Link Modeling

Posted on Edited on In

Time domain modeling

While many different analysis methods exist, including frequency and statistical analysis, time domain results remain the final sign-off

image-20250824223103058

image-20250807000316790

serdespy

Richard Barrie. serdespy — A python library for system-level SerDes modelling and simulation [https://github.com/richard259/serdespy]

python 3.10, samplerate

ifft of sampling continuous-time transfer function

freq2impulse.drawio

1
2
3
4
5
6
7
8
9
10
11
12
13
14
def freq2impulse(H, f):
    #Returns the impulse response, h, and (optionally) the step response,
    #hstep, for a system with complex frequency response stored in the array H
    #and corresponding frequency vector f.  The time array is
    #returned in t.  The frequency array must be linearly spaced.

    Hd = np.concatenate((H,np.conj(np.flip(H[1:H.size-1]))))
    h = np.real(np.fft.ifft(Hd))
    #hstep = sp.convolve(h,np.ones(h.size))
    #hstep = hstep[0:h.size]
    t= np.linspace(0,1/f[1],h.size+1)
    t = t[0:-1]

    return h,t

Maybe, the more straightforward method is sampling impulse response of continuous-time transfer function directly

image-20251127200007941

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
import numpy as np
from scipy import signal
import matplotlib.pyplot as plt

fbw = 20e9
wbw = fbw * 2 * np.pi
samples_per_symbol = 64
UI = 1/50e9
Ts = UI/samples_per_symbol
fs = 1/Ts
ws = fs * 2 * np.pi
ttot = 4/fbw # heuristic
N = int(0.5 * fs * ttot)+1
w, H = signal.freqs([1], [1/wbw, 1], np.linspace(0, 0.5*ws, N))

## freq2impulse(H, f), ifft - using sample of continuous-time tranfer function
f = w/(2*np.pi)
Hd = np.concatenate((H,np.conj(np.flip(H[1:H.size-1]))))
hd = np.real(np.fft.ifft(Hd))
t= np.linspace(0,1/f[1],hd.size+1)
t = t[0:-1]

## continuous-time transfer function - impulse
t, hc = signal.impulse(([1], [1/wbw, 1]), T = t)

## hd(kTs) = Ts*hc(kTs)
plt.figure(figsize = (14,10))
plt.plot(t, hc, t, hd/Ts, '--r', linewidth=3)
plt.grid(); plt.legend(['impulse response by continuous-time transfer function','impulse response/Ts by ifft of sampling transfer function'])
plt.ylabel('Mag'); plt.xlabel('time (s)'); plt.show()

JLSD

Kevin Zheng. JLSD — Julia SerDes [https://github.com/kevjzheng/JLSD]

boundary conditions internally, remembering states from the previous (sub-)block

1
2
out = conv(ir, vbits)*tui/osr
lines(tt, out[1:length(vbits)])

image-20251201003243976

Kronecker product to create oversampled waveform

1
2
3
4
5
6
7
8
9
function gen_wvfm(bits; tui, osr)
	#helper function used to generate oversampled waveform
	
	Vbits = kron(bits, ones(osr)) #Kronecker product to create oversampled waveform
	dt = tui/osr
	tt = 0:dt:(length(Vbits)-1)*dt

	return tt, Vbits
end

normalized to the time step \[ \frac{\alpha}{s+\alpha} \overset{\mathcal{L}^{-1}}{\longrightarrow} \alpha\cdot e^{-\alpha t} \]

The integral of impulse response of low pass RC filter \(\int_{0}^{+\infty} \alpha\cdot e^{-\alpha t}dt = 1\) — sum(ir*dt)

1
2
3
4
5
6
7
8
9
10
11
function gen_ir_rc(dt,bw,t_len)
	#helper function that directly calculates a first order RC response, normalized to the time step
    tt = [0:dt:t_len-dt;]

	#checkout the intuitive symbols!
    ω = (2*π*bw)
    ir = ω*exp.(-tt*ω)
    ir .= ir/sum(ir*dt)

    return ir
end
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
using Plots
using LaTeXStrings

tui = 1/10e9;
tlen_ir = 20*tui;


osr_list = [4, 8, 16, 32, 64, 128, 256, 512, 1024];
bw_ir = 8e9;

ω = (2*π*bw_ir);

ir_dt_sum_list = [];

for osr_cur in osr_list
    dt = tui/osr_cur; # Simulation time step
    tt = [0:dt:tlen_ir-dt;]
    ir = ω*exp.(-tt*ω);
    ir_dt_sum = sum(ir*dt);
    push!(ir_dt_sum_list, ir_dt_sum)
end

println(ir_dt_sum_list)
# Any[1.756575097878581, 1.3468434968519964, 1.1652908056870317, 1.0805951388547221, 1.0397838972257087, 1.0197634612560418, 1.0098496044545267, 1.0049167704129547, 1.0024563772359665]

p = plot(osr_list, ir_dt_sum_list, label = "OSR")
gui(p)

image-20251008131325508

Elastic Buffer

the elastic buffer approach would be the most general for modeling say frequency offsets between TX and RX (will be addressed in future development)

An_example_of_elastic_buffer

generate PAM symbols

here Big Endian

1
2
3
4
5
#generate PAM symbols
   fill!(So, zero(Float64)) #reset So to all 0
   for n = 1:bits_per_sym
       @. So = So + 2^(bits_per_sym-n)*So_bits[n:bits_per_sym:end]
   end
1
2
3
4
5
6
function int2bits(num, nbit)
    return [Bool((num>>k)%2) for k in nbit-1:-1:0]
end


 Si_bits .= vec(stack(int2bits.(Si, bits_per_sym)))

Detailed Transmitter

tx blk diagram

DaVE

DaVE — tools regarding on analog modeling, validation, and generation, [https://github.com/StanfordVLSI/DaVE]

TODO 📅

Statistical Eye

Sanders, Anthony, Michael Resso and John D'Ambrosia. “Channel Compliance Testing Utilizing Novel Statistical Eye Methodology.” (2004).

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.

HSPICE® User Guide: Signal Integrity Modeling and Analysis, Version Q-2020.03, March 2020

IA Title: Common Electrical I/O (CEI) - Electrical and Jitter Interoperability agreements for 6G+ bps, 11G+ bps, 25G+ bps I/O and 56G+ bps IA # OIF-CEI-04.0 December 29, 2017 [pdf]

J. Park and D. Kim, "Statistical Eye Diagrams for High-Speed Interconnects of Packages: A Review," in IEEE Access, vol. 12, pp. 22880-22891, 2024 [pdf]

StatOpt

Savo Bajic, ECE1392, Integrated Circuits for Digital Communications: StatOpt in Python [https://savobajic.ca/projects/academic/statopt] [https://www.eecg.utoronto.ca/~ali/statopt/main.html]

TODO 📅

pystateye

Chris Li. pystateye - A Python Implementation of Statistical Eye Analysis and Visualization [https://github.com/ChrisZonghaoLi/pystateye]

TODO 📅

noise generator

White Noise + Digital Low-Pass filter

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

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

image-20251207222557118

Colouring Noise

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

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

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

The process of generating coloured noise is relatively simple:

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

helper function

int2bits

[https://github.com/kevjzheng/JLSD/blob/4ac92476b67ce78e01820502eb3b4afb6d31bcd7/src/blks/BlkBIST.jl#L126-L128]

1
2
3
function int2bits(num, nbit)
	return [Bool((num>>k)%2) for k in nbit-1:-1:0]
end

Analog Signals Representation

Ben Yochret Sabrine, 2020, "BEHAVIORAL MODELING WITH SYSTEMVERILOG FOR MIXED-SIGNAL VALIDATION" [https://di.uqo.ca/id/eprint/1224/1/Ben-Yochret_Sabrine_2020_memoire.pdf]

image-20250913234701819

image-20250914115332274

PRBS Generator

image-20251008232419923

[https://opencpi.gitlab.io/releases/latest/rst/comp_sdr/components/generator/prbs_generator_b.comp/prbs_generator_b-index.html]

1
2
3
4
5
6
7
8
9
10
11
12
13
# Julia

function bist_prbs_gen(;poly, inv, Nsym, seed)
    seq = Vector{Bool}(undef,Nsym)
    for n = 1:Nsym
        seq[n] = inv
        for p in poly
            seq[n] ⊻= seed[p]
        end
        seed .= [seq[n]; seed[1:end-1]]
    end
    return seq, seed
end
1
2
3
4
5
6
7
8
9
10
11
12
%% Matlab

function [seq, seed] = bist_prbs_gen(poly,inv, Nsym, seed)
    seq = zeros(1,Nsym);
    for n = 1:Nsym
        seq(n) = inv;
        for p = poly
            seq(n) = xor(seq(n), seed(p));
        end
       seed = [seq(n), seed(1:end-1)];
    end
end

image-20251003110528405

[https://github.com/kevjzheng/JLSD/blob/main/Pluto%20Notebooks/pdf/JLSD_pt1_background.pdf]

PRBS Checker

[https://github.com/kevjzheng/JLSD/blob/4ac92476b67ce78e01820502eb3b4afb6d31bcd7/Pluto%20Notebooks/jl/JLSD_pt2_framework.jl#L198-L235]

previous bit determine current bit

bert.drawio

  1. current LSFR generate btst
  2. compare bst with current brcv
  3. push current brcv into LSFR

Reference

MATLAB® and Simulink® RF and Mixed Signal [https://www.mathworks.com/help/overview/rf-and-mixed-signal.html]

Lim, Byong Chan, M. Horowitz, "Error Control and Limit Cycle Elimination in Event-Driven Piecewise Linear Analog Functional Models," in IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 63, no. 1, pp. 23-33, Jan. 2016 [https://sci-hub.se/10.1109/TCSI.2015.2512699]

—, Ph.D. Dissertation 2012. "Model validation of mixed-signal systems" [https://stacks.stanford.edu/file/druid:xq068rv3398/bclim-thesis-submission-augmented.pdf]

—, J. -E. Jang, J. Mao, J. Kim and M. Horowitz, "Digital Analog Design: Enabling Mixed-Signal System Validation," in IEEE Design & Test, vol. 32, no. 1, pp. 44-52, Feb. 2015 [http://iot.stanford.edu/pubs/lim-mixed-design15.pdf]

— , Mao, James & Horowitz, Mark & Jang, Ji-Eun & Kim, Jaeha. (2015). Digital Analog Design: Enabling Mixed-Signal System Validation. Design & Test, IEEE. 32. 44-52. [https://iot.stanford.edu/pubs/lim-mixed-design15.pdf]

S. Liao and M. Horowitz, "A Verilog piecewise-linear analog behavior model for mixed-signal validation," Proceedings of the IEEE 2013 Custom Integrated Circuits Conference, San Jose, CA, USA, 2013 [https://sci-hub.se/10.1109/CICC.2013.6658461]

—, M. Horowitz, "A Verilog Piecewise-Linear Analog Behavior Model for Mixed-Signal Validation," in IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 61, no. 8, pp. 2229-2235, Aug. 2014 [https://sci-hub.se/10.1109/TCSI.2014.2332265]

—,Ph.D. Dissertation 2012. Verilog Piecewise Linear Behavioral Modeling For Mixed-Signal Validation [https://stacks.stanford.edu/file/druid:pb381vh2919/Thesis_submission-augmented.pdf]

Ji-Eun Jang et al. “True event-driven simulation of analog/mixed-signal behaviors in SystemVerilog: A decision-feedback equalizing (DFE) receiver example”. In: Proceedings of the IEEE 2012 Custom Integrated Circuits Conference. 2012 [https://sci-hub.se/10.1109/CICC.2012.6330558]

—, Si-Jung Yang, and Jaeha Kim. “Event-driven simulation of Volterra series models in SystemVerilog”. In: Proceedings of the IEEE 2013 Custom Integrated Circuits Conference. 2013 [https://sci-hub.se/10.1109/CICC.2013.6658460]

—, Ph.D. Dissertation 2015. Event-Driven Simulation Methodology for Analog/Mixed-Signal Systems [file:///home/anon/Downloads/000000028723.pdf]

"Creating Analog Behavioral Models VERILOG-AMS ANALOG MODELING" [https://www.eecis.udel.edu/~vsaxena/courses/ece614/Handouts/CDN_Creating_Analog_Behavioral_Models.pdf]

Rainer Findenig, Infineon Technologies. "Behavioral Modeling for SoC Simulation Bridging Analog and Firmware Demands" [https://www.coseda-tech.com/files/Files/Dokumente/Behavioral_Modeling_for_SoC_Simulation_COSEDA_UGM_2018.pdf]

CC Chen. Why Efficient SPICE Simulation Techniques for BB CDR Verification? [https://youtu.be/Z54MV9nuGUI]

T. Wen and T. Kwasniewski, "Phase Noise Simulation and Modeling of ADPLL by SystemVerilog," 2008 IEEE International Behavioral Modeling and Simulation Workshop, San Jose, CA, USA, 2008 [slides, paper]

Jaeha Kim,Scientific Analog. UCIe PHY Modeling and Simulation with XMODEL [pdf]

S. Katare, "Novel Framework for Modelling High Speed Interface Using Python for Architecture Evaluation," 2020 IEEE REGION 10 CONFERENCE (TENCON), Osaka, Japan, 2020 [https://sci-hub.se/10.1109/TENCON50793.2020.9293846]