Digital Phase-Locked Loops

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Hunting Jitter

Hunting jitter is often referred to as dithering jitter, the periodic time error between data clock and input data, which exhibits a limit-cycle behavior

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BB PD

Youngdon Choi, Deog-Kyoon Jeong and W. Kim, "Jitter transfer analysis of tracked oversampling techniques for multigigabit clock and data recovery," in IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing, vol. 50, no. 11, pp. 775-783, Nov. 2003 [https://sci-hub.st/10.1109/TCSII.2003.819070]

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]

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

notice the transition density = 1 in digital PLL

Linearization

The effective PD gain is a function of the input jitter pdf, it enables one to anticipate the effects of input jitter on loop characteristics

BB Gain is the slope of average BB output \(\mu\), versus phase offset \(\phi\), i.e. \(\frac {\partial \mu}{\partial \phi}\),

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

\[ K_{BB} = \frac{1}{2}\frac {\partial \mu}{\partial \phi} \]

where \(\mu = (1)\times \mathrm{P}(\text{late}|\phi) + (-1)\times \mathrm{P}(\text{early}|\phi)\)

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

\[ \sigma_{BB}^2 = 1^2 \cdot \mathrm{P}(\text{trans}) + 0^2\cdot (1-\mathrm{P}(\text{trans})) = 0.5 \]

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Input referred jitter from BB PD is proportional to incoming jitter

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gain simulation

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]

T. -K. Kuan and S. -I. Liu, "A Bang Bang Phase-Locked Loop Using Automatic Loop Gain Control and Loop Latency Reduction Techniques," in IEEE Journal of Solid-State Circuits, vol. 51, no. 4, pp. 821-831, April 2016 [https://sci-hub.st/10.1109/JSSC.2016.2519391]

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import matplotlib.pyplot as plt
import numpy as np

N = 2**10
sigma = 0.1
dt = np.random.normal(sigma,size=N)
et = np.sign(dt)

# Eq-(2)
coef_form = np.mean(np.abs(dt)) / np.mean(np.power(dt, 2))
print(f'coef_form: {coef_form}')

# Eq-(9)
coef_gauss = (2/np.pi)**0.5/sigma
print(f'coef_gauss: {coef_gauss}')

# polyfit
coef_fit = np.polyfit(dt, et, 1)
print(f'coef_fit: {coef_fit}')

x = np.linspace(-3.5, 3.5, 1000)
y = coef_fit[0]*x + coef_fit[1]

plt.figure(figsize=(12,6))
plt.plot(dt, et, 'o')
plt.plot(x, y, linewidth=2, linestyle='--')

# Calculate histogram counts and bin edges
counts, bin_edges = np.histogram(dt, bins=100)
# Find the maximum count
max_count = counts.max()
# Create weights to normalize the maximum height to 1
weights = np.ones_like(dt) / max_count
plt.hist(dt, bins=100, weights=weights)

plt.xlabel(r'$\Delta t$')
plt.grid(True)
plt.legend([r'$\Delta t \sim \varepsilon $', r'$x_{fit} \sim y_{fit}$', r'$ \text{hist}_{\Delta t}$'])
plt.show()


# coef_form: 0.7910794009505085
# coef_gauss: 7.978845608028654
# coef_fit: [0.79013999 0.0204332 ]

Pavan, Shanthi, Richard Schreier, and Gabor Temes. (2016). Understanding Delta-Sigma Data Converters. 2nd ed. Wiley. - 2.2.1 Quantizer Modeling

image-20250902212931083 \[ \frac{\mathrm{d}\sigma_e^2}{\mathrm{d}k} =0\space\space\Rightarrow\space\space k=\frac{\left\langle v,y\right\rangle}{\left\langle y,y \right\rangle} \]

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DCO Quantization Noise

TODO 📅

TDC Quantization Noise

TODO 📅

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CDR Loop Latency

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

CC Chen. Why A Low Loop Latency in A CDR Design? [https://youtu.be/io9WZbhlahU]

—. Why Understanding and Optimizing Loop Latency for A CDR Design? [https://youtu.be/Jyy18865jv8]

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loop latency is represented as \(e^{-sD}\) in linear model

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Sensitivity to Loop Latency

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Optimizing Loop Latency

TODO 📅

CC Chen. Circuit Image: Why Understanding and Optimizing Loop Latency for A CDR Design? [https://youtu.be/Jyy18865jv8]

DT & CT Spectral Density

Sam Palermo, ECEN620: Network Theory Broadband Circuit Design Fall 2025 Lecture 9: Digital PLLs [https://people.engr.tamu.edu/spalermo/ecen620/lecture09_ee620_digital_PLLs.pdf]

Michael Perrott, August 14, 2008. Short Course On Phase-Locked Loops and Their Applications Day 4, AM Lecture Digital Frequency Synthesizers [https://www.cppsim.com/PLL_Lectures/day4_am.pdf]

—, "A modeling approach for Sigma Delta fractional-N frequency synthesizers allowing straightforward noise analysis," in IEEE Journal of Solid-State Circuits, vol. 37, no. 8, pp. 1028-1038, Aug. 2002 [https://www.cppsim.com/Publications/JNL/perrott_jssc02.pdf]

—. "Techniques for high data rate modulation and low power operation of fractional-N frequency synthesizers." 1997. [https://www.cppsim.com/Publications/Theses/perrott_phdthesis.pdf]

Hsu, Chun-Ming, Ph. D. Massachusetts Institute of Technology. "Techniques for high-performance digital frequency synthesis and phase control." 2008. [http://hdl.handle.net/1721.1/45870]

J. R. Barry, E. A. Lee, and D. G. Messerschmitt, Digital Communication, 3rd ed., Boston, MA: Kluwer Academic Publishers, 2003.

psd_ct_dt.drawio

  • \(\color{red}\frac{1}{T}\) of CT-DT originates from CT to sampled sequence to impulse train Fourier transform, which is explicit in frequency-domain model
  • \(\color{red}\frac{1}{T}\) originates from DT spectrum to CT impulse spectrum is implicit in frequency-domain model
  • \(\color{red}T\) of DT-CT originates from ZOH approximation following impulse train, which is explicit in frequency-domain model

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Divider Sampling Operation & ITM

Impulse Train Modulator (ITM)

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The double outline of the box in the figure is meant to serve as a reminder that a sampling operation is taking place

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DT -> CT

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\(\boxed{S_x(e^{j2\pi fT}) = S_d(f)\cdot \textcolor{blue}{\frac{1}{T}}=S_c(f)\cdot \textcolor{blue}{\frac{1}{T}}}\), For example, the quantization noise spectrum is given by \(S_x(e^{j2\pi fT}) = \frac{1}{12f_s}\cdot \frac{1}{T} = \frac{1}{12}\). To convert the sequence spectrum into a continuous impulse train spectrum, we multiply by \(\color{blue}\frac{1}{T^2}\) \[ S_y(f) = S_x(e^{j2\pi fT}) \cdot \textcolor{blue}{T\cdot \frac{1}{T^2}\cdot |H(f)|^2 } = S_x(e^{j2\pi fT}) \cdot \textcolor{blue}{\frac{1}{T}|H(f)|^2} \]

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CT -> DT -> CT

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reference

Topics in IC (Wireline Transceiver Design) [https://ocw.snu.ac.kr/sites/default/files/NOTE/Lec%203%20-%20ADPLL.pdf]

Michael H. Perrott, ISSCC 2008 Tutorial on Digital Phase-Locked Loops

—, CICC 2009 Tutorial on Digital Phase-Locked Loops [https://www.cppsim.com/PLL_Lectures/digital_pll_cicc_tutorial_perrott.pdf]

Robert Bogdan Staszewski, CICC 2020: Beyond All-Digital PLL for RF and Millimeter-Wave Frequency Synthesis [link]

Akihide Sai, ISSCC 2023 T5: All-digital PLLs From Fundamental Concepts to Future Trends

Mike Shuo-Wei Chen, CICC 2020 ES2-3: Low-Spur PLL Architectures and Techniques [https://youtu.be/sgPDchYhN-4]

S. Levantino, "Digital phase-locked loops," 2018 IEEE Custom Integrated Circuits Conference (CICC), San Diego, CA, USA, 2018

Saurabh Saxena, IIT Madras. Phase-Locked Loops: Noise Analysis in Digital PLL [https://youtu.be/mddtxcqfiKU]

Y. Hu, T. Siriburanon and R. B. Staszewski, "Multirate Timestamp Modeling for Ultralow-Jitter Frequency Synthesis: A Tutorial," in IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 69, no. 7, pp. 3030-3036, July 2022

Neil Robertson. Digital PLL's -- Part 1 [https://www.dsprelated.com/showarticle/967.php]

—. Digital PLL's -- Part 2 [https://www.dsprelated.com/showarticle/973.php]

—. Digital PLL's -- Part 3 [https://www.dsprelated.com/showarticle/1177.php]

Daniel Boschen. GRCon24 - Quick Start on Control Loops with Python Workshop [video, slides]

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]

M. Zanuso, D. Tasca, S. Levantino, A. Donadel, C. Samori and A. L. Lacaita, "Noise Analysis and Minimization in Bang-Bang Digital PLLs," in IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 56, no. 11, pp. 835-839, Nov. 2009 [https://sci-hub.st/10.1109/TCSII.2009.2032470]

N. Da Dalt, "Linearized Analysis of a Digital Bang-Bang PLL and Its Validity Limits Applied to Jitter Transfer and Jitter Generation," in IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 55, no. 11, pp. 3663-3675, Dec. 2008 [https://sci-hub.st/10.1109/TCSI.2008.925948]

—, "Markov Chains-Based Derivation of the Phase Detector Gain in Bang-Bang PLLs," in IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 53, no. 11, pp. 1195-1199, Nov. 2006 [https://sci-hub.st/10.1109/TCSII.2006.883197]

—, "A design-oriented study of the nonlinear dynamics of digital bang-bang PLLs," in IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 52, no. 1, pp. 21-31, Jan. 2005 [https://sci-hub.se/10.1109/TCSI.2004.840089]

—, "Theory and Implementation of Digital Bang-Bang Frequency Synthesizers for High Speed Serial Data Communications", PhD Dissertation, RWTH Aachen University, Aachen, North Rhine-Westphalia, Germany, 2007 [pdf]

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

hunting jitter

S. Jang, S. Kim, S. -H. Chu, G. -S. Jeong, Y. Kim and D. -K. Jeong, "An Optimum Loop Gain Tracking All-Digital PLL Using Autocorrelation of Bang–Bang Phase-Frequency Detection," in IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 62, no. 9, pp. 836-840, Sept. 2015 [https://sci-hub.st/10.1109/TCSII.2015.2435691] [phd thesis]

Deog-Kyoon Jeong. Topics in IC (Wireline Transceiver Design). Lec 3 - All-Digital PLL [https://ocw.snu.ac.kr/sites/default/files/NOTE/Lec%203%20-%20ADPLL.pdf]

—. Topics in IC (Wireline Transceiver Design). Lec 6 - Clock and Data Recovery [https://ocw.snu.ac.kr/sites/default/files/NOTE/Lec%206%20-%20Clock%20and%20Data%20Recovery.pdf]

Lee Hae-Chang.: ‘An estimation approach to clock and data recovery’, PhD Thesis, Stanford University, November 2006 [pdf]

J. Kim, Design of CMOS Adaptive-Supply Serial Links, Ph.D. Thesis, Stanford University, December 2002. [pdf]

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

CC Chen. Why Hunting Jitter Happens in CDR: The Role of Input Jitter and Latency? [https://youtu.be/hPDielPsFgY]