KNOW-HOW

Rd & Rs Tradeoff

TODO 📅

reference

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

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.

Delta Modulator

\[ D_{out} = s V_F \] attenuates the low-frequency content of the signal, and amplifies high-frequency noise.

Delta-Sigma Modulator

The average output of DSM tracks the input signal

integrator

TODO 📅

opamp bandwidth & slewing

reference

B. Razavi, "The Delta-Sigma Modulator [A Circuit for All Seasons]," IEEE Solid-State Circuits Magazine, Volume. 8, Issue. 20, pp. 10-15, Spring 2016. [http://www.seas.ucla.edu/brweb/papers/Journals/BRSpring16DeltaSigma.pdf]

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

R.S. Ashwin Kumar. EE698I: Mixed-signal IC design [https://home.iitk.ac.in/~ashwinrs/2023_EE698I.html]

R.S. Ashwin Kumar. EE698W: Analog circuits for signal processing [https://home.iitk.ac.in/~ashwinrs/2022_EE698W.html]

Vishal Saxena. ECE 697 Delta-Sigma Data Converter Design [https://www.eecis.udel.edu/~vsaxena/courses/ece697A/s10/ECE697.htm]

-, ECE 615 Mixed Signal IC Design Spring 2016, Boise State University [https://www.eecis.udel.edu/~vsaxena/courses/ece615/s16/ECE615.htm]

David A. Johns. ECE1371H - Advanced Analog Circuits - 2014 [https://www.eecg.toronto.edu/~johns/ece1371/ece1371.html]

Richard E. Schreier, ECE 1371 Advanced Analog Circuits - 2015 [http://individual.utoronto.ca/schreier/ece1371-2015.html]

Trevor Caldwell. ECE 1371S Advanced Analog Circuits - 2020 [http://individual.utoronto.ca/trevorcaldwell/ece1371.html]

Noise Analysis

TODO 📅

Noise Simulation

SNR during sampling region and decison region increase

SNR during regeneration region is constant, where noise is critical

\[ \text{SNR} = \frac{V_{o,sig}^2}{V_{o,n}^2} = \frac{V_{i,sig}^2}{V_{i,n}^2} \]

we can get \(V_{i,n}^2 = \frac{V_{i,sig}^2}{\text{SNR}}\), which is constant also

That is \[ V_{i,n}^2 = \frac{V_{i,sig}^2}{V_{o,sig}^2}V_{o,n}^2 = \frac{V_{o,n}^2}{A_v^2} \] where \(V_{i,sig}\) is constant signal is applied to input of comparator

offset

reference

Xu, H. (2018). Mixed-Signal Circuit Design Driven by Analysis: ADCs, Comparators, and PLLs. UCLA. ProQuest ID: Xu_ucla_0031D_17380. Merritt ID: ark:/13030/m5f52m8x. Retrieved from [https://escholarship.org/uc/item/88h8b5t3]

A. Abidi and H. Xu, "Understanding the Regenerative Comparator Circuit," Proceedings of the IEEE 2014 Custom Integrated Circuits Conference, San Jose, CA, 2014, pp. 1-8.

T. Sepke, P. Holloway, C. G. Sodini and H. -S. Lee, "Noise Analysis for Comparator-Based Circuits," in IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 56, no. 3, pp. 541-553, March 2009 [https://dspace.mit.edu/bitstream/handle/1721.1/61660/Speke-2009-Noise%20Analysis%20for%20Comparator-Based%20Circuits.pdf]

Sepke, Todd. "Comparator design and analysis for comparator-based switched-capacitor circuits." (2006). [https://dspace.mit.edu/handle/1721.1/38925]

P. Nuzzo, F. De Bernardinis, P. Terreni and G. Van der Plas, "Noise Analysis of Regenerative Comparators for Reconfigurable ADC Architectures," in IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 55, no. 6, pp. 1441-1454, July 2008 [https://picture.iczhiku.com/resource/eetop/SYirpPPPaAQzsNXn.pdf]

J. Kim, B. S. Leibowitz, J. Ren and C. J. Madden, "Simulation and Analysis of Random Decision Errors in Clocked Comparators," in IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 56, no. 8, pp. 1844-1857, Aug. 2009, doi: 10.1109/TCSI.2009.2028449. URL:https://people.engr.tamu.edu/spalermo/ecen689/simulation_analysis_clocked_comparators_kim_tcas1_2009.pdf

Jaeha Kim, Lecture 12. Aperture and Noise Analysis of Clocked Comparators URL:https://ocw.snu.ac.kr/sites/default/files/NOTE/7038.pdf

Rabuske, Taimur & Fernandes, Jorge. (2017), Chapter 5 Noise-Aware Synthesis and Optimization of Voltage Comparators, "Charge-Sharing SAR ADCs for Low-Voltage Low-Power Applications"

Y. Luo, A. Jain, J. Wagner and M. Ortmanns, "Input Referred Comparator Noise in SAR ADCs," in IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 66, no. 5, pp. 718-722, May 2019. [https://sci-hub.se/10.1109/TCSII.2019.2909429]

Art Schaldenbrand, Senior Product Manager, Keeping Things Quiet: A New Methodology for Dynamic Comparator Noise Analysis URL:https://www.cadence.com/content/dam/cadence-www/global/en_US/videos/tools/custom-_ic_analog_rf_design/NoiseAnalyisposting201612Chalk%20Talk.pdf

X. Tang et al., "An Energy-Efficient Comparator With Dynamic Floating Inverter Amplifier," in IEEE Journal of Solid-State Circuits, vol. 55, no. 4, pp. 1011-1022, April 2020

swpuseprevic

discrete-time frequency: \(\hat{\omega}=\omega T_s\), units are radians per sample

Below diagram show the windowing effect and sampling

For general window function, we know \(W(e^{j\hat{\omega}})=\frac{1}{T_s}W(j\omega)\),

and \[ \frac{W(j\omega|\omega=0)}{T_s} = \frac{T_sW(e^{j\hat{\omega}}|\hat{\omega}=0)}{T_s} =W(e^{j\hat{\omega}}|\hat{\omega}=0)= \sum_{n=-N_w}^{+N_w}w[n] \]

e.g. \(\frac{W(j\omega|\omega=0)}{T_s} = N\) for Rectangular Window, shown in above figure

warmup

Continuous Time Fourier Series (CTFS)

\[\begin{align} a_k &= \frac{1}{T}\int_T x(t)e^{-jk(2\pi/T)) t}dt \\ x(t) &= \sum_{k=-\infty}^{+\infty}a_ke^{jk(2\pi/T) t} \end{align}\]

Continuous-Time Fourier transform (CTFT)

\[\begin{align} X(j\omega) &=\int_{-\infty}^{+\infty}x(t)e^{-j\omega t}dt \\ x(t)&= \frac{1}{2\pi}\int_{-\infty}^{+\infty}X(j\omega)e^{j\omega t}d\omega \end{align}\]

[https://www.rose-hulman.edu/class/ee/yoder/ece380/Handouts/Fourier%20Transform%20Tables%20w.pdf]

Discrete-Time Fourier Transform (DTFT)

\[\begin{align} X(e^{j\hat{\omega}}) &=\sum_{n=-\infty}^{+\infty}x[n]e^{-j\hat{\omega} n} \\ x[n] &= \frac{1}{2\pi}\int_{2\pi}X(e^{j\hat{\omega}})e^{j\hat{\omega} n}d\hat{\omega} \end{align}\]

DTFT is defined for infinitely long signals as well as finite-length signal

DTFT is continuous in the frequency domain

We could verify that is the correct inverse DTFT relation by substituting the definition of the DTFT and rearranging terms

Discrete-Time Fourier Series (DTFS)

TODO 📅

Discrete Fourier Series (DFS)

TODO 📅

Discrete Fourier Transform (DFT)

Two steps are needed to change the DTFT sum into a computable form:

1. the continuous frequency variable \(\hat{\omega}\) must be sampled
2. the limits on the DTFT sum must be finite

\[\begin{align} X[k] &= \sum_{n=0}^{N-1}x[n]e^{-j(2\pi/N)kn}\space\space\space k=0,1,...,N-1 \\ x[n] &= \frac{1}{N}\sum_{k=0}^{N-1}X[k]e^{j(2\pi/N)kn} \space\space\space n=0,1,...,N-1 \end{align}\]

Part of the proof is given by the following step:

DFT \(X[k]\) is a sampled version of the DTFT \(X(e^{j\hat{\omega}})\), where \(\hat{\omega_k} = \frac{2\pi k}{N}\)

impulse train

CTFT:

using time-sampling property

DTFT:

Given \(x[n]=\sum_{k=-\infty}^{\infty}\delta(n-k)\)

\[\begin{align} X(e^{j\hat{\omega}}) &= X_s(j\frac{\hat{\omega}}{T}) \\ &= \frac{2\pi}{T}\sum_{k=-\infty}^{\infty}\delta(\frac{\hat{\omega}}{T}-\frac{2\pi k}{T}) \\ &= \frac{2\pi}{T}\sum_{k=-\infty}^{\infty}T\delta(\hat{\omega}-2\pi k) \\ &= 2\pi\sum_{k=-\infty}^{\infty}\delta(\hat{\omega}-2\pi k) \end{align}\]

[http://courses.ece.ubc.ca/359/notes/notes_part1_set4.pdf]

impulse scaling

\[ \delta(\alpha t)= \frac{1}{\alpha}\delta( t) \]

where \(\alpha\) is scaling ratio

impulse invariance

\[ h[n] = Th_c(nT) \]

When \(h[n]\) and \(h_c(t)\) are related through the above equation, i.e., the impulse response of the discrete-time system is a scaled, sampled version of \(h_c(t)\), the discrete-time system is said to be an impulse-invariant version of the continuous-time system

we have \[ H(e^{j\hat{\omega}}) = H_c\left(j\frac{\hat{\omega}}{T}\right) \]

Multiplication

aka Modulation or Windowing Theorem

CTFT: \[ x_1(t)x_2(t)\overset{FT}{\longrightarrow}\frac{1}{2\pi}X_1(\omega)*X_2(\omega) \]

DTFT:

Duality

Conjugate Symmetry

Parseval's Relation

CTFT:

DTFT:

DFT:

Eigenfunctions & frequency response

Complex exponentials are eigenfunctions of LTI systems, that is,

continuous time: \(e^{j\omega t}\to H(j\omega)e^{j\omega t}\)

discrete time: \(e^{j\hat{\omega}n} \to H(e^{j\hat{\omega}})e^{j\hat{\omega}n}\)

where \(H(j\omega)\), \(H(e^{j\hat{\omega}})\) is frequency response of continuous-time systems and discrete-time systems, which is the function of \(\omega\) and \(\hat{\omega}\) \[\begin{align} H(j\omega) &= \int_{-\infty}^{+\infty}h(t)e^{-j\omega t}dt \\ \\ H(e^{j\hat{\omega}}) &= \sum_{n=-\infty}^{+\infty}h[n]e^{-j\hat{\omega} n} \end{align}\]

The frequency response of discrete-time LTI systems is always a periodic function of the frequency variable \(\hat{\omega}\) with period \(2\pi\)

Sampling Theorem

time-sampling theorem: applies to bandlimited signals

spectral sampling theorem: applies to timelimited signals

Aliasing

The frequencies \(f_{\text{sig}}\) and \(Nf_s \pm f_{\text{sig}}\) (\(N\) integer), are indistinguishable in the discrete time domain.

Given below sequence \[ X[n] =A e^{j\omega T_s n} \]

1. \(kf_s + \Delta f\)

​ \[\begin{align} x[n] &= Ae^{j\left( kf_s+\Delta f \right)2\pi T_sn} + Ae^{j\left( -kf_s-\Delta f \right)2\pi T_sn} \\ &= Ae^{j\Delta f\cdot 2\pi T_sn} + Ae^{-j\Delta f\cdot 2\pi T_sn} \end{align}\]

2. \(kf_s - \Delta f\)

​ \[\begin{align} x[n] &= Ae^{j\left( kf_s-\Delta f \right)2\pi T_sn} + Ae^{j\left( -kf_s+\Delta f \right)2\pi T_sn} \\ &= Ae^{-j\Delta f\cdot 2\pi T_sn} + Ae^{j\Delta f\cdot 2\pi T_sn} \end{align}\]

complex signal

\[\begin{align} A e^{j(\omega_s + \Delta \omega) T_s n} &= A e^{j(k\omega_s + \Delta \omega) T_s n} \\ A e^{j(\omega_s - \Delta \omega) T_s n} &= A e^{j(k\omega_s - \Delta \omega) T_s n} \end{align}\]

CTFS & CTFT

Fourier transform of a periodic signal with Fourier series coefficients \(\{a_k\}\) can be interpreted as a train of impulses occurring at the harmonically related frequencies and for which the area of the impulse at the \(k\)th harmonic frequency \(k\omega_0\) is \(2\pi\) times the \(k\)th Fourier series coefficient \(a_k\)

spectral sampling

spectral sampling by \(\omega_0\), and \(\frac{2\pi}{\omega_0} \gt \tau\) \[ X_{n\omega_0}(\omega) = \sum_{n=-\infty}^{\infty}X(n\omega_0)\delta(\omega - n\omega_0) \] Periodic repetition of \(x(t)\) is \[ x_{n\omega_0}(t) = \frac{1}{\omega_0}\sum_{n=-\infty}^{\infty}x(t -n\frac{2\pi}{\omega_0})=\frac{T_0}{2\pi}\sum_{n=-\infty}^{\infty}x(t -nT_0) \]

Then, if \(x_{T_0} (t)\), a periodic signal formed by repeating \(x(t)\) every \(T_0\) seconds (\(T_0 \gt \tau\)​), its CTFT is \[ X_{T_0}(\omega) = \frac{2\pi}{T_0} \cdot X_{n\omega_0}(\omega) = \frac{2\pi}{T_0}\sum_{n=-\infty}^{\infty}X(n\omega_0)\delta(\omega - n\omega_0) \] Then \(x_{T_0} (t)\) can be expressed with inverse CTFT as \[\begin{align} x_{T_0} (t) &= \frac{1}{2\pi}\int_{-\infty}^{\infty}X_{T_0}(\omega)e^{j\omega t}d\omega \\ &= \frac{1}{T_0}\sum_{n=-\infty}^{\infty}X(n\omega_0)e^{jn\omega_0 t} =\sum_{n=-\infty}^{\infty}\frac{1}{T_0}X(n\omega_0)e^{jn\omega_0 t} \end{align}\]

i.e. the coefficients of the Fourier series for \(x_{T_0} (t)\) is \(D_n =\frac{1}{T_0}X(n\omega_0)\)

alternative method by direct Fourier series

Why DFT ?

We can use DFT to compute DTFT samples and CTFT samples

\[ \overline{x}(t) = \sum_{n=0}^{N_0-1}x(nT)\delta(t-nT) \] applying the Fourier transform yieds \[ \overline{X}(\omega) = \sum_{n=0}^{N_0-1}x[n]e^{-jn\omega T} \] But \(\overline{X}(\omega)\), the Fourier transform of \(\overline{x}(t)\) is \(X(\omega)/T\), assuming negligible aliasing. Hence, \[ X(\omega) = T\overline{X}(\omega) = T\sum_{n=0}^{N_0-1}x[n]e^{-jn\omega T} \] and \[ X(k\omega_0) = T\sum_{n=0}^{N_0-1}x[n]e^{-jn k\omega_0 T} \] with \(\hat{\omega}_0 = \omega_0 T\) \[ X(k\omega_0) = T\sum_{n=0}^{N_0-1}x[n]e^{-jn k\hat{\omega}_0} \] i.e. the relationship between CTFT and DFT is \(X(k\omega_0) = T\cdot X[k]\), DFT is a tool for computing the samples of CTFT

C/D

Sampling with a periodic impulse train, followed by conversion to a discrete-time sequence

The periodic impulse train is \[ s(t) = \sum_{n=-\infty}^{\infty}\delta(t-nT) \] \(x_s(t)\) can be expressed as \[ x_s(t) = \sum_{n=-\infty}^{\infty}x_c(nT)\delta(t-nT) \] i.e., the size (area) of the impulse at sample time \(nT\) is equal to the value of the continuous-time signal at that time.

\(x_s(t)\)​ is, in a sense, a continuous-time signal (specifically, an impulse train)

samples of \(x_c(t)\) are represented by finite numbers in \(x[n]\) rather than as the areas of impulses, as with \(x_s(t)\)

Frequency-Domain Representation of Sampling

The relationship between the Fourier transforms of the input and the output of the impulse train modulator \[ X_s(j\omega) = \frac{1}{T}\sum_{k=-\infty}^{\infty}X_c(j(\omega -k\omega_s)) \] where \(\omega_s\) is the sampling frequency in radians/s

\(X(e^{j\hat{\omega}})\), the discrete-time Fourier transform (DTFT) of the sequence \(x[n]\), in terms of \(X_s(j\omega)\) and \(X_c(j\omega)\)

\[ X(e^{j\omega T}) = \frac{1}{T}\sum_{k=-\infty}^{\infty}X_c(j(\omega-k\omega_s)) \] or equivalently, \[ X(e^{j\hat{\omega}}) = \frac{1}{T}\sum_{k=-\infty}^{\infty}X_c(j(\frac{\hat{\omega}}{T}-\frac{2\pi k}{T})) \]

\(X(e^{j\hat{\omega}})\) is a frequency-scaled version of \(X_s(j\omega)\) with the frequency scaling specified by \(\hat{\omega} =\omega T\)

Ref. 9.5 DTFT connection with the CTFT

Here, \(\Omega = \omega T\)

The factor \(\frac{1}{T}\) in \(X(e^{j\hat{\omega}})\) is misleading, actually \(x[n]\) is not scaled by \(\frac{1}{T}\) when taking \(\hat{\omega}\) variable of integration into account \[\begin{align} x_r[n] &= \frac{1}{2\pi} \int_{2\pi}X(e^{j\hat{\omega}})e^{j\hat{\omega} n}d\hat{\omega} \\ &= \frac{1}{2\pi}\int_{2\pi}\frac{1}{T}\sum_{k=-\infty}^{+\infty}X_c \left[ j\left(\frac{\hat{\omega}}{T} - \frac{2\pi k}{T}\right)\right] e^{j\hat{\omega} n}d\hat{\omega} \\ &\approx \frac{1}{2\pi}\frac{1}{T}\int_{2\pi}X_c (\frac{\hat{\omega}}{T} ) e^{j\hat{\omega} n} d\hat{\omega} \\ &=\frac{1}{2\pi} \frac{1}{T}\int_{2\pi} \left[ \int_{\infty}X_c(\Phi)\delta (\Phi - \frac{\hat{\omega}}{T} )d\Phi \right] e^{j\hat{\omega} n} d\hat{\omega} \\ &=\frac{1}{2\pi} \frac{1}{T} \int_{\infty}X_c(\Phi)d\Phi \int_{2\pi}\delta (\Phi - \frac{\hat{\omega}}{T} )e^{j\hat{\omega} n} d\hat{\omega} \\ &=\frac{1}{2\pi} \frac{1}{T} \int_{\infty}X_c(\Phi)d\Phi \int_{2\pi}T\cdot \delta (\Phi T - \hat{\omega} )e^{j\hat{\omega} n} d\hat{\omega} \\ &=\frac{1}{2\pi} \int_{\infty}X_c(\Phi) e^{j\Phi T n}d\Phi \end{align}\]

That is \[\begin{align} x_r[n] &= \frac{1}{2\pi}\int_{2\pi} \frac{1}{T}X_c (\frac{\hat{\omega}}{T} ) e^{j\hat{\omega} n} d\hat{\omega} \\ &= \frac{1}{2\pi} \int_{\infty}X_c(\omega) e^{j\omega T n}d\omega \tag{31} \end{align}\]

assuming Nyquist–Shannon sampling theorem is met

\[\begin{align} x_r[n] &= \frac{1}{2\pi} \int_{\infty}X_c(\omega) e^{j\omega T n}d\omega \\ &= \frac{1}{2\pi} \int_{\infty}X_c(\omega) e^{j\omega t_n}d\omega \\ &= x_c(t_n) \end{align}\]

where \(t_n = T n\), then \(x_r[n] = x_c(nT)\)

Assuming \(x_c(t) = \cos(\omega_0 t)\), \(x_s(t)= \sum_{n=-\infty}^{\infty}x_c(nT)\delta(t-nT)\) and \(x[n]=x_c(nT)\), that is \[\begin{align} x_c(t) & = \cos(\omega_0 t) \\ x_s(t) &= \sum_{n=-\infty}^{\infty}\cos(\omega_0 nT)\delta(t-nT) \\ x[n] &= \cos(\omega_0 nT) \end{align}\]

• \(X_c(j\omega)\), the Fourier Transform of \(x_c(t)\) \[ X_c(j\omega) = \pi[\delta(\omega - \omega_0) + \delta(\omega + \omega_0)] \]

• \(X(e^{j\hat{\omega}})\), the the discrete-time Fourier transform (DTFT) of the sequence \(x[n]\) \[ X(e^{j\hat{\omega}}) =\sum_{k=-\infty}^{+\infty}\pi[\delta(\hat{\omega} - \hat{\omega}_0-2\pi k) + \delta(\hat{\omega} + \hat{\omega}_0-2\pi k)] \]

• \(X_s(j\omega)\), the Fourier Transform of \(x_s(t)\) \[ X_s(j\omega)= \frac{1}{T}\sum_{k=-\infty}^{+\infty}\pi[\delta(\omega - \omega_0-k\omega_s) + \delta(\omega + \omega_0-k\omega_s)] \]

Express \(X(e^{j\hat{\omega}})\) in terms of \(X_s(j\omega)\) and \(X_c(j\omega)\) \[ X(e^{j\hat{\omega}}) = \frac{1}{T}\sum_{k=-\infty}^{+\infty}\pi[\delta(\frac{\hat{\omega}}{T} - \omega_0-k\omega_s) + \delta(\frac{\hat{\omega}}{T} + \omega_0-k\omega_s)] \] Inverse \(X(e^{j\hat{\omega}})\) \[\begin{align} x_r[n] &= \frac{1}{2\pi} \int_{2\pi}X(e^{j\hat{\omega}}) e^{j\hat{\omega} n} d\hat{\omega} \\ &= \frac{1}{2\pi}\int_{2\pi} \pi[\delta(\frac{\hat{\omega}}{T} - \omega_0) + \delta(\frac{\hat{\omega}}{T} + \omega_0)]e^{j\hat{\omega} n} d\frac{\hat{\omega}}{T} \\ &= \frac{1}{2\pi}\int_{2\pi} \pi[\delta(\frac{\hat{\omega}}{T} - \omega_0)e^{j\hat{\omega}_0 n} + \delta(\frac{\hat{\omega}}{T} + \omega_0)e^{-j\hat{\omega}_0 n}] d\frac{\hat{\omega}}{T} \\ &= \frac{1}{2}[ e^{j\hat{\omega}_0 n}\int_{2\pi} [\delta(\frac{\hat{\omega}}{T} - \omega_0)d\frac{\hat{\omega}}{T} + e^{-j\hat{\omega}_0 n}\int_{2\pi} [\delta(\frac{\hat{\omega}}{T} + \omega_0)d\frac{\hat{\omega}}{T}] \\ &= \frac{1}{2}[ e^{j\hat{\omega}_0 n} + e^{-j\hat{\omega}_0 n} ] \\ &= \cos(\hat{\omega}_0 n) \end{align}\]

or follow EQ.(31)

\[\begin{align} x_r[n] &= \frac{1}{2\pi} \int_{\infty}X_c(\omega) e^{j\omega T n}d\omega \\ &= \frac{1}{2\pi} \int_{\infty} \pi[\delta(\omega - \omega_0) + \delta(\omega + \omega_0)]e^{j\omega T n}d\omega \\ &= \frac{1}{2}(e^{j\omega_0 T n}+e^{-j\omega_0 T n}) \\ &= \cos(\hat{\omega}_0 n) \end{align}\]

where \(\hat{\omega}_0 = \omega_0 T\)

D/C

zero padding

This option increases \(N_0\), the number of samples of \(x(t)\), by adding dummy samples of 0 value. This addition of dummy samples is known as zero padding.

We should keep in mind that even if the fence were transparent, we would see a reality distorted by aliasing.

Zero padding only allows us to look at more samples of that imperfect reality

Transfer function

sampled impulse response

The below equation demonstrates how to obtain continuous Fourier Transform from DTFT . \[ X_c(\omega) = T \cdot X(\omega) \]

\(T\) is sample period, follow previous equation

useful functions

• using fft

  The outputs of the DFT are samples of the DTFT

• using freqz

  modeling as FIR filter, and the impulse response sequence of an FIR filter is the same as the sequence of filter coefficients, we can express the frequency response in terms of either the filter coefficients or the impulse response

  fft is used in freqz internally

Example

Question:

How to obtain continuous system transfer function from sampled impulse

Answer:

using above mentioned functions

First order lowpass filter with 3-dB frequency 1Hz

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clear all;
clc;

%% continuous system
s = tf('s');
h = 2*pi/(2*pi+s);
[mag, phs, wout] = bode(h);
fct = wout(:)/2/pi;
Hct_dB = 20*log10(mag(:));


fstep = 0.01;           % freq resolution
fnyqst = 100;
Ts = 1/(2*fnyqst);
Fs = 1/Ts;              % sampling freq
Ns = ceil(Fs/fstep);    % samping points
fstep = Fs/Ns;          % update fstep
t = (0:Ns-1)*Ts;        % sampling time points

y = impulse(h, t);      % impulse resp

%% modelling as discrete system
Y = fft(y);                 % dft
Hfft = Y * Ts;              % !!! multiply Ts
Hfft_dB = 20*log10(abs(Hfft(1:Ns/2+1)));
ffft = (1:Ns/2+1)*fstep - fstep;


[Hfir, ffir] = freqz(y, 1, [], 1/Ts);   % modelling as FIR
Hfir = Hfir * Ts;           % !!! multiply Ts
Hfir_dB = 20*log10(abs(Hfir));

%% plot
semilogx(fct, Hct_dB, 'k', ffft, Hfft_dB, 'r.-', ffir, Hfir_dB, 'b--');
legend('bode(s)', 'fft', 'FIR model')
xlabel('Freq(Hz)');
ylabel('dB');
xlim([1e-2 1e2]);
grid on;
title('frequency response of different methods');

Gotcha

A remarkable fact of linear systems is that the complex exponentials are eigenfunctions of a linear system, as the system output to these inputs equals the input multiplied by a constant factor.

• Both amplitude and phase may change
• but the frequency does not change

For an input \(x(t)\), we can determine the output through the use of the convolution integral, so that with \(x(t) = e^{st}\) \[\begin{align} y(t) &= \int_{-\infty}^{+\infty}h(\tau)x(t-\tau)d\tau \\ &= \int_{-\infty}^{+\infty} h(\tau) e^{s(t-\tau)}d\tau \\ &= e^{st}\int_{-\infty}^{+\infty} h(\tau) e^{-s\tau}d\tau \\ &= e^{st}H(s) \end{align}\]

Take the input signal to be a complex exponential of the form \(x(t)=Ae^{j\phi}e^{j\omega t}\)

\[\begin{align} y(t) &= h(t)*x(t) \\ &= H(j\omega)Ae^{j\phi}e^{j\omega t} \end{align}\]

The frequency response at \(-\omega\) is the complex conjugate of the frequency response at \(+\omega\), given \(h(t)\) is real

\[\begin{align} H^*(t) &= \left(\int_{-\infty}^{+\infty}h(t)e^{-j\omega t}dt\right)^* \\ &= \int_{-\infty}^{+\infty}h^*(t)e^{+j\omega t}dt \\ &= \int_{-\infty}^{+\infty}h(t)e^{-j(-\omega t)}dt \\ &= H(-j\omega) \end{align}\]

The real cosine signal is actually composed of two complex exponential signals: one with positive frequency and the other with negative \[ cos(\omega t + \phi) = \frac{e^{j(\omega t + \phi)} + e^{-j(\omega t + \phi)}}{2} \]

The sinusoidal response is the sum of the complex-exponential response at the positive frequency \(\omega\) and the response at the corresponding negative frequency \(-\omega\) because of LTI systems's superposition property

• input: \[\begin{align} x(t) &= A cos(\omega t + \phi) \\ &= \frac{1}{2}Ae^{\phi}e^{\omega t} + \frac{1}{2}Ae^{-\phi}e^{-\omega t} \end{align}\]

• output with \(H(j\omega)=Ge^{j\theta}\): \[\begin{align} y(t) &= H(j\omega)\frac{1}{2}Ae^{\phi}e^{\omega t} + H(-j\omega)\frac{1}{2}Ae^{-\phi}e^{-\omega t} \\ &= Ge^{j\theta}\frac{1}{2}Ae^{\phi}e^{\omega t} + Ge^{-j\theta}\frac{1}{2}Ae^{-\phi}e^{-\omega t} \\ &= GAcos(\omega t + \phi + \theta) \end{align}\]

Its phase shift is \(\theta\) and gain is \(G\), which is same with \(H(j\omega)\).

reference

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

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

Alan V. Oppenheim, Alan S. Willsky, and S. Hamid Nawab. 1996. Signals & systems (2nd ed.)

James H. McClellan, Ronald Schafer, and Mark Yoder. 2015. DSP First (2nd. ed.). Prentice Hall Press, USA

assign the amount of time based on the time needed for each step

pipeline ADC

\[ V_{N-1} = \sum_{i=0}^{N-2} B[i]\frac{1}{2^{N-i}}+ B[N-1]\frac{1}{2} \] Compare with \(\frac{1}{2}\), and determine \(B[N-1]\) \[\begin{align} V_{N-2} &= \left(V_{N-1} - B[N-1]\frac{1}{2}\right)\cdot 2 \\ &= \left(\sum_{i=0}^{N-1} B[i]\frac{1}{2^{N-i}} - B[N-1]\frac{1}{2}\right) \cdot 2 \\ &= \sum_{i=0}^{N-2} B[i]\frac{1}{2^{N-i}} \cdot 2 \\ &= \sum_{i=0}^{N-3} B[i]\frac{1}{2^{N-i}}\cdot 2 + B[N-2]\frac{1}{2} \end{align}\]

On your intuition, input can be expressed as below \[\begin{align} V_{N-1} &= \sum_{i=0}^{N-1} B[i] \frac{1}{2}\cdot 2^{i-N+1} \\ &= B[N-1]\frac{1}{2} + B[N-2]\frac{1}{2}\cdot 2^{-1} + B[N-3]\frac{1}{2}\cdot 2^{-2 }... \end{align}\]

It divides the process into several comparison stages, the number of which is proportional to the number of bits

Due to the pipeline structure of both analog and digital signal path, inter-stage residue amplification is needed which consumes considerable power and limits high speed operation

reduced residue amplification gain

Synchronous SAR ADC

It also divides a full conversion into several comparison stages in a way similar to the pipeline ADC, except the algorithm is executed sequentially rather than in parallel as in the pipeline case.

However, the sequential operation of the SA algorithm has traditionally been a limitation in achieving high-speed operation

• a clock running at least \((N + 1) \cdot F_s\) is required for an \(N\)-bit converter with conversion rate of \(F_s\)
• every clock cycle has to tolerate the worst case comparison time
• every clock cycle requires margin for the clock jitter

The power and speed limitations of a synchronous SA design comes largely from the high-speed internal clock

Non-Binary Successive Approximation

The overlapped search range compensates for wrong decisions made in earlier stages as long as they are within the error tolerance range

Kuttner, Franz. “A 1.2V 10b 20MSample/s non-binary successive approximation ADC in 0.13/spl mu/m CMOS.” 2002 IEEE International Solid-State Circuits Conference. Digest of Technical Papers (Cat. No.02CH37315) 1 (2002): 176-177 vol.1.

Split Arrary CDAC

Split capacitor, double-array cap

attenuation capacitance \(C_a\)

\[\begin{align} \Delta V_{dac} &= \frac{1}{2}b_3+\frac{1}{4}b_2+\frac{1}{4}\left(\frac{1}{2}b_1+\frac{1}{4}b_0 \right) \\ &= \frac{1}{2}b_3+\frac{1}{4}b_2 + \frac{1}{8}b_1+\frac{1}{16}b_0 \end{align}\]

Asynchronous processing

a global clock running at the sample rate is still used for an uniform sampling

The concept of asynchronous processing is to trigger the internal comparison from MSB to LSB like dominoes.

reference

S. -W. M. Chen and R. W. Brodersen, "A 6-bit 600-MS/s 5.3-mW Asynchronous ADC in 0.13-μm CMOS," in IEEE Journal of Solid-State Circuits, vol. 41, no. 12, pp. 2669-2680, Dec. 2006

Shuo-Wei Chen, Power Efficient System and A/D Converter Design for Ultra-Wideband Radio [http://www2.eecs.berkeley.edu/Pubs/TechRpts/2006/EECS-2006-71.pdf]

Andrea Baschirotto, "T6: SAR ADCs" ISSCC2009

Pieter Harpe, ISSCC 2016 Tutorial: "Basics of SAR ADCs Circuits & Architectures"

Single-Pole Filter & Complex Conjugate Pole pair

Single-Pole Filter and Complex Conjugate Pole pair in Event-Driven PWL model

Real number modeling of analog circuits in hardware description languages (HDLs) has become more common as a part of mixed-signal SoC validation. Piecewise linear (PWL) waveform approximation represent analog signals and dynamically schedule the events for approximating the signal waveform to PWL segments with a well controlled error bound.

Definition of a piecewise liner (PWL) waveform using struct in Systemverilog

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typedef struct {
    real y; // signal offset
    real slope; // signal slope
    real t0;	// time offset
} pwl; // pwl datatype

When to update piecewise model

1. model parameter update once new input come in
2. error is greater than user-define tolerance \(e_{tol}\), trigger by \(\Delta T\)

Dynamic Time Step Control

When approximating a function \(y(t)\) to a piecewise linear segment for the interval \(t_0 \le t_0 + \Delta t\), the approximation error \(err\) is bounded by \[ \left| err \right| \le \frac{1}{8}\cdot \Delta t^2 \cdot \max(\left| \ddot{y(t)} \right|) \] Using Rolle's theorem for the interval \(t_0 \le t_0 + \Delta t\), the needed time step \(\Delta t\) is givend by \[ \Delta t(t=t_0) = \sqrt{\frac{8\cdot e_{tol}}{\max(\left| \ddot{y(t)} \right|)}} \]

Single-Pole Filter Model

The ramp input \(X(s)\), the single pole system Laplace s-domain \(H(s)\) and the output response \(Y(s)\), \[\begin{align} X(s) &= \frac{a}{s} +\frac{b}{s^2} \\ H(s) &= \frac{Y(s)}{X(s)} = \frac{1}{1+\frac{s}{\omega_{1}}} \\ Y(s) &= X(s) \cdot H(s) \end{align}\]

Time domain of ramp input shown as below \[ x(t) = a +b \cdot t \]

1. The output transfer function \[\begin{align} Y(s) &= X(s) \cdot H(s) \\ &= \frac{\omega_1}{\omega_1+s}\cdot X \end{align}\]

\[ Y\omega_1 + sY = \omega_1 X \]

2. its differential equation

\[ y(t) \cdot \omega_1+\frac{d y(t)}{dt} = \omega_1 \cdot x(t) \]

3. Laplace transfrom two side of the above equation, \(y_0\) is initial conditon of output, \(x_0=0\)

\[ Y\omega_1 + sY-y_0 = \omega_1 \cdot X \]

Solving \(Y(s)\) \[\begin{align} Y &= \frac{y_0}{\omega_1+s}+\frac{\omega_1}{\omega_1+s}\cdot X \\ &= \frac{y_0}{\omega_1+s}+\frac{\omega_1}{\omega_1+s}\cdot (\frac{a}{s}+\frac{b}{s^2}) \\ &= \frac{y_0}{\omega_1+s}+\frac{\omega_1}{\omega_1+s}\cdot \frac{a}{s}+\frac{\omega_1}{\omega_1+s}\cdot\frac{b}{s^2} \end{align}\]

4. inverse Laplace transform

\[ y(t) = y_0e^{-\omega_1t}+(a-a\cdot e^{-\omega_1t})+(b\cdot t-\frac{b}{\omega_1}+\frac{b}{\omega_1}\cdot e^{-\omega_1 t}) \]

step-1 transfer function in Laplace s-domain, which don't initial conditon and is only steady response.

step-2 differential equation

step-3 Laplace transform of \(Y(s)\), (the initial conditon of input \(X(s)\) is zero, that of \(Y(s)\) is explicit)

step-4 inverse Laplace transform, with the help of Laplace transform table or matlab syms and ilaplace function

\(y(t)\) has a continuous second derivative \(\ddot{y(t)}\) \[ \ddot{y(t)} =(-a+\frac{b}{\omega_1}+y_0)\cdot \omega_1^2\cdot e^{-\omega_1t} \] It's obvious \(\left| \ddot{y(t)} \right|\) is a decaying function and thus the maximum value is \(\left| \ddot{y(t_0)} \right|\) for the interval \(t_0 \le t_0 + \Delta t\). The time step \(\Delta t\) for the error tolerance \(e_{tol}\): \[ \Delta t(t=t_0) = \sqrt{\frac{8\cdot e_{tol}}{\left| \ddot{y(t_0)} \right|}} \]

Complex Conjugate Pole pair

\[ H(s) = \frac{r}{s+\omega_p} + \frac{r^*}{s+\omega_p^*} \]

where \(\omega_p\) and \(r\) are complex numbers, \(r=r_r+jr_i\), \(\omega_p=\omega_{pr}+j\omega_{pi}\)

Follow the procedure as above single pole \[ \frac{Y(s)}{X(s)} = \frac{s\cdot r_{cs}+e}{s^2+s\cdot \omega_{p\_cs}+f} \] where \(r_{cs}=r+r^*\), \(\omega_{p\_cs}=\omega_p+\omega_p^*\) and \(e=r\omega_p^*+r^*\omega_p\), \(f=\omega_p\omega_p^*\) implies \[ s^2Y(s)+s\omega_{p\_cs}Y(s)+fY(s)=(s\cdot r_{cs}+e)X(s) \] or a differential equation \[ \frac{d^2y(t)}{dt^2}+\omega_{p\_cs}\frac{dy(t)}{dt}+fy(t)=r_{cs}\frac{dx(t)}{dt}+e\cdot x(t) \] Taking Laplace transform with initial conditions \(y_0\), \(\dot{y_0}\) and \(x_0=0\), \[ s^2-sy_0-\dot{y_0}+\omega_{p\_cs}(sY(s)-y_0)+f\cdot y(t) = r_{cs}\cdot (sX(s)-0)+e\cdot X(s) \] Solving for \(Y(s)\) \[ Y(s)=\frac{s\cdot y_0+\dot{y_0}+\omega_{p\_cs}y_0}{s^2+s\cdot{\omega_{p\_cs}}+f}+\frac{s\cdot{r_{cs}}+e}{s^2+s\cdot{\omega_{p\_cs}}+f}X(s) \] With an ramp input, height \(a\), slope \(b\), i.e. \(X(s)=\frac{a}{s}+\frac{b}{s^2}\) \[ Y(s)=\frac{s\cdot y_0+\dot{y_0}+\omega_{p\_cs}y_0}{s^2+s\cdot{\omega_{p\_cs}}+f}+\frac{s\cdot{r_{cs}}+e}{s^2+s\cdot{\omega_{p\_cs}}+f}(\frac{a}{s}+\frac{b}{s^2}) \] After inverse Laplace transform, we can get total response \[ y(t)=e^{-\omega_{pr}t}\cdot \left[ y_0\cdot \cos(\omega_{pi}t)+\frac{\dot{y_0}+y_0\omega_{pr}}{\omega_{pi}}\sin(\omega_{pi}t)+D\cdot \cos(\omega_{pi}t)+\frac{C-D\cdot{\omega_{pr}}}{\omega_{pi}}\sin(\omega_{pi}t) \right]+B+A\cdot{t} \] where \[\begin{align} A &= \frac{e\cdot{b}}{f} \\ B &= \frac{r_{cs}\cdot{b}+a\cdot{e}-A\cdot{\omega_{p\_{cs}}}}{f} \\ C &= a\cdot{r_{cs}}-A-B\cdot{\omega_{p\_cs}} \\ D &= -B \end{align}\]

As a double check, note that at \(t=0\), \[ y(0)=\left[ y_0 + D \right]+B=y_0 \]

To derive derivative, we first assume \[ y_0\cdot \cos(\omega_{pi}t)+\frac{\dot{y_0}+y_0\omega_{pr}}{\omega_{pi}}\sin(\omega_{pi}t)+D\cdot \cos(\omega_{pi}t)+\frac{C-D\cdot{\omega_{pr}}}{\omega_{pi}}\sin(\omega_{pi}t) = \alpha \cdot{\cos(\omega_{pi}t+\phi)} \] The above equation implies \[\begin{align} y_0+D &= \alpha\cdot{\cos(\phi)} \\ \frac{\dot{y_0}+y_0\omega_{pr}}{\omega_{pi}}+\frac{C-D\cdot{\omega_{pr}}}{\omega_{pi}} &= -\alpha\cdot{\sin(\phi)} \end{align}\] Then \[ \alpha^2=(y_0+D)^2+\left(\frac{\dot{y_0}+y_0\omega_{pr}}{\omega_{pi}}+\frac{C-D\cdot{\omega_{pr}}}{\omega_{pi}} \right)^2 \] And \(\alpha\) can be used to estimate time step size. The total response is \[ y(t)=e^{-\omega_{pr}t}\cdot \alpha \cdot{\cos(\omega_{pi}t+\phi)}+B+A\cdot{t} \] It's second derivative is \[ \ddot{y(t)} = \alpha\left[ (\omega_{pr}^2-\omega_{pi}^2)e^{-\omega_{pr}t}\cos(\omega_{pi}t+\phi)+2\cdot \omega_{pr}\omega_{pi}e^{-\omega_{pr}t}\sin(\omega_{pi}t+\phi) \right] \] Absolute value \[ \left| \ddot{y(t)} \right| = \left| \alpha \right| \left| (\omega_{pr}^2-\omega_{pi}^2)e^{-\omega_{pr}t}\cos(\omega_{pi}t+\phi)+2\cdot \omega_{pr}\omega_{pi}e^{-\omega_{pr}t}\sin(\omega_{pi}t+\phi) \right| \] Define new function \(g_0(t)\) \[ g_0(t) = \left| \alpha \right| \left| (\omega_{pr}^2-\omega_{pi}^2)e^{-\omega_{pr}t}\cos(\omega_{pi}t+\phi) \right|+2\cdot |\alpha| \left| \omega_{pr}\omega_{pi}e^{-\omega_{pr}t}\sin(\omega_{pi}t+\phi) \right| \] another new funtion \(g_1(t)\), by equating \(\sin(\omega_{pi}t+\phi)\) and \(\cos(\omega_{pi}t+\phi)\) to one \[ g_1(t) = \left| \alpha \right| \left| (\omega_{pr}^2-\omega_{pi}^2)e^{-\omega_{pr}t} \right|+2\cdot |\alpha| \left| \omega_{pr}\omega_{pi}e^{-\omega_{pr}t} \right| \]

By triangular inequality, \(g_0(t)\) is the upper bound of \(\left| \ddot{y(t)} \right|\), and \(g_1(t)\) is the upper bound of \(g_0(t)\)

Because \(g_1(t)\) is a decaying exponential function, Therefore, a conservative time step can be obtained, for inteval \(t_0 \le t_0 + \Delta t\), \[ \Delta t(t=t_0) = \sqrt{\frac{8\cdot e_{tol}}{\left| g_1(t_0) \right|}} \]

One Fixed-time step SystemVerilog model example

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timeunit 1ns;
timeprecision 1fs;

initial #0.1 forever begin
    Ir4 = (Vm - Vout) / R4;
    Ir3 = (Vcp - Vm) / R3;
    Ir2 = (Vcp - Vz) / R2;

    Vout = Vout + Ir4 * 0.1 * (1e-9) / C4;
    Vm = Vm + (Ir3 - Ir4) * 0.1 * (1e-9) / C3;
    Vz = Vz + Ir2 * 0.1 * (1e-9) / C2;
    Vcp = Vcp + (Icp - Ir2 - Ir3) * 0.1 * (1e-9) / C1;
    last_time = $realtime;
    #0.1;
end

Acknowledgement

My colleague, Zhang Wenpian help me a lot in understanding this modeling method. Lots of content here are copied from Zhang's note.

Reference

B. C. Lim and 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, doi: 10.1109/TCSI.2015.2512699.

S. Liao and M. Horowitz, "A Verilog piecewise-linear analog behavior model for mixed-signal validation," Proceedings of the IEEE 2013 Custom Integrated Circuits Conference, 2013, pp. 1-5, doi: 10.1109/CICC.2013.6658461.

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

B. C. Lim, 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]

Liao Sabrina, Verilog piecewise linear behavioral modeling for mixed-signal validation [https://stacks.stanford.edu/file/druid:pb381vh2919/Thesis_submission-augmented.pdf]

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

Bash

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for file in prefix*; do mv "$file" "${file#prefix}"; done;

Here is an example to remove "bla_" form the following files:

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bla_1.txt
bla_2.txt
bla_3.txt
blub.txt

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for file in bla_*; do mv "$file" "${file#bla_}";done;

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1.txt
2.txt
3.txt
blub.txt

[https://gist.github.com/guisehn/5438bbc22138435665c6e996493fe02b]

kick-back

kick-back increases CDAC settling time

add pre-amp (more power)

bootstrap T/H Switch

A. Abo et al., "A 1.5-V, 10-bit, 14.3-MS/s CMOS Pipeline Analog-toDigital Converter," IEEE J. Solid-State Circuits, pp. 599, May 1999

Dessouky and Kaiser, "Input switch configuration suitable for rail-to-rail operation of switched opamp circuits," Electronics Letters, Jan. 1999.

Quantization Noise

Quantization noise is less with higher resolution as the input range is divided into a greater number of smaller ranges

This error can be considered a quantization noise with RMS

Bottom plate sampling

Sample signal at the "grounded" side of the capacitor to achieve signal independent sampling

[https://indico.cern.ch/event/1064521/contributions/4475393/attachments/2355793/4078773/esi_sampling_and_converters2022.pdf]

EE 435 Spring 2024 Analog VLSI Circuit Design - Switched-Capacitor Amplifiers Other Integrated Filters, https://class.ece.iastate.edu/ee435/lectures/EE%20435%20Lect%2044%20Spring%202008.pdf

Hold Mode Feedthrough

P. Schvan et al., "A 24GS/s 6b ADC in 90nm CMOS," 2008 IEEE International Solid-State Circuits Conference - Digest of Technical Papers, San Francisco, CA, USA, 2008, pp. 544-634

B. Sedighi, A. T. Huynh and E. Skafidas, "A CMOS track-and-hold circuit with beyond 30 GHz input bandwidth," 2012 19th IEEE International Conference on Electronics, Circuits, and Systems (ICECS 2012), Seville, Spain, 2012, pp. 113-116

Tania Khanna, ESE 568: Mixed Signal Circuit Design and Modeling [https://www.seas.upenn.edu/~ese5680/fall2019/handouts/lec11.pdf]

CDAC intuition

The charge redistribution capacitor network is used to sample the input signal and serves as a digital-to-analog converter (DAC) for creating and subtracting reference voltages

sampling charge \[ Q = V_{in} C_{tot} \] conversion charge \[ Q = -C_{tot}V_c + V_{ref}C_\Delta \] That is \[ V_c = \frac{C_\Delta}{C_{tot}}V_{ref} - V_{in} \]

CDAC is actually working as a capacitive divider during conversion phase, the charge of internal node retain (charge conservation law)

assuming \(\Delta V_i\) is applied to series capacitor \(C_1\) and \(C_2\)

\[ (\Delta V_i - \Delta V_x) C_1 = \Delta V_x \cdot C_2 \] Then \[ \Delta V_x = \frac{C_1}{C_1+C_2}\Delta V_i \]

\(V_x= V_{x,0} + \Delta V_x\)

CDAC settling time

\[\begin{align} V_x(s) &= \frac{C_1+C_2}{RC_1C_2}\cdot \frac{1}{s+\frac{C_1+C_2}{RC_1C_2}}\cdot V_i(s) \\ &= \frac{1}{\tau}\cdot \frac{1}{s+\frac{1}{\tau}}\cdot \frac{1}{s}\\ &= \frac{1}{\tau}\cdot \tau(\frac{1}{s} - \frac{1}{s+\frac{1}{\tau}})=\frac{1}{s} - \frac{1}{s+\frac{1}{\tau}} \end{align}\]

inverse Laplace Transform is \(V_x(t) = 1 - e^{-t/\tau}\)

\[\begin{align} V_y(s) &= V_x\frac{C_1}{C_1+C_2} \\ &= \frac{C_1}{C_1+C_2} \left(\frac{1}{s} - \frac{1}{s+\frac{1}{\tau}}\right)\\ \end{align}\]

inverse Laplace Transform is \(V_y(t) = \frac{C_1}{C_1+C_2}\left(1 - e^{-t/\tau}\right)\)

\(V_x(t)\) and \(V_y(t)\) prove that the settling time is same

\(\tau = R\frac{C_1C_2}{C_1+C_2}\), which means usually worst for MSB capacitor (largest)

CDAC Energy Consumption

\[ E_{Vref} = \int P(t)dt = \int V_{ref} I(t) dt = V_{ref}\int I(t)dt = V_{ref}\cdot \Delta Q \]

Given \(V_{c,0}=\frac{1}{2}V_{ref}-V_{in}\) and \(V_{c,1}=\frac{3}{4}V_{ref}-V_{in}\) \[\begin{align} Q_{b0,0} &= \left(V_{ref} - V_{c,0} \right)\cdot 2C = \left(\frac{1}{2}V_{ref}+V_{in} \right)\cdot 2C \\ Q_{b1,0} &= (0 - V_{c,0})\cdot C = \left(-\frac{1}{2}V_{ref}+V_{in} \right)\cdot C \\ Q_{b0,1} &= \left(V_{ref} - V_{c,1} \right)\cdot 2C = \left(\frac{1}{4}V_{ref}+V_{in} \right)\cdot 2C \\ Q_{b1,1} &= \left(V_{ref} - V_{c,1} \right)\cdot C = \left(\frac{1}{4}V_{ref}+V_{in} \right)\cdot C \end{align}\]

Therefore \[ E_{Vref} = V_{ref}\cdot (Q_{b0,1}+Q_{b1,1} - Q_{b0,0}-Q_{b1,0}) = \frac{1}{4}C V_{ref}^2 \]

CDAC total energy change \[\begin{align} \Delta E_{tot} &= \frac{1}{2}\cdot 2C \cdot (U_{2c,1}^2 - U_{2c,0}^2) + \frac{1}{2}\cdot C \cdot (U_{c,1}^2 - U_{c,0}^2) + \frac{1}{2}\cdot C \cdot (U_{c1,1}^2 - U_{c1,0}^2) \\ &= \left(-\frac{3}{16}V_{ref}^2 - \frac{1}{2}V_{ref}V_{in} - \frac{3}{32}V_{ref}^2+\frac{3}{4}V_{ref}V_{vin} + \frac{5}{32}V_{ref}^2-\frac{1}{4}V_{ref}V_{in}\right)C \\ &= -\frac{1}{8}CV_{ref}^2 \end{align}\]

alternative method

\[ \Delta E_{tot} = \frac{1}{2}\cdot\frac{3}{4}C\cdot V_{ref}^2 - \frac{1}{2}\cdot C\cdot V_{ref}^2 = -\frac{1}{8}CV_{ref}^2 \]

The total energy decreases by \(-\frac{1}{8}CV_{ref}^2\), though \(V_{ref}\) provides \(\frac{1}{4}C V_{ref}^2\)

The charge redistribution change the CDAC energy

\[ E_{c,0} = \frac{1}{2}CV^2 \] After charge redistribution \[ E_{c,1} = \frac{1}{2}\cdot 2C\cdot \left(\frac{1}{2}\right)^2 = \frac{1}{2}CV^2 \]

That make sense, charge redistribution consume energy

Comparator input capacitance

\[ -V_{in}\cdot 2^N C = V_c (2^N C + C_p) \] Then \(V_c = -\frac{2^N C}{2^N C + C_p}V_{in}\), i.e. this capacitance reduce the voltage amplitude by the factor

During conversion \[\begin{align} V_c &= -\frac{2^N C}{2^N C + C_p}V_{in} +V_{ref}\sum_{n=0}^{N-1} \frac{b_n\cdot2^n C}{2^N C + C_p} \\ &= \frac{2^N C}{2^N C + C_p}\left(-V_{in} + V_{ref}\sum_{n=0}^{N-1}\frac{b_n }{2^{N-n}} \right) \end{align}\]

That is, it does not change the sign

SNR of an ADC Due to Clock Jitter

TODO 📅

Chembiyan T, "SNR of an ADC in the presence of clock jitter"[https://www.linkedin.com/posts/chembiyan-t-0b34b910_adcsnrjitter-activity-7171178121021304833-f2Wd?utm_source=share&utm_medium=member_desktop]

K. Tyagi and B. Razavi, "Performance Bounds of ADC-Based Receivers Due to Clock Jitter," in IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 70, no. 5, pp. 1749-1753, May 2023 [https://www.seas.ucla.edu/brweb/papers/Journals/KT_TCAS_2023.pdf]

N. Da Dalt, M. Harteneck, C. Sandner and A. Wiesbauer, "On the jitter requirements of the sampling clock for analog-to-digital converters," in IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 49, no. 9, pp. 1354-1360, Sept. 2002 [https://sci-hub.se/10.1109/TCSI.2002.802353]

reference

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

How to choose FFT depth for ADC performance analysis (SINAD, ENOB). URL:https://dsp.stackexchange.com/a/38201

Computation of Effective Number of Bits, Signal to Noise Ratio, & Signal to Noise & Distortion Ratio Using FFT. URL:https://cdn.teledynelecroy.com/files/appnotes/computation_of_effective_no_bits.pdf

Kester, Walt. (2009). Understand SINAD, ENOB, SNR, THD, THD + N, and SFDR so You Don't Get Lost in the Noise Floor. URL:https://www.analog.com/media/en/training-seminars/tutorials/MT-003.pdf

T. C. Hofner: Dynamic ADC testing part I. Defining and testing dynamic ADC parameters, Microwaves & RF, 2000, vol. 39, no. 11, pp. 75-84,162

T. C. Hofner: Dynamic ADC testing part 2. Measuring and evaluating dynamic line parameters, Microwaves & RF, 2000, vol. 39, no. 13, pp. 78-94

AN9675: A Tutorial in Coherent and Windowed Sampling with A/D Converters https://www.renesas.com/us/en/document/apn/an9675-tutorial-coherent-and-windowed-sampling-ad-converters

APPLICATION NOTE 3190: Coherent Sampling Calculator (CSC) https://www.stg-maximintegrated.com/en/design/technical-documents/app-notes/3/3190.html

Coherent Sampling (Very Brief and Simple) https://www.dsprelated.com/thread/469/coherent-sampling-very-brief-and-simple

Signal Chain Basics #160: Making sense of coherent and noncoherent sampling in data-converter testing https://www.planetanalog.com/signal-chain-basics-160-making-sense-of-coherent-and-noncoherent-sampling-in-data-converter-testing/

Signal Chain Basics #104: Understanding noise in ADCs https://www.planetanalog.com/signal-chain-basics-part-104-understanding-noise-in-adcs/

Signal Chain Basics #101: ENOB Degradation Analysis Over Frequency Due to Jitter https://www.planetanalog.com/signal-chain-basics-part-101-enob-degradation-analysis-over-frequency-due-to-jitter/

Clock jitter analyzed in the time domain, Part 1, Texas Instruments Analog Applications Journal (slyt379), Aug 2010 https://www.ti.com/lit/an/slyt379/slyt379.pdf

Clock jitter analyzed in the time domain, Part 2 https://www.ti.com/lit/slyt389

Measurement of Total Harmonic Distortion and Its Related Parameters using Multi-Instrument [pdf]

Application Note AN-4: Understanding Data Converters' Frequency Domain Specifications [pdf]

Belleman, J. (2008). From analog to digital. 10.5170/CERN-2008-003.131. [pdf]

HandWiki. Coherent sampling [link]

Luis Chioye, TI. Leverage coherent sampling and FFT windows when evaluating SAR ADCs (Part 1) [link]

Coherent Sampling vs. Window Sampling | Analog Devices https://www.analog.com/en/technical-articles/coherent-sampling-vs-window-sampling.html

Understanding Effective Number of Bits https://robustcircuitdesign.com/signal-chain-explorer/understanding-effective-number-of-bits/

ADC Input Noise: The Good, The Bad, and The Ugly. Is No Noise Good Noise? [https://www.analog.com/en/resources/analog-dialogue/articles/adc-input-noise.html]

Walt Kester, Taking the Mystery out of the Infamous Formula, "SNR = 6.02N + 1.76dB," and Why You Should Care [https://www.analog.com/media/en/training-seminars/tutorials/MT-001.pdf]

Dan Boschen, "How to choose FFT depth for ADC performance analysis (SINAD, ENOB)", [https://dsp.stackexchange.com/a/38201]

Least-Mean-Square (LMS)

minimum mean square error (MMSE)

This simplified version of LMS algorithm is identical to the zero-forcing algorithm which minimizes the ISI at data samples

Sign-Sign LMS (SS-LMS)

T11: Basics of Equalization Techniques: Channels, Equalization, and Circuits, 2022 IEEE International Solid-State Circuits Conference

V. Stojanovic et al., "Autonomous dual-mode (PAM2/4) serial link transceiver with adaptive equalization and data recovery," in IEEE Journal of Solid-State Circuits, vol. 40, no. 4, pp. 1012-1026, April 2005, doi: 10.1109/JSSC.2004.842863.

Jinhyung Lee, Design of High-Speed Receiver for Video Interface with Adaptive Equalization; Phd thesis, August 2019. thesis link

Paulo S. R. Diniz, Adaptive Filtering: Algorithms and Practical Implementation, 5th edition

E. -H. Chen et al., "Near-Optimal Equalizer and Timing Adaptation for I/O Links Using a BER-Based Metric," in IEEE Journal of Solid-State Circuits, vol. 43, no. 9, pp. 2144-2156, Sept. 2008

DFE h0 Estimator

summer output \[ r_k = a_kh_0+\left(\sum_{n=-\infty,n\neq0}^{+\infty}a_{k-n}h_n-\sum_{n=1}^{\text{ntap}}\hat{a}_{k-n}\hat{h}_n\right) \] error slicer analog output \[ e_k=r_k-\hat{a}_k \hat{h}_0 \] error slicer digital output \[ \hat{e}_k=|e_k| \] It's NOT possible to implement \(e_k\), which need to determine \(\hat{a}_k=|r_k|\) in no time. One method to approach this problem is calculate \(e_k^{a_k=1}=r_k-\hat{a}_k \hat{h}_0\) and \(e_k^{a_k=-1}=r_k+\hat{a}_k \hat{h}_0\), then select the right one based on \(\hat{a}_k\)

The update equation based on Sign-Sign-Least Mean square (SS-LMS) and loss function \(L(\hat{h}_{\text{0~ntap}})=E(e_k^2)\) \[ \hat{h}_n(k+1) = \hat{h}_n(k)+\mu \cdot |e_k|\cdot \hat{a}_{k-n} \] Where \(n \in [0,...,\text{ntap}]\). This way, we can obtain \(\hat{h}_0\), \(\hat{h}_1\), \(\hat{h}_2\), ...

\(\hat{h}_0\) is used in AFE adaptation

We may encounter difficulty if the first tap of DFE is unrolled, its \(e_k\) is modified as follow \[ r_k = a_kh_0+\left(\sum_{n=-\infty,n\neq0}^{+\infty}a_{k-n}h_n-\sum_{n=2}^{\text{ntap}}\hat{a}_{k-n}\hat{h}_n\right) \] Where there is NO \(\hat{h}_1\)

To find \(\hat{h}_1\), we shall use different pattern for even and odd error slicer

Maximum Likelihood Sequence Estimation (MLSE)

[IBIS-AMI Modeling and Correlation Methodology for ADC-Based SerDes Beyond 100 Gb/s https://static1.squarespace.com/static/5fb343ad64be791dab79a44f/t/63d807441bcd266de258b975/1675102025481/SLIDES_Track02_IBIS_AMI_Modeling_and_Correlation_Tyshchenko.pdf]

M. Emami Meybodi, H. Gomez, Y. -C. Lu, H. Shakiba and A. Sheikholeslami, "Design and Implementation of an On-Demand Maximum-Likelihood Sequence Estimation (MLSE)," in IEEE Open Journal of Circuits and Systems, vol. 3, pp. 97-108, 2022, doi: 10.1109/OJCAS.2022.3173686.

Zaman, Arshad Kamruz (2019). A Maximum Likelihood Sequence Equalizing Architecture Using Viterbi Algorithm for ADC-Based Serial Link. Undergraduate Research Scholars Program. Available electronically from [https://hdl.handle.net/1969.1/166485]

There are several variants of MLSD (Maximum Likelihood Sequence Detection), including:

• Viterbi Algorithm
• Decision Feedback Sequence Estimation (DFSE)
• Soft-Output MLSD

[Evolution Of Equalization Techniques In High-Speed SerDes For Extended Reaches. https://semiengineering.com/evolution-of-equalization-techniques-in-high-speed-serdes-for-extended-reaches/]

Mueller-Muller CDR

MMPD infers the channel response from baud-rate samples of the received data, the adaptation aligns the sampling clock such that pre-cursor is equal to the post-cursor in the pulse response

Faisal A. Musa. "HIGH-SPEED BAUD-RATE CLOCK RECOVERY" [https://www.eecg.utoronto.ca/~tcc/thesis-musa-final.pdf]

Faisal A. Musa."CLOCK RECOVERY IN HIGH-SPEED MULTILEVEL SERIAL LINKS" [https://www.eecg.utoronto.ca/~tcc/faisal_iscas03.pdf]

Eduardo Fuentetaja. "Analysis of the M&M Clock Recovery Algorithm" [https://edfuentetaja.github.io/sdr/m_m_analysis/]

Liu, Tao & Li, Tiejun & Lv, Fangxu & Liang, Bin & Zheng, Xuqiang & Wang, Heming & Wu, Miaomiao & Lu, Dechao & Zhao, Feng. (2021). Analysis and Modeling of Mueller-Muller Clock and Data Recovery Circuits. Electronics. 10. 1888. 10.3390/electronics10161888.

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.

Baud-Rate CDRs [https://ocw.snu.ac.kr/sites/default/files/NOTE/Lec%206%20-%20Clock%20and%20Data%20Recovery.pdf]

F. Spagna et al., "A 78mW 11.8Gb/s serial link transceiver with adaptive RX equalization and baud-rate CDR in 32nm CMOS," 2010 IEEE International Solid-State Circuits Conference - (ISSCC), San Francisco, CA, USA, 2010, pp. 366-367, doi: 10.1109/ISSCC.2010.5433823.

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, vol. 3, pp. 216-227, 2022

SS-MM CDR

\(h_1\) is necessary

• without DFE

  SS-MMPD locks at the point (\(h_1=h_{-1}\)​)

• With a 1-tap DFE

  1-tap adaptive DFE that forces the \(h_1\) to be zero, the SS-MMPD locks wherever the \(h_{-1}\)​ is zero and drifts eventually.

  Consequently, it suffers from a severe multiple-locking problem with an adaptive DFE

Kwangho Lee, "Design of Receiver with Offset Cancellation of Adaptive Equalizer and Multi-Level Baud-Rate Phase Detector" [https://s-space.snu.ac.kr/bitstream/10371/177584/1/000000167211.pdf]

Pattern filter

During adapting, we make

• \(s_{011}\) & \(s_{110}\) are approaching to each other
• \(s_{100}\) & \(s_{001}\) are approaching to each other

Then, \(h_{-1}\) and \(h_1\) are same, which is desired

Bang-Bang CDR

alexander PD or !!PD

The alexander PD locks that edge clock (clkedge) is located at zero crossings of the data. The \(h_{-0.5}\) and \(h_{0.5}\) are equal at the lock point, where the \(h_{-0.5}\) and \(h_{0.5}\) are the cursors located at -0.5 UI and 0.5 UI.

Kwangho Lee, "Design of Receiver with Offset Cancellation of Adaptive Equalizer and Multi-Level Baud-Rate Phase Detector" [https://s-space.snu.ac.kr/bitstream/10371/177584/1/000000167211.pdf]

Shahramian, Shayan, "Adaptive Decision Feedback Equalization With Continuous-time Infinite Impulse Response Filters" [https://tspace.library.utoronto.ca/bitstream/1807/77861/3/Shahramian_Shayan_201606_PhD_thesis.pdf]

MENIN, DAVIDE, "Modelling and Design of High-Speed Wireline Transceivers with Fully-Adaptive Equalization" [https://air.uniud.it/retrieve/e27ce0ca-15f7-055e-e053-6605fe0a7873/Modelling%20and%20Design%20of%20High-Speed%20Wireline%20Transceivers%20with%20Fully-Adaptive%20Equalization.pdf]

reference

Stojanovic, Vladimir & Ho, A. & Garlepp, B. & Chen, Fred & Wei, J. & Alon, Elad & Werner, C. & Zerbe, J. & Horowitz, M.A.. (2004). Adaptive equalization and data recovery in a dual-mode (PAM2/4) serial link transceiver. IEEE Symposium on VLSI Circuits, Digest of Technical Papers. 348 - 351. 10.1109/VLSIC.2004.1346611.

A. A. Bazargani, H. Shakiba and D. A. Johns, "MMSE Equalizer Design Optimization for Wireline SerDes Applications," in IEEE Transactions on Circuits and Systems I: Regular Papers, doi: 10.1109/TCSI.2023.3328807.