KNOW-HOW

autocorrelation

The expectation returns the probability-weighted average of the specific function at that specific time over all possible realizations of the process

Ergodicity

ensemble autocorrelation and temporal autocorrelation (time autocorrelation)

ECE438 - Laboratory 7: Discrete-Time Random Processes (Week 2) October 6, 2010 [https://engineering.purdue.edu/VISE/ee438L/lab7/pdf/lab7b.pdf]

Ensemble Averages and Time Averages

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

[https://www.nii.ac.jp/qis/first-quantum/e/forStudents/lecture/pdf/noise/chapter1.pdf]

• time average: time-averaged quantities for the \(i\)-th member of the ensemble
• ensemble average: ensemble-averaged quantities for all members of the ensemble at a certain time

where \(\theta\) is one member of the ensemble; \(p(x)dx\) is the probability that \(x\) is found among \([x, x + dx]\)

sample autocorrelation

[https://engineering.purdue.edu/VISE/ee438L/lab7/pdf/lab7b.pdf]

[http://www.signal.uu.se/Courses/CourseDirs/SignalbehandlingIT/forelas02.pdf]

ergodic vs. stationary

[https://bookdown.org/kevin_davisross/stat350-handouts/stationary.html]

Strict Sense Stationary (SSS) & Wide Sense Stationary (WSS)

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

LTI Filtering of WSS process

mean

autocorrelation

deterministic autocorrelation function

\[ R_{yy}(\tau) = h(\tau)*R_{xx}(\tau)*h(-\tau) =R_{xx}(\tau)*h(\tau)*h(-\tau) \]

why \(\overline{R}_{hh}(\tau) \overset{\Delta}{=} h(\tau)*h(-\tau)\) is autocorrelation ? the proof is as follows:

\[\begin{align} \overline{R}_{hh}(\tau) &= h(\tau)*h(-\tau) \\ &= \int_{-\infty}^{\infty}h(x)h(-(\tau - x))dx \\ &= \int_{-\infty}^{\infty}h(x)h(-\tau + x)dx \\ &=\int_{-\infty}^{\infty}h(x+\tau)h(x)dx \end{align}\]

PSD

Topic 6 Random Processes and Signals [https://www.robots.ox.ac.uk/~dwm/Courses/2TF_2021/N6.pdf]

Alan V. Oppenheim, Introduction To Communication, Control, And Signal Processing [https://ocw.mit.edu/courses/6-011-introduction-to-communication-control-and-signal-processing-spring-2010/a6bddaee5966f6e73450e6fe79ab0566_MIT6_011S10_notes.pdf]

Balu Santhanam, Probability Theory & Stochastic Process 2020: LTI Systems and Random Signals [https://ece-research.unm.edu/bsanthan/ece541/LTI.pdf]

Time Reversal \[ x(-t) \overset{FT}{\longrightarrow} X(-j\omega) \]

if \(x(t)\) is real, then \(X(j\omega)\)​ has conjugate symmetry \[ X(-j\omega) = X^*(j\omega) \]

Derivatives of Random Processes

since \(x(t)\) is stationary process, and \(y(t) = \frac{dx(t)}{dt}\)

Using \(R_{yy}(\tau) = h(\tau)*R_{xx}(\tau)*h(-\tau)\)

\[\begin{align} R_{yy}(\tau) &= \mathcal{F}^{-1}[H(j\omega)\Phi_{xx}(j\omega)H(-j\omega)] \\ &= \mathcal{F}^{-1}[-(j\omega)^2\Phi_{xx}(j\omega)] \end{align}\]

we obtain the autocorrelation function of the output process as \[ R_{yy}(\tau) = -\frac{d^2}{d\tau^2}R_{xx}(\tau) \]

Liu Congfeng, Xidian University. Random Signal Processing: Chapter 5 Linear System: Random Process [https://web.xidian.edu.cn/cfliu/files/20121125_153218.pdf]

[https://sharif.ir/~bahram/sp4cl/PapoulisLectureSlides/lectr14.pdf]

Periodogram

The periodogram is in fact the Fourier transform of the aperiodic correlation of the windowed data sequence

estimating continuous-time stationary random signal

The sequence \(x[n]\) is typically multiplied by a finite-duration window \(w[n]\), since the input to the DFT must be of finite duration. This produces the finite-length sequence \(v[n] = w[n]x[n]\)

\[\begin{align} \hat{P}_{ss}(\Omega) &= \frac{|V(e^{j\omega})|^2}{LU} \\ &= \frac{|V(e^{j\omega})|^2}{\sum_{n=0}^{L-1}(w[n])^2} \tag{1}\\ &= \frac{L|V(e^{j\omega})|^2}{\sum_{k=0}^{L-1}(W[k])^2} \tag{2} \end{align}\]

That is, by \((1)\) \[ \hat{P}_{ss}(\Omega) = T_s\hat{P}_{xx(\omega)} = \frac{T_s|V(e^{j\omega})|^2}{\sum_{n=0}^{L-1}(w[n])^2}=\frac{|V(e^{j\omega})|^2}{f_{res}L\sum_{n=0}^{L-1}(w[n])^2} \]

That is, by \((2)\) \[ \hat{P}_{ss}(\Omega) = T_s\hat{P}_{xx(\omega)} = \frac{T_sL|V(e^{j\omega})|^2}{\sum_{k=0}^{L-1}(W[k])^2} = \frac{|V(e^{j\omega})|^2}{f_{res}\sum_{k=0}^{L-1}(W[k])^2} \]

!! ENBW

Wiener-Khinchin theorem

Norbert Wiener proved this theorem for the case of a deterministic function in 1930; Aleksandr Khinchin later formulated an analogous result for stationary stochastic processes and published that probabilistic analogue in 1934. Albert Einstein explained, without proofs, the idea in a brief two-page memo in 1914

\(x(t)\), Fourier transform over a limited period of time \([-T/2, +T/2]\) , \(X_T(f) = \int_{-T/2}^{T/2}x(t)e^{-j2\pi ft}dt\)

With Parseval's theorem \[ \int_{-T/2}^{T/2}|x(t)|^2dt = \int_{-\infty}^{\infty}|X_T(f)|^2df \] So that \[ \frac{1}{T}\int_{-T/2}^{T/2}|x(t)|^2dt = \int_{-\infty}^{\infty}\frac{1}{T}|X_T(f)|^2df \]

where the quantity, \(\frac{1}{T}|X_T(f)|^2\) can be interpreted as distribution of power in the frequency domain

For each \(f\) this quantity is a random variable, since it is a function of the random process \(x(t)\)

The power spectral density (PSD) \(S_x(f )\) is defined as the limit of the expectation of the expression above, for large \(T\): \[ S_x(f) = \lim _{T\to \infty}\mathrm{E}\left[ \frac{1}{T}|X_T(f)|^2 \right] \]

The Wiener-Khinchin theorem ensures that for well-behaved wide-sense stationary processes the limit exists and is equal to the Fourier transform of the autocorrelation \[\begin{align} S_x(f) &= \int_{-\infty}^{+\infty}R_x(\tau)e^{-j2\pi f \tau}d\tau \\ R_x(\tau) &= \int_{-\infty}^{+\infty}S_x(f)e^{j2\pi f \tau}df \end{align}\]

Note: \(S_x(f)\) in Hz and inverse Fourier Transform in Hz (\(\frac{1}{2\pi}d\omega = df\))

[https://www.robots.ox.ac.uk/~dwm/Courses/2TF_2011/2TF-L5.pdf]

Frank R. Kschischang. The Wiener-Khinchin Theorem [https://www.comm.utoronto.ca/~frank/notes/wk.pdf]

Example

Remember: impulse scaling

\[ \cos(2\pi f_0t) \overset{\mathcal{F}}{\longrightarrow} \frac{1}{2}[\delta(f -f_0)+\delta(f+f_0)] \]

due to \(\cos(\omega_0 t +\phi_0) = \sin(\omega_0 t +\phi_0 + \frac{\pi}{2})\)

Energy Signal

Discrete time

SIMG-713 Noise and Random Processes Spring 2002 . Lecture 15 Power Spectrum Estimation [https://www.cis.rit.edu/class/simg713/Lectures/Lecture713-15-4.pdf]

Properties of the Fourier Transform for Discrete-Time Signals [https://www.comm.utoronto.ca/dkundur/course_info/362/EmanHammadDTFT2.pdf]

\[ \frac{1}{2\pi}F^{-1}\{R_{xx}\}d\omega = \frac{1}{2\pi}F^{-1}\{R_{xx}\}d(2\pi f T)=T\cdot F^{-1}\{R_{xx}\}df = P_{xx}(f)df \] power spectral density of a discrete-time random process \(\{x(n)\}\) is given by \[ P_{xx}(f) =T\cdot F^{-1}\{R_{xx}\} \]

Periodic and Cyclostationary Processes

Multirate Systems & Random Sequences

Balu Santhanam. ece541 Probability Theory & Stochastic Process: Random Signals and Multirate Systems [http://ece-research.unm.edu/bsanthan/ece541/rand.pdf]

WSS Random Sequences

autocorrelation function of WSS random sequences

psd of WSS of WSS random sequences

Interpretation of the psd

Equation 8.4-4 is permissible for cyclostationary waveforms

decimation

\[ \frac{1}{M}\int_{-Mx_0}^{Mx_0}f(\frac{x}{M})dx = \int_{-Mx_0}^{Mx_0}f(\frac{x}{M}) d\frac{x}{M} = \int_{-x_0}^{x_0} f(\acute{x}) d\acute{x} \]

where \(\acute{x} = \frac{x}{M}\)

interpolation

The resulting expanded random sequence is clearly nonstationary, because of the zero insertions.

This random sequences and processes is classified as being cyclostationary

psd of \(X_e[n]\), the expansion or the upsampled version of \(X[n]\)

\[ S_{X_eX_e}(\omega) = \frac{1}{L}S_{XX}(L\omega) \]

where \(L\) is upsampling factor

apply \(X_e[n]\) to an ideal lowpass filter with bandwidth \([-\pi/2 , +\pi/2]\) and gain of \(2\) or \(L\)

\[ S_{YY}(\omega) = \left\{ \begin{array}{cl} L \cdot S_{XX}(L\omega), &\ |\omega|\leq \pi/L \\ 0, &\ \pi/L \lt |\omega| \leq \pi \end{array} \right. \]

Stark H, Woods JW. Probability, Statistics, and Random Processes for Engineers, 3rd [pdf]

—, 3th Ed Solution Manual [https://www.scribd.com/document/353335818/Stark-Woods-3th-Ed-Manual#page=212]

General case with upsampling factor \(L\)

\[\begin{align} &\mathrm{E}\{|\sum_{n=-N}^N X_e[n]e^{-j\omega n}|^2\}\\ &= \sum_{n=-N}^N\sum_{l=-N}^NX_e[n]\cdot X_e^*[l]\cdot e^{-j\omega(n-l)}\\ &= \sum_{n=-N}^N\sum_{l=-N}^NX[\frac{n}{L}]\cdot X^*[\frac{l}{L}]\cdot e^{-j\omega(n-l)}\\ &= \sum_{k=-N/L}^{N/L}\sum_{m=-N/L}^{N/L}X[k]\cdot X^*[m]\cdot e^{-jL\omega (k-m)}\\ &= \sum_{k=-\infty}^{\infty}\sum_{m=-\infty}^{\infty}R_{XX}[k-m]\cdot e^{-j\omega L(k-m)}\cdot \mathcal{rect}[\frac{k}{2N/L}]\cdot \mathcal{rect}[\frac{m}{2N/L}]\\ &= \sum_{i=-\infty}^{\infty}\sum_{m=-\infty}^{\infty}R_{XX}[i]\cdot e^{-jL\omega i}\cdot \mathcal{rect}[\frac{i+m}{2N/L}]\cdot \mathcal{rect}[\frac{m}{2N/L}]\\ &= \sum_{i=-\infty}^{\infty}R_{XX}[i]\cdot e^{-jL\omega i} \cdot \frac{2N}{L}\left[1 -\frac{|i|}{2N/L} \right] \end{align}\]

Then \[\begin{align} S_{X_eX_e}(\omega) &= \lim_{N\to \infty}\frac{1}{2N+1}\cdot \mathrm{E}\{|\sum_{n=-N}^N X_e[n]e^{-j\omega n}|^2\} \\ &= \frac{1}{L}\sum_{i=-\infty}^{\infty}R_{XX}[i]\cdot e^{-jL\omega i}\\ &= \frac{1}{L}S_{XX}(L\omega) \end{align}\]

Wiener Process (Brownian Motion)

Dennis Sun, Introduction to Probability: Lesson 49 Brownian Motion [https://dlsun.github.io/probability/brownian-motion.html]

Wiener process (also called Brownian motion)

NRZ PSD

Lecture 26 Autocorrelation Functions of Random Binary Processes [https://bpb-us-e1.wpmucdn.com/sites.gatech.edu/dist/a/578/files/2003/12/ECE3075A-26.pdf]

Lecture 32 Correlation Functions & Power Density Spectrum, Cross-spectral Density [https://bpb-us-e1.wpmucdn.com/sites.gatech.edu/dist/a/578/files/2003/12/ECE3075A-32.pdf]

Normal Distribution and Input-Referred Noise [https://a2d2ic.wordpress.com/2013/06/05/normal-distribution-and-input-referred-noise/]

reference

L.W. Couch, Digital and Analog Communication Systems, 8th Edition, 2013 [pdf]

Alan V Oppenheim, George C. Verghese, Signals, Systems and Inference, 1st edition [pdf]

R. Ziemer and W. Tranter, Principles of Communications, Seventh Edition, 2013 [pdf]

Stark H, Woods JW. Probability, Statistics, and Random Processes for Engineers, 4th ed. 2012 [pdf]

Kuchler, Ryan J. Theory of multirate statistical signal processing and applications. Monterey, California.: Naval Postgraduate School, 2005. [pdf]

Balu Santhanam. Fall 2020 ECE541 Probability Theory & Stochastic Process [https://ece-research.unm.edu/bsanthan/ece541/]

PSS and HB Overview

The steady-state response is the response that results after any transient effects have dissipated.

The large signal solution is the starting point for small-signal analyses, including periodic AC, periodic transfer function, periodic noise, periodic stability, and periodic scattering parameter analyses.

Designers refer periodic steady state analysis in time domain as "PSS" and corresponding frequency domain notation as "HB"

Harmonic Balance Analysis

The idea of harmonic balance is to find a set of port voltage waveforms (or, alternatively, the harmonic voltage components) that give the same currents in both the linear-network equations and nonlinear-network equations

that is, the currents satisfy Kirchoff's current law

Define an error function at each harmonic, \(f_k\), where \[ f_k = I_{\text{LIN}}(k\omega) + I_{\text{NL}}(k\omega) \] where \(k=0, 1, 2,...,K\)

Note that each \(f_k\) is implicitly a function of all voltage components \(V(k\omega)\)

Newton Solution of the Harmonic-Balance Equation

Iterative Process and Jacobian Formulation

The elements of the Jacobian are the derivatives \[ \frac{\partial F_{\text{n,k}}}{\partial _{V_\text{m,l}}} \] where \(n\) and \(m\) are the port indices \((1,N)\), and \(k\) and \(l\) are the harmonic indices \((0,...,K)\)

Number of Harmonics & Time Samples

Initial Estimate

One important property of Newton's method is that its speed and reliability of convergence depend strongly upon the initial estimate of the solution vector.

Conversion Matrix Analysis

Large-signal/small-signal analysis, or conversion matrix analysis, is useful for a large class of problems wherein a nonlinear device is driven, or "pumped" by a single large sinusoidal signal; another signal, much smaller, is applied; and we seek only the linear response to the small signal.

The most common application of this technique is in the design of mixers and in nonlinear noise analysis

1. First, analyzing the nonlinear device under large-signal excitation only, where the harmonic-balance method can be applied
2. Then, the nonlinear elements in the device's equivalent circuit are then linearized to create small-signal, linear, time-varying elements
3. Finally, a small-signal analysis is performed

Element Linearized

Below shows a nonlinear resistive element, which has the \(I/V\) relationship \(I=f(V)\). It is driven by a large-signal voltage

Assuming that \(V\) consists of the sum of a large-signal component \(V_0\) and a small-signal component \(v\), with Taylor series \[ f(V_0+v) = f(V_0)+\frac{d}{dV}f(V)|_{V=V_0}\cdot v+\frac{1}{2}\frac{d^2}{dV^2}f(V)|_{V=V_0}\cdot v^2+... \] The small-signal, incremental current is found by subtracting the large-signal component of the current \[ i(v)=I(V_0+v)-I(V_0) \] If \(v \ll V_0\), \(v^2\), \(v^3\),... are negligible. Then, \[ i(v) = \frac{d}{dV}f(V)|_{V=V_0}\cdot v \]

\(V_0\) need not be a DC quantity; it can be a time-varying large-signal voltage \(V_L(t)\) and that \(v=v(t)\), a function of time. Then \[ i(t)=g(t)v(t) \] where \(g(t)=\frac{d}{dV}f(V)|_{V=V_L(t)}\)

The time-varying conductance \(g(t)\), is the derivative of the element's \(I/V\) characteristic at the large-signal voltage

By an analogous derivation, one could have a current-controlled resistor with the \(V/I\) characteristic \(V = f_R(I)\) and obtain the small-signal \(v/i\) relation \[ v(t) = r(t)i(t) \] where \(r(t) = \frac{d}{dI}f_R(I)|_{I=I_L(t)}\)

A nonlinear element excited by two tones supports currents and voltages at mixing frequencies \(m\omega_1+n\omega_2\), where \(m\) and \(n\) are integers. If one of those tones, \(\omega_1\) has such a low level that it does not generate harmonics and the other is a large-signal sinusoid at \(\omega_p\), then the mixing frequencies are \(\omega=\pm\omega_1+n\omega_p\), which shown in below figure

A more compact representation of the mixing frequencies is \[ \omega_n=\omega_0+n\omega_p \] which includes only half of the mixing frequencies:

• the negative components of the lower sidebands (LSB)
• and the positive components of the upper sidebands (USB)

For real signal, positive- and negative-frequency components are complex conjugate pairs

Shooting Newton

TODO 📅

Nonlinearity & Linear Time-Varying Nature

Nonlinearity Nature

The nonlinearity causes the signal to be replicated at multiples of the carrier, an effect referred to as harmonic distortion, and adds a skirt to the signal that increases its bandwidth, an effect referred to as intermodulation distortion

It is possible to eliminate the effect of harmonic distortion with a bandpass filter, however the frequency of the intermodulation distortion products overlaps the frequency of the desired signal, and so cannot be completely removed with filtering.

Time-Varying Linear Nature

linear with respect to \(v_{in}\) and time-varying

Given \(v_{in}(t)=m(t)\cos (\omega_c t)\) and LO signal of \(\cos(\omega_{LO} t)\), then \[ v_{out}(t) = \text{LPF}\{m(t)\cos(\omega_c t)\cdot \cos(\omega_{LO} t)\} \] and \[ v_{out}(t) = m(t)\cos((\omega_c - \omega_{LO})t) \]

A linear periodically-varying transfer function implements frequency translation

Periodic small signal analyses

Analysis in Simulator

1. LPV analyses start by performing a periodic analysis to compute the periodic operating point with only the large clock signal applied (the LO, the clock, the carrier, etc.).
2. The circuit is then linearized about this time-varying operating point (expand about the periodic equilibrium point with a Taylor series and discard all but the first-order term)
3. and the small information signal is applied. The response is calculated using linear time-varying analysis

Versions of this type of small-signal analysis exists for both harmonic balance and shooting methods

PAC is useful for predicting the output sidebands produced by a particular input signal

PXF is best at predicting the input images for a particular output

Linear Time Varying

The response of a relaxed LTV system at a time \(t\) due to an impulse applied at a time \(t − \tau\) is denoted by \(h(t, \tau)\)

The first argument in the impulse response denotes the time of observation.

The second argument indicates that the system was excited by an impulse launched at a time \(\tau\) prior to the time of observation.

Thus, the response of an LTV system not only depends on how long before the observation time it was excited by the impulse but also on the observation instant.

The output \(y(t)\) of an initially relaxed LTV system with impulse response \(h(t, \tau)\) is given by the convolution integral \[ y(t) = \int_0^{\infty}h(t,\tau)x(t-\tau)d\tau \] Assuming \(x(t) = e^{j2\pi f t}\) \[ y(t) = \int_0^{\infty}h(t,\tau)e^{j2\pi f (t-\tau)}d\tau = e^{j2\pi f t}\int_0^{\infty}h(t,\tau)e^{-j2\pi f\tau}d\tau \] The (time-varying) frequency response can be interpreted as \[ H(j2\pi f, t) = \int_0^{\infty}h(t,\tau)e^{-j2\pi f\tau}d\tau \] Linear Periodically Time-Varying (LPTV) Systems, which is a special case of an LTV system whose impulse response satisfies \[ h(t, \tau) = h(t+T_s, \tau) \] In other words, the response to an impulse remains unchanged if the time at which the output is observed (\(t\)) and the time at which the impulse is applied (denoted by \(t_1\)) are both shifted by \(T_s\) \[ H(j2\pi f, t+T_s) = \int_0^{\infty}h(t+T_s,\tau)e^{-j2\pi f\tau}d\tau = \int_0^{\infty}h(t,\tau)e^{-j2\pi f\tau}d\tau = H(j2\pi f, t) \] \(H(j2\pi f, t)\) of an LPTV system is periodic with timeperiod \(T_s\), it can be expanded as a Fourier series in \(t\), resulting in \[ H(j2\pi f, t) = \sum_{k=-\infty}^{\infty} H_k(j2\pi f)e^{j2\pi f_s k t} \] The coefficients of the Fourier series \(H_k(j2\pi f)\) are given by \[ H_k(j2\pi f) = \frac{1}{T_s}\int_0^{T_s} H(j2\pi f, t) e^{-j2\pi k f_s t}dt \]

reference

K. S. Kundert, "Introduction to RF simulation and its application," in IEEE Journal of Solid-State Circuits, vol. 34, no. 9, pp. 1298-1319, Sept. 1999, doi: 10.1109/4.782091. [pdf]

Stephen Maas, Nonlinear Microwave and RF Circuits, Second Edition , Artech, 2003. [pdf]

Karti Mayaram. ECE 521 Fall 2016 Analog Circuit Simulation: Simulation of Radio Frequency Integrated Circuits [pdf1, pdf2]

The Value Of RF Harmonic Balance Analyses For Analog Verification: Frequency domain periodic large and small signal analyses. [https://semiengineering.com/the-value-of-rf-harmonic-balance-analyses-for-analog-verification/]

Shanthi Pavan, "Demystifying Linear Time Varying Circuits"

S. Pavan and G. C. Temes, "Reciprocity and Inter-Reciprocity: A Tutorial— Part I: Linear Time-Invariant Networks," in IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 70, no. 9, pp. 3413-3421, Sept. 2023, doi: 10.1109/TCSI.2023.3276700.

S. Pavan and G. C. Temes, "Reciprocity and Inter-Reciprocity: A Tutorial—Part II: Linear Periodically Time-Varying Networks," in IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 70, no. 9, pp. 3422-3435, Sept. 2023, doi: 10.1109/TCSI.2023.3294298.

S. Pavan and R. S. Rajan, "Interreciprocity in Linear Periodically Time-Varying Networks With Sampled Outputs," in IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 61, no. 9, pp. 686-690, Sept. 2014, doi: 10.1109/TCSII.2014.2335393.

Prof. Shanthi Pavan. Introduction to Time - Varying Electrical Network. [https://youtube.com/playlist?list=PLyqSpQzTE6M8qllAtp9TTODxNfaoxRLp9]

Shanthi Pavan. EE5323: Advanced Electrical Networks (Jan-May. 2015) [https://www.ee.iitm.ac.in/vlsi/courses/ee5323/start]

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

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Wereley, Norman. (1990). Analysis and control of linear periodically time varying systems.

Hameed, S. (2017). Design and Analysis of Programmable Receiver Front-Ends Based on LPTV Circuits. UCLA. ProQuest ID: Hameed_ucla_0031D_15577. Merritt ID: ark:/13030/m5gb6zcz. Retrieved from https://escholarship.org/uc/item/51q2m7bx

Matt Allen. Introduction to Linear Time Periodic Systems. [https://youtu.be/OCOkEFDQKTI]

Fivel, Oren. "Analysis of Linear Time-Varying & Periodic Systems." arXiv preprint arXiv:2202.00498 (2022).

RF Harmonic Balance Analysis for Nonlinear Circuits [https://resources.pcb.cadence.com/blog/2019-rf-harmonic-balance-analysis-for-nonlinear-circuits]

Steer, Michael. Microwave and RF Design (Third Edition, 2019). NC State University, 2019.

Steer, Michael. Harmonic Balance Analysis of Nonlinear RF Circuits - Case Study Index: CS_AmpHB [link]

Vishal Saxena, "SpectreRF Periodic Analysis" URL:https://www.eecis.udel.edu/~vsaxena/courses/ece614/Handouts/SpectreRF%20Periodic%20Analysis.pdf

Josh Carnes,Peter Kurahashi. "Periodic Analyses of Sampled Systems" URL:https://slideplayer.com/slide/14865977/

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

Article (20482538) Title: Why is pnoise sampled(jitter) different than pnoise timeaverage on a driven circuit?

Article (20467468) Title: The mathematics behind choosing the upper frequency when simulating pnoise jitter on an oscillator

Andrew Beckett. "Simulating phase noise for PFD-CP". URL:https://groups.google.com/g/comp.cad.cadence/c/NPisXTElx6E/m/XjWxKbbfh2cJ

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Jeffrey Wong. Math 353, Fall 2020 Ordinary and Partial Differential Equations [https://services.math.duke.edu/~jtwong/math353-2020/main.html]

Tip of the Week: Please explain in more practical (less theoretical) terms the concept of "oscillator line width." [https://community.cadence.com/cadence_blogs_8/b/rf/posts/please-explain-in-more-practical-less-theoretical-terms-the-concept-of-quot-oscillator-line-width-quot]

Rubiola, E. (2008). Phase Noise and Frequency Stability in Oscillators (The Cambridge RF and Microwave Engineering Series). Cambridge: Cambridge University Press. doi:10.1017/CBO9780511812798

Dr. Ulrich L. Rohde, Noise Analysis, Then and Today [https://www.microwavejournal.com/articles/29151-noise-analysis-then-and-today]

Nicola Da Dalt and Ali Sheikholeslami. 2018. Understanding Jitter and Phase Noise: A Circuits and Systems Perspective (1st. ed.). Cambridge University Press, USA.

Kester, Walt. (2005). Converting Oscillator Phase Noise to Time Jitter. [https://www.analog.com/media/en/training-seminars/tutorials/MT-008.pdf]

Drakhlis, B.. (2001). Calculate oscillator jitter by using phase-noise analysis: Part 2 of two parts. Microwaves and Rf. 40. 109-119.

Explanation for sampled PXF analysis. [https://community.cadence.com/cadence_technology_forums/f/custom-ic-design/45055/explanation-for-sampled-pxf-analysis/1376140#1376140]

模拟放大器的低噪声设计技术2-3实践 아날로그 증폭기의 저잡음 설계 기법2-3실습 [https://youtu.be/vXLDfEWR31k?si=p2gbwaKoTxtfvrTc]

2.2 How POP Really Works [https://www.simplistechnologies.com/documentation/simplis/ast_02/topics/2_2_how_pop_really_works.htm]

Jaeha Kim. Lecture 11. Mismatch Simulation using PNOISE [https://ocw.snu.ac.kr/sites/default/files/NOTE/7037.pdf]

Hueber, G., & Staszewski, R. B. (Eds.) (2010). Multi-Mode/Multi-Band RF Transceivers for Wireless Communications: Advanced Techniques, Architectures, and Trends. John Wiley & Sons. https://doi.org/10.1002/9780470634455

B. Boser,A. Niknejad ,S.Gambin 2011 EECS 240 Topic 6: Noise Analysis [https://mixsignal.files.wordpress.com/2013/06/t06noiseanalysis_simone-1.pdf]

Phase Noise on \(log\) scale

How to Identify the Source of Phase Jitter through Phase Noise Plots [https://www.sitime.com/company/newsroom/blog/how-identify-source-phase-jitter-through-phase-noise-plots]

AN10072 Determine the Dominant Source of Phase Noise, by Inspection [https://www.sitime.com/support/resource-library/application-notes/an10072-determine-dominant-source-phase-noise-inspection]

4-minute Clinic: Determine the Dominant Source of Jitter by Inspection of Phase Noise Plot [https://youtu.be/2elHk3v45Pk]

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

a -10 dB/decade reference line can be used to pinpoint the location in a phase noise curve that dominates its integral

Reference-Clock Phase Noise in PLL

Gary Giust. How to Evaluate Reference-Clock Phase Noise in High-Speed Serial Links [expanded version], [compact version]

G. Richmond, "Refclk Fanout Best Practices for 8GT/s and 16GT/s Systems," PCI-SIG Developers Conference, June 7, 2017

Knowing how input phase noise aliases when sampled by a PLL

An alternate view of phase noise aliasing during the sampling process

• Instead of mirroring the jitter-transfer function located below \(F_S/2\) across spectral boundaries located at integer multiples of \(F_S/2\) (i.e. 50 MHz) as shown in Figure 2 (a)
• we could alternatively fold the portion of the Raw Data curve located above \(F_S/2\) across these spectrum boundaries to appear below \(F_S/2\) as shown in Figure 2 (b)

Integrating the combined area under each Filtered Data curve shown in Figure 2 (b) is mathematically equivalent to integrating the entire Filtered Data curve shown in Figure 2 (a)

Phase Noise Analyzer vs TIE jitter using Real-time Oscilloscope

Since an oscilloscope observes jitter similar to a real system, we regard its result as the gold standard against which other methods may be judged

Flat Phase Noise Extension to twice the clock frequency

Phase Noise Aliasing & Integration Limits

These two types of measurements deliver the same rms jitter of \(f_{CK}\)

• both rising and falling: integrated from \(-f_{CK}\) to \(+f_{CK}\)
• only the rising (or falling) edges: integrated from \(-f_{CK}/2\) to \(+f_{CK}/2\)

temporal autocorrelation and Wiener-Khinchin theorem is more appropriate to arise rms value

Y. Zhao and B. Razavi, "Phase Noise Integration Limits for Jitter Calculation,"[https://www.seas.ucla.edu/brweb/papers/Conferences/YZ_ISCAS_22.pdf]

G. Giust, "Phase Noise Aliases as TIE Jitter," Signal Integrity Journal, July 23, 2018 [https://www.signalintegrityjournal.com/articles/912-phase-noise-aliases-as-tie-jitter]

Jitter and Edge phase noise

[https://community.cadence.com/cadence_technology_forums/f/custom-ic-design/56929/how-to-derive-edge-phase-noise-from-output-noise-in-sampled-pnoise-simulation/1388888]

Shawn Logan, Summary of Study of Cadence Sampled Phase Noise and Jitter Definitions with a Comparison to Conventional Time Interval Error (TIE) for a Driven Circuit [www.dropbox.com/s/3m531dl4fl7bwbr/jee_computation_example_sml_032823v1p0.pdf]

timeaverage noise (phase-noise) & sampled noise (edge-phase noise or jitter) spectrum

Time-average noise analysis

\(S_{\Delta f}(f)\) between \([\Delta f, F_0/2]\) may be less than that of other harmonic window

Sampled noise analysis

correlation

VCO Phase Noise

pnoise - timeaverage

1. Direct Plot/Pnoise/Phase Noise or

   

2. manually calculate by definition

   

3. output noise with unit dBc

   Direct Plot/Pnoise/Output Noise Units:dBc/Hz and Noise convention: SSB

   

The above method 2 and 3 only apply to timeaveage pnoise simulation,

pnoise - sampled(jitter)/Edge Crossing

Direct Plot/Pnoise/Edge Phase Noise or

Another way, the following equation can also be used for sampled(jitter)/Edge Crossing

1

PhaseNoise(dBc/Hz) = dB20( OutputNoise(V/sqrt(Hz)) / slopeCrossing / Tper*twoPi ) - dB10(2)

where dB10(2) is used to obtain SSB from DSB

Output Noise of sampled(jitter) pnoise

The last section's Output Noise (V**2/Hz) can be obtained by transient noise simulation

The idea is that sample waveform with ideal clock, subtract DC offset, then fft(psd)

• samplesRaw = sample(wv)
• samplePost = samplesRaw - average(samplesRaw)
• Output Noise (V**2/Hz) = psd(samplePost)

Expression:

The computation cost is typically very high, and the accuracy is lesser as compared to PSS/Pnoise

Pnoise Sampled(jitter): Sampled Phase Option

• Identical to noisetype=timedomain in old GUI
• Use model:
  • Sampleds Per Period: number of ponits
  • Add Specific Points: specific time point, still time points

pss beat freq = 5GHz

pnoise sweeptype: absolute, from 100k to 2.5GHz

Transient noise

phase noise from transient noise analysis

1. The Phase Noise function is now available in the Direct Plot form (Results-Direct Plot-Main Form) after Transient Analysis is run
   • Absolute jitter Method
   • Direct Power Spectral Density Method
2. PN phase noise function
   • Absolute jitter Method
   • Direct Power Spectral Density Method

Absolute jitter Method: Phase noise is defined as the power spectral density of the absolute jitter of an input waveform

and absolute jitter method is the default method

In below discussion, we only think about the absolute jitter method

PSD & Phase Noise

• phase noise is single-sideband
• psd is double-sideband
• Then the ratio is 2

By PSS_Pnoise

jee

1

rfEdgePhaseNoise(?result "pnoise_sample_pm0" ?eventList 'nil) + 10 * log10(2)

convert single-sideband phase noise to psd by multiplying 2 or 10 * log10(2)

By trannoise PN function

1

PN(clip(VT("/Out1") 2.60417e-08 0.000400052) "rising" 1.65 ?Tnom (1 / 3.84e+07) ?windowName "Rectangular" ?smooth 1 ?windowSize 15000 ?detrending "None" ?cohGain 1 ?methodType "absJitter")

double-sideband psd

By trannoise psd and abs_jitter function

1

dB10(psd(abs_jitter(clip(VT("/Out1") 2.60417e-08 0.000400052) "rising" 1.65 ?Tnom (1 / 3.84e+07)) 2.60417e-08 0.000400052 15360 ?windowName "Rectangular" ?smooth 1 ?windowSize 15000 ?detrending "None" ?cohGain 1))

double-sideband psd

abs_jitter Y-Unit default is rad

Comparison

PN's result is same with psd's

RMS value

• build the abs_jitter function with seconds as the Y axis and add the stddev function to determine the Jee jitter value
• or integrate psd

The RMS \(x_{\text{RMS}}\) of a discrete domain signal \(x(n)\) is given by \[ x_{\text{RMS}}=\sqrt{\frac{1}{N}\sum_{n=0}^{N-1}|x(n)|^2} \] Inserting Parseval's theorem given by \[ \sum_{n=0}^{N-1}|x(n)|^2=\frac{1}{N}\sum_{n=0}^{N-1}|X(k)|^2 \] allows for computing the RMS from the spectrum \(X(k)\) as \[ x_{\text{RMS}}=\sqrt{\frac{1}{N^2}\sum_{n=0}^{N-1}|X(k)|^2} \]

Cadence Spectre's PN function may call abs_jitter and psd function under the hood.

Phase Noise in vsource

Suppose pnoise result of one block is shown as below, and the result is stimulus of following block

First export Output Noise and Edge Phase Noise， then select noiseModelType and noisefile respectively

Under vsource (Source type: pulse) with different amplitude & rising/falling time, simulation result demonstrate that Edge Phase Noise(dBc) maintain jitter or phase noise by tweaking voltage noise at edge under the hoods, however Noise Voltage(V^2/Hz) maintain voltage noise

In the conclusion, Edge Phase Noise(dBc) is preferred for phase noise evaluation

notice:

@(#)$CDS: spectre version 21.1.0 64bit 12/01/2023 07:24 (csvcm36c-1) $

@(#)$CDS: virtuoso version ICADVM20.1-64b 10/11/2023 09:26 (cpgbld01) $

SSB Phase Noise (dBc)

Divider PN simulation

Cadence Support. "How to set up pss/pnoise when simulating a driven circuit or a VCO, both containing dividers"

Modeling Oscillators with Arbitrary Phase Noise Profiles

TODO 📅

reference

Article (11514536) Title: How to obtain a phase noise plot from a transient noise analysis

Article (20500632) Title: How to simulate Random and Deterministic Jitters

Cadence, Application Note: Understanding the relations between time-average noise (phase-noise) and sampled noise (edge-phase noise or jitter) in Pnoise analysis

Tutorial on Scaling of the Discrete Fourier Transform and the Implied Physical Units of the Spectra of Time-Discrete Signals Jens Ahrens, Carl Andersson, Patrik Höstmad, Wolfgang Kropp URL: https://appliedacousticschalmers.github.io/scaling-of-the-dft/AES2020_eBrief/

Tawna, "Modeling Oscillators with Arbitrary Phase Noise Profiles"[https://community.cadence.com/cadence_blogs_8/b/rf/posts/modeling-oscillators-with-arbitrary-phase-noise-profiles]

—, "How to Specify Phase Noise as an Instance Parameter in Spectre Sources (e.g. vsource, isource, Port)" [https://community.cadence.com/cadence_blogs_8/b/rf/posts/how-to-specify-phase-noise-as-an-instance-parameter-in-spectre-sources-e-g-vsource-isource-port]

Load-Transient Response

TODO 📅

bleeding current

A bleeding resistor (or a dedicated bleeding current source) is typically used in a Low Dropout (LDO) voltage regulator or a power supply circuit to ensure stability and proper operation under light load or no-load conditions [Google AI Mode]

resistor: affect both DC and AC (small signal)

current source: affect DC bias only (assuming infinite output impedance of current source)

Kelvin connection

Kelvin connection in IC design [https://analoghub.ie/category/Circuits/article/kelvinConnection]

The entire idea behind Kelvin connection is to separate the nodes that are carrying high currents from the sensing nodes to the feedback

Error Amplifier

Using type B amplifier to drive the NMOS power stage will enhance the NMOS’s PSRR performance

PSRR (Power Supply Rejection Ratio)

A good PSRR is important when an LDO is used as a sub-regulator in cascade with a switching regulator

The LDO would need to have a sufficiently high rejection at the switching frequency of the switching converter to filter out the ripples at that frequency

Mid-Frequency PSRR

High frequency PSRR

Open-Loop PSRR

Chen, Feng & Lu, Yasu & Mok, Philip. (2022). Transfer Function Analysis of the Power Supply Rejection Ratio of Low-Dropout Regulators and the Feed-Forward Ripple Cancellation Scheme. IEEE Transactions on Circuits and Systems I: Regular Papers. [https://sci-hub.se/10.1109/TCSI.2022.3167860]

neglect the contribution of the voltage regulation circuits to the PSRR

HW #3 - "Precision Low-Dropout Regulators" Online Course (2025) - Prof. Yan Lu (Tsinghua University) [https://youtu.be/LXX1Xuhv2kI]

DC output impedance

The output impedance of the LDO at DC is known as its load regulation

DC output impedance of NMOS and PMOS LDO is the same for the same error amplifier gain

\[ R_{\text{out}} = \frac{1}{\beta A_{o1} g_{m2}} \]

The NMOS LDO has a faster response to line transients than the PMOS LDO since it has better (smaller) PSRR

Power MOS gain affect on PMOS LDO

DC gain \[ A_{dc} = g_mR_\text{ota} A_2 \] 3dB bandwidth \[ \omega_p = \frac{1}{R_\text{ota}(C_g+A_2C_c)} \] and GBW \[ \omega_u = \frac{g_m}{\frac{C_g}{A_2}+C_c} \]

Feedforward Compensation with \(C_\text{FF}\)

• Improved Noise
• Improved Stability and Transient Response
• Improved PSRR

\[\begin{align} R_1 \parallel \frac{1}{sC_{FF}} &= \frac{R_1}{1+sR_1C_{FF}} \\ Z_o &= \left( R_1\parallel \frac{1}{sC_{FF}}+R_2\right)\parallel \frac{1}{sC_L} \\ &=\frac{R_1+R_2+sR_1R_2C_{FF}}{s^2R_1R_2C_{FF}C_L + s[(R_1+R_2)C_L+R_1C_{FF}]+1} \\ A_{V2} &= g_m Z_o \\ &= g_m \frac{R_1+R_2+sR_1R_2C_{FF}}{s^2R_1R_2C_{FF}C_L + s[(R_1+R_2)C_L+R_1C_{FF}]+1} \\ \beta &= \frac{R_2}{\frac{R_1}{1+sR_1C_{FF}}+R_2} \\ &= \frac{R_2(1+sR_1C_{FF})}{R_1+R_2+sR_1R_2C_{FF}} \\ A_{V2}\beta &= \frac{g_mR_2(1+sR_1C_{FF})}{s^2R_1R_2C_{FF}C_L+s[(R_1+R_2)C_L+R_1C_{FF}]+1} \end{align}\]

That is, adding a \(C_{FF}\) also introduces a zero (\(\omega_z\)) and pole (\(\omega_p\)) into the LDO feedback loop

\[\begin{align} \omega_{po} &= \frac{1}{(R_1+R_2)C_L} \\ \omega_z &= \frac{1}{R_1C_{FF}} \\ \omega_{p} &= \frac{1}{(R_1 \parallel R_2)C_{FF}} \end{align}\]

Application Report SBVA042–July 2014, Pros and Cons of Using a Feedforward Capacitor with a Low-Dropout Regulator [https://www.ti.com/lit/an/sbva042/sbva042.pdf]

LDO Basics: Noise – How a Feed-forward Capacitor Improves System Performance [https://www.ti.com/document-viewer/lit/html/SSZTA13]

LDO Basics: Noise – How a Noise-reduction Pin Improves System Performance [https://www.ti.com/document-viewer/lit/html/SSZTA40]

NMOS Slave LDO

\[\begin{align} \frac{V_g}{V_i} &=\frac{R||\frac{1}{s(C_g+C_{gs})}}{R||\frac{1}{s(C_g+C_{gs})}+\frac{1}{sC_{gd}}} \\ &= \frac{sRC_{gd}}{sR(C+C_{gd}+C_{gs})+1} \\ \frac{V_g}{V_i} &= -\frac{sC_{ds}+g_{ds}}{sC_{gs}+g_m} \end{align}\]

That is, \[ \omega_{z,d} \approx \frac{1}{R(\frac{g_m}{g_{ds}}C_{gd}+C)} \]

To calculate PSRR pole is similar with above PSRR zero, though \(V_o/V_i=0\), i.e. set \(V_0\) 0 potential \[ \omega_{p,d} \approx \frac{1}{R(\frac{C_s}{g_mR}+C)} \]

DC PSRR \[ \text{PSRR} = \frac{1}{g_mr_o} \]

PSRR @Vgate

KCL at output node

\[ g_m(-V_o\beta A_{E} - V_o) + \frac{V_i - V_o}{r_o} = \frac{V_o}{R_1+R_2} \]

Hence \[ \frac{V_o}{V_i} = \frac{1}{A_E\beta g_mr_o+g_mr_o +\frac{r_o}{R_1+R_2}+1} \approx \frac{1}{A_E\beta g_m r_o} \]

Through feedback loop, we derive \[ V_g = V_o \beta (-A_E) \approx \frac{V_i}{A_E\beta g_m r_o} \beta (-A_E) = -\frac{V_i}{g_mr_o} \]

That is \[ \frac{V_g}{V_i} \approx -\frac{1}{g_mr_o} \]

Due to closed loop, \(V_g\) and \(V_o\) is not source follower

feedback resistor divider noise

assuming \(\text{LG} \gg 1\)

\[\begin{align} I_\text{t} &= \frac{V_\text{ref} - v_\text{n2}}{R_\text{2}} \\ V_\text{o} &= V_\text{ref} +v_\text{n1} + I_\text{t}R_\text{1} \\ \end{align}\]

Then \[ V_\text{o} = \frac{R_1+R_2}{R_2}V_\text{ref} + v_\text{n1} - \frac{R_1}{R_2}v_\text{n2} \] that is

\[ v_\text{no}^2 = v_\text{n1}^2 + \left(\frac{R_1}{R_2}\right)^2 v_\text{n2}^2 \]

\[ \text{vno1}^2= \text{vn1}^2+\text{vn2}^2/6^2=16.5758 + 99.45453/6^2 = 19.338425833 \]

reference

Hinojo, J.M., Martinez, C.I., & Torralba, A.J. (2018). Internally Compensated LDO Regulators for Modern System-on-Chip Design.

Chen, K.-H. (2016). Power Management Techniques for Integrated Circuit Design. Wiley-IEEE Press.

Morita, B.G. (2014). Understand Low-Dropout Regulator ( LDO ) Concepts to Achieve Optimal Designs.

H. -S. Kim, "Exploring Ways to Minimize Dropout Voltage for Energy-Efficient Low-Dropout Regulators: Viable approaches that preserve performance," in IEEE Solid-State Circuits Magazine, vol. 15, no. 2, pp. 59-68, Spring 2023, doi: 10.1109/MSSC.2023.3262767.

Ali Sheikholeslami, Circuit Intuitions: Voltage Regulators IEEE Solid-State Circuits Magazine, Vol. 12, Issue 4, to appear, Fall 2020.

Operational Transconductance Amplifier II Multi-Stage Designs [https://people.eecs.berkeley.edu/~boser/courses/240B/lectures/M07%20OTA%20II.pdf]

Toshiba, Load Transient Response of LDO and Methods to Improve it Application Note [https://toshiba.semicon-storage.com/info/application_note_en_20210326_AKX00312.pdf?did=66268]

Carusone, Tony Chan, David Johns, and Kenneth Martin. Analog integrated circuit design. John wiley & sons, 2011. [https://mrce.in/ebooks/Analog%20Integrated%20Circuit%20Design%202nd%20Ed.pdf]

Pavan Kumar Hanumolu. CICC 2015. "Low Dropout Regulators" [https://uofi.app.box.com/v/CICC15-LDO]

Mingoo Seok. ISSCC 2020 T7: "Basics of Digital Low-Dropout (LDO) Integrated Voltage Regulator" [https://www.nishanchettri.com/isscc-slides/2020%20ISSCC/TUTORIALS/T7Visuals.pdf]

Yan Lu, ISSCC2021 T10: "Fundamentals of Fully Integrated Voltage Regulators" [https://www.nishanchettri.com/isscc-slides/2021%20ISSCC/TUTORIALS/ISSCC2021-T10.pdf]

—, (Tsinghua U.) Preview - “Precision Low-Dropout Regulators” Online Course (2025) [https://youtu.be/IgWTou7Ikbs]

Mao, Xiangyu, Yan Lu, and Rui P. Martins. Fully-Integrated Low-Dropout Regulators. Springer, 2025.

Hyun-Sik Kim, Low-Dropout (LDO) Voltage Regulators – From Basics to Recent Design Trends (presented in A-SSCC 2022) [pdf]

A. Raychowdhury. ISSCC 2024 T2: Fundamentals of Digital and Digitally-Assisted Linear Voltage Regulators

Mixed-Mode S-parameter

12 May 2021 Introduction to Mixed-Mode S-parameters [https://blog.teledynelecroy.com/2021/05/introduction-to-mixed-mode-s-parameters.html]

Troy Beukema (IBM Research, Yorktown Heights, NY). 03-Sep-2009. Topics in Design and Analysis of High Data Rate SERDES Systems [https://ewh.ieee.org/r5/denver/sscs/Presentations/2009_09_Beukema.pdf]

Bert Simonovich. A Guide for Single-Ended to Mixed-Mode S-parameter Conversions [https://www.signalintegrityjournal.com/articles/1832-a-guide-for-singleended-to-mixedmode-s-parameter-conversions]

single-ended S-parameters

Mixed-mode S-parameters

Missing Term in KVL

Prof. Kolb/Whites. EE 382 Applied Electromagnetics Lecture 8: Maxwell's Equations and Electrical CIrcuits [http://montoya.sdsmt.edu/ee382/lectures/382Lecture8.pdf]

Transmission-line

Telegrapher’s equations

EECS 723- Microwave Engineering Spring 2.1 -The Lumped Element Circuit Model for Transmission Lines

1/20/2005 [https://www.ittc.ku.edu/~jstiles/723/handouts/2_1_Lumped_Element_Circuit_Model_package.pdf]

note [https://www.ittc.ku.edu/~jstiles/723/handouts/section_2_1_The_Lumped_Element_Circuit_Model_package.pdf]

present [https://www.ittc.ku.edu/~jstiles/723/handouts/section_2_1_The_Lumped_Element_Circuit_Model_present.pdf]

Transmission Line Wave Equation

Characteristic Impedance (\(Z_0\))

Remember, S-parameters don't mean much unless you know the value of the reference impedance (it's frequently called Z0).

simulator will read sp file's Z0 parameter

The default Z0 exported by EMX is 50

Complex Propagation Constant \(\gamma\)

TODO 📅

Input impedance (Line Impedance)

Reflection Coefficient

TODO 📅

Steady-State Solution (DC voltage division)

Sam Palermo. [https://people.engr.tamu.edu/spalermo/ecen689/lecture3_ee689_tlines.pdf]

Kyoung-Jae Chung. Special Topics in Radiation Engineering (High-voltage pulsed power engineering) [https://ocw.snu.ac.kr/sites/default/files/NOTE/Lecture_03_Transmission%20line%20theory.pdf]

David R. Jackson. [https://courses.egr.uh.edu/ECE/ECE3317/SectionJackson/Class%20Notes/Notes%208%203317%20Transmission%20Lines%20(Bounce%20Diagram).pdf]

Shouri Chatterjee [https://web.iitd.ac.in/~shouri/ell112/material/txline.pdf]

How can I go from transmission line model to lumped elements model? [https://physics.stackexchange.com/a/386603]

E157 Introduction to Radio Frequency Circuit Design [https://pages.hmc.edu/mspencer/e157/fa24/]

Shen Lin. On-Chip Inductance and Coupling Effects [http://eda.ee.ucla.edu/pub/asic.pdf]

A. Deutsch et al., "When are transmission-line effects important for on-chip interconnections?," in IEEE Transactions on Microwave Theory and Techniques, vol. 45, no. 10, pp. 1836-1846, Oct. 1997

Ho, Ron. “Chip Wires: Scaling and Efficiency.” (2003). [https://www-vlsi.stanford.edu/people/alum/pdf/0303_Ho_Wires.pdf]

—. ISSCC 2007 T3: Dealing with Issues in VLSI Interconnect Scaling, by Ron Ho

Tony Chan Carusone. ISSCC 2017 T6: Signal Integrity Analysis for Gb/s Links

Byungsub Kim ISSCC 2022 T11: "Basics of Equalization Techniques: Channels, Equalization, and Circuits"

Voltage scattering

transmitted voltage \[ V= \frac{2Z_l}{Z_l+R_0}\frac{V_s}{2}= \frac{Z_l}{Z_l+R_0}\cdot V_s \]

CHAPTER 6 Transmission-Line Essentials for Digital Electronics [https://ws.engr.illinois.edu/sitemanager/getfile.asp?id=178]

CHAPTER 7 Transmission-Line Analysis [https://ws.engr.illinois.edu/sitemanager/getfile.asp?id=199]

Voltage Transfer Function

Troy Beukema (IBM Research, Yorktown Heights, NY). 03-Sep-2009. Topics in Design and Analysis of High Data Rate SERDES Systems [https://ewh.ieee.org/r5/denver/sscs/Presentations/2009_09_Beukema.pdf]

Impulse Response from S-Parameters (channel)

David Banas. A comparison of different techniques (i.e. - windowing, vector fitting, etc.) for extracting the impulse response from S-parameters. [https://github.com/capn-freako/ImpulseResponseFromSparameters/tree/main]

Sam Palermo. ECEN720: High-Speed Links Circuits and Systems Spring 2025 - Lecture 3: Time-Domain Reflectometry & S-Parameter Channel Models [https://people.engr.tamu.edu/spalermo/ecen689/lecture3_ee720_tdr_spar.pdf]

Troy Beukema (IBM Research, Yorktown Heights, NY). 03-Sep-2009. Topics in Design and Analysis of High Data Rate SERDES Systems [https://ewh.ieee.org/r5/denver/sscs/Presentations/2009_09_Beukema.pdf]

Causality

causality-passivity correction

P. Triverio, S. Grivet-Talocia, M. S. Nakhla, F. G. Canavero and R. Achar, "Stability, Causality, and Passivity in Electrical Interconnect Models," in IEEE Transactions on Advanced Packaging, vol. 30, no. 4, pp. 795-808, Nov. 2007 [https://sci-hub.ru/10.1109/TADVP.2007.901567]

S. Sercu, C. Kocuba, J. Nadolny, "Causality Demystified", in DesignCon 2015, Jan. 2015 [pdf]

Vinod Arjun Huddar. Causality Problems in Power Delivery Networks [https://www.signalintegrityjournal.com/articles/1217-causality-in-power-delivery-network-in-package-board]

Tyler Huddleston, Signal Edge Solutions. Causality in Practice: How Frequency Sampling and Bandwidth Shape Time-Domain Fidelity [https://www.signalintegrityjournal.com/articles/4061-causality-in-practice-how-frequency-sampling-and-bandwidth-shape-time-domain-fidelity]

\(H_e(f) = \mathcal{Re}\{H(f)\}\), \(H_o(f) = j\cdot\mathcal{Im}\{H(f)\}\) and \(\enclose{circle}{1}\) , \(\enclose{circle}{2}\)

Rational Fit

Use the rational function to fit data defined in the frequency domain with an equivalent Laplace transfer function. Using rational function fitting you can create simple models for a required accuracy, model order reduction, zero phase on extrapolation to DC, and causal modeling system among other advantages

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filename  = 'touchstone/ISI.S4P';
s4p = read(rfdata.data, filename);
sdd_params = s2sdd(s2p.S_Parameters, 2);
sdd21 = squeeze(sdd_params(2, 1, :));   % s21
freq = s4p.Freq;

% rational fitting
weight = ones(size(sdd21));
weight(floor(end*3/4):end) = 0.2;
weight(2:10) = 0;

[hfit, errb] = rationalfit(freq, sdd21, 'IterationLimit', [4, 16], 'Delayfactor', 0.98, ...
    'Weight', weight, 'Tolerance', -38, 'NPoles', 32);
[sdd21_fit, ff] = freqresp(hfit, freq);

figure(1)
plot(freq/1e9, db(sdd21), 'b-'); hold on;
plot(ff/1e9, db(sdd21_fit), 'r-'); hold off; grid on;
legend('sdd21', 'sdd21\_fit');
xlabel('Freq (GHz)');
ylabel('magnitude (dB)');


ts = 1e-12;
n = 2^18;
trise = 4e-14;
[yout, tout] = stepresp(hfit, ts, n, trise);
figure(2)
plot(tout*1e12, yout, 'b-'); grid on;
xlabel('Time (ps)');
ylabel('V');
title('Step Response');


% write verilog-A
writeva(hfit, 'channel_32poles.va');

Using S Parameters to Estimate Q

TODO 📅

Jeff Walling. ECE 5984 Using S Parameters to Estimate Q [https://youtu.be/PXgM6pGIRvk?si=YDeh-COQEBXKUiw-]

Reading S-parameters

teledynelecroy. Reading S-parameters [https://blog.teledynelecroy.com/2020/05/]

keysight. How to Interpret Ripple in an S Parameters Measurement [https://docs.keysight.com/kkbopen/how-to-interpret-ripple-in-an-s-parameters-measurement-849642201.html]

You Measured What? Four Must-Know Checks Before Trusting Your Trace S-Parameters [https://www.signalintegrityjournal.com/articles/4083-you-measured-what-four-must-know-checks-before-trusting-your-trace-s-parameters]

TODO 📅

Ripple in an S Parameters

S11 vs TDR

[https://ewh.ieee.org/r5/denver/sscs/Presentations/2004_04_Quach.pdf]

TODO 📅

Spar in Tran simulation

Spar in AC simulation

reference

microwaves101, S-parameters (https://www.microwaves101.com/encyclopedias/s-parameters)

Pupalaikis, P. (2020). S-Parameters for Signal Integrity. Cambridge: Cambridge University Press. doi:10.1017/9781108784863

Coelho, C. P., Phillips, J. R., & Silveira, L. M. (n.d.). Robust rational function approximation algorithm for model generation. Proceedings 1999 Design Automation Conference (Cat. No. 99CH36361). [https://sci-hub.ru/10.1109/dac.1999.781313]

Cadence IEEE IMS 2023, Introducing the Spectre S-Parameter Quality Checker and Rational Fit Model Generator

The Complex Art Of Handling S-Parameters: The importance of extraction and fitting to circuit simulation involving S-parameters [https://semiengineering.com/the-complex-art-of-handling-s-parameters]

Dr. John Choma. EE 541, Fall 2006: Course Notes #2 Scattering Parameters: Concept, Theory, and Applications [https://www.ieee.li/pdf/essay/scattering_parameters_concept_theory_applications.pdf]

Dr. Ray Kwok . Network Techniques: Conversion between Filter Transfer Function and Filter Scattering (SMatrix) Parameters [https://www.sjsu.edu/people/raymond.kwok/docs/project172/FTF%20to%20S-Matrix%20Spring%202011.pdf]

田庆诚教授 台湾中华大学 射频电路基础（公司培训）[https://www.bilibili.com/video/BV1LA41177wr/?p=3&share_source=copy_web&vd_source=5a095c2d604a5d4392ea78fa2bbc7249]

Bogatin, E. (2018). Signal and power integrity, simplified. Prentice Hall. [pdf]

Three fast time-domain system simulation techniques:

• single-bit response method
• double-edge response method
• multiple-edge response method

Single-Bit Response (SBR) Method

Overlapping portions of a pulse response from neighboring bits are referred to as intersymbol interference (ISI). A received waveform is formed by superimposing, in time, the pulse responses of each bit in the sequence, as illustrated in Figure 9, assuming symmetric positive and negative pulses are transmitted for 1s and 0s

To avoid spurious glitches between consecutive ones, rising and falling edge responses shall be symmetric. This is the limitation of SBR method.

Let \(p(t)\) be the SBR of the channel, \(t_s\) be the data sampling phase, \(T\) be the bit time, \(N_c\) is the number of UI in stored pulse response and \(b_m\) be the \(m\)th transmitted symbol. The voltage seen by the receiver's data sampler at the \(m\)th data sample is determined by \[ y_m = \sum_{k=m-N_c+1}^{m}b_kp(t_s+(m-k)T) \] where \(b_k \in [0, 1]\) and \(p(t) \ge 0\)

We always prepend \(Nc-1\) 0s in random bit stream for consistency.

For computation convenient, the pulse need to be positive. For differential signal and amplitude \(V_{peak}\), the peak to peak is \(-V_{peak}\) to \(+V_{peak}\). After pulse added by \(V_{peak}\), peak to peak is \(0\) to \(+2V_{peak}\).

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hold on;

yy_sum = zeros(OSR*Ns, Ns);
for idxBit = 1:Ns
    bs_split = zeros(1, Ns+Nc-1);
    bs_split(idxBit) = bs(idxBit);
    yy = zeros(OSR, Ns);
    for ii = Nc:Nc+Ns-1
        bb = bs_split(ii:-1:ii-Nc+1);
        yy(:,ii-Nc+1) = sum(bb.*yrps, 2);
    end
    yy_cont2 = reshape(yy, [], 1);
    h = plot(yy_cont2);
    h.Annotation.LegendInformation.IconDisplayStyle = 'off';
    yy_sum(:, idxBit) = yy_cont2;
end
yy_sum = sum(yy_sum, 2);    % merge
plot(yy_sum, 'k--');
plot(yy_cont, 'm-.');
grid on;
legend('sum', 'syn');
title('merge all single bit');
ylabel('mag');
xlabel('Time (\times Ts)');

The pulse response contain rising and falling edge. The 1 bit first rise from -1 to 1, then fall to -1; The 0 bit just do nothing for synthesized waveform with the help of falling edge of 1 bit.

The DC shift help deal with continuous 0 bits.

another SBR example

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A = zeros(10,21);
n = [1:10];

% post cursor
for m = 1:3
    A(m, 11+n(m)) = 0.5;
    A(m, 11-n(m)) = 0.5;
end

% one
for m = 4:10
    A(m, 11) = 1;
end

% h0 or main cursor
h0 = zeros(1, 21);
h0(1, 1) = 0.5;
h0(1, 21) = 0.5;
out = h0;

for m = 1:10
    out = conv(out, A(m, :), "full");
end


stem(out)

S-Parameter to Single Bit Response (SBR)

Mike Li, "S-Parameter to Single Bit Response (SBR) Transformation and Convergence Study" [https://ieee802.org/3/bj/public/may12/li_01_0512.pdf]

TODO 📅

Double-Edge Response (DER) Method

To handle the more general cases, with asymmetric rising and falling edges, the system response can be constructed in terms of edge transitions instead of bit responses.

The DER method decomposes the input data pattern, in terms of rising and falling edge transitions. The system response can be calculated by superimposing the shifted versions of the rising and falling edge responses : \[ y_m = \sum_{k=m-N_c+1}^{m}(b_k-b_{k-1})s_k(t_s+(m-k)T) + y_{int} \] where

\[\begin{align} s_i(t) &= r(t) -V_{low} \quad \text{if} \: (b_i\gt b_{i-1}) \\ &= V_{high}-f(t) \quad \text{otherwise} \end{align}\]

\(r(t)\) and \(f(t)\) are the rising and falling edge responses,respectively. \(V_{high}\) and \(V_{low}\) are the steady state DC levels, in response to a constant stream of ones and zeros, respectively. \(y_{int}\) is the initial DC state (either \(V_{high}\) or \(V_{low}\) ).

We always prepend \(Nc\) 0s in random bit stream for consistency.

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figure(1)
subplot(3, 1, 1)
plot(yrc);
hold on;
plot(yfc);
hold off;
legend('rising', 'falling')
grid on;
ylabel('mag');
xlabel('Time (\times Ts)');
title('step response');

subplot(3, 1, 2)
stem(bs, 'k'); grid on;
hold on;
stem((idxPreRspStart:idxPreRspEnd), bs(idxPreRspStart:idxPreRspEnd), "filled", 'r');
stem(idxPreRspStart, bs(idxPreRspStart), 'go');
stem(idxCurData, bs(idxCurData), "filled", 'm');
stem((idxPreRspStart:idxPreRspEnd)+0.5, 0.1.*bd(idxPreRspStart:idxPreRspEnd), 'bd-.');
hold off;
legend('', 'Nc bits', 'y_{int}', 'Current bit', 'Edge Transitions');
ylabel('mag');
xlabel('Time (\times UI)');
title('input stream');

subplot(3, 1, 3)
yy_cont = reshape(yy, [], 1);   % continuous version
plot(yy_cont); grid on;
title('continuous yout')
ylabel('mag');
xlabel('Time (\times Ts)');

figure(2)
hold on;
for idx = idxPreRspStart+1-Nc:idxCurData+32-Nc
    ys = yy(:, idx);
    tt = ((idx-1)*OSR+1:idx*OSR);
    h = plot(tt(:), ys(:), 'LineWidth',3);
    h.Annotation.LegendInformation.IconDisplayStyle = 'off';
end
plot(yy_cont, 'm--', 'LineWidth',1);
hold off;
legend('syn')
ylabel('mag');
xlabel('Time (\times Ts)');
title('synthesize with step response');
grid on;

Reference

T. C. Carusone, "Introduction to Digital I/O: Constraining I/O Power Consumption in High-Performance Systems," in IEEE Solid-State Circuits Magazine, vol. 7, no. 4, pp. 14-22, Fall 2015

Oh, Kyung Suk Dan, and Xing Chao Chuck Yuan. High-Speed Signaling: Jitter Modeling, Analysis, and Budgeting. Prentice Hall, 2011.

Ren, Jihong and Kyung Suk Oh. “Multiple Edge Responses for Fast and Accurate System Simulations.” IEEE Transactions on Advanced Packaging 31 (2008): 741-748.

Shi, Rui. “Off-chip wire distribution and signal analysis.” (2008).

X. Chu, W. Guo, J. Wang, F. Wu, Y. Luo and Y. Li, "Fast and Accurate Estimation of Statistical Eye Diagram for Nonlinear High-Speed Links," in IEEE Transactions on Very Large Scale Integration (VLSI) Systems, vol. 29, no. 7, pp. 1370-1378, July 2021, doi: 10.1109/TVLSI.2021.3082208.

Phase Noise to Jitter

Note that \(L(f )\) is defined over positive frequencies only \((f \ge 0)\)

for simple PLL

For small \(N\): \(\sigma_{p(N)}^2 \approx \frac{\mathcal{L}_0 f_{3dB}}{2\pi f_0^2} \cdot \frac{2\pi f_{3dB}N}{f_0}=\frac{\mathcal{L}_0f_{3dB}^2}{f_0^3}=\sigma_{PER}^2\)

For large \(N\): \(\sigma_{p(N)}^2 \approx \frac{\mathcal{L}_0f_{3dB}^2}{f_0^3} \cdot \frac{f_0}{2\pi f_{3dB}}=\sigma_{PER}^2\cdot \frac{f_0}{2\pi f_{3dB}}\)

Free Running OSC	OSC in simple PLL
\(\mathcal{L}(f) = \frac{\mathcal{L}_0 f_{3dB}^2}{f^2}\)	\(\mathcal{L}(f) = \frac{\mathcal{L}_0 f_{3dB}^2}{f^2 + f_{3dB}^2}\)

\[\begin{align} S_{jACC(N)}(f) &= |1-z^{-N}|^2\cdot S_{jABS}(f) \\ &= |1-\cos\theta +j\sin\theta|^2\cdot S_{jABS}(f) = ((1-\cos\theta)^2 + \sin^2\theta)\cdot S_{jABS}(f) \\ &= 2(1-\cos\theta)\cdot S_{jABS}(f) = 4\sin^2(\theta/2)\cdot S_{jABS}(f) \end{align}\]

where \(\theta = 2\pi f N/f_0\)

As EQ(3.44), EQ(3.45)

the autocorrelation is the inverse Fouer transform of the PSD

\[ R_{\varphi}(t) = \int_{-\infty}^{+\infty} S_{\varphi} (f) e^{j2\pi f t} df \]

Then, \[\begin{align} R_{\varphi}(0) &= \int_{-\infty}^{+\infty} S_{\varphi} (f) df \\ R_{\varphi}(NT_0) &= \int_{-\infty}^{+\infty} S_{\varphi} (f) e^{j2\pi f NT_0} df \end{align}\]

Thus, yield EQ(3.48)

Simplified PLL Phase Noise Profile

Absolute Jitter

TODO 📅

Period Jitter

a random-walk DCO - \(1/f^2\) Phase Noise Profile

L. Avallone, M. Mercandelli, A. Santiccioli, M. P. Kennedy, S. Levantino and C. Samori, "A Comprehensive Phase Noise Analysis of Bang-Bang Digital PLLs," in IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 68, no. 7, pp. 2775-2786, July 2021 [https://sci-hub.st/10.1109/TCSI.2021.3072344]

Mozhgan Mansuri “Low-Power Low-Jitter On-Chip Clock Generation” thesis UCLA [https://people.engr.tamu.edu/spalermo/ecen689/pll_thesis_mansuri_ucla_2003.pdf]

[https://people.engr.tamu.edu/spalermo/ecen689/PRBS_&_PLL_model.pdf]

Intersymbol interference (ISI)

\[ \color{red}\phi = 2\pi D \cdot f \]

Even-odd Jitter (EOJ)

Even-odd jitter, also known as F/2 jitter, arises from a clock signal's duty cycle not being perfectly 50%

Even-odd jitter has been referred to as duty cycle distortion by other Physical Layer specifications for operation over electrical backplane or twinaxial copper cable assemblies

Comparing DCD and F/2 Jitter Using a BERTScope® Bit Error Rate Testing Application Note [https://download.tek.com/document/65W_26040_0_Letter.pdf]

Pulse Width Jitter (PWJ)

Jeff Morriss Updated 10/25/07. Analysis of 8G PCIe Pulse Width Jitter (UI to UI Jitter_10_25.ppt)

Duty Cycle Distortion – DCD

Jitter fundamental & How Isolating Root Causes of Jitter [https://picture.iczhiku.com/resource/eetop/ShKgzTEiUfdFOcvn.pdf]

There are two primary causes of DCD jitter which are usually generated within a transmitter

• If the data input to a transmitter is theoretically perfect, but if the transmitter sampling threshold is offset from its ideal level, then the output of transmitter will have duty cycle distortion as a function of the slew rate of the data signal
• Another cause of duty cycle distortion can be a mismatch/asymmetry in rising and falling edge speeds

Unfortunately, other sources such as ISI almost always exist making it sometimes difficult to isolate the DCD component. One technique to test for DCD is to stimulate your system/components with a repeating 1-0-1-0… data pattern. This technique will eliminate inter-symbol interference (ISI) jitter and make viewing the DCD within the spectrum display much easier

Why clock pattern? That's because all symbols experience same inter-symbol interference, which are canceled out

[https://scdn.rohde-schwarz.com/ur/pws/dl_downloads/dl_application/application_notes/1td03/1TD03_2e_RTO_Jitter_Analysis.pdf]

Correlated vs. Uncorrelated

If the PDF of one jitter source changes when the PDF of another source is changed, then those two sources are dependent or correlated

Inter-Symbol Interference (ISI)

The primary cause of Data Dependent Jitter

Jitter measurements can be classified into three categories: cycle-to-cycle jitter, period jitter, and long-term jitter

Jitter is a key performance parameter. Need to know what matters in each case:

• PJ for digital timing
• LTJ for data converters and serial data
• Phase noise for communications (not all bandwidths matter)

The above Cycle-Cycle Jitter equation is wrong, \(\tau_1\) and \(\tau_2\) are not independent

Short Term Jitter

Period jitter, Jper is the short term variation in clock period compared to the average (mean) clock period.

Cycle-to-Cycle, Jcc is the time difference of two adjacent clock periods

Long Term Jitter (LTJ)

[https://people.engr.tamu.edu/spalermo/ecen689/PRBS_&_PLL_model.pdf]

absolute jitter is also known as long-term jitter

measuring LTJ

Jitter Calculation Examples

Jcc vs Jper

Estimating the RMS cycle-to-cycle jitter if all you have available is the RMS period jitter.

• Cycle-to-cycle jitter - The short-term variation in clock period between adjacent clock cycles. This jitter measure, abbreviated here as \(J_{CC}\), may be specified as either an RMS or peak-to-peak quantity.
• Period jitter - The short-term variation in clock period over all measured clock cycles, compared to the average clock period. This jitter measure, abbreviated here as \(J_{PER}\), may be specified as either an RMS or peak-to-peak quantity.

Let the variable below represent the variance of a single edge's timing jitter, i.e. the difference in time of a jittery edge versus an ideal edge, \(\sigma^2_j\)

If each edge's jitter is independent then the variance of the period jitter can be written as \[\begin{align} \sigma^2_\text{jper} &= (\sigma_\text{j(n+1)}-\sigma_\text{j(n)})^2 \\ &= \sigma_\text{j(n+1)}^2-2\sigma_\text{j(n+1)}\sigma_\text{j(n)})+\sigma_\text{j(n)})^2\\ &= \sigma_\text{j(n+1)}^2+\sigma_\text{j(n)})^2 \\ &=2\sigma^2_j \end{align}\]

In every cycle-to-cycle measurement we use one "interior" clock edge twice and therefore we must account for this

\[\begin{align} \sigma^2_\text{jcc} &= (\sigma_\text{jper(n+1)}-\sigma_\text{jper(n)})^2 \\ &=(\sigma_\text{j(n+2)}-2\sigma_\text{j(n+1)}+\sigma_\text{j(n)})^2 \end{align}\]

Since each edge's jitter is assumed to be independent and have the same statistical properties we can drop the cross correlation terms and write:

\[\begin{align} \sigma^2_\text{jcc} &=(\sigma_\text{j(n+2)}-2\sigma_\text{j(n+1)}+\sigma_\text{j(n)})^2 \\ &=\sigma_\text{j(n+2)}^2+4\sigma_\text{j(n+1)}^2+\sigma_\text{j(n)}^2 \\ &=6\sigma_\text{j}^2 \end{align}\]

The ratio of the variances is therefore \[ \frac{\sigma^2_\text{jcc}}{\sigma^2_\text{jper}} = \frac{6\sigma_\text{j}^2} {2\sigma_\text{j}^2}=3 \] Then \[ \sigma_\text{jcc} = \sqrt{3}\sigma_\text{per} \]

[Timing 101 #8: The Case of the Cycle-to-Cycle Jitter Rule of Thumb, Silicon Labs]

references

AN10007 Clock Jitter Definitions and Measurement Methods, SiTime [pdf]

SERDES Design and Simulation Using the Analog FastSPICE Platform, Silicon Creations [pdf]

Flexible clocking solutions in advanced processes from 180nm to 5nm, Silicon Creations [pdf]

One-size-fits-all PLLs for Advanced Samsung Foundry Processes, Silicon Creations [pdf]

Circuit Design and Verification of 7nm LowPower, Low-Jitter PLLs, Silicon Creations, [pdf]

Lecture 10: Jitter, ECEN720: High-Speed Links Circuits and Systems Spring 2023 [pdf]

Jitter 360° Knowledge Series [pdf, slides]

N. Da Dalt, "Tutorial: Jitter: Basic and Advanced Concepts, Statistics, and Applications," 2012 IEEE International Solid-State Circuits Conference, San Francisco, CA, USA, 2012 [slides, transcript ]

Two-port parameters of MOS

\(Y_{11},Y_{12},Y_{21},Y_{22}\) are same with four AC simulation, but more elegant and concise

Gate (thermal) noise

Two-Side Poly Contact & folding

Both scheme yield a total distributed resistance of \(R_G/4\) for gate noise calculation

folding

finger 0 \[\begin{align} \overline{i_{tot,0}^2} &= \left(\frac{g_m}{2} \right)^2(4kT\frac{R_G/2}{3}) \\ &= g_m^2\left(4kT\frac{R_G}{3}\right)\frac{1}{2^2\cdot 2} \end{align}\]

similarly finger 1 \[ \overline{i_{tot,1}^2} = g_m^2\left(4kT\frac{R_G}{3}\right)\frac{1}{2^2\cdot 2} \] Assuming uncorrelated \[ \overline{i_{tot}^2} = \sum_{N=0}^1\overline{i_{tot,N}^2} =g_m^2\left(4kT\frac{R_G}{3}\right)\frac{1}{2^2\cdot 2} \cdot 2 = g_m^2\left(4kT\frac{R_G}{3}\right)\frac{1}{2^2} \] Generally \[ \overline{i_{tot}^2} = g_m^2\left(4kT\frac{R_G}{3}\right)\frac{1}{N^2} \] where the gate is decomposed into \(N\) parallel fingers

two-side poly contact

We fracture Gate poly at the center point, then we obtain 2 segments, both have same \(\frac{g_m}{2}\) and \(R_G/2\).

The derivation procedure is same with folding structure, i.e. plug \(N=2\) into \(\overline{i_{tot}^2} = g_m^2\left(4kT\frac{R_G}{3}\right)\frac{1}{N^2}\)

That is \[ \overline{i_{tot}^2} = g_m^2\left(4kT\frac{R_G}{12}\right) \] The input referred noise of gate resistance \[ \overline{V_{nRG}^2} = 4kT\frac{R_G}{12} \]

four equal gate fingers

\[ \overline{i_{tot}^2} = g_m^2\left(4kT\frac{R_G}{3}\right)\frac{1}{4^2} \] Then \[ \overline{V_{nRG}^2} = \frac{\overline{i_{tot}^2}}{g_m^2} =4kT\frac{R_G}{48} \]

Gate Res noise contribution

\(R_s\): previous stage output impedance

\(R_G\), \(I_n\): gate resistance and its noise current

\(R_L\): next stage input impedance

with \(R_SI_o + (I_o - I_n)R_G + I_oR_L=0\), we have \(I_o=I_n\frac{R_G}{R_S+R_G+R_L}\), then \[\begin{align} V_S &= -I_n\frac{R_GR_S}{R_S+R_G+R_L} \overset{R_L\to \infty}{\longrightarrow} 0\\ V_G &= I_n\frac{R_GR_L}{R_S+R_G+R_L} \overset{R_L\to \infty}{\longrightarrow} I_nR_G \end{align}\]

The above equation show that gate resistance noise don't contribute to previous stage output noise, however contribute to next stage

Gate resistance in PEX

They fracture the poly line at the intersection with the active (diffusion) layer, breaking it into "gate poly"(poly over active) and "field poly" (poly outside active)

gploy, fpoly

Gate poly is also fractured at the center point. Gate instance pin of the MOSFET (SPICE model) is connected to the center point of the gate poly. Gate poly is described by two parasitic resistors, connecting the fracture points.

MOSFET extrinsic parasitic capacitance between gate poly and source / drain diffusion and contacts is calculated by parasitic extraction tools, and assigned to the nodes of the resistive networks.

Different extraction tools do this differently - some tools connect these parasitic capacitances to the center point of the gate poly, while some other tools connect them to the end points of the gate poly resistors.

\(\Delta\) gate model

This distributed network has a different AC and transient response than a simple lumped one-R and one-C circuit.

It was shown [B. Razavi] that such RC network behaves approximately the same as a network with one R and one C element, where C is the total capacitance, and R=1/3 * W/L rsh for single-side connected poly, and R=1/12 W/L * rsh for double-sided connected poly.

These coefficients - 1/3 and 1/12 - effectively enable an accurate reduced order model for the gate, reducing a large number of R and C elements to two (or three) resistors and one capacitor.

Gate Delta Model: where a gate is described by two positive and one negative resistors

only applicable to contacts on gate overhangs

invalid for self-aligned gate contacts, where gate contact land directly on top of gate, not gate overhang

1. 1-side gate contact \[ R_{eq,1side} =R_1 \parallel (R_2+R_1)= \frac{R_G}{6}\parallel (-\frac{R_G}{2}+\frac{R_G}{6})=\frac{R_G}{3} \]

2. 2-side contact \[ R_{eq,2side}= R_1 \parallel R_1 = \frac{R_G}{6}\parallel \frac{R_G}{6} = \frac{R_G}{12} \]

Some SPICE simulators have problems handling negative resistors, that's possibly why this model did not get a wide adoption. Some foundries and PDKs support delta gate model, while some others don't.

Vertical component of gate resistance

In "old" technologies (pre-16nm), gate resistance was dominated by lateral resistance. However, in advanced technologies, multiple interfaces between gate material layers lead to a large vertical gate resistance.

It's very easy to check this in DSPF file - if gate instance pin is connected directly to the center of the gate poly - vertical resistance is not accounted for. If it is connected by a positive resistor to the center of the gate poly - that resistors represents the vertical gate resistance.

decap

leakage current is determined by \(R_s + R_p\), and \(R_p \gg R_s\)

• low freq: \(Z=R_s + R_p\)
• high freq: \(Z=R_s\)

An example: \(R_s=200\space \Omega\), \(R_p = 8 \space M\Omega\) and \(C_\text{gate}=10\space fF\)

reference

⭐ B. Razavi, Y. Ran, and K. F. Lee, “Impact of Distributed Gate Resistance on the Performance of MOS Devices,” IEEE Trans. Circuits and Systems, Part I, pp. 750–754, Nov. 1994.

⭐ Maxim Ershov, Diakopto. "Gate Resistance in IC design flow", [link, pdf]

A.J.Sholten et al., "FinFET compact modelling for analogue and RF applications", IEDM'2010 [https://sci-hub.se/10.1109/IEDM.2010.5703322]

Saha, Samar K.. “FinFET Devices for VLSI Circuits and Systems.” (2020).

Harpe, Pieter J. A., Andrea Baschirotto and Kofi A. A. Makinwa. “Hybrid ADCs, Smart sensors for the IoT, and Sub-1V and Advanced node analog circuit design: Advances in Analog Circuit Design 2017.” (2018).

Chauhan, Yogesh Singh. FinFET Modeling for IC Simulation and Design: Using the BSIM-CMG Standard. London, UK: Academic Press, 2015.

A.J.Sholten et al., "FinFET compact modelling for analogue and RF applications", IEDM'2010, p.190.

W. Wu and M. Chan, “Gate resistance modeling of multifin MOS devices,” IEEE Electron Device Letters, vol. 27, no. 1, pp. 68-70, Jan. 2006.

A. L. S. Loke, C. K. Lee and B. M. Leary, "Nanoscale CMOS Implications on Analog/Mixed-Signal Design," 2019 IEEE Custom Integrated Circuits Conference (CICC), Austin, TX, USA, 2019, pp. 1-57, doi: 10.1109/CICC.2019.8780267.

Nonzero FFT bins

everynanocounts. Memos on FFT With Windowing. [https://a2d2ic.wordpress.com/2018/02/01/memos-on-fft-with-windowing]

The problem with sine-wave scaling is that the noise power is, on average, evenly distributed over all FFT bins, whereas the sine-wave power is concentrated in only a few bins. With sine-wave scaling, the power of individual sine-wave components can be read directly from the spectral plot, but in order to determine the noise power, the powers of all the noise bins must be added together.

We can't use Fourier transform of random signal

[https://raytroop.github.io/2023/11/10/random/#lti-systems-on-wss-processes]

\[\begin{align} \mathrm{SNR} &= \frac{X_\text{sig}^2}{X_\text{n}^2N} \\ &= \frac{X_\text{sig}^2\cdot \sum_k W_k^2}{X_\text{n}^2N\cdot \sum_k W_k^2} \\ &= \frac{\sum_\text{nb} X_\text{w,sig}^2}{N X_\text{w,n}^2} \end{align}\]

where \(\text{nb}\) is number of non-zero FFT bins

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subplot(3,1,1);
for N = [16 256 1024]
    wrect = rectwin(N);
    Wrect = fftshift(fft(wrect));
    Wrect_mag = abs(Wrect)/sum(wrect);
    nb_rect = sum(Wrect_mag > 0.1);
    fprintf('Number of nonzero FFT bin(rect window N=%d): %d\n', N, nb_rect);
    stem(1:N, Wrect_mag, LineWidth=2)
    hold on
end
grid on
legend('16', '256', '1024')
title('rect window')

subplot(3,1,2);
for N = [16 256 1024]
    whann = hann(N);
    Whann = fftshift(fft(whann));
    Whann_mag = abs(Whann)/sum(whann);
    nb_hann = sum(Whann_mag > 0.1);
    fprintf('Number of nonzero FFT bin(hann window N=%d): %d\n', N, nb_hann);
    stem(1:N, Whann_mag, LineWidth=2)
    hold on
end
grid on
legend('16', '256', '1024')
title('Hanning window')

subplot(3,1,3);
for N = [16 256 1024]
    whann2 = (1-cos(2*pi*(0:N-1)/N)).^2/2^2;
    Whann2 = fftshift(fft(whann2));
    Whann2_mag = abs(Whann2)/sum(whann2);
    nb_hann2 = sum(Whann2_mag > 0.1);
    fprintf('Number of nonzero FFT bin(hann2 window N=%d): %d\n', N, nb_hann2);
    stem(1:N, Whann2_mag, LineWidth=2)
    hold on
end
grid on
legend('16', '256', '1024')
title('Hann2 window')


% Number of nonzero FFT bin(rect window N=16): 1
% Number of nonzero FFT bin(rect window N=256): 1
% Number of nonzero FFT bin(rect window N=1024): 1
% Number of nonzero FFT bin(hann window N=16): 3
% Number of nonzero FFT bin(hann window N=256): 3
% Number of nonzero FFT bin(hann window N=1024): 3
% Number of nonzero FFT bin(hann2 window N=16): 5
% Number of nonzero FFT bin(hann2 window N=256): 5
% Number of nonzero FFT bin(hann2 window N=1024): 5

[https://web.engr.oregonstate.edu/~temes/ece627/Lecture_Notes/FFT_for_delta_sigma_spectrum_estimation.pdf]

where \(\mathrm{NBW} = \frac{\sum_{n}|w[n]|^2}{ \left| \sum_{n} w[n] \right|^2}\)

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% excerpt of A.4 An Example in Pavan, Schreier and Temes, "Understanding Delta-Sigma Data Converters, Second Edition" ISBN 978-1-119-25827-8
%
% Compute modulator output and actual NTF
%
OSR = 32;
ntf0 = synthesizeNTF(5,OSR,1);
N = 64*OSR;
fbin = 11;
u = 1/2*sin(2*pi*fbin/N*[0:N-1]);
[v tmp1 tmp2 y] = simulateDSM(u,ntf0);
k = mean(abs(y)/mean(y.^2))
ntf = ntf0 / (k + (1-k)*ntf0);
%
% Compute windowed FFT and NBW
%
w = hann(N); % or ones(1,N) or hann(N).^2
nb = 3; % 1 for Rect; 5 for Hann^2
w1 = norm(w,1);
w2 = norm(w,2);
NBW = (w2/w1)^2
V = fft(w.*v)/(w1/2);
%
% Compute SNR
%
signal_bins = fbin + [-(nb-1)/2:(nb-1)/2];
inband_bins = 0:N/(2*OSR);
noise_bins = setdiff(inband_bins,signal_bins);
snr = dbp(sum(abs(V(signal_bins+1)).^2)/sum(abs(V(noise_bins+1)).^2))

Noise Floor

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

General Formula

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

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

where white noise, \(X_n(i) = X_n(j)\) for any \(i \neq j\)

Therefore, 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 \(\mathrm{SNR}\)

\[ \text{PS(k)} = \text{PSD(k)} \cdot \text{ENBW} \]

where Effective Noise BandWidth \(\text{ENBW} =f_{\text{res}} \cdot \frac{N S_2}{S_1^2}\)

yield \[ \text{PS(k)} = \text{PSD(k)} \cdot f_{\text{res}} \cdot \frac{N S_2}{S_1^2} = \left\{\text{PSD(k)} \cdot f_{\text{s}} \right\} \cdot \frac{S_2}{S_1^2} \] where \(\left\{\text{PSD(k)} \cdot f_{\text{s}} \right\} =\text{const}\), i.e. \(x_\text{n,rms}\)

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ratio_list = [];
for len = 6:12
 wnd = hanning(2^len);
 S2 = sum(wnd.^2);
 S1 = sum(wnd);
 ratio = S2./S1.^2;
 ratio_list(end+1) = ratio;
end

ratio_db = 10*log10(ratio_list);
plot(2.^[6:1:12], ratio_db, 'ro-', LineWidth=4)
grid on; grid minor;

xlabel('N'); ylabel("$10\log(S_2/S_1^2)$",'Interpreter', 'latex')

FFT Noise Floor

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N = 2048;
cycles = 67;
fs = 1000;
fx = fs*cycles/N;
LSB = 2/2^10;
%generate signal, quantize (mid-tread) and take FFT
x = cos(2*pi*fx/fs*[0:N-1]);
x = round(x/LSB)*LSB;
s = abs(fft(x));
s = s(1:end/2)/N*2;
% calculate SNR
sigbin = 1 + cycles;
noise = [s(1:sigbin-1), s(sigbin+1:end)];
snr = 10*log10( s(sigbin)^2/sum(noise.^2) );

sdb = 20*log10(s);

% How to plot a series of numbers which some of them are inf?
% https://www.mathworks.com/matlabcentral/answers/476643-how-to-plot-a-series-of-numbers-which-some-of-them-are-inf
plot([0:N/2-1]/N, max(sdb, -120), LineWidth=4)
hold on;
plot([0 0.5], [-61.9 -61.9], 'r--', LineWidth=2)
plot([0 0.5], [-92 -92], 'm--', LineWidth=2)
grid on; grid minor;
ylim([-120 0]); xlim([0 0.5]);
xlabel('Frequency [f/fs]'), ylabel('DFT Magnitude [dBFS]');
title('2048 point FFT, SNR=61.90dB')

Noise Spectral Density (NSD)

Understanding Key Parameters for RF-Sampling Data Converters White Paper (WP509) [https://docs.amd.com/v/u/en-US/wp509-rfsampling-data-converters]

Boris Murmann, ISSCC2022 SC1: Introduction to ADCs/DACs: Metrics, Topologies, Trade Space, and Applications [https://www.nishanchettri.com/isscc-slides/2022%20ISSCC/SHORT%20COURSE/SC1.pdf]

Rectangular Window

DFT bin's output noise standard deviation (rms) value is proportional to \(\sqrt{N}\), and the DFT's output magnitude for the bin containing the signal tone is proportional to \(N\)

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

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

The displayed SNR \[ \mathrm{SNR} = \mathrm{SNR}' - 10\log_{10}(2/N) \] If we increase a DFT's size from \(N\) to \(2N\), the DFT's output SNR increased by 3dB. So we say that a DFT's processing gain increases by 3dB whenever \(N\) is doubled.

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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

How spectrum analyzer work?

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

PS and PSD

The spectral density format is appropriate for random or noise signals but inappropriate for discrete frequency components because the latter theoretically have zero bandwidth

Amplitude Correction

1. A finite-duration window \(w[n]\)

   DTFT is \(W[e^{j\omega}]\) and the maximum magnitude is is at DC frequency, which \(\sum_n w_n\)

2. Sinusoidal signal \(x[n]\)

   DFS is \(X_k\), and DTFT shall be \(\frac{2\pi}{N}X_k(e^{j\omega})\)

3. the windowed sequence \(v[n] = x[n]w[n]\)

   with multiplication property, DTFT of \(v[n]\) shall be \(\frac{X_k(e^{j\omega})}{N}\sum_n w_n\)

   As we know, DFT of \(v[n]\) is samples of its DTFT, that is \[ \frac{X_k(e^{j\omega})}{N}\sum_n w_n = X_v[k] \] Therefore, \[ \frac{X_k(e^{j\omega})}{N} = \frac{X_v[k]}{\sum_n w_n} \]

Effective Noise BandWidth (ENBW)

General derivation

The relationship between a power spectrum (\(PS, V^2\)) and a power spectral density (\(PSD, V^2/Hz\)) is given by the effective noise bandwidth (ENBW), which can easily be determined at the time when the DFT is computed.

ENBW should always be recorded when a spectrum or spectral density is computed, such that the result can be converted to the other form at a later stage, when the information about the frequency resolution \(f_{res}\) and the window that was used is normally not easily available any more.

The normalized equivalent noise bandwidth (NENBW) of the window is given by

\[ \text{NENBW} = \frac{NS_2}{S_1^2} \] where \(S_1 = \sum _{k=0}^{N-1}w_k\) and \(S_2 = \sum _{k=0}^{N-1}w_k^2\)

The ENBW is given by

\[ \text{ENBW} = \text{NENBW}\cdot f_{res} = \text{NENBW}\cdot \frac{f_s}{N} = f_s\frac{S_2}{S_1^2} \]

Equivalent noise bandwidth (ENBW) compares a window to an ideal, rectangular time-window. It is the bandwidth of the rectangular window's frequency-domain shape that passes the same amount of white noise energy as the frequency-domain shape defined by the other window.

Therefore, the equivalent noise bandwidth \(B_{enbw}\) is given by

\[ B_{enbw} = \frac{\int_{-f}^{f} |W(f)|^2 df}{|W(f_0)|^2} \]

Translating to discrete domain, the equivalent noise bandwidth can be computed using DFT samples as

\[ B_{enbw} =\frac{\sum_{k=0}^{N-1}|W[k]|^2}{|W[k_0]|^2} \]

where \(k_0\) is the index at which the magnitude of FFT output is maximum and \(N\) is the window length, i.e. \(k_0=0\).

Applying Parseval's theorem and \(W[0]=\sum_{n}w[n]\), \(B_{enbw}\) can also be computed using time domain samples as

\[ B_{enbw} = N \frac{\sum_{n}|w[n]|^2}{ \left| \sum_{n} w[n] \right|^2} \]

scale \(w[n]\) don't change \(B_{enbw}\)

Noise power inside window: \(\int_{-f}^{f} |W(f)|^2 df \to N\cdot\sum_{n}|w[n]|^2\)

peak amplitude: \(|W(f_0)|^2 \to \left| \sum_{n} w[n] \right|^2\)

An alternative derivation

Assuming the windowed sequence \(v[n] = x[n]w[n]\)

• \(W[k]\): Fourier Transform of finite sequence window

• \(X_{sig}\): Fourier Transform of signal

• \(X_{n}\): Fourier Transform of noise

• \(X_{v,sig}\): Fourier Transform of windowed signal

• \(X_{v,n}\): Fourier Transform of windowed noise

From Fig. 6,, we observe that the amplitude of the harmonic estimate at a given frequency is biased by the accumulated broad-band noise included in the bandwidth of the window.

In this sense, the window behaves as a filter, gathering contributions for its estimate over its bandwidth

The Fourier Transform of windowed signal can be expressed as

\[\begin{align} X_{v,sig} &= W_{max}\cdot X_{sig} \\ &= W[0]\cdot X_{sig} \end{align}\]

For a typical window, \(W_{max}\) occurs at \(\omega = 0\)

And the Fourier Transform of windowed noise can be expressed as

\[ X_{v,n}^2 = \sum_k (W[k])^2 \cdot X_n^2 \]

divided by \((W[0])^2\) on both sides of the above equation

\[ \frac{X_{v,n}^2}{(W[0])^2} = \frac{\sum_k (W[k])^2}{(W[0])^2} \cdot X_n^2 \]

By Parseval's theorem

\[ \frac{X_{v,n}^2}{\left(\sum_n w[n]\right)^2} = \frac{N\sum_n w^2[n]}{\left(\sum_n w[n]\right)^2} \cdot X_n^2 \]

where \(X^2_n\) is what is deserved and

\[ X_n^2 = \frac{PS_{n}}{B_{enbw}} \]

where \(B_{enbw} = N \frac{\sum_{n}|w[n]|^2}{ \left| \sum_{n} w[n] \right|^2}\) and \(PS_{n}=\left| \frac{X_{v,n}}{\sum_n w[n]}\right|^2\)

example

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lw = 128;
win = hann(lw);
lt = 2048;
windft = fftshift(fft(win,lt));

ad = abs(windft).^2;
mg = max(ad);

fs = 1000;

bw = enbw(win,fs);

bdef = sum((win).^2)/sum(win)^2*fs;
fprintf("bw: %.3f\n", bw);
fprintf("bdef: %.3f\n", bdef);

freq = -fs/2:fs/lt:fs/2-fs/lt;

plot(freq,ad, bw/2*[-1 -1 1 1],mg*[0 1 1 0],'--')
xlim(bw*[-1 1])

Adiff = trapz(freq,ad)-bw*mg;
fprintf("Adiff: %.3e\n", Adiff);

Verify that the area of the rectangle contains the same total power as the window.

1

Adiff = trapz(freq,ad)-bw*mg

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bw: 11.811
bdef: 11.811
Adiff: 7.276e-12

Unpingco, José. Python for Signal Processing: Featuring IPython Notebooks. Cham: Springer, 2013. [pdf]

[https://github.com/unpingco/Python-for-Signal-Processing/blob/master/Windowing_Part2.ipynb]

demo

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%https://aaronscher.com/Course_materials/Communication_Systems/documents/PSD_Autocorrelation_Noise.pdf

%Clear variables. clear command window, close all figures:
clc;
clear all;
close all;
%%%Setup and define variables
f0=10; %frequency of sinusoidal signal (Hz)
fs=100; %sampling frequency (Hz)
Ts=1/fs; %sampling period (seconds)
N0=3000; %number of samples
t=[0:Ts:Ts*(N0-1)]; %Sample times
noise_PSD=.5; %This is the desired noise power spectral density in W/Hz.
variance=noise_PSD*fs;% Variance = sigma^2
sigma=sqrt(variance);
noise=transpose(sigma*randn(N0,1));%create sampled white Gaussian noise.
xsignal=20*sin(2*pi*f0*t); %create sampled sinusoidal signal
x=xsignal+noise; %Add signal to noise
figure(1)
histogram(noise,30) %plot histogram
set(gca,'FontSize',14) %set font size of axis tick labels to 18
xlabel('Noise amplitude','fontsize',14)
ylabel('Frequency of occurance','fontsize',14)
title('Simulated histogram of white Gaussian noise','fontsize',14)
SNR_try1=snr(xsignal,noise); %calculate SNR using built in "snr" function.
SNR_try2=10*log10(sum(xsignal.^2)/sum(noise.^2)); %manually calculate SNR.
%If everything is correct, the two SNR calculations above should agree.
%Plot noise in time-domain
figure(2)
plot(t,x)
set(gca,'FontSize',14) %set font size of axis tick labels to 18
xlabel('Time (s)','fontsize',14)
ylabel('Amplitude','fontsize',14)
title('Noisey sinusoid','fontsize',14)
grid on
%Plot power spectral density (PSD) of noise using three different methods:
%
%Method 1. Calcululate PSD from amplitude spectrum
N=2^16; %Number of discrete points in the FFT
y=fft(x,N)/fs; %fft of noise
z=fftshift(y);%center noise spectrum
f_vec=[0:1:N-1]*fs/N-fs/2; %designate sample frequencies
amplitude_spectrum=abs(z); %compute two-sided amplitude spectrum
ESD1=amplitude_spectrum.^2; %ESD = |F(w)|^2;
PSD1=ESD1/((N0-1)*Ts);% PSD=ESD/T where T = total time of sample
figure(3)
plot(f_vec,10*log10(PSD1));
xlabel('Frequency [Hz]','fontsize',14)
ylabel('dB/Hz','fontsize',14)
title('Power spectral density - method 1','fontsize',14)
grid on
set(gcf,'color','w'); %set background color from grey (default) to white
axis tight
%calculate average power using PSD calclated from method 1:
Average_power_method_1=sum(PSD1)*fs/N; % Pav=sum(PSD)*delta_f where delta_f=fs/N;
%
%Method 2 - Calculate PSD from autocorrelation
time_lag=((-length(x)+1):1:(length(x)-1))*Ts;
auto_cor=xcorr(x,x)/fs; %Use xcorr function to find PSD
y=1/fs*fft(auto_cor,N); %fft of auto correlation function
PSD2=abs(1/(N0-1)*fftshift(fft(auto_cor,N)));
figure(4)
plot(f_vec,10*log10(PSD2));%use convolution
xlabel('Frequency [Hz]','fontsize',14)
ylabel('dB/Hz','fontsize',14)
title('Power spectral density - method 2','fontsize',14)
grid on
set(gcf,'color','w'); %set background color from grey (default) to white
axis tight
%calculate average power using PSD calclated from method 1:
Average_power_method_2=sum(PSD2)*fs/N; %Pav=sum(PSD)*delta_f where delta_f=fs/N;
%
%Method 3 - Calculate PSD using built in pwelch function
figure(5)
PSD3=periodogram(x,[],N,fs,'centered');
plot(10*log10(PSD3))
xlabel('Frequency [Hz]','fontsize',14)
ylabel('dB/Hz','fontsize',14)
title('Power spectral density - method 3','fontsize',14)
grid on
set(gcf,'color','w'); %set background color from grey (default) to white
axis tight
Average_power_method_3=sum(PSD3)*fs/N; %Pav=sum(PSD)*delta_f where delta_f=fs/N;
%
%Calculate mean and average PSD of noise:
PSD_noise=periodogram(noise,[],N,fs,'centered');
Average_noise_PSD=mean(PSD_noise);
Mean_noise=mean(noise);

The power spectral density plots for methods 2 and 3 exactly match that for method 1 (shown above).

reference

enbw Matlab URL:https://www.mathworks.com/help/signal/ref/enbw.html

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N = 256;
wn = hanning(N);
s1 = sum(wn)^2;
s2 = sum(wn.^2);

(N*s2)/s1
enbw(wn)

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ans =

    1.4942


ans =

    1.4942

Schmid, H. (2012). How to use the FFT and Matlab's pwelch function for signal and noise simulations and measurements. URL:http://www.schmid-werren.ch/hanspeter/publications/2012fftnoise.pdf

Bonnie C.Baker. Reading and Using Fast Fourier Transforms (FFT) URL:http://ww1.microchip.com/downloads/en/appnotes/00681a.pdf

FFT analysis: SNR and Noise level vary with FFT Window size URL:https://www.virtins.com/forum/viewtopic.php?f=7&t=1382

Lyons, R. G. (2011). Understanding digital signal processing (3rd ed.). Prentice Hall

Stefan Scholl, "Exact Signal Measurements using FFT Analysis",Microelectronic Systems Design Research Group, TU Kaiserslautern, Germany. [pdf]

Aaron Scher. PSD, Autocorrelation, and Noise in MATLAB [pdf]

Aaron Scher. FFT, total energy, and energy spectral density computations in MATLAB [pdf]

Amplitude Estimation and Zero Padding URL:https://www.mathworks.com/help/signal/ug/amplitude-estimation-and-zero-padding.html

Harris, F. (1978). On the use of windows for harmonic analysis with the discrete Fourier transform. Proceedings of the IEEE, 66, 51-83. [pdf]

Harris, F. J. . (1976). Windows, harmonic analysis and the discrete fourier transform. [pdf]

Wang Hongwei. Virtins. Evaluation of Various Window Functions using Multi-Instrument [pdf]

Properties of FFT Windows Used in Stable32 [pdf]

Solomon, Jr, O M. PSD computations using Welch's method. [Power Spectral Density (PSD)]. United States. https://doi.org/10.2172/5688766 [pdf]

Measure Power of Deterministic Periodic Signals [https://www.mathworks.com/help/signal/ug/measuring-the-power-of-deterministic-periodic-signals.html]

Mathuranathan. Equivalent noise bandwidth (ENBW) of window functions. [https://www.gaussianwaves.com/2020/09/equivalent-noise-bandwidth-enbw-of-window-functions/]

recordingblogs, Equivalent noise bandwidth [https://www.recordingblogs.com/wiki/equivalent-noise-bandwidth]

unpingco, Python for Signal Processing [https://github.com/unpingco/Python-for-Signal-Processing/blob/master/Windowing_Part2.ipynb]

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

A finite-length data record = an infinite record multiplied by a rectangular window

Windowing is unavoidable

Applying the Hanning window (or any window) to a periodic signal creates leakage.

leakage:

​ The component at one frequency leaks into the vicinity of another compnent owing to the spectral smearing introdued by window.

Notice side lobes adding out of phase can reduce the heights of the peaks

Windowed Signal

Short transient signals in the time domain produce high, broadband frequency content.

To reduce leakage, a mathematical function called a window is applied to the data. Windows are designed to reduce the sharp transient in the re-created signal as much as possible.

Because the sharp transients are reduced and smoothed, the broadband frequency of the spectral leakage is also reduced.

Periodic versus Non-Periodic Background

When performing a Fourier Transform on measurement data, a window affects periodic and non-periodic data differently:

• Periodic (No Window needed): A signal captured in a periodic manner does not require a window, and a resulting Fourier Transform has no leakage. Applying a window alters the resulting Fourier transform, and even creates spectral leakage where there would have been no leakage otherwise.

• Non-periodic (Window needed): Windows are used on signals that are captured in a non-periodic manner to reduce spectral leakage and get closer to the periodic results. A window can minimize the leakage present in a non-periodic signal, but cannot eliminate it.

The signal is repeated and appended mathematically because the measured data is assumed to be representative of the entire original signal

Periodic

When a measurement signal is captured in a periodic manner, the Fourier Transform of the captured signal will have no leakage in the frequency domain.

A window is not recommended for a periodic signal as it will distort the signal in an unnecessary manner, and actually creates spectral leakage.

Maloberti, F. Data Converters. Dordrecht, Netherlands: Springer, 2007.

Non-periodic

The same sine wave, with a different measurement time, results in a non-periodic captured signal. Here, when the captured signal is repeated, the original sine wave signal is not re-created.

In fact, several broadband transient events (circled in red) are introduced. These transients create a broadband response, or leakage.

Windows are used to minimize this leakage effect in the frequency domain.

Hanning

When doing operational noise and vibration measurements, the Hanning window is commonly used.

Random data has spectral leakage due to the abrupt cutoff at the beginning and end of the time block. It is non-periodic.

There is no way to ensure that the captured random signal is periodic by varying the measurement time.

Hanning windows are often used with random data because they have moderate impact on the frequency resolution and amplitude accuracy of the resulting frequency spectrum.

• The maximum amplitude error of a Hanning window is 15%

  In the cited article, all spectral data had an amplitude correction factor applied.

• while the frequency leakage is typically confined to 1.5 spectral lines to each side of the original sine wave signal

periodic signal

Applying the Hanning window (or any window) to a periodic signal creates leakage.

The periodically captured sine wave with the Hanning window (blue) is wider in frequency than the original signal (red)

In the figure, the sine wave with the Hanning window (blue) is wider in frequency than the original signal (red).

non-periodic signal

When a Hanning window is applied to a non-periodic signal, the leakage is greatly reduced and the amplitude is higher.

A non-periodically captured sine wave (magenta) has a spectral leakage over the entire bandwidth, applying a Hanning window minimized the leakage (green)

RMS calculation

A RMS calculation sums up the energy within a frequency range.

• both the RMS of the periodic and non-periodic signals with a Hanning window are equal to the RMS of the leakage-free sine wave.

• Only the RMS of the non-periodic sine wave without a window applied is not equal to the others

With the leakage spread over a smaller frequency range, doing analysis calculations like RMS yields more accurate results.

Flattop

• The Flattop window has a better amplitude accuracy in frequency domain compared to the Hanning window,

  The maximum amplitude error of a Flattop window is less than 0.01%. By contrast, the Hanning window maximum amplitude error is 15%.

• A Flattop window confines leakage to 3.43 spectral lines to each side of the original signal.

amplitude errors

These maximum amplitude errors assume that amplitude correction factors are applied to the frequency spectrums. These amplitude correction factors compensate for any reduction caused by applying a window.

leakage

The frequency accuracy of the Flattop window is more coarse compared to a Hanning window. As a result, the Flattop window is typically employed on data where frequency peaks are distinct and well separated from each other.

When the frequency peaks are not guaranteed to be well separated, the Hanning window is preferred because it is less likely to cause individual peaks to be lost in the spectrum

Spectrum of two periodically captured tones that are \(4Hz\) apart with a \(1Hz\) frequency resolution. The spectrum with a Hanning window (green) shows two peaks while the spectrum with a Flattop window (blue) shows one peak.

Note that at the original frequencies of the tones the amplitude is correct and equal to one for both windows.

One common application for a flattop window is performing calibration. For example, a sound pistonphone only produces one single and distinct frequency during microphone calibration.

Uniform

A Uniform window has a value of 1.0 across the entire measurement time. In reality, a Uniform window could be called no window.

Depending on the data acquisition system used, sometimes the term Rectangular window is also used.

• A Uniform window creates no frequency or amplitude distortion when the measured signal is periodic.

• When a measured signal is not periodic, the amplitude is reduced by a maximum of 36% and the frequency content is spread over the entire bandwidth of the measurement.

  This is due to sharp transients that are created by repeating and appending the measured signal.

Whenever a measurement signal is periodic, a Uniform window is preferred.

Applying a Hanning or Flattop window to a periodic signal will actually create amplitude and frequency distortion.

Benefit of Reducing Leakage

The benefit is not that the captured signal is perfectly replicated.

The main benefit is that the leakage is now confined over a smaller frequency range, instead of affecting the entire frequency bandwidth of the measurement.

With the leakage spread over a smaller frequency range, doing analysis calculations like RMS yields more accurate results.

It is impossible to calculate the proper RMS amplitude estimate over a limited frequency range of the un-windowed sine wave, since the leakage is over the full frequency range. Therefore the RMS amplitude is not correct.

Two tones

In the case of two closely spaced sine tones, without a window being applied, two tones frequencies would leak into each other, which make determining the true amplitude of individual peaks very difficult.

The window makes it easier to separate and distinguish each tone so a proper analysis could be performed.

window function in frequency domain

The transfer function \(a(f)\) of a window \(w_j, j \in [0, N-1]\) expresses the response of the window to a sinusoidal signal at an offset of \(f\) frequency bins, i.e. DFT .

real part: \[ a_r(f)=\sum_{j=0}^{N-1}w_j\cos (2\pi f j/N) \]

imaginary part: \[ a_i(f)=\sum_{j=0}^{N-1}w_j\sin (2\pi f j/N) \]

frequency response can be obtained as \[ a(f) = \frac{\sqrt{a_r^2+a_i^2}}{S_1} \] where \(S_1 = \sum _{k=0}^{N-1}w_k\)

Rectangular window example

aka. Uniform window, "Rectangular" window, "no window"

Whenever a measurement signal is periodic, a Uniform window is preferred. Applying a Hanning or Flattop window to a periodic signal will actually create amplitude and frequency distortion.

1. When \(f=0\)

\[ a_r(f) + ja_i(f) = \sum_{k=0}^{N-1}w_k = N \]

3. When \(f \neq 0\)

\[\begin{align} a_r(f) + ja_i(f) &= \sum_{k=0}^{N-1} e^{\frac{j2\pi k f}{N}} \\ &= \sum_{k=0}^{N/2} e^{\frac{j2\pi k f}{N}} + e^{\frac{j2\pi (k+N/2) f}{N}} \\ &= \sum_{k=0}^{N/2} e^{\frac{j2\pi k f}{N}} + e^{j\pi} e^{\frac{j2\pi k f}{N}} \\ &= \sum_{k=0}^{N/2} e^{\frac{j2\pi k f}{N}} - e^{\frac{j2\pi k f}{N}} \\ &= 0 \end{align}\]

A Uniform window creates no frequency or amplitude distortion when the measured signal is periodic.

However, if the signal cannot be guaranteed to be periodic, a Uniform window should be avoided.

Window Properties

There is no possibility of trade-off between main-lobe width and sied-lobe amplitude, since the window length is the only variable parameter.

The rectangular window has the narrowest main lobe for a given length, i.e. \(\Delta _{ml}=4\pi/L\)

Other windows include the Bartlett, Hann, and Hamming windows. The DTFTs of all these windows have main-lobe width \(\Delta _{ml}=8\pi/(L-1)\), which is approximately twice that of the rectangular window, but they have significantly smaller side-lobe amplitudes.

Demo

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clc;
clear all;

N = 512;
fs = 40*1e3; % 40kHz
fres = fs/N; % 78.125
tt = (0:N-1)*1/fs;
ff = (0:N/2)*fres;
fin = 390.625;

whan = hanning(N); % hanning window
wrect = rectwin(N); % rect window


% fin/fs = 5/N, periodically captured sine wave
wv = cos(2*pi*fin*tt);
power = periodogram(wv, whan, N, fs, 'power');
X = (power).^0.5*2^0.5;
psd = periodogram(wv, whan, N, fs, 'psd');
rms = sum(psd*fres)^0.5;
fprintf('RMS@periodic & hanning: %.5f\n', rms);

power_rect = periodogram(wv, wrect, N, fs, 'power');
X_rect = (power_rect).^0.5*2^0.5;
psd_rect = periodogram(wv, wrect, N, fs, 'psd');
rms_rect = sum(psd_rect*fres)^0.5;
fprintf('RMS@periodic & rect: %.5f\n', rms_rect);


figure(1)
plot(ff, X, 'r-o', ff, X_rect, 'b-s');
xlabel('Frequency(Hz)');
ylabel('|X|')
title('Amplitude spectrum of periodically captured sine wave');
legend('w/ hanning', 'w/ rect');
grid on
grid minor
% rectangular window provide higher frequency resolution
% hanning window induce leakage for the periodically captured sine wave



% fin - 0.5fres
fin_lkg0d5 = fin - 0.5*fres;
wv_lkg0d5 = cos(2*pi*fin_lkg0d5*tt);
power_lkg0d5 = periodogram(wv_lkg0d5, whan, N, fs, 'power');
X_lkg0d5 = (power_lkg0d5).^0.5*2^0.5;
psd_lkg0d5 = periodogram(wv_lkg0d5, whan, N, fs, 'psd');
rms_lkg0d5 = sum(psd_lkg0d5*fres)^0.5;
fprintf('RMS@-0.5fres & hanning: %.5f\n', rms_lkg0d5);

power_lkg0d5_rect = periodogram(wv_lkg0d5, wrect, N, fs, 'power');
X_lkg0d5_rect = (power_lkg0d5_rect).^0.5*2^0.5;
psd_lkg0d5_rect = periodogram(wv_lkg0d5, wrect, N, fs, 'psd');
rms_lkg0d5_rect = sum(psd_lkg0d5_rect*fres)^0.5;
fprintf('RMS@-0.5fres & rect: %.5f\n', rms_lkg0d5_rect);

figure(2)
plot(ff, X_lkg0d5, 'r-o', ff, X_lkg0d5_rect, 'b-s');
xlabel('Frequency(Hz)');
ylabel('|X|')
title('Amplitude spectrum of -0.5fres');
legend('w/ hanning', 'w/ rect');
grid on
grid minor
% hanning reduce leakage and max amplitude error 15%



% fin - 0.25fres
fin_lkg0d25 = fin - 0.25*fres;
wv_lkg0d25 = cos(2*pi*fin_lkg0d25*tt);
power_lkg0d25 = periodogram(wv_lkg0d25, whan, N, fs, 'power');
X_lkg0d25 = (power_lkg0d25).^0.5*2^0.5;
psd_lkg0d25 = periodogram(wv_lkg0d25, whan, N, fs, 'psd');
rms_lkg0d25 = sum(psd_lkg0d25*fres)^0.5;
fprintf('RMS@-0.25fres & hanning: %.5f\n', rms_lkg0d25);

power_lkg0d25_rect = periodogram(wv_lkg0d25, wrect, N, fs, 'power');
X_lkg0d25_rect = (power_lkg0d25_rect).^0.5*2^0.5;
psd_lkg0d25_rect = periodogram(wv_lkg0d25, wrect, N, fs, 'psd');
rms_lkg0d25_rect = sum(psd_lkg0d25_rect*fres)^0.5;
fprintf('RMS@-0.25fres & rect: %.5f\n', rms_lkg0d25_rect);

figure(3)
plot(ff, X_lkg0d25, 'r-o', ff, X_lkg0d25_rect, 'b-s');
xlabel('Frequency(Hz)');
ylabel('|X|')
title('Amplitude spectrum of -0.25fres');
legend('w/ hanning', 'w/ rect');
grid on
grid minor
% hanning reduce leakage

output

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RMS@periodic & hanning: 0.70711
RMS@periodic & rect: 0.70711
RMS@-0.5fres & hanning: 0.70711
RMS@-0.5fres & rect: 0.70711
RMS@-0.25fres & hanning: 0.70711
RMS@-0.25fres & rect: 0.70780

rectangular window provide higher frequency resolution

hanning reduce leakage and max amplitude error 15%

hanning reduce leakage and reduce amplitude error

reference

Windows and Spectral Leakage. URL:https://community.sw.siemens.com/s/article/windows-and-spectral-leakage

Article (20416822) Title: How to Utilize a Windowing Technique for Accurate DFT

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

Window Types: Hanning, Flattop, Uniform, Tukey, and Exponential URL:https://community.sw.siemens.com/s/article/window-types-hanning-flattop-uniform-tukey-and-exponential

Window Correction Factors URL:https://community.sw.siemens.com/s/article/window-correction-factors

Root Mean Square (RMS) and Overall Level. URL:https://community.sw.siemens.com/s/article/root-mean-square-rms-and-overall-level

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

Stefan Scholl, "Exact Signal Measurements using FFT Analysis",Microelectronic Systems Design Research Group, TU Kaiserslautern, Germany. [ pdf ]

Harris, F. (1978). On the use of windows for harmonic analysis with the discrete Fourier transform. Proceedings of the IEEE, 66, 51-83. [pdf]

Equivalent noise bandwidth (ENBW) of window functions URL:https://www.gaussianwaves.com/2020/09/equivalent-noise-bandwidth-enbw-of-window-functions/

Why should I zero-pad a signal before taking the Fourier transform? URL:https://dsp.stackexchange.com/q/741

enbw function in MATLAB URL:https://www.mathworks.com/help/signal/ref/enbw.html

Window function – figure of merits URL:https://www.gaussianwaves.com/2020/09/window-function-figure-of-merits/

Memos on FFT With Windowing URL:https://a2d2ic.wordpress.com/2018/02/01/memos-on-fft-with-windowing/

Jens Ahrens, "Some Notes on Windows in Spectral Analysis," Tech. Report, Chalmers Univeristy of Technology, 2020. URL:https://appliedacousticschalmers.github.io/scaling-of-the-dft/notes_on_windows/