Noise Analysis

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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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\[ X(j\Omega)d\Omega = \frac{1}{T_c}X(e^{j\omega})d\omega \]

ref. [Frequency-Domain Representation of Sampling] of EQ.(31) in the blog

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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.

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

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]

spectrum analyzer

We tried to plot a power spectral density together with something that we want to interpret as a power spectrum

  • spectrum of a periodic signal
  • spectral density of a broadband signal such as noise

Sine-wave components are located in individual FFT bins, but broadband signals like noise have their power spread over all FFT bins!

The noise floor depends on the length of the FFT

[http://individual.utoronto.ca/schreier/lectures/2015/1.pdf]

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signal tone power \[ P_{\text{sig}} = 2 \frac{X_{w,sig}^2}{S_1^2} \]

noise power \[ P_n = \frac{X_{w,n}^2}{S_2} \]

Then, displayed SNR is obtained \[\begin{align} \mathrm{SNR} &= 10\log10\left(\frac{X_{w,sig}^2}{X_{w,n}^2}\right) \\ &= 10\log_{10}\left(\frac{P_{\text{sig}}}{P_n}\right) + 10\log_{10}\left(\frac{S_1^2}{2S_2}\right) \\ &= \mathrm{SNR}'-10\log_{10}\left(\frac{2S_2}{S_1^2}\right) \\ &= \mathrm{SNR}'-10\log_{10}(2\cdot\mathrm{NBW}) \\ \end{align}\]

DFT's output \(\mathrm{SNR}\)

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for N=[2^6 2^8 2^10 2^12]
  wd = rectwin(N);
  nbw = enbw(wd)/N;
  snr_shift = 10*log10(nbw * 2);
  disp(snr_shift);
end

output:

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-15.0515

-21.0721

-27.0927

-33.1133

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The solution to the scaling problem in the case of a PSD obtained from a sine-wave scaled FFT is similarly simple. All we need do is provide the value of NBW

APPENDIX A - SPECTRAL ESTIMATION - A.2 Scaling and Noise Bandwidth

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

  • For a filter with infinitely steep roll-off, the noise bandwidth (NBW) is equal to the filter's bandwidth,
  • while for a filter with a single-pole roll-off, NBW is 2 times the 3-dB bandwidth

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