Noise Analysis

Posted on 2024-04-27 Edited on 2025-11-03 In noise

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 📅

Sampling Noise

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

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

TODO 📅

1/f Noise - Fourier Transform

Steve Smith. An Interesting Fourier Transform - 1/f Noise [https://www.dsprelated.com/showarticle/40.php]

TODO 📅

Pulsed Noise Signals

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

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

banlimited input

wideband white noise input

flicker noise input

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

Assuming \(\Delta f \ll f_0\)

ADC SNR & clock jitter

cyclostationary random process

\[\begin{align} \text{SNR}_\text{ADC}[\text{dB}] &= -20\cdot \log \sqrt{\left(10^{-\frac{\text{SNR}_\text{Quantization Noise}}{20}}\right)^2 + \left(10^{-\frac{\text{SNR}_\text{Jitter}}{20}}\right)^2} \\ &= -10\cdot \log \left(\left(10^{-\frac{\text{SNR}_\text{Quantization Noise}}{20}}\right)^2 + \left(10^{-\frac{\text{SNR}_\text{Jitter}}{20}}\right)^2\right) \\ &= -10\cdot \log \left(\left(10^{-\frac{10\log(\frac{3\times2^{2N}}{2})}{20}}\right)^2 + \left(10^{-\frac{-20\log{(2\pi f_\text{in}\sigma_\text{jitter})}}{20}}\right)^2\right) \\ &= -10\cdot \log \left( \frac{2}{3\times 2^{2N}} + (2\pi f_\text{in}\sigma_\text{jitter})^2 \right) \end{align}\]

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]

CC Chen, Why Absolute Jitter Matters for ADCs & DACs? [https://youtu.be/jBgDDFFDq30?si=XFyTEfApN86Ef-RG]

Thomas Neu, TIPL 4704. Jitter vs SNR for ADCs [https://www.ti.com/content/dam/videos/external-videos/en-us/2/3816841626001/5529003238001.mp4/subassets/TIPL-4704-Jitter-vs-SNR.pdf]

import numpy as np
import matplotlib.pyplot as plt

N = 12  # ADC bit
fin = 100e6 # the frequency of the sinusoidal input signal
jrms = np.linspace(0, 10, 1000)*1e-12 #ps

# ADC (SNR) with Quantization Noise & Jitter degradation
SNR_ADC = -10 * np.log10(10**(-np.log10(3*2**(2*N)/2)) + (2*np.pi*fin*jrms)**2)
ENOB = (SNR_ADC - 1.76) / 6.02

plt.plot(jrms*1e12, ENOB, label='100 MHZ input')
plt.plot([0, 10], [12, 12], '--', label='12-bit limit')
plt.plot([0, 10], [6, 6], '--', label='6-bit limit')

plt.xscale('linear')
plt.xlim([0, 10])
plt.ylim([5, 15])
plt.xlabel('RMS Jitter (ps)')
plt.ylabel('Effective Number of Bits (ENOB')
plt.grid(which='both')
plt.title('ENOB vs. RMS Clock Jitter (100 MHz)')
plt.legend()
plt.show()

Chun-Hsien Su (蘇純賢). Design of Oversampled Sigma-Delta Data Converters. July, 2006 [pdf]

Chembian Thambidurai, "SNR of an ADC in the presence of clock jitter" [https://www.linkedin.com/posts/chembiyan-t-0b34b910_adcsnrjitter-activity-7171178121021304833-f2Wd/]

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

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

For sinusoid input:

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()

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]

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]

待学芯. ADC量化结果反推采样时钟抖动（Jitter） [https://mp.weixin.qq.com/s/55xfVQMe_N8zUGpI8ZvmsQ]

—. 关于时钟抖动(Jitter)与ADC的一些讨论 [https://mp.weixin.qq.com/s/GW1keHhfq7zrd036lyG0CQ]

DAC SNR & clock jitter

ampling Jitter Effects for ADC/DAC

In both DAC or ADC cases, doubling the timing jitter doubles the noise level
Also, doubling the frequency or amplitude doubles the jitter induced noise - SNR is not improved

Boris Murmann ISSCC 2022 SC1: Introduction to ADCs/DACs: Metrics, Topologies, Trade Space, and Applications [pdf]

S. Kim, K. -Y. Lee and M. Lee, "Modeling Random Clock Jitter Effect of High-Speed Current-Steering NRZ and RZ DAC," in IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 65, no. 9, pp. 2832-2841, Sept. 2018 [https://sci-hub.se/10.1109/TCSI.2018.2821198]

Martin Clara. High-Performance D/A-Converters - Application to Digital Transceivers, 2013 [pdf]

Chun-Hsien Su (蘇純賢). Design of Oversampled Sigma-Delta Data Converters. July, 2006 [pdf]

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

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

\(v_s[n]\) 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 \(\frac{d\omega}{T_C}= d\Omega\) \[ S_s(j\Omega) = \sum_{k=-\infty}^{\infty}S_{RC}(j(\Omega-k\Omega_s)) \cdot d\Omega \]

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

where \(m\) is the duty cycle

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

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.

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]

Below analysis focusing on sampled noise

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

—. EE315A VLSI Signal Conditioning Circuits [pdf]

—. EE315B VLSI Data Conversion Circuits, Autumn 2013 [pdf]

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

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]

Cyclostationary Noise (Modulated Noise)

[https://ece-research.unm.edu/bsanthan/ece541/cyclo.pdf]

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

White Noise Modulation

Noisy Resistor & Clocked Switch

\[ v_t (t) = v_i(t)\cdot m_t(t) \]

where \(v_i(t)\) is input white noise, whose autocorrelation is \(A\delta(\tau)\), and \(m_t(t)\) is periodically operating switch, then autocorrelation of \(v_t(t)\) \[\begin{align} R_t (t_1, t_2) &= E[v_t(t_1)\cdot v_t(t_2)] \\ &= R_i(t_1, t_2)\cdot m_t(t_1)m_t(t_2) \end{align}\]

Then \[\begin{align} R_t(t, t-\tau) &= R_i(\tau)\cdot m_t(t)m_t(t-\tau) \\ & = A\delta(\tau) \cdot m_t(t)m_t(t-\tau) \\ & = A\delta(\tau) \cdot m_t(t) \end{align}\] Because \(m_t(t)=m_t(t+T)\), \(R_t(t, t-\tau)\) is is periodic in the variable \(t\) with period \(T\)

The time-averaged ACF is denoted as \(\tilde{R_t}(\tau)\)

\[ \tilde{R}_{t}(\tau) = m\cdot A\delta(\tau) \] That is, \[ S_t(f) = m\cdot S_{A}(f) \]

Colored Noise Modulation

tavg_factor.drawio \[ \tilde{R_t}(\tau) = R_i(\tau)\cdot m_{tac}(\tau) \]

where \(m_t(t)m_t(t-\tau)\) averaged on \(t\) is denoted as \(m_{tac}(\tau)\) or \(\overline{m_t(t)m_t(t-\tau)}\)

The DC value of \(m_{tac}(\tau)\) can be calculated as below

for \(m\le 0.5\), the DC value of \(m_{tac}(\tau)\) \[ \frac{m\cdot mT}{T} = m^2 \]
for \(m\gt 0.5\), the DC value of \(m_{tac}(\tau)\) \[ \frac{(m+2m-1)(1-m)T + (2m-1)\{mT -(1-m)T\}}{T} = m^2 \]

Therefore, time-average power spectral density and total power are scaled by \(m^2\) in fundamental frequency sideband

Switched-Capacitor Track signal

track signal pnoise (sc)

zoom in first harmonic by linear step of pnoise

decreasing the rising/falling time of clock, the harmonics still retain

equivalent circuit for pnoise (eq)

thermal noise of R is modulated at first
then filtered by ideal filter

sc vs eq

sc: harmonic distortion
eq: no harmonic distortion

Non-Stationary Processes (Comparator)

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]

WSS white noise input
white noise step input

Wide-Sense-Stationary Noise

Much like sinusoidal-steady-state signal analysis, steady-state noise analysis methods assume an input \(x(t)\) of infinite duration, which is a Wide-Sense Stationary (WSS) random process

Frequency-domain Analysis

Time-domain Analysis

The output \(y(t)\) of a linear time-invariant (LTI) system \(h(t)\) \[\begin{align} R_{yy}(\tau) &= R_{xx}(\tau)*[h(\tau)*h(-\tau)] \\ &= S_{xx}(0)\delta(\tau) * [h(\tau)*h(-\tau)] \\ &= S_{xx}(0)[h(\tau)*h(-\tau)] \\ &= S_{xx}(0) \int_\alpha h(\alpha)h(\alpha-\tau)d\alpha \end{align}\]

with WSS white noise input \(x(t)\), \(R_{xx}(\tau)=S_{xx}(0)\delta(\tau)\), therefore

\[ V_o(s) = \frac{1}{C}\frac{RC}{1+sRC}\cdot I_n(s) \overset{\mathcal{L}} {\longrightarrow} \frac{1}{C}e^{-t/\tau_0} \]

where \(\tau_0 = RC\)

Non-stationary Noise

Assuming the noise applied duration is much less than the time constant, the output voltage does not reach steady-state and WSS noise analysis does not apply

Time-domain Analysis

The step noise input \(x(t) = \nu(t)u(t)\) where an underlying WSS process \(\nu(t)\) \[ R_{xx}(t_1,t_2) = E[x(t_1)x(t_2)] = R_{\nu\nu}(t_1, t_2)u(t_1)u(t_2)=R_{\nu\nu}(t_1, t_2) \tag{3.28} \]

\[ R_{xy}(t_1, t_2) = E[x(t_1)y(t_2)] = E[x(t_1)(x(t_2)*h(t_2))] = E(x(t_1)x(t_2))*h(t_2) = R_{xx}(t_1,t_2)*h(t_2) \]

\[ R_{yy}(t_1,t_2) = E[y(t_1)y(t_2)] = E[(x(t_1)*h(t_1))y(t_2)] = E[x(t_1)y(t_2)]*h(t_1)=R_{xy}(t_1,t_2)*h(t_1) \]

the absolute value of each time index is important for a non-stationary signal, and only the time difference was important for WSS signals

\[\begin{align} R_{yy}(t_1,t_2) &= h(t_1)*R_{\nu\nu}(t_1, t_2)*h(t_2) \\ &= h(t_1)*S_{xx}(0)\delta(t_2-t_1)*h(t_2) \\ &=S_{xx}(0) h(t_1)*(\delta(t_2-t_1)*h(t_2)) \\ &= S_{xx}(0)h(t_1)*h(t_2-t_1) \\ &= S_{xx}(0)\int_\tau h(\tau)h(t_2-t_1+\tau))d\tau \end{align}\]

That is \[ \sigma^2_y (t)= R_{yy}(t_1,t_2)|_{t_1=t_2=t}=S_{xx}(0)\int_{-\infty}^t |h(\tau)|^2d\tau \tag{3.33} \]

\(t\), the upper limit of integration is just intuitive, which lacks strict derivation

Because stable systems have impulse responses that decay to zero as time goes to infinity, the output noise variance approaches the WSS result as time approaches infinity

Richard Schreier. ECE1371 Advanced Analog Circuits Lecture 8 - COMPARATOR & FLASH ADC DESIGN [http://individual.utoronto.ca/schreier/lectures/2015/8-6.pdf]

\[ R_{yy}(0) = \frac{1}{2\pi}\int_{-\infty}^{\infty}|H(\omega)|^2S_{xx}(\omega)d\omega = S \cdot \frac{1}{2\pi}\int_{-\infty}^{\infty}|H(\omega)|^2d\omega \overset{\text{Parseval's Relation}}{=} S\cdot \int_{-\infty}^{\infty}|h(t)|^2dt \]

Frequency-domain Analysis

Because the definition of the PSD assumes that the variance of the noise process is independent of time, the PSD of a non-stationary process is not very meaningful

Input Referred Noise

Noise Voltage to Timing Jitter Conversion & noise gain

\[\begin{align} \overline{v_n^2}(t_i) &= \frac{\overline{v_{on}^2}}{|A_N|^2} \\ &= \frac{G_n}{G_m}\frac{kT}{C}\frac{1}{A_0}\frac{1+e^{-t_i/\tau_o}}{1-e^{-t_i/\tau_o}} \\ &=4kT\frac{G_n}{G_m^2}\frac{1}{4R_oC} \coth(\frac{t_i}{2\tau_o}) \\ &= 4kTR_n\frac{1}{4\tau_o} \coth(\frac{t_i}{2\tau_o}) \end{align}\]

where \(R_n = \frac{G_n}{G_m^2}\), the equivalent thermal noise resistance

Suppose \(t_i \ll \tau_0\) \[ \overline{v_n^2}(t_i) = 4kTR_n\frac{1}{4\tau_o} \coth(\frac{t_i}{2\tau_o}) \approx 4kTR_n\frac{1}{4\tau_o} \frac{1 + (1-\frac{t_i}{\tau_0})}{1 - (1-\frac{t_i}{\tau_0})} =4kTR_n \cdot \frac{1}{2t_i} \]

Suppose \(t_i \gg \tau_0\) \[ \overline{v_n^2}(t_i) = 4kTR_n\frac{1}{4\tau_o} \coth(\frac{t_i}{2\tau_o}) \approx 4kTR_n\frac{1}{4\tau_o} = \frac{G_n}{G_m}\frac{kT}{C}\frac{1}{A_0} \] As expected, the input referred noise voltage is \(kT/C\) noise

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