Noise Analysis

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noise power at filter output

Chembian Thambidurai, "Comparison Of Noise Power At Lowpass Filter Output" [link]

—, "On Noise Power At The Bandpass Filter Output" [link]

—, "Integrated Power of Thermal and Flicker Noise" [link]

TODO 📅

Pulsed Noise Signals

Chembian Thambidurai, "Power Spectral Density of Pulsed Noise Signals" [link]

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Above, the output of the multiplier be \(y(t)\) is passed through a ideal brick wall low pass filter with a bandwidth of \(f_0/2\)

When a random signal is multiplied by a pulse function, the resulting signal becomes a cyclo-stationary random process.

As rule of thumb, the spectrum of such a pulsed noise signal

  • thermal noise is multiplied by \(D\)

  • flicker noise is multiplied by \(D^2\),

where \(D\) is the duty cycle of the pulse signal

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banlimited input

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wideband white noise input

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flicker noise input

with \(S_x(f)=\frac{K_f}{f}\)

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Assuming \(\Delta f \ll f_0\)

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Sampling Noise

Chembian Thambidurai, "Noise, Sampling and Zeta Functions" [link]

A random signal \(v_n(t)\) is sampled using an ideal impulse sampler

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TODO 📅

Aperture Jitter & ADC SNR

Chembian Thambidurai, "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]

Unlike the quantization noise and the thermal noise, the impact of the clock jitter on the ADC performance depends on the input signal properties like its PSD

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The error between the ideal sampled signal and the sampling with clock jitter can be treated as noise and it results in the degradation of the SNR of the ADC

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For sinusoid input:

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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]

M. Shinagawa, Y. Akazawa and T. Wakimoto, "Jitter analysis of high-speed sampling systems," in IEEE Journal of Solid-State Circuits, vol. 25, no. 1, pp. 220-224, Feb. 1990 [https://sci-hub.se/10.1109/4.50307]

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import numpy as np
import matplotlib.pyplot as plt

ENOB = 8
fin = np.logspace(8, 11, 60)

# quantization noise: SNR = 6.02*ENOB + 1.76 dB
Ps_PnQ = 10**((6.02*ENOB + 1.76)/10)
PnQ = 1/Ps_PnQ

# jitter noise: SNR = 6 - 20log10(2*pi*fin*Jrms) dB @ref. Chembiyan T
Jrms_list = [25e-15, 50e-15, 100e-15, 250e-15, 500e-15, 1000e-15]
for Jrms in Jrms_list:
    # Ps_PnJ_lcl = 10**((6-20*np.log10(2*np.pi*fin*Jrms))/10)     # ref. Chembiyan T
    Ps_PnJ_lcl = 10**((0 - 20 * np.log10(2 * np.pi * fin * Jrms)) / 10)  # ref. Nicola Da Dalt
    PnJ_lcl = 1/Ps_PnJ_lcl
    SNR_lcl = 10*np.log10(1/(PnQ+PnJ_lcl))
    plt.plot(fin, SNR_lcl, label=r'$\sigma_{jitter}$'+'='+str(int(Jrms*1e15))+'fs')

plt.xscale('log')
plt.ylim([0, 55])
plt.xlabel(r'$f_{in}$ [Hz]')
plt.ylabel(r'SNR [dB]')
plt.grid(which='both')
# plt.title(r'ref. Chembiyan T')
plt.title(r'ref. Nicola Da Dalt')
plt.legend()
plt.show()

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i.e. N. Da Dalt's

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Ayça Akkaya, "High-Speed ADC Design and Optimization for Wireline Links" [https://infoscience.epfl.ch/server/api/core/bitstreams/96216029-c2ff-48e5-a675-609c1e26289c/content]

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ISF for Oscillators

TODO 📅

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Sampled Thermal Noise

The aliasing of the noise, or noise folding, plays an important role in switched-capacitor as it does in all switched-capacitor filters

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Assume for the moment that the switch is always closed (that there is no hold phase), the single-sided noise density would be

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\(v_s[n]\) is the sampled version of \(v_{RC}(t)\), i.e. \(v_s[n]= v_{RC}(nT_C)\) \[ S_s(e^{j\omega}) = \frac{1}{T_C} \sum_{k=-\infty}^{\infty}S_{RC}(j(\frac{\omega}{T_C}-\frac{2\pi k}{T_C})) \cdot d\omega \] where \(\omega \in [-\pi, \pi]\), furthermore \(\frac{d\omega}{T_C}= d\Omega\) \[ S_s(j\Omega) = \sum_{k=-\infty}^{\infty}S_{RC}(j(\Omega-k\Omega_s)) \cdot d\Omega \]

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The noise in \(S_{RC}\) is a stationary process and so is uncorrelated over \(f\) allowing the \(N\) rectangles to be combined by simply summing their noise powers

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where \(m\) is the duty cycle

Below analysis focus on sampled noise

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  • Calculate autocorrelation function of noise at the output of the RC filter
  • Calculate the spectrum by taking the discrete time Fourier transform of the autocorrelation function

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Kundert, Ken. (2006). Simulating Switched-Capacitor Filters with SpectreRF [https://designers-guide.org/analysis/sc-filters.pdf]

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

Boris Murmann, EE315B VLSI Data Conversion Circuits, Autumn 2013

- Noise Analysis in Switched-Capacitor Circuits, ISSCC 2011 / tutorials [slides, transcript]

Tania Khanna, ESE568 Fall 2019, Mixed Signal Circuit Design and Modeling URL: https://www.seas.upenn.edu/~ese568/fall2019/

Matt Pharr, Wenzel Jakob, and Greg Humphreys. 2016. Physically Based Rendering: From Theory to Implementation (3rd. ed.). Morgan Kaufmann Publishers Inc., San Francisco, CA, USA.

Bernhard E. Boser . Advanced Analog Integrated Circuits Switched Capacitor Gain Stages [https://people.eecs.berkeley.edu/~boser/courses/240B/lectures/M05%20SC%20Gain%20Stages.pdf]

R. Gregorian and G. C. Temes. Analog MOS Integrated Circuits for Signal Processing. Wiley-Interscience, 1986

Trevor Caldwell, Lecture 9 Noise in Switched-Capacitor Circuits [http://individual.utoronto.ca/trevorcaldwell/course/NoiseSC.pdf]

Christian-Charles Enz. High precision CMOS micropower amplifiers [pdf]

reference

David Herres, The difference between signal under-sampling, aliasing, and folding URL: https://www.testandmeasurementtips.com/the-difference-between-signal-under-sampling-aliasing-and-folding-faq/

Pharr, Matt; Humphreys, Greg. (28 June 2010). Physically Based Rendering: From Theory to Implementation. Morgan Kaufmann. ISBN 978-0-12-375079-2. Chapter 7 (Sampling and reconstruction)

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