Time-Interleaved ADCs

Posted on 2025-06-07 Edited on 2026-10-04 In adc-dac

Frequency Domain Model

f-mdl.drawio

Why (8.14) means "sampled at \(T_s\)" \[ X_s(f) = \frac{1}{T_s}\sum_{k=-\infty}^{\infty} X(f - kf_s) \]

This spectrum is periodic with period \(f_s\): copies of \(X(f)\) sit at every multiple of \(f_s\), with weight \(1/T_s\). That is exactly the spectrum of uniform sampling at \(T_s\)

Interleaver Architectures

Direct Interleaver

similar to increase the resolution of the flash ADC with more parallel comparators

De-multiplexing Interleaver

it is the front-end samplers that determine timing/bandwidth mismatch errors

only one front-end channel \(L=1\) eliminate any timing/bandwidth mismatch errors to the first order

Re-sampling Interleaver

back-end re-sampling occur after the front-end, two \(\frac{KT}{C}\) contribution in total noise (De-multiplexing Interleaver only one \(\frac{KT}{C}\))

without buffer, charging distribution reduce signal and reduce SNR, but buffers give excess noise

Interleaver Model

Interchannel Crosstalk

Interleaving Errors

Offset Mismatch Errors

Gain Mismatch Errors

Timing Mismatch Errors

In the frequency domain,

\[ \mathcal{L}\left\{ \frac{dV(t)}{dt} \right\} \;\longrightarrow\; sV(s) \quad\text{and, for }s=j\omega,\quad j\omega V(j\omega). \]

The factor

\[ j=e^{j\pi/2} \]

introduces a \(+\pi/2\) (90°) phase shift, while \(\omega\) scales the magnitude proportionally to frequency.

In the time domain,

\[ \frac{d}{dt}\sin(\omega t) =\omega\cos(\omega t) =\omega\sin\left(\omega t+\frac{\pi}{2}\right). \]

Thus, differentiation produces two effects: magnitude scaling by \(\omega\) and a \(90^\circ\) phase advance.

Frequency-dependent: the higher frequency input signal \(f_\text{in}\), the larger error becomes

Bandwidth Mismatch Errors

Clock Generation for Interleaved ADCs

Note that the falling edge of \(\phi_1\) is determined by the rising edge of CK and is thus free from jitter and mismatch in \(X_1\), which accumulates jitter and phase mismatch

reset mechanism to the latches for nominal order

a clock generator for an eight-channel ADC

NORed		\(\div 2\)	\(\div 2 \div 2\)
\(\phi_1\)	\(\text{CK}\)	\(I_1\)	\(I_2\)
\(\phi_2\)	\(\overline{\text{CK}}\)	\(Q_1\)	\(I_3\)
\(\phi_3\)	\(\text{CK}\)	\(\overline{I_1}\)	\(Q_2\)
\(\phi_4\)	\(\overline{\text{CK}}\)	\(\overline{Q_1}\)	\(Q_3\)
\(\phi_5\)	\(\text{CK}\)	\(I_1\)	\(\overline{I_2}\)
\(\phi_6\)	\(\overline{\text{CK}}\)	\(Q_1\)	\(\overline{I_3}\)
\(\phi_7\)	\(\text{CK}\)	\(\overline{I_1}\)	\(\overline{Q_2}\)
\(\phi_8\)	\(\overline{\text{CK}}\)	\(\overline{Q_1}\)	\(\overline{Q_3}\)

\(\text{CK}\), \(I/Q_1\) and \(I/Q_{2,3}\) shift by 1UI

resync (alignment)

TODO 📅

Calibration Techniques

Autocorrelation-based Skew Calibration

S. Chen, L. Wang, H. Zhang, R. Murugesu, D. Dunwell, A. Chan Carusone, “All-Digital Calibration of Timing Mismatch Error in Time-Interleaved Analog-to-Digital Converters,” IEEE Transactions on VLSI Systems, Sept. 2017. [PDF, slides]

B. Razavi, "Problem of timing mismatch in interleaved ADCs," Proceedings of the IEEE 2012 Custom Integrated Circuits Conference, San Jose, CA, USA, 2012 [pdf]

Binary-Search Calibration Method & its limitations

M. Gu, Y. Tao, X. He, Y. Zhong, L. Jie and N. Sun, "A 1-GS/s 11-b Time-Interleaved SAR ADC With Robust, Fast, and Accurate Autocorrelation-Based Background Timing-Skew Calibration," in IEEE Journal of Solid-State Circuits, vol. 60, no. 2, pp. 421-431, Feb. 2025

—. "Timing-Skew Calibration Techniques in Time-Interleaved ADCs," in IEEE Open Journal of the Solid-State Circuits Society, vol. 5, pp. 1-10, 2025 [https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=10804623]

An autocorrelation-based background timing-skew calibration method, which uses the correlations between adjacent channels to extract timing-skew errors, which relaxes the input bandwidth limitation up to the Nyquist frequency

Analyses Of The Derivative of The Autocorrelation

MAD (Mean Absolute Difference) vs. correlation

H. Wei, P. Zhang, B. Datta Sahoo and B. Razavi, "An 8-Bit 4-GS/s 120-mW CMOS ADC," Proceedings of the IEEE 2013 Custom Integrated Circuits Conference, San Jose, CA, USA, 2013 [pdf]

—, "An 8 Bit 4 GS/s 120 mW CMOS ADC," in IEEE Journal of Solid-State Circuits, vol. 49, no. 8, pp. 1751-1761, Aug. 2014 [pdf]

M. Gu, Y. Tao, X. He, Y. Zhong, L. Jie and N. Sun, "A 1-GS/s 11-b Time-Interleaved SAR ADC With Robust, Fast, and Accurate Autocorrelation-Based Background Timing-Skew Calibration," in IEEE Journal of Solid-State Circuits, vol. 60, no. 2, pp. 421-431, Feb. 2025

TODO 📅

approximate the absolute value operation by a squaring function

Overlapping versus Non-overlapping track time

tracking accuracy stay same, Cin (2Cs) counteract the longer tracking

Summing Interleaved Alias

The sampling function - impulse train is \[ s(t) = \sum_{n=-\infty}^{\infty}\left[ \delta(t-n4T_s) + \delta(t-n4T_s-T_s) + \delta(t-n4T_s-2T_s) + \delta(t-n4T_s-3T_s)\right] \]

Its Fourier transform is \[\begin{align} S(f) &= \frac{2\pi}{4T}\sum_{k=-\infty}^{\infty}\left[\delta(f-k\frac{f_s}{4}) + e^{-j2\pi f\cdot T_s}\delta(f-k\frac{f_s}{4}) + e^{-j2\pi f\cdot 2T_s}\delta(f-k\frac{f_s}{4}) + e^{-j2\pi f\cdot 3T_s}\delta(f-k\frac{f_s}{4}) \right] \\ &= \frac{2\pi}{4T}\sum_{k=-\infty}^{\infty}\left(1+e^{-j2\pi\frac{f}{f_s}} + e^{-j4\pi\frac{f}{f_s}} + e^{-j6\pi\frac{f}{f_s}} \right) \delta(f-k\frac{f_s}{4}) \\ &= \frac{2\pi}{4T}\sum_{k=-\infty}^{\infty}\left(1+e^{-jk\frac{\pi}{2}} + e^{-jk\pi} + e^{-jk\frac{3\pi}{2}} \right) \delta(f-k\frac{f_s}{4}) \end{align}\]

We define \(M[k] = 1+e^{-jk\frac{\pi}{2}} + e^{-jk\pi} + e^{-jk\frac{3\pi}{2}}\), which is periodic, i.e. \(M[k]=M[k+4]\) \[ M[k]=\left\{ \begin{array}{cl} 4 & : \ k = 4m \\ 0 & : \ k=4m+1 \\ 0 & : \ k=4m+2 \\ 0 & : \ k=4m+3 \\ \end{array} \right. \]

That is \[ S(f) = \frac{2\pi}{T}\sum_{k=-\infty}^{\infty} \delta(f-kf_s) \]

Alias has same frequency for each slice but different phase: Alias terms sum to zero if all slices match exactly

Random Chopping in TI-ADC

\[ D_n(kT) = (G_n R(kT) V(kT) + O_n)R(kT)= C_n V(kT) + R(kT)O_n \]

ADC buffers & memory effect

Y. Shifman, Y. Krupnik, U. Virobnik, A. Khairi, Y. Sanhedrai and A. Cohen, "A 1.64mW Differential Super Source-Follower Buffer with 9.7GHz BW and 43dB PSRR for Time-Interleaved ADC Applications in 10nm," 2019 IEEE Asian Solid-State Circuits Conference (A-SSCC), Macau, Macao, 2019 [pdf]

E. -H. Chen et al., "7.1 A 212.5Gb/s DSP-Based PAM-4 Transceiver with 50dB Loss Compensation for Large AI System Interconnects in 4nm FinFET," 2025 IEEE International Solid-State Circuits Conference (ISSCC), San Francisco, CA, USA, 2025

TODO 📅

DNL/INL Benefit

Each sub-ADC has its own physical capacitor array, so each slice has its own independent, zero-mean mismatch error

Because the sub-ADC are sampled round-robin, a busy input signal visits all \(N\) sub-ADC with equal probability over the whole input range. The transfer curve that the composite output presents is therefore the ensemble average \[ e_{\mathrm{eff}}(k) = \frac{1}{N}\sum_{i=1}^{N} e_i(k) \quad\Rightarrow\quad \sigma\!\left\{e_{\mathrm{eff}}(k)\right\} = \frac{\sigma\{e_i(k)\}}{\textcolor{red}{\sqrt{N}}} \] the measured static DNL/INL genuinely improves by \(\sqrt{N}\) at code-density (histogram) test

The composite result \[ \boxed{\sigma_{DNL,\max}^{TI} = \frac{\sigma_{\text{sub},DNL}}{\sqrt{N}}, \qquad \sigma_{INL,\max}^{TI} = \frac{\sigma_{\text{sub},INL}}{\sqrt{N}}} \] where \(N\) is channel number

Conventional binary array sub-SARADC \[ \sigma_{DNL,\max}=\sqrt{2^n-1}\,\frac{\sigma_u}{C_u}\,[\mathrm{LSB}], \qquad \sigma_{INL,\max}=\frac{\sqrt{2^n}}{2}\,\frac{\sigma_u}{C_u}\,[\mathrm{LSB}] \] VCM-based (\(n-1\) bit array) sub-SARADC

In a VCM-based (top-plate-sampled, tri-level) SAR, the MSB decision needs no capacitor switching at all — it's a direct comparison against \(V_{CM}\) \[ \sigma^{VCM}_{DNL,\max} = \sqrt{2^{n-1}-1}\,\frac{\sigma_u}{C_u}, \qquad \sigma^{VCM}_{INL,\max} = \frac{\sqrt{2^{n-1}}}{2}\,\frac{\sigma_u}{C_u} \qquad \text{at } k=2^{n-2},\,3\cdot 2^{n-2} \]

Paper from industry

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 [https://sci-hub.st/10.1109/ISSCC42614.2022.9731650]

P. Liu et al., "A 128Gb/s ADC/DAC Based PAM-4 Transceiver with >45dB Reach in 3nm FinFET," 2025 Symposium on VLSI Technology and Circuits (VLSI Technology and Circuits), Kyoto, Japan, 2025

8way-interleaving-Marvell-ISSCC2022.drawio

RX-Clocking-Marvell-VLSI2025.drawio

reference

Poulton, Ken. ISSCC2009 "Time-Interleaved ADCs, Past and Future" (slides)

—. CICC2010 "GHz ADCs: From Exotic to Mainstream", tutorial session, (slides)

—. ISSCC2015 "Interleaved ADCs Through the Ages", (slides)

ISSCC2015 F1: High-Speed Interleaved ADCs

Ron Kapusta, Analog Devices, CICC2015 ED007: SAR ADCs in parallel [time-interleaved] converter arrays

Samuel Palermo, ISSCC 2018 T10: ADC-Based Serial Links: Design and Analysis

Ewout Martens. ESSCIRC 2019 Tutorials: Advanced Techniques for ADCs for 5G Massive MIMO [https://youtu.be/7hYichGGU6k]

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

John P. Keane, ISSCC2020 T5: "Fundamentals of Time-Interleaved ADCs"

Athanasios Ramkaj. January 26, 2022, IEEE SSCS Santa Clara Valley Section Technical Talk: Design Considerations Towards Optimal High-Resolution Wide-Bandwidth Time-Interleaved ADCs [video] [slides]

V. Chen, "Tutorial: High-Speed Analog-to-Digital Converters," 2025 IEEE International Solid-State Circuits Conference (ISSCC), San Francisco, CA, USA, 2025, pp. 1-1, doi: 10.1109/ISSCC49661.2025.11076112.

Ahmed M. A. Ali 2016, "High Speed Data Converters"

Razavi, B. (2025). Analysis and design of data converters. Cambridge University Press.

S. Jang, J. Lee, Y. Choi, D. Kim, and G. Kim, "Recent advances in ultra-high-speed wireline receivers with ADC-DSP-based equalizers," IEEE Open Journal of the Solid-State Circuits Society (OJ-SSCS), vol. 4, pp. 290-304, Nov. 2024.

Yida Duan. Design Techniques for Ultra-High-Speed Time-Interleaved Analog-to-Digital Converters (ADCs) [http://www2.eecs.berkeley.edu/Pubs/TechRpts/2017/EECS-2017-10.pdf]

Preview Lecture #1 - "Extreme SAR ADCs" Online Course (2024) - Prof. Chi-Hang Chan (U. of Macau) [https://youtu.be/rgMRL4QZ-wA]

oscarmattia. Data Converter Toolbox [https://github.com/oscarmattia/data_converter_toolbox]