Noise Analysis

Posted on Edited on In

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White noise

David Murray. Topic 6: Random Processes and Signals [slides notes]

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Mathuranathan. White Noise : Simulation and Analysis using Matlab [https://www.gaussianwaves.com/2013/11/simulation-and-analysis-of-white-noise-in-matlab/]

Wiener Khinchin Theorem

image-20260205202253286

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clear all; clc; close all;
L=2^20; %Sample length for the random signal
mu=0;
sigma=2;
X=sigma*randn(L,1)+mu;

figure();
subplot(5,1,1)
plot(X);
title(['White noise : \mu_x=',num2str(mu),' \sigma^2=',num2str(sigma^2)])
xlabel('Samples')
ylabel('Sample Values')
grid on

subplot(5,1,2)
n=100; %number of Histrogram bins
[f,x]=hist(X,n);
bar(x,f/trapz(x,f)); hold on;
%Theoretical PDF of Gaussian Random Variable
g=(1/(sqrt(2*pi)*sigma))*exp(-((x-mu).^2)/(2*sigma^2));
plot(x,g, LineWidth=3);hold off; grid on;
title('Theoretical PDF and Simulated Histogram of White Gaussian Noise');
legend('Histogram','Theoretical PDF');
xlabel('Bins');
ylabel('PDF f_x(x)');


subplot(5,1,3)
Rxx=1/L*conv(flipud(X),X);
lags=(-L+1):1:(L-1);

%Alternative method
%[Rxx,lags] =xcorr(X,'biased'); 
%The argument 'biased' is used for proper scaling by 1/L
%Normalize auto-correlation with sample length for proper scaling

plot(lags,Rxx, LineWidth=4); 
title('Auto-correlation Function of white noise');
xlabel('Lags')
ylabel('Correlation')
grid on; hold on;

ss = X.^2;
ssm = sum(ss)/L; % 4.0
stem([0], [ssm], '--o', LineWidth=2)


subplot(5,1,4)
[pxx_norm, f] = pwelch(X, 1024,[],[],1, 'centered');
plot(f, 10*log10(pxx_norm))
axis([-0.5 0.5 0 10]); grid on;
ylabel('PSD (dB/Hz)');
xlabel('Normalized Frequency');
title('Power spectral density of white noise with inbuilt pwelch');

subplot(5,1,5)
%Verifying the constant PSD of White Gaussian Noise Process
%with arbitrary mean and standard deviation sigma

mu=0; %Mean of each realization of Noise Process
sigma=2; %Sigma of each realization of Noise Process

L = 1000; %Number of Random Signal realizations to average
N = 1024; %Sample length for each realization set as power of 2 for FFT

%Generating the Random Process - White Gaussian Noise process
MU=mu*ones(1,N); %Vector of mean for all realizations
Cxx=(sigma^2)*diag(ones(N,1)); %Covariance Matrix for the Random Process
R = chol(Cxx); %Cholesky of Covariance Matrix
%Generating a Multivariate Gaussian Distribution with given mean vector and
%Covariance Matrix Cxx
z = repmat(MU,L,1) + randn(L,N)*R;
%By default, FFT is done across each column - Normal command fft(z)
%Finding the FFT of the Multivariate Distribution across each row
%Command - fft(z,[],2)
Z = 1/sqrt(N)*fft(z,[],2);  % Scaling by sqrt(N);
Pzavg = mean(Z.*conj(Z));   % Computing the **mean power** from fft

normFreq= [-N/2:N/2-1]/N;
Pzavg=fftshift(Pzavg); %Shift zero-frequency component to center of spectrum
plot(normFreq,10*log10(Pzavg),'r');
axis([-0.5 0.5 0 10]); grid on;
ylabel('PSD (dB/Hz)');
xlabel('Normalized Frequency');
title('Power spectral density of white noise');

Julius Orion Smith III. Why an Impulse is Not White [https://www.dsprelated.com/freebooks/sasp/Why_Impulse_Not_White.html]

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I.I.D. Noise

Independent and Identically Distributed

Kevin Zheng. The Frequency Domain Trap – Beware of Your AC Analysis [https://circuit-artists.com/the-frequency-domain-trap-beware-of-your-ac-analysis/]

Mathuranathan. White Noise : Simulation and Analysis using Matlab [https://www.gaussianwaves.com/2013/11/simulation-and-analysis-of-white-noise-in-matlab/]

white noise doesn't mean it has a Gaussian/normal distribution

The only criteria for a (discrete) signal to be "white" is for each sample to be independently taken from the same probability distribution

img

By understanding input signal's statistical nature, we can gather more insights about certain requirements for our circuits than just from frequency domain

img

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fs = 1;
N = 2^20;
Nhist = 100;
segmentLength = 512;

x_norm = randn(1, N,1); % Generate random data (e.g., Gaussian white noise)
[pxx_norm, f] = pwelch(x_norm, segmentLength, [], [], fs);

x_uniform = 2*rand(N,1) - 1; % uniform random -1 ~ 1;
[pxx_uniform, ~] = pwelch(x_uniform, segmentLength, [], [], fs);

x_sin = sin(2*pi*(rand(N,1) - 0.5)); % sinusoidal-like
[pxx_sin, ~] = pwelch(x_sin, segmentLength, [], [], fs);

x_bin = 2*randi(2,N,1) -3; % binomial distribution, -1, 1
[pxx_bin, ~] = pwelch(x_bin, segmentLength, [], [], fs);

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val = randn(1, 2^20,1);
val_abs = abs(val);
val_abs_avg = mean(val_abs);
subplot(2,2,1)
histogram(val);
title('x_{norm} distribution'); grid on

subplot(2,2,2)
histogram(val_abs);
title('|x_{norm}| distribution')

subplot(2,2,[3 4])
[pxx, f] = pwelch(val_abs - val_abs_avg, 512, [], [], 1);
plot(f, 10*log10(pxx), LineWidth=4); grid on
title('psd (dB)')

Additive White Gaussian Noise (AWGN)

Qasim Chaudhari. Additive White Gaussian Noise (AWGN) [https://wirelesspi.com/additive-white-gaussian-noise-awgn/]

Mathuranathan. Simulate additive white Gaussian noise (AWGN) channel [https://www.gaussianwaves.com/2015/06/how-to-generate-awgn-noise-in-matlaboctave-without-using-in-built-awgn-function/]

Does PSD (dBm/Hz) of white noise depend on sampling rate?. Does PSD (dBm/Hz) of white noise depend on sampling rate? [https://dsp.stackexchange.com/a/87654/59253]

Matthew Schubert. Colouring Noise - Generating coloured noise to simulate physical processes [https://blog.ioces.com/matt/posts/colouring-noise/]

\[ \text{PSD}_s = \frac{\sigma^2}{f_s} \space\space \text{for two-sided PSD} \space\space\space \text{or}\space\space\space =\frac{\sigma^2}{f_s/2}\space\space \text{for one-sided PSD} \]

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that total power will always be the same for any sampling rate used (as long as the histogram of those samples is still \(\sigma_x\)). Thus the power spectral density will be \(\sigma_x^2/f_s\).

dt_wn.drawio

How to generate white noise signal from a given PSD? [https://www.mathworks.com/matlabcentral/answers/1968-how-to-generate-white-noise-signal-from-a-given-psd#answer_1498849]

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White noise in CONTINUOUS-time blocks

MathWorks Support Team. [https://www.mathworks.com/matlabcentral/answers/100763-why-have-a-band-limited-white-noise-generator-block-instead-of-just-a-white-noise-generator-block#answer_110112]

white_noise_sampling.drawio

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Colored Noise Generation

Allen B. Downey. Think DSP - Digital Signal Processing in Python [book repo]

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

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

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

Mathuranathan. Generate correlated Gaussian sequence (colored noise) [https://www.gaussianwaves.com/2019/10/generating-correlated-gaussian-sequences/]

Jeruchim et., al, Simulation of communication systems – modeling, methodology, and techniques, second edition, Kluwer academic publishers, 2002

Power spectral densities of white noise and colored noise

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Two methods to generate colored Gaussian noise for given mean and PSD shape

  • Spectral Factorization Method
  • Auto-Regressive (AR) Model

Filtering in Time domain

Alessandro Cudazzo. noise generator developed in C99 (white, brown) [https://github.com/alessandrocuda/noise_generator]

with Derivatives approximation, \(H_p(s) = \frac{1}{s\tau + 1} \to H_p(z)=\frac{\alpha}{1 +(\alpha -1)z^{-1}}\), where \(\alpha = \frac{T}{\tau+T}\)

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get_bnoise(){
    RawData = (randq64_double()*2.0 -1.0)/VOLUME_BN;
	SmoothData = SmoothData - (LPF_Beta * (SmoothData - RawData)); // RC Filter
    return SmoothData;
}

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%Colored Noise Generation
x=whiteNoise;
%First Order Low pass filter y(n)=a*y(n-1)+(1-a)*x(n)
%Filter Trasfer function Y(Z) = X(Z)*(1-a)/(1-aZ^-1)
[y zf]=filter(1-a,[1 -a],x);
coloredNoise = y;

Julius Orion Smith III. Spectral Audio Signal Processing [https://www.dsprelated.com/freebooks/sasp/Filtered_White_Noise.html]

Example: Synthesis of 1/F Noise (Pink Noise)

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Nx = 2^16;  % number of samples to synthesize
B = [0.049922035 -0.095993537 0.050612699 -0.004408786];
A = [1 -2.494956002   2.017265875  -0.522189400];
nT60 = round(log(1000)/(1-max(abs(roots(A))))); % T60 est.
v = randn(1,Nx+nT60); % Gaussian white noise: N(0,1)
x = filter(B,A,v);    % Apply 1/F roll-off to PSD
x = x(nT60+1:end);    % Skip transient response

%{
octave:1> sum(B)/sum(A)
ans = 1.0991
%}

Filtering in Frequency domain

Matthew Schubert. Colouring Noise - Generating coloured noise to simulate physical processes [https://blog.ioces.com/matt/posts/colouring-noise/] [https://gist.github.com/m-schubert/45c562146c6607b8990f1e8f34ff87b0]

A dark plot with 5 data series on it for violet, blue, white, pink and brownian noise, each coloured in their respective colours.

White, Pink \(1/f\), Brownian \(1/f^2\), Blue \(f\), Violet \(f^2\)

The process of generating coloured noise is relatively simple:

  1. Start with a representation of white noise signal in the frequency domain
  2. Shape the signal in the frequency domain according to the PSD you'd like to achieve
  3. Take the inverse Fourier transform of the shaped signal

Sampling shaping transfer function (not scaled by \(1/T_s\)) make sense in time domain convolution between sampled noise and sampled impulse response of filter

AR(N) Model Method

Generate color noise using Auto-Regressive (AR) model [https://www.gaussianwaves.com/2019/12/color-noise-generation-using-auto-regressive-ar-model-power-law-noises/]

Konstantinos Chatzilygeroudis. Lecture 9 & 10: Noise Models & Linear Filters [notes, slides]

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Autoregressive (AR) modeling in signal processing represents a signal as a linear combination of its own previous values plus a white noise error term, effectively acting as an all-pole IIR filter

Google AI Mode [https://share.google/aimode/QAuvWdwJFng54cYZo]

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% https://github.com/vineel49/colored_noise/blob/master/RUN_ME.m


% IIR FILTER PARAMETERS (USED TO GENERATE COLOURED NOISE)
a = 0.99;
B = sqrt(1-a^2); % INPUT COEFFICIENTS IN THE DIFFERENCE EQUATION
A = [1 -a]; % OUTPUT COEFFICIENTS IN THE DIFFERENCE EQUATION

% ADDITIVE COLOURED GAUSSIAN NOISE
% AWGN
AWGN = normrnd(0,NOISE_STD_DEV,1,1000+FRAME_SIZE)+1i*normrnd(0,NOISE_STD_DEV,1,1000+FRAME_SIZE);
ACGN = filter(B,A,AWGN);
ACGN(1:1000) = []; % DISCARDING TRANSIENT SAMPLES

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sampling bandlimited white 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

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]

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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

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]

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impulse 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 📅

ADC SNR & clock jitter

cyclostationary random process

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

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]

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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]

image-20250809182751263

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

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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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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]

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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]

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]

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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

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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]

Cyclostationary Noise (Modulated Noise)

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

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

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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, \[ S_t(f) = m\cdot S_{A}(f) \]

image-20241118212505205

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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

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

  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

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Switched-Capacitor Track signal

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track signal pnoise (sc)

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zoom in first harmonic by linear step of pnoise

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

equivalent circuit for pnoise (eq)

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

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sc vs eq

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  • 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]

image-20251020204611858

WSS white noise input image-20251022084714201 image-20251022084805639
white noise step input image-20251022084648664 image-20251022084948197

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

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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

image-20251020210923501

image-20251020211524360

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

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

image-20251020212837224

Time-domain Analysis

image-20251020213929095

The step noise input \(x(t) = \nu(t)u(t)\) where an underlying WSS process \(\nu(t)\) \[ R_{xx}(t_1,t_2) = E[x(t_1)x(t_2)] = R_{\nu\nu}(t_1, t_2)u(t_1)u(t_2)=R_{\nu\nu}(t_1, t_2) \tag{3.28} \] image-20251020213729837

image-20241123005644828

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

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

image-20241123011304449

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

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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

image-20251020221312637

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

image-20250710221018596

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

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

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Input Referred Noise

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Noise Voltage to Timing Jitter Conversion & noise gain

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

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

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

image-20251021004235631

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

Windowed Integrals of Noise

A. A. Abidi, "Phase Noise and Jitter in CMOS Ring Oscillators," in IEEE Journal of Solid-State Circuits, vol. 41, no. 8, pp. 1803-1816, Aug. 2006 [pdf]

image-20260212231450725

Lossless Integral

a noise current integrates onto a capacitor only

image-20260212231256772

With white Gaussian noise at the input, the output spectrum is no longer white although its distribution remains Gaussian

image-20260213001029878

Lossy Integral

a noise current integrates onto a capacitor which is shunted by a finite loss resistance

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

Laplace transform of Truncated Impulse Response \(0 \to t_d\) \[ \int_{0}^{t_d}\frac{1}{C}\cdot e^{-\frac{t}{RC}} \cdot e^{-st}dt = \frac{1}{C}\cdot \frac{1}{s+\frac{1}{RC}}\left(1-e^{-(st_d+\theta)}\right)=\frac{1}{C}\cdot \color{red}\frac{t_d}{st_d+\theta}\left(1-e^{-(st_d+\theta)}\right) \]

Google AI Mode [https://share.google/aimode/O6rjWM1YWel0NudI7]

image-20260213011010297

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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

Mathuranathan Viswanathan. Digital Modulations using Matlab: Build Simulation Models from Scratch