KNOW-HOW

Assuming Target \(T\) ( for example, the total resistance) is function of \(x_1,x_2,...,x_N\), then total variation can be expressed as

\[\begin{align} dT &= \sum_{n=1}^N\frac{\partial T}{\partial x_n}dx_n \\ &= \sum_{n=1}^N\frac{\partial T}{\partial x_n}x_n\cdot \frac{dx_n}{x_n} \end{align}\]

Then, we obtain relative variation \[\begin{align} \frac{dT}{T} &= \sum_{n=1}^N\frac{\partial T}{\partial x_n}\frac{x_n}{T}\cdot \frac{dx_n}{x_n} \\ &= \sum_{n=1}^N S_{x_n}^T \cdot \frac{dx_n}{x_n} \end{align}\]

⭐ where \(S_{x_n}^T=\frac{\partial T}{\partial x_n}\frac{x_n}{T}\) is relative sensitivity

relative sensitivity connect \(\frac{dx_n}{x_n}\) with total relative variation \(\frac{dT}{T}\)

And \(dT\) can be expressed as \[ dT =\sum_{n=1}^N S_{x_n}^T T\cdot \frac{dx_n}{x_n} = \sum_{n=1}^N x_n'\cdot \frac{dx_n}{x_n} \] ⭐ where \(x_n'= S_{x_n}^T T\) is the contribution of \(x_n\) in \(T\)

⭐ For parallel or series resistors, it can prove \(\sum_{n=1}^N S_{x_n}^T = 1\) and $ _{n=1}^N x_n'=T$

Here \(T= R_1 \parallel R_2 = \frac{R_1R_2}{R_1+R_2}\), and \(T|_{R_1=8000, R_2=2000} = 1600\)

We obtain relative sensitivity: \[\begin{align} S_{R_1}^T & = \frac{R_2}{R_1+R_2} \\ S_{R_2}^T & = \frac{R_1}{R_1+R_2} \end{align}\]

The contribution of \(R_1\) and \(R_2\) to \(T\) \[\begin{align} R_1' &= S_{R_1}^T T | _{R_1=8000, R_2=2000} = 320 \\ R_2' &= S_{R_2}^T T | _{R_1=8000, R_2=2000} = 1280 \end{align}\]

scholar

Normalized sensitivity captures relative sensitivity

change in objective per change in design variable

reference

Olivier de Weck, Karen Willcox. MIT, Gradient Calculation and Sensitivity Analysis [pdf]

Karti Mayaram, ECE 521 Fall 2016 Analog Circuit Simulation, Sensitivity and noise analyses [https://web.engr.oregonstate.edu/~karti/ece521/lec16_11_09.pdf]

Transit frequency \(f_T\)

aka cut-off frequency

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 resistance handling by parasitic extraction tools

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]

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.

The repository visibility shall be public

realtime vs time

• $realtime round the current time to timeprecision

• $time round the current time to integer

• %t will scale the rounded value to represent timeprecision,

  i.e. \([\$\text{realtime}, \$\text{time}]\cdot \$\text{timeunit} / \$\text{timeprecision}\)

https://verificationacademy.com/forums/systemverilog/time-vs-realtime#answer-94062 https://verificationacademy.com/forums/systemverilog/time-vs-realtime#answer-94096

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module tb;
  timeunit 10ns;
  timeprecision 1ps;

  initial begin
    $display("$realtime = %g", $realtime);
    $display("$time = %g", $time);
    // $realtime round to timeprecision
    // $time round to integer
    #1.10016
    $display("$realtime = %g", $realtime);
    $display("$time = %g", $time);

    // %t format will scale the rounded value to represent timeprecision
    // 1.1002*10e-9/1e-12 = 11002
    $display("$realtime %%t = %t", $realtime);
    $display("$time %%t = %t", $time);
  end
endmodule

output

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$realtime = 0
$time = 0
$realtime = 1.1002
$time = 1
$realtime %t =                11002
$time %t =                10000

timeunit, timeprecision

The time unit and time precision can be specified in the following two ways: - Using the compiler directive `timescale - Using the keywords timeunit and timeprecision

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module D (...);
  timeunit 100ps;
  timeprecision 10fs;
  ...
endmodule

module E (...);
  timeunit 100ps / 10fs; // timeunit with optional second argument
  ...
endmodule

The minimum of timeprecision determine %t output, the nearest timeunit and timeprecision determine the round of $realtime and $time. Of course, the simulator follow the time tick shown by $realtime.

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`timescale 1ns/1ns

// Due to minimum of timeprecision is 10ps
module rawplant;
  timeunit 1ns;
  timeprecision 100ps;

  task print;
    /*
    1ns: 1
    100ps:  6
    10ps: 6
    1ps: 6
    */
    #1.666
    $display("raw: $realtime = %g", $realtime);     // 1.7
    $display("raw: $time = %g", $time);             // 2
    $display("raw: $realtime %%t = %t", $realtime); // 1.7*1ns/10ps=170
    $display("raw: $time %%t = %t", $time);         // 2*1ns/10ps = 200
  endtask

endmodule

module fineplant;
  timeunit 100ps;
  timeprecision 10ps;

  task print;
    /*
    100ps: 2
    10ps: 6
    1ps: 6
    */
    #2.66
    $display("fine: $realtime = %g", $realtime);     // 2.7
    $display("fine: $time = %g", $time);             // 3
    $display("fine: $realtime %%t = %t", $realtime); // 2.7*100ps/10ps = 27
    $display("fine: $time %%t = %t", $time);         // 3*100ps/10ps = 30
  endtask

endmodule

module tb;
  rawplant rawblock();
  fineplant fineblock();

  initial begin
  fork
    rawblock.print();
    fineblock.print();
  join
  end
endmodule

output

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fine: $realtime = 2.7
fine: $time = 3
fine: $realtime %t =                   27
fine: $time %t =                   30
raw: $realtime = 1.7
raw: $time = 2
raw: $realtime %t =                  170
raw: $time %t =                  200

questasim cmd

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vlib work
vlog tb.v
vsim -c -do "run -all;exit" work.tb

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 \(PS_{n}=\left| \frac{X_{v,n}}{\sum_n w[n]}\right|^2\)

code 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.

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

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

Noise Floor

General Formula

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

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

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

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

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

+/- frequency spectrum and apply Parseval's theorem

noise power \[\begin{align} P_n &= \sum_{k=0}^{N-1}{\frac{(X_{w,n}(k))^2}{N^2}} \\ &= \frac{X_{w,n}^2}{N^2}\cdot N \\ &= \frac{X_{w,n}^2}{N} \end{align}\]

note white noise, that is \(X_n(i) = X_n(j)\) for any \(i \neq j\)

displayed SNR \[\begin{align} \mathrm{SNR} &= 10\log10\left(\frac{X_{w,sig}^2}{X_{w,n}^2}\right) \\ &= 10\log_{10}\left(\frac{P_{\text{sig}}N^2}{P_n2N}\right) \\ &= 10\log_{10}\left(\frac{P_{\text{sig}}}{P_n}\right) + 10\log_{10}\left(\frac{N}{2}\right) \\ &= \mathrm{SNR}' - 10\log_{10}(2/N) \end{align}\]

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.

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

No. of nonzero FFT bins

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]

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

The number of nonzero signal bins for the Hann window is \(\mathrm{nb} = 3\)

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))

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

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]

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

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

Substrate noise

A variety of techniques can be used to minimize this noise, for example by keeping analog devices surrounded by guard rings, or using a separate supply for the substrate/well taps.

However guard rings alone cannot prevent noise coupling deep in the substrate, only surface currents.

PMOS are less noisy than NMOS since PMOS has its nwell which isolates the substrate noise, but such is not valid for NMOS .

DNW

The N-channel devices built directly into the P-type substrate are not as effectively isolated as P-channel devices in their N-wells. This is because despite creating a P+ guard ring around the devices, there remains an electrical path below the guard ring for charge to flow.

To overcome this issue, a deep N-well can be used to more effectively isolate these N-channel devices.

pwdnw: PW/DNW diode

dnwpsub: DNW/PSUB diode

Together At Last – Combining Netlist and Layout Data for Power-Aware Verification

• the P-well is separated, allowing the voltage to be controlled
• because the circuit within the deep N-well is separated from the p-substrate in this structure, there is the benefit that this circuitry is less susceptible to noise that propagates through the p-substrate.

reference

Chew, K.W., Zhang, J., Shao, K., Loh, W., & Chu, S.F. (2002). Impact of Deep N-well Implantation on Substrate Noise Coupling and RF Transistor Performance for Systems-on-a-Chip Integration. 32nd European Solid-State Device Research Conference, 251-254. URL:[slides, paper]

Mark Waller, Analog layout: Why wells, taps, and guard rings are crucial

KEITH SABINE Using Deep N Wells in Analog Design

Faricelli, J. (2010). Layout-dependent proximity effects in deep nanoscale CMOS. IEEE Custom Integrated Circuits Conference 2010, 1-8.

cmos_processing, URL:http://users.ece.utexas.edu/~athomsen/cmos_processing.pdf

Kuo-Tsai LiPaul ChangAndy Chang, TSMC, US20120053923A1, "Methods of designing integrated circuits and systems thereof"

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.

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 \(S_1 = \sum _{k=0}^{N-1}w_k\)

rectangle 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 URL: https://support.cadence.com/apex/ArticleAttachmentPortal?id=a1Od000000050UrEAI

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/

Traditional RC Corners

DPT effect

When using two masks per layer (Double Patterning Technology, DPT) there is an issue of mask alignment where any mis-alignment will cause layer spacing values to change, therefore changing the parasitic coupling capacitance values.

Misalignment scale and direction are not deterministic facts: coupling cap and total cap may be increased or decreased.

Five new corners are added in a DPT flow to account for RC variations accurately:

sapced-dependent side-wall dielectric constant also affect coupling cap

and CC_worst means to increase both K1 and K2

CC_best means to decrease both K1 and K2

• Setup time sign-off would use:

  Cworst_CCworst / RCworst_CCworst

• Hold time sign-off would use:

  Cbest_CCbest / RCbest_CCbest / Cworst_CCworst / RCworst_CCworst

Signoff corner

BEOL Target: typical

The recommended RC corner:

​ cworst_CCworst, cbest_CCbest, rcworst_CCworst, rcbest_CCbest and typical

The others are for pre-color RC calculation purpose

**_T** stands for "Tighten DPT corner"; these are less pessimistic 1.5 sigma corners

Below table is caputre of Aragio's TSMC16: LVDS datasheet

BEOL corner

Spacing variation is implicitly defined by \(\Delta W_m\).

We denote the conductor width and thickness of the layer m by \(W_m\) and \(T_m\), respectively.

Similarly, we denote the thickness of the layer's interlayer dielectric (i.e., the distance between layer m and layer m +1) by \(H_m\)

• C-based means worst and best caps
• RC-based means worst and best R in adjustment with C (RC product)

Based on experience, it was found that C-based extraction provides worst and best case over RC for internal timing paths because Capacitance dominates short wire.

However, for large design, inter-block timing paths were often worst with RC worst parasitic since R dominates for long wires.

signoff corners for setup and hold

reference

Modeling Sub-90nm On-Chip Variation Using Monte Carlo Method for DFM https://www.aspdac.com/aspdac2007/pdf/archive/2D-1.pdf

Double Patterning for IC Design, Extraction and Signoff https://semiwiki.com/eda/synopsys/1974-double-patterning-for-ic-design-extraction-and-signoff/

抽刀断水水更流，RC Corner不再愁：STA之RC Corner URL: http://mp.weixin.qq.com/s?__biz=MzUzODczODg2NQ==&mid=2247484115&idx=1&sn=de99f27aadf58ea316c284dad9000b7c&chksm=fad26b0dcda5e21b8c9750f738b55053f695843a66c3c202ff0ba586c738f45aa270254c3722&scene=21#wechat_redirect

一曲新词酒一杯，RC Corner继续飞: STA之RC Corner拾遗 URL:https://mp.weixin.qq.com/s?__biz=MzUzODczODg2NQ==&mid=2247484135&idx=2&sn=bddc632850bd10c32b5688fd7af46218&chksm=fad26b39cda5e22f1c3970f8c8c2e1287c9492c526c4caf02b61f61faffdf829381c392d6ea1&scene=21#wechat_redirect

且将新火试新茶，深究趁年华：STA之RC Corner再论 URL:https://mp.weixin.qq.com/s?__biz=MzUzODczODg2NQ==&mid=2247484144&idx=1&sn=059843381e77cd4008d25166db388d02&chksm=fad26b2ecda5e23816b33b3a949f34d4ca09118bad76f1089dfcb228b7da6491f423a5f4e703&cur_album_id=1326356275000705025&scene=189#wechat_redirect

LDP_IN_800_25V_DN: 1.2GHz LVDS Receiver http://aragiosolutions.com/pdf/rgo_tsmc16_18v25_lvds_product_brief_rev_1a.pdf

Parasitic extraction technologies Advanced node and 3D-IC design https://static.sw.cdn.siemens.com/siemens-disw-assets/public/81845/en-US/Siemens-SW-Parasitic-extraction-technologies-for-advanced-WP-81845-C2.pdf

New Game, New Goal Posts: A Recent History of Timing Closure https://pdfs.semanticscholar.org/9360/5ce48f9bd3b7527ae8979f41a9c7e310efa4.pdf

The Evolution, Pitfalls, and Cargo Cult Engineering of Advanced Digital Timing Sign-off https://www.tauworkshop.com/2021/speaker_slides/christian_l.pdf

T. -B. Chan, S. Dobre and A. B. Kahng, "Improved signoff methodology with tightened BEOL corners," 2014 IEEE 32nd International Conference on Computer Design (ICCD), Seoul, Korea (South), 2014, pp. 311-316, doi: 10.1109/ICCD.2014.6974699.

Chan, T. (2014). Mitigation of Variability and Reliability Margins in IC Implementation /. UC San Diego. ProQuest ID: Chan_ucsd_0033D_14269. Merritt ID: ark:/20775/bb52916761. Retrieved from https://escholarship.org/uc/item/35r1m001

Dr. Adam Temanm, Digital VLSI Design:Lecture 10: Routing https://www.eng.biu.ac.il/temanad/files/2017/02/Lecture-10-Routing.pdf

Article (20487193) Title: Setting Pegasus - LVS to Quantus av_extracted view Flow with TSMC16 packages URL: https://support.cadence.com/apex/ArticleAttachmentPortal?id=a1O0V000009MprZUAS

Clock Gating is defined as: "Clock gating is a technique/methodology to turn off the clock to certain parts of the digital design when not needed".

Clock Gating Overview

AND gate-based clock gating

In simplest form a clock gating can be achieved by using an AND gate as shown in picture below

However, this simplest form of clock gating technique has some problem of generating glitches in the clock provide to the FF, which are not desirable.

Glitches in enable/gated clock

Latch based clock gating

These glitches can be removed by introducing a negative edge triggered FF (assuming downstream FFs are positive edge) or low-level sensitive latch at the output of the clock enable signal.

This will make sure that any glitch in the clock enable signal will not be visible to the gated clock output. The Latch output will only be updated during the negative clock cycle and thus input to AND gate will be stable high.

Glitch Free Gated Clock

reference

The Ultimate Guide to Clock Gating https://anysilicon.com/the-ultimate-guide-to-clock-gating/

To preserve the hand-instantiated cells

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set_dont_touch [get_cells -hierarchical *dont_touch_*]

The instances whose name contain "dont_touch_" shall be preserved during synthesis

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// no performace concerns, rest sync use sync3 is enough

module CN_resetb_sync_cell
(
    input resetb_in,
    input clkdst,
    output resetb_out
);

`ifdef USE_VERILOG
reg [2:0] resetb_dly;
`else
wire [2:0] resetb_dly;
`endif

`ifdef USE_VERILOG
always @(posedge clkdst or negedge resetb_in)
    if (~resetb_in) resetb_dly <= 3'b000;
    else resetb_dly <= {resetb_dly[1:0], 1'b1};
`else
SDFCNQD4 dont_touch_sync_flop0 (
    .SI(1'b0),
    .SE(1'b0),
    .CP(clkdst),
    .CDN(resetb_in),
    .D(1'b1),
    .Q(resetb_dly[0])
);
SDFCNQD4 dont_touch_sync_flop1 (
    .SI(1'b0),
    .SE(1'b0),
    .CP(clkdst),
    .CDN(resetb_in),
    .D(resetb_dly[0]),
    .Q(resetb_dly[1])
);
SDFCNQD4 dont_touch_sync_flop2 (
    .SI(1'b0),
    .SE(1'b0),
    .CP(clkdst),
    .CDN(resetb_in),
    .D(resetb_dly[1]),
    .Q(resetb_dly[2])
);
`endif

assign resetb_out = resetb_dly[2];

endmodule