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Machine Learning with X Parameters for Behavioral Model Synthesis
Jose Schutt-AineUniversity of Illinois
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• Behavioral models are more efficient.• Behavioral models protect the intellectual property• Current behavioral models for nonlinear devices are
not very accurate.
Veeeee AVeeeeeC
Veeeee BFeeeeee Ie Heeee FeeeeeeIe Heeee
Motivation
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• ApplicationsHigh-speed links, power amplifiers, mixed-
signal circuits
• Existing Characterization MethodsLoad pull techniquesIBIS modelsModels are flawed and incomplete
Motivation
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4
Handling NonlinearitiesConceptual Diagram Tri-State Buffer
Pull-Up CurvePull-Down Curve
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• IBIS models can be difficult to generate, especially without revealing IP to the model generator.– s2ibis3 is still the open-source standard for simulated
IBIS generation.• Generate models via X parameters and ML
– X Parametershandle nonlinearities– Machine Learningnavigate through huge data sets
• Value and Relevance to Industry– Approach will protect IP– Models will be more accurate.– Framework will facilitate exchange between vendors
and suppliers
Rationale for Machine Learning
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Why X Parameters?• X parameters:
– Are behavioral, protect IP.– Are the mathematical superset of S
parameters.– Can describe nonlinear effects.– Can be measured with NVNAs.
• Would like for designers to be able to exchange X-parameter files and generate IBIS models from themx2ibis– This research will create the framework
that will facilitate such exchange.
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XA1 B2
X-Parameters*
*X-Parameters is a trademark of Keysight Technologies.
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(11) (11) (12) (12) (11) (11) (12) (12)11 11 11 11 12 12 12 12(11) (11) (12) (12) (11) (11) (12) (12)11 11 11 11 12 12 12 12(21) (21) (22) (22) (21) (21)11 11 11 11 12 12
rr ri rr ri rr ri rr ri
ir ii ir ii ir ii ir ii
rr ri rr ri rr ri
X X X X X X X XX X X X X X X XX X X X X X
=X
(21) (21)12 12
(21) (21) (22) (22) (21) (21) (22) (22)11 11 11 11 12 12 12 12(11) (11) (12) (12) (11) (11) (12) (12)21 21 21 21 22 22 22 22(11) (11) (12) (12)21 21 21 21 2
rr ri
ir ii ir ii ir ii ir ii
rr ri rr ri rr ri rr ri
ir ii ir ii
X XX X X X X X X XX X X X X X X XX X X X X (11) (11) (12) (12)
2 22 22 22(21) (21) (22) (22) (21) (21) (22) (22)21 21 21 21 22 22 22 22(21) (21) (22) (22) (21) (21) (22) (22)21 21 21 21 22 22 22 22
ir ii ir ii
rr ri rr ri rr ri rr ri
ir ii ir ii ir ii ir ii
X X XX X X X X X X XX X X X X X X X
- Need many harmonics- Need wide bandwidth (many frequencies)- Need sufficient input range (many power levels)
*DC term not included
real matrix2 harmonics
X Matrix for 2-Port System*b = Xa
Data Set is Large!
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Generate X parameters for composite systemPower level: 20 dBm, frequency: 1 GHzConstruct X matrixCombine with terminations for simulation
CMOS Driver/Receiver Channel
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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-8
-6
-4
-2
0
2
4
6
8
time(ns)
Volts
Time-Domain Response
VinVout
25.2 25.4 25.6 25.8 26.0 26.2 26.4 26.6 26.8 27.0 27.2 27.4 27.6 27.8 28.0 28.2 28.4 28.6 28.8 29.0 29.2 29.4 29.6 29.825.0 30.0
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
-7
7
time, nsec
Vin
, VV
out,
V
X Parameter (Behavioral) Simulation
Transistor-Level (ADS) Simulation
ValidationX Parameters can easily simulate steady-state behavior…
Transient simulation is a challenge…
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A linear causal system with memory can be described by the convolution representation
( ) ( )( )y t h x t dσ σ σ+∞
−∞= −∫
where x(t) is the input, y(t) is the output, and h(t)the impulse response of the system.
A nonlinear system without memory can be described with a Taylor series as:
[ ]1
( ) ( ) nn
ny t a x t
∞
=
=∑where x(t) is the input and y(t) is the output. The an are Taylor series coefficients.
Volterra Series
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A Volterra series combines the above two representations to describe a nonlinear system with memory
( )1 111
1( ) ... ,..., ( )!
n
n n n rrn
y t du du h u u x t un
∞ ∞∞
== −∞ −∞
= −∑ Π∫ ∫
1 2 2 1
1 1 1
2 1 2
1 2 3 3 1 2 3 1 2 2 1 2 1 2 3
1( 1 ( ) (
1 ( , ) ( ) ( )2
)
!
1 ( , , ) ( ) ( ) ( , ) ( ) ( ) ( )2!.
!
.
)1
.
du du h u u x t u x t u
du du du h u u u x t u x t u h u u x t u x
du h u x ty
t u x t
t u
u
∞ ∞
−∞ −∞
∞ ∞ ∞
−∞ −∞
∞
∞
−
−
∞
+ − −
+ − − − −
= −
−
+
∫
∫ ∫
∫ ∫ ∫
where x(t) is the input and y(t) is the output and the hn(u1,…,un) are called the Volterra kernels
impulse response
higher-order impulse responses
Volterra Series
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• The number of kernels in a Volterra expansion grows dramatically with the number of harmonics
• The data size becomes very large and un-manageable for X parameters and Volterraexpansion
• Machine learning (ML) techniques can help with the rapid extraction of Volterra kernels from X-parameter data
X Parameters and Volterra
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• Frequency-Domain Inverse TransformRational Function Approximation (poles &
residues)Impulse Function Approximation
• Time-Domain High-speed links, power amplifiers, mixed-
signal circuits
Methods for Kernel Extraction
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Neural Network to learn dynamical behavior
• For a subset of p-port dynamical systems of order N, data 𝑿𝑿𝒊𝒊 𝒊𝒊=𝟏𝟏,𝟐𝟐,… , 𝒌𝒌 ∈ ℂ𝒌𝒌×𝑴𝑴×𝒑𝒑𝟐𝟐 is collected (e.g: k sets of S-parameters, M frequency points each)
• Assume ∃𝒇𝒇 ∈ ℂ𝒌𝒌×𝑵𝑵:𝑿𝑿 𝑷𝑷,𝑹𝑹 ∈ ℂ𝑵𝑵,ℂ𝒑𝒑×𝑵𝑵 such that𝒍𝒍𝒊𝒊𝒍𝒍𝒌𝒌→∞
𝑯𝑯𝒇𝒇 − 𝑿𝑿 = 𝟎𝟎
Where, for a given hypothesis 𝒇𝒇,
𝑯𝑯𝒇𝒇 𝒊𝒊,𝒊𝒊=𝟏𝟏,..,𝒑𝒑𝟐𝟐= �
𝒍𝒍=𝟏𝟏
𝑵𝑵𝒓𝒓𝒊𝒊𝒍𝒍
𝒋𝒋𝝎𝝎 − 𝒑𝒑𝒊𝒊𝒍𝒍• For S-parameter, once poles and residues are found, they can fully represent the system• For X-parameter, once poles and residues are found, they only represent the LTI system
behind a non-linearity.
Pole/residue learner
Rational matrix builder
ℋ2 cost Optim
Cost function
�𝑺𝑺
𝑺𝑺
𝑺𝑺, 𝒇𝒇
𝒑𝒑
𝒓𝒓 𝑒𝑒
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Neural Network to learn dynamical behavior
• Tested on multiple sets of S-parameter of interconnect circuits.• Able to extracted poles and residues with 10% error.• Future improvements:
– Improve the speed by vectorizing the rational matrix builder block.– Handle real-valued poles/residues.
Pole/residue learner
Rational matrix builder
ℋ2 cost Optim
Cost function
�𝑺𝑺
𝑺𝑺
𝑺𝑺, 𝒇𝒇
𝒑𝒑𝒓𝒓 𝑒𝑒
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• 3 Technologies Connected to Large DataX parametersVolterra SeriesMachine Learning
• Benefits to IndustryIP ProtectionPowerful Platform for Exchanging Models
Conclusions
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Thank You