TODO 📅

reference

Jiří Lebl. Notes on Diffy Qs: Differential Equations for Engineers [link]

Matt Charnley. Differential Equations: An Introduction for Engineers [link]

Åström, K.J. & Murray, Richard. (2021). Feedback Systems: An Introduction for Scientists and Engineers Second Edition [https://www.cds.caltech.edu/~murray/books/AM08/pdf/fbs-public_24Jul2020.pdf]

Cadence Blog, “Resonant Frequency vs. Natural Frequency in Oscillator Circuits” [link]

TODO 📅

Linear phase

image-20241213232002042

image-20241213233748837
image-20241213233917966

[https://web.ece.ucsb.edu/~yoga/courses/DSP/P10_Linear_phase_FIR.pdf]

Digital DC Offset Correction

image-20241229122711845 XYβz111z1=Y therefore YX=1z11(1β)z1

VDD Droop Mitigation

image-20250105134745277

speed of voltage monitor does matter

reference

B. Farhang-Boroujeny (2013), Adaptive Filters: Theory and Applications (2nd ed.). John Wiley & Sons, Inc.

Simon O. Haykin (2014), “Adaptive Filter Theory” Prentice-Hall, Inc. 5rd edition

Diniz, P. S. R. (2020). Adaptive Filtering: Algorithms and Practical Implementation (5th ed.). Springer

Jiang X, ed. Digitally-Assisted Analog and Analog-Assisted Digital IC Design. Cambridge University Press; 2015.

Albert Jerng. ISSCC2012 T7: Digital Calibration for RF Transceivers [pdf]

Ahmed M. A. Ali. ISSCC2021 T5: Calibration Techniques in ADCs [pdf]

Salvatore Levantino. ISSCC2024 T5: Calibration Techniques in PLLs [pdf]


A. Chan Carusone and D. A. Johns, “Analog Filter Adaptation Using a Dithered Linear Search Algorithm,” IEEE Int. Symp. Circuits and Syst., May 2002. [PDF], [Slides]

—, Ph. D. Thesis, “Digital Algorithms for Analog Adaptive Filters”, Feb. 2002. [http://www.eecg.utoronto.ca/~tcc/thesis.pdf]

—, “Analog Adaptive Filters,” tutorial at the IEEE Int. Symp. Circuits and Syst., Bangkok, Thailand, May 2003. [http://www.eecg.utoronto.ca/~tcc/iscas03_tutorial.pdf]

David Johns, “Integrated Circuits for Digital Communications” [https://www.eecg.toronto.edu/~johns/nobots/courses/ece1392/slides.pdf]

Tai-Haur Kuo “Advanced Analog IC Design for Communications” [http://msic.ee.ncku.edu.tw/course/AdvancedAnalogICDesign/AdvancedAnalogICDesign.html]

Reference

DaVE - tools regarding on analog modeling,validation, and generation, [https://github.com/StanfordVLSI/DaVE]


Lim, Byong Chan,Ph.D. Dissertation 2012. “Model validation of mixed-signal systems” [https://stacks.stanford.edu/file/druid:xq068rv3398/bclim-thesis-submission-augmented.pdf]

—, J. -E. Jang, J. Mao, J. Kim and M. Horowitz, “Digital Analog Design: Enabling Mixed-Signal System Validation,” in IEEE Design & Test, vol. 32, no. 1, pp. 44-52, Feb. 2015 [http://iot.stanford.edu/pubs/lim-mixed-design15.pdf]

— , Mao, James & Horowitz, Mark & Jang, Ji-Eun & Kim, Jaeha. (2015). Digital Analog Design: Enabling Mixed-Signal System Validation. Design & Test, IEEE. 32. 44-52. [https://iot.stanford.edu/pubs/lim-mixed-design15.pdf]

—, M. Horowitz, “Error Control and Limit Cycle Elimination in Event-Driven Piecewise Linear Analog Functional Models,” in IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 63, no. 1, pp. 23-33, Jan. 2016 [https://sci-hub.se/10.1109/TCSI.2015.2512699]


Ben Yochret Sabrine, 2020, “BEHAVIORAL MODELING WITH SYSTEMVERILOG FOR MIXED-SIGNAL VALIDATION” [https://di.uqo.ca/id/eprint/1224/1/Ben-Yochret_Sabrine_2020_memoire.pdf]

“Creating Analog Behavioral Models VERILOG-AMS ANALOG MODELING” [https://www.eecis.udel.edu/~vsaxena/courses/ece614/Handouts/CDN_Creating_Analog_Behavioral_Models.pdf]

Rainer Findenig, Infineon Technologies. “Behavioral Modeling for SoC Simulation Bridging Analog and Firmware Demands” [https://www.coseda-tech.com/files/Files/Dokumente/Behavioral_Modeling_for_SoC_Simulation_COSEDA_UGM_2018.pdf]

Trellis Coding

TODO 📅

Convolutional Code

TODO 📅

Baseline Wander

TODO 📅

Pete Anslow, Ciena. Baseline wander with FEC [https://www.ieee802.org/3/bs/public/17_05/anslow_3bs_03_0517.pdf]

Vladimir Dmitriev-Zdorov. Baseline Wander, its Time Domain and Statistical Analysis [https://ibis.org/summits/feb19/dmitriev-zdorov.pdf]

Pavel Zivny, Tektronix. Baseline Wander: Systematic Approach to Rapid Simulation and Measurement [pdf]

Update on Performance Studies of 100 Gigabit Ethernet Enabled by Advanced Modulation Formats [https://www.ieee802.org/3/bm/public/sep12/wei_01_0912_optx.pdf]

Sampling Front-End (SFE) Pulse Response

image-20250107234500537

sweep the setup time between ideal pulse input and clock, sample the output of SFE at falling edge

ISI & DDJ filtering

image-20250104183820308

Modulation and SNR

Data and noise mutually uncorrelated

xRX, n[p] = d[p]hRX[0] + ∑ISI + n[p]

image-20250101105936807
image-20250101110902006

“ISI cancellation” based equalization is conceptually more straightforward but suffers from SNR penalty or error propagation

Jitter Amplification by Passive Channels

image-20250103215417021

CDR Loop Latency

image-20241102235118149
image-20241102235145417

loop latency is represented as esD in linear model


image-20241102235736432
image-20241103000223470
image-20241103000653906

Sensitivity to Loop Latency

image-20241103142137640

image-20241103142656134
image-20241103142531277
image-20241103142938907

Enhancing Resolution with a ΔΣ Modulator

Sub-Resolution Time Averaging

image-20241103160332995

ΔΣ modulator effectively dithers the LSB bit between zero and one, such that you can get the effective resolution of a much higher resolution DAC in the number of bits

Decimation

how they affect sampling phase

image-20241020140430663

DLF’s input bit-width can be reduced by decimating BBPD’s output. Decimation is typically performed by realizing either majority voting (MV) or boxcar filtering.

Note that deserialization is inherent to both MV and boxcar filtering

image-20241019225016868
  • Decimation is commonly employed to alleviate the high-speed requirement. However, decimation increases loop-latency which causes excessive dither jitter.
  • Decimation is basically, widen the data and slowing it down
  • Decimating by L means frequency register only added once every L UI, thus integral path gain reduced by L in linear model
  • proportional path gain is unchanged
intg_path_decim.drawio

CDR Linear Model

image-20220504101924272

condition:

Linear model of the CDR is used in a frequency lock condition and is approaching to achieve phase lock

Using this model, the power spectral density (PSD) of jitter in the recovered clock Sout(f) is Sout(f) = |HT(f)|2Sin(f) + |HG(f)|2SVCO(f) Here, we assume φin and φVCO are uncorrelated as they come from independent sources.

Jitter Transfer

HT(s)=φout(s)φin(s)|φvco=0=KPDKVCORs+KPDKVCOCs2+KPDKVCORs+KPDKVCOC

Using below notation ωn2=KPDKVCOCξ=KPDKVCO2ωn2

We can rewrite transfer function as follows HT(s)=2ξωns+ωn2s2+2ξωns+ωn2

The jitter transfer represents a low-pass filter whose magnitude is around 1 (0 dB) for low jitter frequencies and drops at 20 dB/decade for frequencies above ωn

image-20220504104202197
  • the recovered clock track the low-frequency jitter of the input data
  • the recovered clock DONT track the high-frequency jitter of the input data

The recovered clock does not suffer from high-frequency jitter even though the input signal may contain high-frequency jitter, which will limit the CDR tolerance to high-frequency jitter.

Jitter Peaking in Jitter Transfer Function

The peak, slightly larger than 1 (0dB) implies that jitter will be amplified at some frequencies in the CDR, producing a jitter amplitude in the recovered clock, and thus also in the recovered data, that is slightly larger than the jitter amplitude in the input data.

This is certainly undesirable, especially in applications such as repeaters.

image-20220504110722442

Jitter Generation

If the input data to the CDR is clean with no jitter, i.e., φin = 0, the jitter of the recovered clock comes directly from the VCO jitter. The transfer function that relates the VCO jitter to the recovered clock jitter is known as jitter generation. HG(s)=φoutφVCO|φin=0=s2s2+2ξωns+ωn2 Jitter generation is high-pass filter with two zeros, at zero frequency, and two poles identical to those of the jitter transfer function

image-20220504110737718

Jitter Tolerance

To quantify jitter tolerance, we often apply a sinusoidal jitter of a fixed frequency to the CDR input data and observe the BER of the CDR

The jitter tolerance curve DONT capture a CDR’s true tolerance to random jitter. Because we are applying “sinusoidal” jitter, which is deterministic signal.

We can deal only with the jitter’s amplitude and frequency instead of the PSD of the jitter thanks to deterministic sinusoidal jitter signal. JTOL(f) = |φin(f)|pp-max  for a fixed BER Where the subscript pp-max indicates the maximum peak-to-peak amplitude. We can further expand this equation as follows JTOL(f)=|φin(f)φe(f)||φe(f)|ppmax image-20220504114650749

Relative jitter, φe must be less than 1UIpp for error-free operation

In an ideal CDR, the maximum peak-to-peak amplitude of |φe(f)| is 1UI, i.e.,|φe(f)|pp − max = 1UI

Accordingly, jitter tolerance can be expressed in terms of the number of UIs as JTOL(f)=|φin(f)φe(f)|[UI] Given the linear CDR model, we can write JTOL(f)=|1+KPDKVCOHLF(f)j2πf|[UI] Expand HLF(f) for the CDR, we can write JTOL(f)=|12ξj(fnf)(fnf)2|[UI] image-20220504120538534

At frequencies far below and above the natural frequency, the jitter tolerance can be approximated by the following JTOL(f)={(fnf)2: ffn1: ffn

the jitter tolerance at very high jitter frequencies is limited to 1UIpp

OJTF

Concepts of JTF and OJTF

Simplified Block Diagram of a Clock-Recovery PLL pll_block_diagram

Jitter Transfer Function (JTF)

  • Input Signal Versus Recovered Clock
  • JTF, by jitter frequency, compares how much input signal jitter is transferred to the output of a clock-recovery’s PLL (recovered clock)
    • Input signal jitter that is within the clock recovery PLL’s loop bandwidth results in jitter that is faithfully transferred (closed-loop gain) to the clock recovery PLL’s output signal. JTF in this situation is approximately 1.
    • Input signal jitter that is outside the clock recovery PLL’s loop bandwidth results in decreasing jitter (open-loop gain) on the clock recovery PLL’s output, because the jitter is filtered out and no longer reaches the PLL’s VCO

Observed Jitter Transfer Function

  • Input Signal Versus Sampled Signal
  • OJTF compares how much input signal jitter is transferred to the output of a receiver’s decision making circuit as effected by a clock recovery’s PLL. As the recovered clock is the reference for detecting the input signal
    • Input signal jitter that is within the clock recovery PLL’s loop bandwidth results in jitter on the recovered clock which reduces the amount of jitter that can be detected. The input signal and clock signal are closer in phase
    • Input signal jitter that is outside the clock recovery PLL’s loop bandwidth results in reduced jitter on the recovered clock which increases the amount of jitter that can be detected. The input signal and clock signal are more out of phase. Jitter that is on both the input and clock signals can not detected or is reduced

JTF and OJTF for 1st Order PLLs

jsa_1st_order_graph
neuhelium-jtf-ojtf

The observed jitter is a complement to the PLL jitter transfer response OJTF=1-JTF (Phase matters!)

OTJF gives the amount of jitter which is tracked and therefore not observed at the output of the CDR as a function of the jitter rate applied to the input.

A-jtf-ojtf

Jitter Measurement

Jmeasured = JTFDUT ⋅ OJTFinstrument

The combination of the OJTF of a jitter measurement device and the JTF of the clock generator under test gives the measured jitter as a function of frequency.

image-20220716094732273

For example, a clock generator with a type 1, 1st order PLL measured with a jitter measurement device employing a golden PLL is Jmeasured=ω1s+ω1ss+ω2

Accurate measurement of the clock JTF requires that the OJTF cutoff of the jitter measurement be significantly below that of the clock JTF and that the measurement is compensated for the instrument’s OJTF.

The overall response is a band pass filter because the clock JTF is low pass and the jitter measurement device OJTF is high pass.

The compensation for the instrument OJTF is performed by measuring the jitter of the reference clock at each jitter rate being tested and comparing the reference jitter with the jitter measured at the output of the DUT.

jtf-ojtf

The lower the cutoff frequency of the jitter measurement device the better the accuracy of the measurement will be.

The cutoff frequency is limited by several factors including the phase noise of the DUT and measurement time.

Digital Sampling Oscilloscope

How to analyze jitter:

  • TIE (Time Interval Error) track
  • histogram
  • FFT

TIE track provides a direct view of how the phase of the clock evolves over time.

histogram provides valuable information about the long term variations in the timing.

FFT allows jitter at specific rates to be measured down to the femto-second range.

Maintaining the record length at a minimum of 1/10 of the inverse of the PLL loop bandwidth minimizes the response error

reference

Dalt, Nicola Da and Ali Sheikholeslami. “Understanding Jitter and Phase Noise: A Circuits and Systems Perspective.” (2018).

neuhelium, 抖动、眼图和高速数字链路分析基础 URL: http://www.neuhelium.com/ueditor/net/upload/file/20200826/DSOS254A/03.pdf

Keysight JTF & OJTF Concepts, https://rfmw.em.keysight.com/DigitalPhotonics/flexdca/FlexPLL-UG/Content/Topics/Quick-Start/jtf-pll-theory.htm?TocPath=Quick%20Start%7C_____4

Complementary Transmitter and Receiver Jitter Test Methodlogy, URL: https://www.ieee802.org/3/bm/public/mar14/ghiasi_01_0314_optx.pdf

SerDesDesign.com CDR_BangBang_Model URL: https://www.serdesdesign.com/home/web_documents/models/CDR_BangBang_Model.pdf

M. Schnecker, Jitter Transfer Measurement in Clock Circuits, LeCroy Corporation, DesignCon 2009. URL: http://cdn.teledynelecroy.com/files/whitepapers/designcon2009_lecroy_jitter_transfer_measurement_in_clock_circuits.pdf

VCO model

TODO 📅

respone to vctrl focus on phase

[https://designers-guide.org/verilog-ams/functional-blocks/vco/vco.va]

ADC Spec

TODO 📅

ENOB - Not sufficient & not accurate enough

  • Based on SNDR
  • Assume unbounded Gaussian distribution

quantization noise is ~ bounded uniform distribution

Using unbounded Gaussian -> pessimistic BER prediction

AFE Nonlinearity

“total harmonic distortion” (THD) in AFE

Relative to NRZ-based systems, PAM4 transceivers require more stringent circuit linearity, equalizers which can implement multi-level inter-symbol interference (ISI) cancellation, and improved sensitivity

image-20240923204055369

Because if it compresses, it turns out you have to use a much more complicated feedback filter. As long as it behaves linearly, the feedback filter itself can remain a linear FIR

image-20240923211841053

Linearity can actually be a critical constraint in these signal paths, and you really want to stay as linear as you can all the way up until the point where you’ve canceled all of the ISI

image-20240923222650556

A. Roshan-Zamir, O. Elhadidy, H. -W. Yang and S. Palermo, “A Reconfigurable 16/32 Gb/s Dual-Mode NRZ/PAM4 SerDes in 65-nm CMOS,” in IEEE Journal of Solid-State Circuits, vol. 52, no. 9, pp. 2430-2447, Sept. 2017 [https://people.engr.tamu.edu/spalermo/ecen689/2017_reconfigurable_16_32Gbps_NRZ_PAM4_SERDES_roshanzamir_jssc.pdf]

Hongtao Zhang, designcon2016. “PAM4 Signaling for 56G Serial Link Applications − A Tutorial”[https://www.xilinx.com/publications/events/designcon/2016/slides-pam4signalingfor56gserial-zhang-designcon.pdf]

Elad Alon, ISSCC 2014, “T6: Analog Front-End Design for Gb/s Wireline Receivers”

BER with Quantization Noise

image-20240804110522955

Var(X) = E[X2] − E[X]2

image-20240804110235178

Impulse Response or Pulse Response

image-20240807221637401

image-20240807224407213image-20240807224505987

TX FFE

TX FFE suffers from the peak power constraint, which in effect attenuates the average power of the outgoing signal - the low-frequency signal content has been attenuated down to the high-frequency level

image-20240727225120002

[https://www.signalintegrityjournal.com/articles/1228-feedforward-equalizer-location-study-for-high-speed-serial-systems]

S. Palermo, “CMOS Nanoelectronics Analog and RF VLSI Circuits,” Chapter 9: High-Speed Serial I/O Design for Channel-Limited and Power-Constrained Systems, McGraw-Hill, 2011.

Eye-Opening Monitor (EOM)

An architecture that evaluates the received signal quality

data slicers, phase slicers, error slicers, scope slicers

image-20240922143125270
image-20240922144605196

Analui, Behnam & Rylyakov, Alexander & Rylov, Sergey & Meghelli, Mounir & Hajimiri, Ali. (2006). A 10-Gb/s two-dimensional eye-opening monitor in 0.13-??m standard CMOS. Solid-State Circuits, IEEE Journal of. 40. 2689 - 2699, [https://chic.caltech.edu/wp-content/uploads/2013/05/B-Analui_JSSC_10-Gbs_05.pdf]

reference

G. Balamurugan, A. Balankutty and C. -M. Hsu, “56G/112G Link Foundations Standards, Link Budgets & Models,” 2019 IEEE Custom Integrated Circuits Conference (CICC), Austin, TX, USA, 2019, pp. 1-95 [https://youtu.be/OABG3u2H2J4?si=CxryBSGbxrUpZNBT]

Paul Muller Yusuf Leblebici École Polytechnique Fédérale de Lausanne (EPFL). Pattern generator model for jitter-tolerance simulation; VHDL-AMS models

Savo Bajic, ECE1392, Integrated Circuits for Digital Communications: StatOpt in Python [https://savobajic.ca/projects/academic/statopt]

Anritsu Company, “Measuring Channel Operating Margin,” 2016. [https://dl.cdn-anritsu.com/en-us/test-measurement/files/Technical-Notes/White-Paper/11410-00989A.pdf]

JLSD - Julia SerDe [https://github.com/kevjzheng/JLSD]

Kiran Gunnam, Selected Topics in RF, Analog and Mixed Signal Circuits and Systems

H. Shakiba, D. Tonietto and A. Sheikholeslami, “High-Speed Wireline Links-Part I: Modeling,” in IEEE Open Journal of the Solid-State Circuits Society, vol. 4, pp. 97-109, 2024 [https://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=10608184]

H. Shakiba, D. Tonietto and A. Sheikholeslami, “High-Speed Wireline Links-Part II: Optimization and Performance Assessment,” in IEEE Open Journal of the Solid-State Circuits Society, vol. 4, pp. 110-121, 2024 [https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=10579874]

G. Souliotis, A. Tsimpos and S. Vlassis, “Phase Interpolator-Based Clock and Data Recovery With Jitter Optimization,” in IEEE Open Journal of Circuits and Systems, vol. 4, pp. 203-217, 2023 [https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=10184121]

loop dynamic

Dithering Jitter in Bang-bang PLL

hunting jitter is also called as dithering jitter the time error between data clock and input data

  • proportional gain
  • loop latency
image-20240924225542342

where the proportional gain (KP), heavily damped systems means that KP ≫ KI

image-20240924234154476

Hanumolu, Pavan Kumar. 2006. Design Techniques for Clocking High Performance Signaling Systems. : Oregon State University. https://ir.library.oregonstate.edu/concern/graduate_thesis_or_dissertations/1v53k219r]

Hae-Chang Lee, “An Estimation Approach To Clock And Data Recovery” [https://www-vlsi.stanford.edu/people/alum/pdf/0611_HaechangLee_Phase_Estimation.pdf]

R. Walker, “Designing Bang-Bang PLLs for Clock and Data Recovery in Serial Data Transmission Systems,” in Phase-Locking in High-Performance Systems, B. Razavi, Ed. New Jersey: IEEE Press, 2003, pp. 34-45. [http://www.omnisterra.com/walker/pdfs.papers/BBPLL.pdf]

J. Kim, Design of CMOS Adaptive-Supply Serial Links, Ph.D. Thesis, Stanford University, December 2002. [https://www-vlsi.stanford.edu/people/alum/pdf/0212_Kim_______Design_Of_CMOS_AdaptiveSu.pdf]

P. K. Hanumolu, M. G. Kim, G. -y. Wei and U. -k. Moon, “A 1.6Gbps Digital Clock and Data Recovery Circuit,” IEEE Custom Integrated Circuits Conference 2006, San Jose, CA, USA, 2006, pp. 603-606 [https://sci-hub.se/10.1109/CICC.2006.320829]

Da Dalt N. A design-oriented study of the nonlinear dynamics of digital bang-bang PLLs. IEEE Transactions on Circuits and Systems I: Regular Papers. 2005;52(1):21–31. [https://sci-hub.se/10.1109/TCSI.2004.840089]

Jang S, Kim S, Chu SH, Jeong GS, Kim Y, Jeong DK. An optimum loop gain tracking all-digital PLL using autocorrelation of bang–bang phasefrequency detection. IEEE Transactions on Circuits and Systems II: Express Briefs. 2015;62(9):836–840. [https://sci-hub.se/10.1109/TCSII.2015.2435691]


ditheringjitter.drawio
image-20240925213924764

CDR Loop Latency

Denoting the CDR loop latency by ΔT , we note that the loop transmission is multiplied by exp(−sΔT) ≃ 1 − sΔT.The resulting right-half-plane zero, fz degrades the phase margin and must remain about one decade beyond the BW fz12πΔT

This assumption is true in practice since the bandwidth of the CDR (few mega Hertz) is much smaller than the data rate (multi giga bits/second).

Fernando , Marvell Italy.”Considerations for CDR Bandwidth Proposal” [https://www.ieee802.org/3/bs/public/16_03/debernardinis_3bs_01_0316.pdf]

Loop Bandwidth

The closed-loop −3-dB bandwidth is sometimes called the “loop bandwidth”

Continuous-Time Approximation Limitations

A rule of thumb often used to ensure slow changes in the loop is to select the loop bandwidth approximately equal to one-tenth of the input frequency.

image-20240806230158367
image-20240928095850580

Gardner, F.M. (1980). Charge-Pump Phase-Lock Loops. IEEE Trans. Commun., 28, 1849-1858.

Homayoun, Aliakbar and Behzad Razavi. “On the Stability of Charge-Pump Phase-Locked Loops.” IEEE Transactions on Circuits and Systems I: Regular Papers 63 (2016): 741-750.

N. Kuznetsov, A. Matveev, M. Yuldashev and R. Yuldashev, “Nonlinear Analysis of Charge-Pump Phase-Locked Loop: The Hold-In and Pull-In Ranges,” in IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 68, no. 10, pp. 4049-4061, Oct. 2021

Deog-Kyoon Jeong, Topics in IC Design - 2.1 Introduction to Phase-Locked Loop [https://ocw.snu.ac.kr/sites/default/files/NOTE/Lec%202%20-%20Charge-Pump%20PLL%2C%20Freuqency%20Synthesizers%2C%20and%20SSCG.pdf]

Limit Cycle Oscillation

limit cycles imply self-sustained oscillators due nonlinear nature

Ouzounov, S., Hegt, H., Van Roermund, A. (2007). SUB-HARMONIC LIMIT-CYCLE SIGMA-DELTA MODULATION, APPLIED TO AD CONVERSION. In: Van Roermund, A.H., Casier, H., Steyaert, M. (eds) Analog Circuit Design. Springer, Dordrecht. [https://sci-hub.se/10.1007/1-4020-5186-7_6]

BB PD

It’s ternary, because early, late and no transition

Linearing BB-PD

BB Gain is the slope of average BB output μ, versus phase offset ϕ, i.e. μϕ,

BB only produces output for a transition and this de-rates the gain. Transition density = 0.5 for random data

KBB=12μϕ

where μ = (1) × P(late|ϕ) + (−1) × P(early|ϕ)

bb-PDF.drawio

Both jitter and amplitude noise distribution are same, just scaled by slope

Self-Noise Term

One price we pay for BB PD versus linear PD is the self-noise term. For small phase errors BB output noise is the full magnitude of the sliced data

The PD output should be almost 0 for small phase errors. i.e. ideal PD output noise should be 0

σBB2 = 12 ⋅ P(trans) + 02 ⋅ (1−P(trans)) = 0.5

image-20241127215947017

Input referred jitter from BB PD is proportional to incoming jitter

image-20241127220933103

John T. Stonick, ISSCC 2011 TUTORIALS T5: DPLL-Based Clock and Data Recovery

Walker, Richard. (2003). Designing Bang-Bang PLLs for Clock and Data Recovery in Serial Data Transmission Systems. [pdf]

- Clock and Data Recovery for Serial Data Communications, focusing on bang-bang CDR design methodology, ISSCC Short Course, February 2002. [slides]

Digital CDR Category

image-20241024221619909
  • DCO part is analogous so that it cannot be perfectly modeled
  • Digital-to-phase converter is well-defined phase output, thus, very good to model real situation

DCO

image-20241024224500048
image-20241024224603927

limit cycle

image-20241026230332655

Z-domain modeling

image-20241027001226490

The difference equation is ϕ[n] = ϕ[n−1] + KDCOVC[n] ⋅ T ⋅ 2π z-transform is Φ(z)VC(z)=2πKDCOT1z1

where KDCO : Δf (Hz/bit)

ΔΣ-dithering in DCO

Quantization noise

image-20241019200102827

Here, αT is data transition density

BBPD quantization noise

DAC quantization noise

M. -J. Park and J. Kim, “Pseudo-Linear Analysis of Bang-Bang Controlled Timing Circuits,” in IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 60, no. 6, pp. 1381-1394, June 2013 [https://sci-hub.st/10.1109/TCSI.2012.2220502]

Time to Digital Converter (TDC)

Digital to Phase Converter (DPC)

IIR low pass filter

image-20241024232055792

simple approximation: z = 1 + sT bilinear-z transform z=

image-20241024232111368

Peak-to-peak jitter in ADPLL with BBPD

image-20241025001015194

Accumulate-and-dump (AAD) decimator

accumulating the input for N cycles and then latching the result and resetting the integrator

image-20241015222205883

It adds up N succeeding input samples at rate 1/T and delivers their sum in a single sample at the output. Therefore, the process comprises a filter (in the accumulation) and a down-sampler (in the dump)

Moving Average and CIC Filters

cascade-integrator-comb (CIC) decimator

TODO 📅

An Intuitive Look at Moving Average and CIC Filters [web, code]

A Beginner’s Guide To Cascaded Integrator-Comb (CIC) Filters [https://www.dsprelated.com/showarticle/1337.php]

FAQ

PLL vs. CDR

PLL CDR
Clock edge periodic Data edge random
Phase & Frequency detecting possible Phase detecting possible ,
Frequency detecting impossible

PLL or FD(Frequency Detector) for frequency detecting in CDR

reference

J. Stonick. ISSCC 2011 “DPLL-Based Clock and Data Recovery” [slides,transcript]

P. Hanumolu. ISSCC 2015 “Clock and Data Recovery Architectures and Circuits” [slides]

Amir Amirkhany. ISSCC 2019 “Basics of Clock and Data Recovery Circuits”

Fulvio Spagna. INTEL, CICC2018, “Clock and Data Recovery Systems” [slides]

M. Perrott. 6.976 High Speed Communication Circuits and Systems (lecture 21). Spring 2003. Massachusetts Institute of Technology: MIT OpenCourseWare, [lec21.pdf]

Akihide Sai. ISSCC 2023, T5 “All Digital Plls From Fundamental Concepts To Future Trends” [T5.pdf]

J. L. Sonntag and J. Stonick, “A Digital Clock and Data Recovery Architecture for Multi-Gigabit/s Binary Links,” in IEEE Journal of Solid-State Circuits, vol. 41, no. 8, pp. 1867-1875, Aug. 2006 [https://sci-hub.se/10.1109/JSSC.2006.875292]

—, “A digital clock and data recovery architecture for multi-gigabit/s binary links,” Proceedings of the IEEE 2005 Custom Integrated Circuits Conference, 2005.. [https://sci-hub.se/10.1109/CICC.2005.1568725]

Fernando De Bernardinis, eSilicon. “Introduction to DSP Based Serial Links” [https://www.corsi.univr.it/documenti/OccorrenzaIns/matdid/matdid835215.pdf]

Yohan Frans, CICC2019 ES3-3- “ADC-based Wireline Transceivers” [pdf]


H. Kang et al., “A 42.7Gb/s Optical Receiver With Digital Clock and Data Recovery in 28nm CMOS,” in IEEE Access, vol. 12, pp. 109900-109911, 2024 [https://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=10630516]

Marinaci, Stefano. “Study of a Phase Locked Loop based Clock and Data Recovery Circuit for 2.5 Gbps data-rate” [https://cds.cern.ch/record/2870334/files/CERN-THESIS-2023-147.pdf]

P. Palestri et al., “Analytical Modeling of Jitter in Bang-Bang CDR Circuits Featuring Phase Interpolation,” in IEEE Transactions on Very Large Scale Integration (VLSI) Systems, vol. 29, no. 7, pp. 1392-1401, July 2021 [https://sci-hub.se/10.1109/TVLSI.2021.3068450]

F. M. Gardner, “Phaselock Techniques”, 3rd Edition, Wiley Interscience, Hoboken, NJ, 2005 [https://picture.iczhiku.com/resource/eetop/WyIgwGtkDSWGSxnm.pdf]

Rhee, W. (2020). Phase-locked frequency generation and clocking : architectures and circuits for modern wireless and wireline systems. The Institution of Engineering and Technology

M.H. Perrott, Y. Huang, R.T. Baird, B.W. Garlepp, D. Pastorello, E.T. King, Q. Yu, D.B. Kasha, P. Steiner, L. Zhang, J. Hein, B. Del Signore, “A 2.5 Gb/s Multi-Rate 0.25μm CMOS Clock and Data Recovery Circuit Utilizing a Hybrid Analog/Digital Loop Filter and All-Digital Referenceless Frequency Acquisition,” IEEE J. Solid-State Circuits, vol. 41, Dec. 2006, pp. 2930-2944 [https://cppsim.com/Publications/JNL/perrott_jssc06.pdf]

M.H. Perrott. CICC 2009 “Tutorial on Digital Phase-Locked Loops” [https://www.cppsim.com/PLL_Lectures/digital_pll_cicc_tutorial_perrott.pdf]

—, Short Course On Phase-Locked Loops and Their Applications Day 4, PM Lecture “Examples of Leveraging Digital Techniques in PLLs” [https://www.cppsim.com/PLL_Lectures/day4_pm.pdf]

—, Short Course On Phase-Locked Loops IEEE Circuit and System Society, San Diego, CA “Digital Frequency Synthesizers” [https://www.cppsim.com/PLL_Lectures/digital_pll.pdf]

Gain Kim, “Equalization, Architecture, and Circuit Design for High-Speed Serial Link Receiver” [pdf]


Deog-Kyoon Jeong Topics in IC(Wireline Transceiver Design) - 3.1. Introduction to All Digital PLL [https://ocw.snu.ac.kr/sites/default/files/NOTE/Lec%203%20-%20ADPLL.pdf]

—, Topics in IC(Wireline Transceiver Design) - 6.1 Introduction to Clock and Data Recovery [https://ocw.snu.ac.kr/sites/default/files/NOTE/Lec%206%20-%20Clock%20and%20Data%20Recovery.pdf]

High-speed Serial Interface Lect. 16 – Clock and Data Recovery 3 [http://tera.yonsei.ac.kr/class/2013_1_2/lecture/Lect16_CDR-3.pdf]

Shiva Kiran. Phd thesis 2018. Modeling and Design of Architectures for High-Speed ADC-Based Serial Links [https://hdl.handle.net/1969.1/192031]

—, et al., “Modeling of ADC-Based Serial Link Receivers With Embedded and Digital Equalization,” in IEEE Transactions on Components, Packaging and Manufacturing Technology, vol. 9, no. 3, pp. 536-548, March 2019 [https://sci-hub.se/10.1109/TCPMT.2018.2853080]

K. Zheng, “System-Driven Circuit Design for ADC-Based Wireline Data Links”, Ph.D. Dissertation, Stanford University, 2018 [https://purl.stanford.edu/hw458fp0168]

S. Cai, A. Shafik, S. Kiran, E. Z. Tabasy, S. Hoyos and S. Palermo, “Statistical modeling of metastability in ADC-based serial I/O receivers,” 2014 IEEE 23rd Conference on Electrical Performance of Electronic Packaging and Systems [pdf]

John M. Cioffi. “Decoding Methods” [https://cioffi-group.stanford.edu/doc/book/chap7.pdf]

—. “Equalization” [https://cioffi-group.stanford.edu/doc/book/chap3.pdf]

Iain. [https://youtu.be/rnjy4_gXLAg?si=PC3aowaon-e_mhXX]

—. [https://youtu.be/IJE94FhyygM?si=BMMQ-GmirBWNf4ep]

Definition of Phase Noise

image-20250104080553842

Eq. (3.25) is widely adopted by industry and academia

image-20250104080619943

using the narrow angle assumption, the two definitions above are equivalent

If the narrow angle condition is not satisfied, however, the two definitions differ

Phase Noise Profile

Power Spectral Density of Brownian Motion despite non-stationary [https://dsp.stackexchange.com/a/75043/59253]

white noise

1/f2 Phase Noise Profile

image-20250104084510063
image-20250104084814395
image-20250104085222610
image-20250104084925644
image-20250104085722649

flicker noise

1/f3 Phase Noise Profile

Sϕn=Kf(KVCO2πf)21f3


image-20250104092711462

[https://dsp.stackexchange.com/a/75152/59253]

Free-running Oscillator

image-20250103224818171

Note that fmin is related to the observation time. The longer we observe the device under test, the smaller fmin must be

image-20250104091109521

image-20250104111025626

Lorentzian spectrum

image-20240720134811859

We typically use the two spectra, Sϕn(f) and Sout(f), interchangeably, but we must resolve these inconsistencies. voltage spectrum is called Lorentzian spectrum


The periodic signal x(t) can be expanded in Fourier series as:

image-20240720141514040

Assume that the signal is subject to excess phase noise, which is modeled by adding a time-dependent noise component α(t). The noisy signal can be written x(t+α(t)), the added excess phase ϕ(t)=α(t)ω0

image-20250103211650043

The autocorrelation of the noisy signal is by definition:

image-20240720141525576

The autocorrelation averaged over time results in:

image-20240720141659415

By taking the Fourier transform of the autocorrelation, the spectrum of the signal x(t+α(t))​ can be expressed as

image-20240720141813256

It is also interesting to note how the integral in Equation 9.80 around each harmonic is equal to the power of the harmonic itself |Xn|2

The integral Sx(f) around harmonic is Px,n=f=|Xn|2ω02n2c14ω04n4c2+(ω+nω0)2df=|Xn|2Δf=2ββ2+(2πΔf)2dΔf=|Xn|21πarctan(2πΔfβ)|=|Xn|2

The phase noise does not affect the total power in the signal, it only affects its distribution

  • Without phase noise, Sv(f) is a series of impulse functions at the harmonics of fo.
  • With phase noise, the impulse functions spread, becoming fatter and shorter but retaining the same total power

Integration Limits

Y. Zhao and B. Razavi, “Phase Noise Integration Limits for Jitter Calculation,”[https://www.seas.ucla.edu/brweb/papers/Conferences/YZ_ISCAS_22.pdf]

TODO 📅

Phase perturbed by a stationary noise with Gaussian PDF

image-20241227233228376
image-20241228022311313

If keep ϕrms in Rx(τ), i.e. Rx(τ)=A22eϕrms2cos(2πf0τ)eRϕ(τ)A22eϕrms2cos(2πf0τ)(1+Rϕ(τ)) The PSD of the signal is Sx(f)=F{Rx(τ)}=Pc2eϕrms2[Sϕ(f+f0)+Sϕ(ff0)]+Pc2eϕrms2[δ(f+f0)+δ(ff0)] ❗❗above Eq isn’t consistent with stationary white noise process - the following section

Phase perturbed by a stationary WHITE noise process

image-20241207091104944

Assuming that the delay line is noiseless

image-20241207100921644

image-20241207091457850

Expanding the cosine function we get Ry(t,τ)=A22{cos(2πf0τ)E[cos(ϕ(t)ϕ(tτ))]sin(2πf0τ)E[sin(ϕ(t)ϕ(tτ))]}+A22{cos(4πf0(t+τ/2TD))E[cos(ϕ(t)+ϕ(tτ))]sin(4πf0(t+τ/2TD))E[sin(ϕ(t)+ϕ(tτ))]}

where, both the process ϕ(t) − ϕ(tτ) and ϕ(t) + ϕ(tτ) are independent of time t, i.e. E[cos(ϕ(t)+ϕ(tτ))] = mcos +(τ), E[cos(ϕ(t)−ϕ(tτ))] = mcos −(τ), E[sin(ϕ(t)+ϕ(tτ))] = msin +(τ) and E[sin(ϕ(t)−ϕ(tτ))] = msin −(τ)

we obtain Ry(t,τ)=A22{cos(2πf0τ)mcos(τ)sin(2πf0τ)msin(τ)}+A22{cos(4πf0(t+τ/2TD))mcos+(τ)sin(4πf0(t+τ/2TD))msin+(τ)}

The second term in the above expression is periodic in t and to estimate its PSD, we compute the time-averaged autocorrelation function Ry(τ)=A22{cos(2πf0τ)mcos(τ)sin(2πf0τ)msin(τ)} image-20241207095906575

After nontrivial derivation

image-20241207104018395
image-20241227205459845

image-20241207103912086

Phase perturbed by a Weiner process

image-20241207103414365
image-20241207105127885

The phase process ϕ(t) is also gaussian but with an increasing variance which grows linearly with time t

image-20241207110524419

Ry(t,τ)=A22{cos(2πf0τ)E[cos(ϕ(t)ϕ(tτ))]sin(2πf0τ)E[sin(ϕ(t)ϕ(tτ))]}+A22{cos(4πf0(t+τ/2TD)E[cos(ϕ(t)+ϕ(tτ))]sin(4πf0(t+τ/2TD)E[sin(ϕ(t)+ϕ(tτ))]}

The spectrum of y(t) is determined by the asymptotic behavior of Ry(t,τ) as t → ∞

❗❗ limt → ∞Ry(t,τ) rather than time-averaged autocorrelation function of cyclostationary process, ref. Demir’s paper

We define ζ(t,τ) = ϕ(t) + ϕ(tτ) = ϕ(t) − ϕ(tτ) + 2ϕ(tτ), the expected value of ζ(t,τ) is 0, the variance is σζ2 = (kσ)2(τ+4(tτ)) = (kσ)2(4t−3τ) E[cos(ζ(t,τ))]=12πσζ2eζ2/2σζ2cos(ζ)dζ=eσζ2/2=e(kσ)2(4tτ) i.e., limt → ∞E[cos(ζ(t,τ))] = limt → ∞e−(kσ)2(4tτ) = 0

For E[sin(ζ(t,τ))], we have E[sin(ζ(t,τ))]=12πσζ2eζ2/2σζ2sin(ζ)dζ i.e., E[sin(ζ(t,τ))] is odd function, therefore E[sin(ζ(t,τ))] = 0

Finally, we obtain

image-20241207114053083
image-20241227210018613
image-20241207114805792

image-20241207174403033
image-20241207181038749
image-20241208100556466

VCO ISF Simulation

PSS + PXF Method

Yizhe Hu, “A Simulation Technique of Impulse Sensitivity Function (ISF) Based on Periodic Transfer Function (PXF)” [https://bbs.eetop.cn/thread-869343-1-1.html]

TODO 📅

Transient Method

David Dolt. ECEN 620 Network Theory - Broadband Circuit Design: “VCO ISF Simulation” [https://people.engr.tamu.edu/spalermo/ecen620/ISF_SIM.pdf]

image-20241016211020230
image-20241016211101204
image-20241016211115630

To compare the ring oscillator and VCO the total injected charge to both should be the same

reference

A. Hajimiri and T. H. Lee, “A general theory of phase noise in electrical oscillators,” in IEEE Journal of Solid-State Circuits, vol. 33, no. 2, pp. 179-194, Feb. 1998 [paper], [slides]

—, “Corrections to”A General Theory of Phase Noise in Electrical Oscillators”” [https://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=678662]

—, RFIC2024 “Noise in Oscillators from Understanding to Design”

Carlo Samori, “Phase Noise in LC Oscillators: From Basic Concepts to Advanced Topologies” [https://www.ieeetoronto.ca/wp-content/uploads/2020/06/DL-VCO-short.pdf]

—, “Understanding Phase Noise in LC VCOs: A Key Problem in RF Integrated Circuits,” in IEEE Solid-State Circuits Magazine, vol. 8, no. 4, pp. 81-91, Fall 2016 [https://picture.iczhiku.com/resource/eetop/whIgTikLswaaTVBv.pdf]

—, ISSCC2016, “Understanding Phase Noise in LC VCOs”

A. Demir, A. Mehrotra and J. Roychowdhury, “Phase noise in oscillators: a unifying theory and numerical methods for characterization,” in IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 47, no. 5, pp. 655-674, May 2000 [https://sci-hub.se/10.1109/81.847872]

Dalt, Nicola Da and Ali Sheikholeslami. “Understanding Jitter and Phase Noise: A Circuits and Systems Perspective.” (2018) [https://picture.iczhiku.com/resource/eetop/WykRGJJoHQLaSCMv.pdf]

F. L. Traversa, M. Bonnin and F. Bonani, “The Complex World of Oscillator Noise: Modern Approaches to Oscillator (Phase and Amplitude) Noise Analysis,” in IEEE Microwave Magazine, vol. 22, no. 7, pp. 24-32, July 2021 [https://iris.polito.it/retrieve/handle/11583/2903596/e384c433-b8f5-d4b2-e053-9f05fe0a1d67/MM%20noise%20-%20v5.pdf]

Poddar, Ajay & Rohde, Ulrich & Apte, Anisha. (2013). How Low Can They Go?: Oscillator Phase Noise Model, Theoretical, Experimental Validation, and Phase Noise Measurements. Microwave Magazine, IEEE. [http://time.kinali.ch/rohde/noise/how_low_can_they_go-2013-poddar_rohde_apte.pdf]

Pietro Andreani, “RF Harmonic Oscillators Integrated in Silicon Technologies” [https://www.ieeetoronto.ca/wp-content/uploads/2020/06/DL-Toronto.pdf]

Chembiyan T, “Brownian Motion And The Oscillator Phase Noise” [link]

—, “Jitter and Phase Noise in Oscillators” [link]

—, “Jitter and Phase Noise in Phase Locked Loops” [link]

—, “PLLs and reference spurs” [link]

Godone, A. & Micalizio, Salvatore & Levi, Filippo. (2008). RF spectrum of a carrier with a random phase modulation of arbitrary slope. [https://sci-hub.se/10.1088/0026-1394/45/3/008]

Bae, Woorham; Jeong, Deog-Kyoon: ‘Analysis and Design of CMOS Clocking Circuits for Low Phase Noise’ (Materials, Circuits and Devices, 2020)

Akihide Sai, Toshiba. ISSCC 2023 T5: All-digital PLLs From Fundamental Concepts to Future Trends [https://www.nishanchettri.com/isscc-slides/2023%20ISSCC/TUTORIALS/T5.pdf]

  • proportional term (P) depends on the present error
  • integral term (I) depends on past errors
  • derivative term (D) depends on anticipated future errors

PID controller makes use of linear extrapolation of the measured output

PI controller does not make use of any prediction of the future state of the system

The prediction by linear extrapolation (D) can generate large undesired control signals because measurement noise is amplified, that’s why D is not used widely


limit cycle

image-20250105094709373

reference

Gene F. Franklin, J. David Powell, and Abbas Emami-Naeini. Feedback Control of Dynamic Systems, Global Edition (8th Edition). Pearson. [https://mrce.in/ebooks/Feedback%20Control%20of%20Dynamic%20Systems%208th%20Ed.pdf]

Åström, K.J. & Murray, Richard. (2021). Feedback Systems: An Introduction for Scientists and Engineers Second Edition [https://www.cds.caltech.edu/~murray/books/AM08/pdf/fbs-public_24Jul2020.pdf]

Yan Lu, ISSCC2021 T10: Fundamentals of Fully Integrated Voltage Regulators [https://www.nishanchettri.com/isscc-slides/2021%20ISSCC/TUTORIALS/ISSCC2021-T10.pdf]

image-20241004163356709

charge pumps are capacitive DC-DC converters. The two most common switched capacitor voltage converters are the voltage inverter and the voltage doubler circuit


image-20241014211627207

voltage doubler

image-20241019092038444

output buffer capacitor

To achieve a stable DC output voltage

Step-Wise Ramp-Up

without load

VinCp + Vout, n − 1Co = (Vout, nVin)Cp + Vout, nCo

We derive a recursive equation that describes the output voltage Vout, n after the nth clock cycle Vout,n=2VinCp+Vout,n1CoCp+Co

Voltage Ripple & Droop

ripple_droop.drawio

(VtVh)(Cp+Co)=Iload2fsw(VhVb)Co=Iload2fsw

we obtain VtVb=IloadfswCo(1Cp2(Cp+Co)) That is, peak-to-peak ripple ΔVout,p2pIloadfswCo    if  CoCp

Then, with aforementioned Step-Wise Ramp-Up equation, Vt=2VinCp+VbCoCp+Co Vb=2VinIloadfswCp(1+Cp2Co)Vt=2VinIloadfswCp(1Cp2(Cp+Co))

Therefore, average output voltage Vout in steady-state is Vout=Vt+Vb2=2VinIloadfswCp(1+Cp24Co(Cp+Co))2VinIloadfswCp which results in a simple expression for the output voltage droop

ΔVout=IloadfswCp

The charge pump can be modeled as a voltage source with a source resistance Rout. Therefore, ΔVout can be seen as the voltage drop across Rout due to the load current:

Rout=ΔVoutIload=1fswCp image-20241015072846141

multiphase CP

multiphaeCP.drawio

(VtVb)(Cp+Co) = IloadΔt

Therefore peak-to-peak ripple ΔVout,p2p=IloadΔtCp+Co=IloadΔtCtot

where Ctot = Cp + Co

with {Vb=2VinIloadΔtCpVt=2VinIloadΔtCp+IloadΔtCp+Co

Then Vout=Vt+Vb2=2VinIloadΔtCpCp+2Co2Cp+2Co2VinIloadΔtCp That is output voltage droop ΔVout=IloadΔtCp

reference

Bernhard Wicht, “Design of Power Management Integrated Circuits”. 2024 Wiley-IEEE Press

Breussegem, T. v., & Steyaert, M. (2013). CMOS integrated capacitive DC-DC converters. Springer

Zhang, Milin, Zhihua Wang, Jan van der Spiegel and Franco Maloberti. “Advanced Tutorial on Analog Circuit Design.” (2023).

Anton Bakker, Tim Piessens., ISSCC2014 T9: Charge Pump and Capacitive DC-DC Converter Design

Wicht, B., ISSCC2020 T2: Analog Building Blocks of DC-DC Converters [https://www.nishanchettri.com/isscc-slides/2020%20ISSCC/TUTORIALS/T2Visuals.pdf]

Hoi Lee, ISSCC2018 T8: Fundamentals of Switched-Mode Power Converter Design [slides,transcript]

image-20241019142915175

alternative view of sampling, assuming DC value is A

sampling-c2d-d2d.drawio
  • xc(t) and xs(t)

    xc=A; xs=AT: therefore Xs(j0)=1TXc(j0)

  • x[n] and xd[n]

    x=A; xd=A2: therefore Xd(ej0)=12X(ej0)

expander

sampling-expander.drawio
  • x[n] and xe[n]

    x=A; xe=A: therefore Xe(ej0) = X(ej0)

    Fourier transform of the output of the expander is a frequency-scaled version of the Fourier transform of the input


Subsampling or Downsampling

image-20241004151215993
image-20241004151308422
image-20241004151434477
  • Eqs. (4.72)

    the superposition of an infinite set of amplitude-scaled copies of Xc(jΩ), frequency scaled through ω = ΩTd and shifted by integer multiples of 2π

  • Eq. (4.77)

    the superposition of M amplitude-scaled copies of the periodic Fourier transform X(ejω), frequency scaled by M and shifted by integer multiples of 2π


downsampled by a factor of M = 2

image-20241004161805974

image-20241005073349726
image-20241005073534041

Upsampling or Zero Insertion

image-20241006072426572
image-20241006074425704
image-20241006075854246

image-20241006074604512

sampling identities

sampling-ID.drawio

downsampling identity

image-20241007085509889
image-20241007090624888

upsampling identity

image-20241007085527233
image-20241007090939701

Polyphase Decomposition

image-20241020122709610
image-20241020122726153

where ek[n] = h[nM+k]


Polyphase Implementation of Decimation Filters & Interpolation Filters

Decimation system Interpolation system
image-20241020123035001 image-20241020123043829
image-20241020123027067 image-20241020123101780
sampling identity image-20241020123345371 image-20241020123355113

LPTV Implementation

TODO 📅

The interpolation filter following an up-sampler generally is time varying and cannot be represented by a simple transfer function. The equivalent filter in a zero-order hold is an exception, perhaps unique, that can be represented with a time-invariant transfer function

Dr. Deepa Kundur, Multirate Digital Signal Processing: Part I [pdf, https://www.comm.utoronto.ca/dkundur/course/discrete-time-systems/]

ZOH interpolator

The interpolation filter following an up-sampler generally is time varying and cannot be represented by a simple transfer function. The equivalent filter in a Zero-Order Hold is an exception, perhaps unique, that can be represented with a time-invariant transfer function


zoh.drawio F1(z)=X(zLM)1zLM1z1

Split the 1 : LM hold process into a 1 : L hold followed by a 1 : M hold Y(η)=X(ηL)1ηL1η1 then F2(z)=Y(zM)1zM1z1=X(zLM)1zLM1zM1zM1z1=X(zLM)1zLM1z1

That is F1(z) = F2(z), i.e. they are equivalent


image-20241103180315919

Random Signals & Multirate Systems

Balu Santhanam, Probability Theory & Stochastic Process 2020: Random Signals & Multirate Systems [https://ece-research.unm.edu/bsanthan/ece541/rand.pdf]

Decimation by Summing

proportional path

The loop gain of a proportional path is unchanged

phug_loop.drawio

In (a), the loop gain is ϕo(z)ϕe(z), which is LGa(z)=ϕo(z)ϕe(z)=11z1

In (b), Accumulate-and-dump (AAD) is 1zL1z1, then ϕm(η) can be expressed as ϕm(η)=1η11η1/L1L Hence ϕo(η)=ϕm(η)11η1=1η11η1/L1L11η1

After zero-order hold process, we obtain ϕf(z), which is ϕf(z)=ϕo(zL)1zL1z1=1zL1z11L11zL1zL1z1 i.e., LGb(z)=11z11L1zL1z1

When bandwidth is much less than sampling rate (data rate), 1L1zL1z11

Therefore LGb(z)11z1

In the end LGa(z) ≈ LGb(z)


Assume PD output is constant

phug_seq.drawio

integral path

integral path gain reduced by L

frug_loop.drawio

In (a), ϕo(z)=1(1z1)2, i.e. LGa(z)=1(1z1)2

In (b), after Accumulate-and-dump (AAD), ϕ(η) is ϕm(η)=1η11η1/L1L

After frequency integrator and phase integrator ϕo(η)=ϕm(η)1(1η1)2=1η11η1/L1L1(1η1)2 Then ϕf(z) is shown as below ϕf(z)=ϕo(zL)1zL1z1=1zL1z11L1(1zL)21zL1z1=1L1(1z1)2

That is, LGb(z)=1L1(1z1)2=1LLGa(z)


Assume PD output is constant

frug_seq.drawio

limn+ΔP1ΔP0=limn+n+2LnL+αL+βL2=1L

Decimation by Voting

image-20241126211307012

In above screenshot

  1. KD is just relative value
  2. frug shall not be scaled by decimator factor

proved as below

DC gain KB of summing (boxcar filter) is decimation factor M , voting gain KV is about 0.54Kb = 0.54M

  1. downsampling 1M and ZOH 1zM1z1 can be can be cancelled out at low frequency
  2. decimation gain: accumulator 1zM1z1 replaced with linearizing gain KB and majority voting replaced with KV

proportional path: LGph=KBB1zM1z11M11zM1zM1z1KBB1zM1z111zM=KBBKD11zM

integral path: LGfr=KBB1zM1z11M1(1zM)21zM1z1KBB1zM1z11(1zM)2=KBBKD1(1zM)2

J. Stonick. ISSCC 2011 “DPLL-Based Clock and Data Recovery” [slides,transcript]

J. L. Sonntag and J. Stonick, “A Digital Clock and Data Recovery Architecture for Multi-Gigabit/s Binary Links,” in IEEE Journal of Solid-State Circuits, vol. 41, no. 8, pp. 1867-1875, Aug. 2006 [https://sci-hub.se/10.1109/JSSC.2006.875292]

J. Sonntag and J. Stonick, “A digital clock and data recovery architecture for multi-gigabit/s binary links,” Proceedings of the IEEE 2005 Custom Integrated Circuits Conference, 2005.. [https://sci-hub.se/10.1109/CICC.2005.1568725]

Y. Xia et al., “A 10-GHz Low-Power Serial Digital Majority Voter Based on Moving Accumulative Sign Filter in a PS-/PI-Based CDR,” in IEEE Transactions on Microwave Theory and Techniques, vol. 68, no. 12 [https://sci-hub.se/10.1109/TMTT.2020.3029188]

J. Liang, A. Sheikholeslami, “On-Chip Jitter Measurement and Mitigation Techniques for Clock and Data Recovery Circuits” [https://tspace.library.utoronto.ca/bitstream/1807/91138/3/Liang_Joshua_201706_PhD_thesis.pdf]

J. Liang, A. Sheikholeslami. ISSCC2017. “A 28Gbps Digital CDR with Adaptive Loop Gain for Optimum Jitter Tolerance” [slides,paper]

J. Liang, A. Sheikholeslami,, “Loop Gain Adaptation for Optimum Jitter Tolerance in Digital CDRs,” in IEEE Journal of Solid-State Circuits [https://sci-hub.se/10.1109/JSSC.2018.2839038]

M. M. Khanghah, K. D. Sadeghipour, D. Kelly, C. Antony, P. Ossieur and P. D. Townsend, “A 7-Bit 7-GHz Multiphase Interpolator-Based DPC for CDR Applications,” in IEEE Transactions on Circuits and Systems I: Regular Papers [https://cora.ucc.ie/bitstreams/7ae5bfaa-8dd9-45a7-8276-99676b7b6078/download]

[CDR CIRCUIT-BLOCKS: DESIGN AND VERIFICATION USING VERILOG - 2.6. DECIMATOR]

Michael H. Perrott, Tutorial on Digital Phase-Locked Loops, CICC 2009, San Jose, CA, Sept. 13, 2009 [https://www.cppsim.com/PLL_Lectures/digital_pll_cicc_tutorial_perrott.pdf]

Liu, Tao, Tiejun Li, Fangxu Lv, Bin Liang, Xuqiang Zheng, Heming Wang, Miaomiao Wu, Dechao Lu, and Feng Zhao. 2021. “Analysis and Modeling of Mueller-Muller Clock and Data Recovery Circuits” Electronics 10 [https://www.mdpi.com/2079-9292/10/16/1888/pdf?version=1628492599]

Gu, Youzhi & Feng, Xinjie & Chi, Runze & Chen, Yongzhen & Wu, Jiangfeng. (2022). Analysis of Mueller-Muller Clock and Data Recovery Circuits with a Linearized Model. 10.21203/rs.3.rs-1817774/v1. [https://assets-eu.researchsquare.com/files/rs-1817774/v1_covered.pdf?c=1664188179]

Chen, Junkun, Youzhi Gu, Xinjie Feng, Runze Chi, Jiangfeng Wu, and Yongzhen Chen. 2024. “Analysis of Mueller–Muller Clock and Data Recovery Circuits with a Linearized Model” Electronics [https://mdpi-res.com/electronics/electronics-13-04218/article_deploy/electronics-13-04218-v2.pdf?version=1730106095]

K. Yadav, P. -H. Hsieh and A. C. Carusone, “Loop Dynamics Analysis of PAM-4 Mueller–Muller Clock and Data Recovery System,” in IEEE Open Journal of Circuits and Systems [https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=9910561]

TODO 📅

Tristate: α = 1

XOR: α = 1

1T in Divider

image-20240928004526381
image-20240928004308700

Michael H. Perrott, PLL Design Using the PLL Design Assistant Program. [https://designers-guide.org/forum/Attachments/pll_manual.pdf]


1T & T come from CT-DT & DT-CT

image-20240928203714450

H. Kang et al., “A 42.7Gb/s Optical Receiver With Digital Clock and Data Recovery in 28nm CMOS,” in IEEE Access, vol. 12, pp. 109900-109911, 2024 [https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=10630516]

Sonntag JSSC 2006

image-20241129222258061
image-20241129223706720
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clear;
close all;
clc;


Tb = 200e-12;
Ts = Tb*8; % the decimation factor was 8
z = tf('z', Ts);

Kdpc = 1/2^9;
Kv = 8*0.54;
Kpd = 10.6;
phug = 2^-3;
frug = 2^-12;
Nel = 18;

options = bodeoptions;
options.FreqUnits = 'MHz';
options.XLim = [1e-2, 1e1];
options.YLim = [-10, 5];

L = Kpd*Kv*Kdpc/(1-z^-1)*(phug + frug/(1-z^-1))*z^-Nel;
TF = L/(1+L);
bodemag(TF,options);

hold on;
frug = 2^-11;
L = Kpd*Kv*Kdpc/(1-z^-1)*(phug + frug/(1-z^-1))*z^-Nel;
TF = L/(1+L);
bodemag(TF,options);

hold on;
frug = 2^-10;
L = Kpd*Kv*Kdpc/(1-z^-1)*(phug + frug/(1-z^-1))*z^-Nel;
TF = L/(1+L);
bodemag(TF,options);

legend('frug=2^{-12}','frug=2^{-11}', 'frug=2^{-10}', 'FontSize',10)
grid on;
title('phase transfer function', 'FontSize', 12)
xlabel('frequency', 'FontSize',10)
ylabel('frequency response', 'FontSize',10)

Full View

image-20241129223734870

Kpd, Kb, Kv

Both decimation factor and factor for voting are 4

image-20241130162850467
  • Kpd formula: 12.467; Kpd_bb_0 12.465
  • Kpd_Kb: 49.860; Kpd_Kv 27.265
  • Kb: 4.00; Kv 2.19

That is

  1. gain of BoxCar is the decimation factor
  2. Voting across 4 inputs had a 54% reduced gain relative to boxcar filter
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import numpy as np
from scipy.stats import norm
import itertools
from collections import defaultdict
import matplotlib.pyplot as plt

sigmai = 0.032 #UI, input jitter
Ptrans = 0.5 # Transition density
deci_factor = 4

phase_error = np.linspace(-0.1, 0.1, 201) #UI, phase offset
pd_late = norm.cdf(phase_error/sigmai)
pd_early = 1.0 - pd_late
pd_avg = pd_late*1.0 - 1.0*pd_early

Kpd_bb = (pd_avg[1:] - pd_avg[:-1])/(phase_error[1:] - phase_error[:-1])*Ptrans
Kpd_bb_0 = np.max(Kpd_bb)

## by formula
Kpd_calc = 1.0/(sigmai*np.sqrt(2*np.pi))

print(f'Kpd formula: {Kpd_calc:.3f}; Kpd_bb_0 {Kpd_bb_0:.3f}') # Kpd formula: 12.467; Kpd_bb_0 12.465

plt.figure()
plt.plot(phase_error, pd_avg, color='r', linewidth=3)
plt.title('!! PD average output vs timing offset(UI)')
plt.grid()
plt.show()


prob = np.zeros((phase_error.shape[0],3))
prob[:,0] = pd_early*Ptrans # -1
prob[:,1] = 1.0 - Ptrans # 0
prob[:,2] = pd_late*Ptrans # 1

pd_out = np.array([-1.0,0.0,1.0])
idxs = list([[0,1,2] for _ in range(deci_factor)])
boxcar_avg = []
voting_avg = []
for i in range(phase_error.shape[0]):
prob_i = prob[i,:]
boxcar_tmp = 0.0
voting_tmp = 0.0
for idxs_tmp in itertools.product(*idxs):
pd_list = pd_out[[idxs_tmp]]
prob_list = prob_i[[idxs_tmp]]
pd_sum = np.sum(pd_list)
pd_vote = 1.0 if pd_sum > 0.0 else -1.0 if pd_sum <0.0 else 0.0
prob_prod = np.prod(prob_list)
boxcar_tmp += pd_sum*prob_prod
voting_tmp += pd_vote*prob_prod
boxcar_avg.append(boxcar_tmp)
voting_avg.append(voting_tmp)

boxcar_avg = np.array(boxcar_avg)
voting_avg = np.array(voting_avg)

plt.figure()
plt.plot(phase_error,boxcar_avg, label='FIR BoxCar', color='r', linewidth=3)
plt.plot(phase_error,voting_avg, label='Voting', color='b', linewidth=3, linestyle='--')
plt.legend()
plt.title('!!PD+BoxCar / !!PD+Voting vs timing offset(UI)')
plt.grid()
plt.show()


Kpd_Kb = (boxcar_avg[1:] - boxcar_avg[:-1])/(phase_error[1:] - phase_error[:-1])
Kpd_Kv = (voting_avg[1:] - voting_avg[:-1])/(phase_error[1:] - phase_error[:-1])
Kpd_kb_0 = np.max(Kpd_Kb)
Kpd_kv_0 = np.max(Kpd_Kv)
print(f'Kpd_Kb: {Kpd_kb_0:.3f}; Kpd_Kv {Kpd_kv_0:.3f}') # Kpd_Kb: 49.860; Kpd_Kv 27.265

plt.figure()
plt.plot(phase_error[:-1], Kpd_Kb, color='r', linewidth=3)
plt.plot(phase_error[:-1], Kpd_Kv, color='b', linewidth=3, linestyle='--')
plt.legend(['Kpd_Kb', 'Kpd_Kv'])
plt.title('Kpd*Kb / Kpd*Kv vs timing offset(UI)')
plt.grid()
plt.show()

Kb = Kpd_kb_0 / Kpd_bb_0
Kv = Kpd_kv_0 / Kpd_bb_0
print(f'Kb: {Kb:.2f}; Kv {Kv:.2f}') # Kb: 4.00; Kv 2.19

reference

Alan V Oppenheim, Ronald W. Schafer. 2010. Discrete-Time Signal Processing, 3rd edition

R. E. Crochiere and L. R. Rabiner, “Multirate Digital Signal Processing”, Prentice Hall, 1983.

John G. Proakis and Dimitris G. Manolakis, Digital Signal Processing: Principles, Algorithms, and Applications, 4th edition, 2007.

D. Sundararajan. 2024. Digital Signal Processing: An Introduction 2nd Edition

F. M. Gardner, “Phaselock Techniques”, 3rd Edition, Wiley Interscience, Hoboken, NJ, 2005 [https://picture.iczhiku.com/resource/eetop/WyIgwGtkDSWGSxnm.pdf]

Rhee, W. (2020). Phase-locked frequency generation and clocking : architectures and circuits for modern wireless and wireline systems. The Institution of Engineering and Technology

Integrator

TODO 📅

[https://www.eecg.utoronto.ca/~johns/ece1371/slides/10_switched_capacitor.pdf]

[https://www.seas.ucla.edu/brweb/papers/Journals/BRWinter17SwCap.pdf]

[https://class.ece.iastate.edu/ee508/lectures/EE%20508%20Lect%2029%20Fall%202016.pdf]

Push-Pull

TODO 📅

Rinaldo Castello, “LINEARIZATION TECHNIQUES FOR PUSH-PULL AMPLIFIERS” [https://www.ieeetoronto.ca/wp-content/uploads/2020/06/AMPLIFIERS_Stanf_Tor_2016_Last.pdf]

MOS parasitic Rd&Rs, Cd&Cs

Decrease the parasitic R&C

priority: Rs > Rd, Cs > Cd

XCP as Negative Impedance Converter (NIC)

The Cross-Coupled Pair (XCP) can operate as an impedance negator [a.k.a. a negative impedance converter (NIC)]

A common application is to create a negative capacitance that can cancel the positive capacitance seen at a port, thereby improving the speed

image-20240922174319496 INIC=VimVip2gm+1sCc=2Vip2gm+1sCc Therefore ZNIC=VipVimINIC=2VipINIC=2gm1sCc half-circuit

If Cgd is considered, and apply miller effect. half equivalent circuit is shown as below

nic.drawio

B. Razavi, “The Cross-Coupled Pair - Part III [A Circuit for All Seasons],” IEEE Solid-State Circuits Magazine, Issue. 1, pp. 10-13, Winter 2015. [https://www.seas.ucla.edu/brweb/papers/Journals/BR_Magzine3.pdf]

S. Galal and B. Razavi, “10-Gb/s Limiting Amplifier and Laser/Modulator Driver in 0.18um CMOS Technology,” IEEE Journal of Solid-State Circuits, vol. 38, pp. 2138-2146, Dec. 2003. [https://www.seas.ucla.edu/brweb/papers/Journals/G&RDec03_2.pdf]

Flipped Voltage Follower (FVF)

image-20240921110019881
image-20240921113630249

T&H buffer in ADC

image-20240923200147070

[https://www.linkedin.com/posts/chembiyan-t-0b34b910_flipped-voltage-follower-fvf-basics-activity-7118482840803020800-qwyX?utm_source=share&utm_medium=member_desktop]

Z. Guo et al., “A 112.5Gb/s ADC-DSP-Based PAM-4 Long-Reach Transceiver with >50dB Channel Loss in 5nm FinFET,” 2022 IEEE International Solid-State Circuits Conference (ISSCC), San Francisco, CA, USA, 2022, pp. 116-118, doi: 10.1109/ISSCC42614.2022.9731650.

Super-source follower (SSF)

image-20240924213742877
image-20240924213845608
image-20240924213853954

A. Sheikholeslami, “Voltage Follower, Part III [Circuit Intuitions],” in IEEE Solid-State Circuits Magazine, vol. 15, no. 2, pp. 14-26, Spring 2023, doi: 10.1109/MSSC.2023.3269457

Paul R. Gray. 2009. Analysis and Design of Analog Integrated Circuits (5th. ed.). Wiley Publishing. [pdf]

Double differential Pair

Vip and Vim are input, Vrp and Vrm are reference voltage Vo=Av(VipVimVrpVrm)

2diffpair.drawio

In differential comparison mode, the feedback loop ensure Vip = Vrp, Vim = Vrm in the end

assume input and reference common voltage are same

Pros of (b)

  • larger input range i.e., >±2Vov of (a), it works even one differential is off due to lower voltage
  • larger gm (smaller input difference of pair)

Cons of (b)

  • sensitive to the difference of common voltage between Vip, Vim and Vrp, Vrm

common-mode voltage difference

doublepair_cm.drawio

copy aforementioned formula here for convenience Vo=Av(VipVimVrpVrm)

at sample phase Vip = Vim = Vcmi and Vrp = Vrm = Vcmr

  • Iip0 = Iim0 = Ii0
  • Irp0 = Irm0 = Ir0

i.e. Iip+IrmIim+Irp=0

at compare start

  • Vip = Vim = Vcmi and Vrp = Vcmr + Δ, Vrp = Vcmr − Δ

  • Iip < Iip0, Irp > Irp0

  • Iim > Iim0, Irm < Irm0

i.e. Iip+IrmIim+Irp<0, we need to increase Vip and decrease Vim.

at the compare finish

Vip=Vcmi+ΔVim=VcmiΔ

and Iip0 = Iim0 = Ii0, Irp0 = Irm0 = Ir0

i.e. Iip+IrmIim+Irp=0


If VcmrVcmi=2VOV+δ, and δ > 0. one transistor carries the entire tail current

  • Iip = 0 and Irp = ISS, all the time

At the end, Vim = Vcmi − (Δδ), the error is δ

In closing, VcmrVcmi<2VOV for normal work

Furthermore, the difference between Vcmr and Vcmi should be minimized due to limited impedance of current source and input pair offset

In the end VcmrVcmi<2VOVVOS

Under the condition, every transistor of pairs are on in equilibrium

pair mismatch

diff_mismatch_connect.drawio

ISE=gm(σvth,0+σvth,1)IDE=gm(σvth,0+σvth,1)

The input equivalient offset voltage Vos,SE=ISE2gm=σvth,0+σvth,12Vos,DE=IDEgm=σvth,0+σvth,1

Then σvos,SE=2σvth24=σvth2σvos,DE=2σvth2=2σvth

We obtain σvos, DE = 2σvos, SE

peaking without inductor

TODO 📅

How to generate complex poles without inductor? [https://a2d2ic.wordpress.com/2020/02/19/basics-on-active-rc-low-pass-filters/]

Input Diff-Pair

DM Distortion

image-20241027095213326

CM Distortion

image-20241027095248946

Resistive Degeneration

Resistive degeneration in differential pairs serves as one major technique for linear amplifier

image-20240824132739726

The linear region for CMOS differential pair would be extended by  ± ISSR/2 as all of ISS/2 flows through R. Vin+Vin=VOV+VTH+ISS2RVTH=2ISSμnCOXWL+ISSR2

Jri Lee, “Communication Integrated Circuits.” https://cc.ee.ntu.edu.tw/~jrilee/publications/Comm_IC.pdf

Figure 14.12, Design of Analog CMOS Integrated Circuits, Second Edition [https://electrovolt.ir/wp-content/uploads/2014/08/Design-of-Analog-CMOS-Integrated-Circuit-2nd-Edition-ElectroVolt.ir_.pdf]

Biasing Tradeoffs in Resistive-Degenerated Diff Pair

image-20241027095520556

Todd Brooks, Broadcom “Input Programmable Gain Amplifier (PGA) Design for ADC Signal Conditioning” [https://classes.engr.oregonstate.edu/eecs/spring2021/ece627/Lecture%20Notes/OSU%20Classroom%20Presentaton%20042511.ppt]

Source-Degenerated Differential Pairs

TODO 📅

reference

Elad Alon, ISSCC 2014, “T6: Analog Front-End Design for Gb/s Wireline Receivers” [https://picture.iczhiku.com/resource/eetop/wHKfZPYpAleAKXBV.pdf]

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

Minsoo Choi et al., “An Approximate Closed-Form Channel Model for Diverse Interconnect Applications,” IEEE Transactions on Circuits and Systems-I: Regular Papers, vol. 61, no. 10, pp. 3034-3043, Oct. 2014.

K. Yadav, P. -H. Hsieh and A. Chan Carusone, “Linearity Analysis of Source-Degenerated Differential Pairs for Wireline Applications,” in IEEE Open Journal of Circuits and Systems, [link]

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