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Machine Learning with X Parameters for Behavioral Model Synthesis

Jose Schutt-AineUniversity of Illinois

• Behavioral models are more efficient.• Behavioral models protect the intellectual property• Current behavioral models for nonlinear devices are

not very accurate.

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Motivation

• ApplicationsHigh-speed links, power amplifiers, mixed-

signal circuits

• Existing Characterization MethodsLoad pull techniquesIBIS modelsModels are flawed and incomplete

Motivation

4

Handling NonlinearitiesConceptual Diagram Tri-State Buffer

Pull-Up CurvePull-Down Curve

• 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

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.

XA1 B2

X-Parameters*

*X-Parameters is a trademark of Keysight Technologies.

(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!

Generate X parameters for composite systemPower level: 20 dBm, frequency: 1 GHzConstruct X matrixCombine with terminations for simulation

CMOS Driver/Receiver Channel

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…

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

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

• 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

• 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

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

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

• 3 Technologies Connected to Large DataX parametersVolterra SeriesMachine Learning

• Benefits to IndustryIP ProtectionPowerful Platform for Exchanging Models

Conclusions

Thank You