_1sC_{12} \\ s &= \frac{g_2}{C_{12}} \end{align}\]

Nonoverlapping clock

Classical

DWC

C2PHIa is important to ensure nonoverlapping and DelayA2B is due to level shifter

[https://youtu.be/G0lzrMll-Ho&t=3114]

Single ended Amplifier Offset Voltage

unity gain buffer

\[\begin{align} V_o &= V_{o,dc}+A(V_p-V_m) \\ V_o' &= V_{o,dc}+A(V_p+V_{os}-V_m') \end{align}\]

Then, we get \[ V_{os}=\frac{V_o'-V_o}{A}+(V_m'-V_m) \] Due to \(V_o=V_m\) and \(V_o'=V_m'\) \[ V_{os}=(1/A+1)\Delta{V_m} \] or \[ V_{os}=(1/A+1)\Delta{V_o} \] if \(A \gg 1\) \[ V_{os}=\Delta{V_o} \]

non-inverting amplifier

\[\begin{align} V_o &= V_{o,dc}+A(V_p-V_m) \\ V_o' &= V_{o,dc}+A(V_p+V_{os}-V_m') \\ V_m &= \beta V_o \\ V_m' &= \beta V_o' \end{align}\]

we get \[ V_{os}=\frac{V_o'-V_o}{A}+(V_m'-V_m) \] or \[ V_{os}=\frac{\Delta V_o}{A}+\beta \Delta V_o \] if \(A \gg 1\) \[ V_{os}=\beta \Delta V_o \] or \[ V_{os}=\Delta V_m \]

Lecture 22 Variability and Mismatch of Dr. Hesham A. Omran's Analog IC Design

URL: https://www.master-micro.com/professional-courses/analog-ic-design/course-resources

Gotcha MOS ron

There is discrepancy between model operating point and \(V_{ds}/I_{ds}\)

I believe that the equation \(V_{ds}/I_{ds}\) is more appropriate where mos is used as switch, though \(V_{ds}=0\) is an outlier.

Schmitt Inverter

transition frequency \(f_T\)

Defined as the frequency at which the small-signal current gain of a device is unity

mag(Ids@ft) = Ig(1mA)

Aditya Varma Muppala. MMIC 08: High Frequency Device Characterization in Cadence - Fmax, Ft, NFmin vs Jd [https://youtu.be/kgEypIA8eus]

fmax

Art3, Measuring Fmax for MOS Transistors [https://community.cadence.com/cadence_blogs_8/b/rf/posts/measuring-fmax-for-mos-transistors]

[https://www.rfinsights.com/concepts/ft-vs-fmax/]

TODO 📅

Miller multiplication of Capacitor

Positive Cap

Negative Cap

gain has limited bandwidth

\(V_o = V_i |A|e^{j\theta}\), and \(A_r = |A|\cos\theta\), \(A_i = |A|\sin\theta\)

Then \(I_i = (V_i - V_o)sC_f= V_i(1-|A|e^{j\theta})sC_f\), impedance is shown as below

\[\begin{align} Z &= \frac{V_i}{I_i} \\ &= \frac{1}{(1-|A|e^{j\theta})j\omega C_f} \\ &= -\frac{j}{\omega C_f\frac{1+|A|^2-2|A|\cos\theta}{1-|A|\cos\theta}} + \frac{|A|\sin\theta}{\omega C_f (1+|A|^2-2|A|\cos\theta)} \\ \end{align}\]

\(C_\text{eq}\) and \(R_\text{eq}\) are obtained \[\begin{align} C_\text{eq} &= \frac{1+|A|^2-2A_r}{1-A_r}\cdot C_f \\ R_\text{eq} &= \frac{A_i}{1+|A|^2-2A_r}\cdot \frac{1}{\omega C_f} \end{align}\]

D/S small signal model

The Drain and Source of MOS are determined in DC operating point, i.e. large signal.

That is, top of \(M_2\) is drain and bottom is source, \[\begin{align} R_\text{eq2} &= \frac{r_\text{o2}+R_L}{1+g_\text{m2}r_\text{o2}} \\ & \simeq \frac{1}{g_\text{m2}} \end{align}\]

PMOS small signal model polarity

The small-signal models of NMOS and PMOS transistors are identical

A negative \(\Delta V_\text{GS}\) leads to a negative \(\Delta I_D\).

Recall that \(I_D\), in the direction shown here, is negative because the actual current of holes flows from the source to the drain.

Conversely, a positive \(\Delta V_\text{GS}\) produces a positive \(\Delta I_D\), as is the case for an NMOS device.

Leakage in MOS

Alvin L.S. Loke. 2004. Introduction to Deep Submicron CMOS Device Technology & Its Impact on Circuit Design [https://ewh.ieee.org/r5/denver/sscs/Presentations/2004_12_Loke.pdf]

• Subthreshold leakage
  • Drain-Induced Barrier Lowering (DIBL)
• Reverse-bias Source/Drain junction leakages
• Gate leakage
• two other leakage mechanisms
  • Gate Induced Drain Leakage (GIDL)
  • Punchthrough

W. M. Elgharbawy and M. A. Bayoumi, "Leakage sources and possible solutions in nanometer CMOS technologies," in IEEE Circuits and Systems Magazine, vol. 5, no. 4, pp. 6-17, Fourth Quarter 2005, doi: 10.1109/MCAS.2005.1550165.

X. Qi et al., "Efficient subthreshold leakage current optimization - Leakage current optimization and layout migration for 90- and 65- nm ASIC libraries," in IEEE Circuits and Devices Magazine, vol. 22, no. 5, pp. 39-47, Sept.-Oct. 2006, doi: 10.1109/MCD.2006.272999.

P. Monsurró, S. Pennisi, G. Scotti and A. Trifiletti, "Exploiting the Body of MOS Devices for High Performance Analog Design," in IEEE Circuits and Systems Magazine, vol. 11, no. 4, pp. 8-23, Fourthquarter 2011, doi: 10.1109/MCAS.2011.942751.

Andrea Baschirotto, ISSCC2015 "ADC Design in Scaled Technologies"

Joachim Assenmacher Infineon Technologies, "BSIM4 Modeling and Parameter Extraction" [https://ewh.ieee.org/r5/denver/sscs/References/2003_03_Assenmacher.pdf]

Stefan Rusu, Intel ISSCC 2008 Tutorial: "Leakage Reduction Techniques" [https://www.nishanchettri.com/isscc-slides/2008%20ISSCC/Tutorials/T06_Pres.pdf]

Drain-Induced Barrier Lowering (DIBL)

As a result of DIBL, threshold voltage is reduced with shorter channel lengths and, consequently, the subthreshold leakage current is increased

impact on output impedance

The principal impact of DIBL on circuit design is the degraded output impedance.

In short-channel devices, as \(V_{DS}\) increases further, drain-induced barrier lowering becomes significant, reducing the threshold voltage and increasing the drain current