Macromodels for simulation-based verification of Systems on Chip and Systems in Package

(1)

Porto Institutional Repository

[Doctoral thesis] Macromodels for simulation-based verification of Systems

on Chip and Systems in Package

Original Citation:

Olivadese, Salvatore Bernardo (2014). Macromodels for simulation-based verification of Systems

on Chip and Systems in Package. PhD thesis

Availability:

This version is available at :

http://porto.polito.it/2535095/

since: March 2014

Published version:

DOI:

10.6092/polito/porto/2535095

Terms of use:

This article is made available under terms and conditions applicable to Open Access Policy Article

("Public - All rights reserved") , as described at

http://porto.polito.it/terms_and_conditions.

html

Porto, the institutional repository of the Politecnico di Torino, is provided by the University Library

and the IT-Services. The aim is to enable open access to all the world. Please

share with us

how

this access benefits you. Your story matters.

(2)

SCUOLA DI DOTTORATO

Dottorato in Ingegneria Elettroni a e delle

Comuni azioni - XXV i lo

Tesi di Dottorato

Ma romodels for simulation-based

veri ation of System on Chip and

System in Pa kage

Candidato

Salvatore Bernardo Olivadese

Tutore

Prof. Stefano Grivet Talo ia

(3)

(4)

In re ent years the fo us on ele troni integration shifted from high

perfor-man e mi ropro essors, whose integration trend is di tated by the famous

Moore law, to System on Chip (SoC) and System inPa kage (SiP) for

mo-bile and embedded appli ations. The most ommon example of SoC an

be found in smartphones and tablets: multi ore CPU (Central Pro essing

Unit) and GPU (Graphi s Pro essing Unit), memory and Radio Frequen y

(RF) trans eivers are often integrated in the same dieor pa kage leading to

tremendous redu tion in size and power onsumption of the devi e.

There-foreSoCs/SiPsarebydenitionheterogeneousele tri alsystems,inthesense

thatanaloganddigital omponentsforRFandBaseBand(BB)appli ations

are losely tied together.

To blend su h a variety of omponents in the same ele troni pa kage

engineers fa enew di ulties both in designand veri ation phases. Signal

and Powerintegrityneedtobe arefullyaddressed in onjun tion withnoise

levelsto addressdevi es onstraints. In the ontext of Analog MixedSignal

(AMS) validation, analog blo ks are still the simulation time bottlene ks.

Themainissuesare: thehuge omplexityoftheparasiti networksextra ted

from omponents layouts and inter onne ts, the need of parametri models

for non-linear omponents for what-if analyses, the need of redu ed order

models for devi es having huge ports ount like Power Delivery Networks

(PDNs) and pa kages and the la k of low omplexity noise omplaint

syn-thesis methodsfor linear ma romodels. Although tremendoussteps forward

were a hieved in the last de ades in the areas of system identi ation and

modelorder redu tion there are still han es for improvement.

Inthisthesisthestateoftheartfromsystemidenti ationofLinearTime

Invariant(LTI)systems isrevised and improved tailoringthe needs of AMS

simulationsfor SoC/SiPappli ations: a newsystem identi ationalgorithm

to opewithlinear omponentshavinghugedynami alorderandports ount

(more than two order of magnitudes) is proposed and passivity onstraints

are veried and imposed by means of parallel algorithms. The

identi a-tion of parametri linear models is extended to parameterized small-signal

models fornon-lineardevi es. Finallyalow- omplexitynoise ompliant

syn-thesisalgorithmisintrodu edinordertoexportthema romodelsinstandard

SPICE-based solvers. The main ontributions of this work are: redu tion of

simulation time for the veri ation of modern SoCs/SiPs, introdu tion of

parameterized small-signalmodels fornon-linear RF omponentsenablinga

(5)

form of noise ompliantnetworks.

We are fa ing the rise of a new era for onsumer ele troni , and

time-to-market is a key feature in the development of new produ ts. Therefore

the availability of ee tive Analog Mixed Signal methodologies be omes a

sustainable ompetitive advantage for ompanies that are willing to lead

these new market segments. The novel algorithms proposed in this work

were proved tobeof pra ti alrelevan e inthat sense.

Most part of the material presented in this work is based on a resear h

a tivity arriedoutatthe Muni hsite ofIntelMobileCommuni ation. Asa

onsequen e the methodologies proposed here, arising from pra ti alneeds,

weretestedonseveral ommer ialben hmarksdemonstratingtheimportan e

(6)

I would like to thank the sta of Intel Mobile Communi ation in Muni h

for the hospitality andfor the toolsoered during my PhD.A spe ial thank

goes to my Intel supervisor Pietro Brenner and to my manager Alexander

Ruehl for making my PhD at Intel possible; to Gianni Signorini (Intel),

Mi helangelo Bandinu (IdemWorks), Alessandro Chinea (IdemWorks) and

Piero Triverio (University of Toronto) for the support and the suggestions

during the development of my work.

Last but not least, the most spe ial thank is reserved tomy PhD

super-visor, Professor Stefano Grivet-Talo ia, for the patien e and the invaluable

(7)

1 System on Chip for mobile appli ations 1

1.1 History and marketperspe tives . . . 1

1.2 Design hallenges . . . 2

1.3 Ma romodeling and Design ow . . . 5

1.3.1 IP reuse . . . 8

1.3.2 Ma romodels taxonomy . . . 10

1.4 Proposed advan es . . . 14

2 Linear Time Invariant ma romodels 17 2.1 State-spa e ma romodels . . . 17

2.2 Sanathanan-Koerner and Ve tor Fitting. . . 20

2.3 Compressed ma romodeling . . . 23

2.3.1 SVD-based ompression . . . 24

2.3.2 Fittingthe basis fun tions . . . 26

2.3.3 Compressed tting examples . . . 30

2.3.4 Passivity of ompressed ma romodels . . . 33

2.3.5 Asymptoti passivity enfor ement . . . 34

2.3.6 Numeri al Results . . . 37

2.4 Global passivity enfor ement . . . 37

2.4.1 Passivity enfor ement. . . 39

2.4.2 A ura y ontrol . . . 41

2.4.3 Passivity enfor ement examples . . . 44

2.4.4 A summary of numeri alresults . . . 46

2.5 Parallel passivity he k . . . 48

2.5.1 A ura y- ontrolled samplingvia eigenve tor tra king . 51 2.5.2 ParallelAdaptiveSampling . . . 52

2.5.3 Lo alpassivity he k . . . 57

2.5.4 Optimizations . . . 58

2.5.5 Parallelpassivity he k results . . . 60

(8)

3 Small-signal and P-LTI ma romodels 66

3.1 DC- orre ted small-signalmodels . . . 67

3.1.1 DC orre tion strategy . . . 69

3.1.2 Results . . . 71

3.2 Parameterized small-signalmodels. . . 75

3.2.1 Linear Transfer Fun tion Models . . . 77

3.2.2 Frequen y and Time-domain ma romodeling . . . 78

3.2.3 Parameterizedma romodeling . . . 80

3.2.4 The need for DC orre tion . . . 82

3.2.5 DC- ompliant parameterized ma romodeling . . . 84

3.2.6 Ma romodelrepresentation. . . 84

3.2.7 Stabilityand passivity . . . 85

3.3 Examples . . . 87

3.3.1 A NMOS transistor . . . 87

3.3.2 A two-stagebuer . . . 88

3.3.3 A LowDropout (LDO) voltage regulator . . . 91

3.3.4 A system-levelappli ation . . . 91

3.4 Con lusions . . . 97

4 Noise- ompliant ma romodel synthesis 102 4.1 Problemstatement . . . 104

4.2 Stati network synthesis . . . 107

4.2.1 Basi assumptions . . . 107

4.2.2 Fixed topology . . . 110

4.2.3 Synthesis with Resistors and ideal Transformers . . . . 112

4.2.4 Stati synthesis results . . . 114

4.3 Dynami network synthesis . . . 116

4.3.1 Preliminaries onstate-spa e models . . . 117

4.3.2 Dire t state-spa esynthesis . . . 120

4.3.3 Youla's rea tan e extra tion . . . 128

4.3.4 Darlington'sresistan e extra tion . . . 140

4.3.5 Dynami synthesis results and omparison . . . 145

4.4 Con lusions . . . 155

Con lusion 156 A Notation, a ronyms and symbols 158 Notation . . . 158

A ronyms . . . 158

(9)

B The Ve tor Fitting algorithm 163

C RC-example state-spa e derivation 165

(10)

System on Chip for mobile

appli ations

Businesses fail either be ause they leave their ustomersor be ausetheir

ustomers leave them![1℄

AndrewS. Grove,Intel orporationsenior advisor

1.1 History and market perspe tives

System on Chip (SoC) denes a highly integrated design pattern for

Inte-grated Cir uits (IC). Sundry levels of integration are grouped by the SoC

denition: starting froma simple hip to memory inter onne tion up to the

integration of a omplete trans eiver 1

hainfor ellphones appli ations. The

SoC paradigm raised naturally in the last de ade to meet the requirements

of a new fast-growing marketsegment, i.e. the so alled mobilemarket.

OnlyafewyearsagoPersonal Computer (PC)userswerealways

demand-ing for an in rease of the omputational power. Central Pro essing Unit

(CPU) evolution was well predi ted by the famous Moore's law [2℄ and the

out ome nowadays are very omplex devi es delivering huge omputing

a-pabilities. The rst step towards mobility was the introdu tion of Laptops.

Thereupon new design onstraints appeared: power onsumption and form

fa tor.

Tele ommuni ationsystemsprotedfromtheele troni evolutionaswell:

internetandtheworldwidewebin reasedinusageandpopularity, ellphones

evolved deliveringa wide range of appli ationsexploitingthe potentiality of

afastgrowingnetworkinfrastru ture. Thestandardsformobile

ommuni a-1

Trans eiver:devi e omprisingbothatransmitterandare eiverwhi hare ombined andshare ommon ir uitry orasinglehousing.

(11)

Figure 1.1: The most ommon system integration te hnologies are grouped

in the gure above as a fun tion of form fa tor and ir uit-to- ir uit

inter- onne t density [3, 4℄.

tion fromthethird generation(3G) onpushed towardanoptimized usageof

the ommuni ation hannelinordertoallowthetransmissionof onsiderable

amountsof data.

Inorderto ombine laptopfeatureswith ellphonesportability,SiP

(Sys-tem in Pa kage) and SoC are nowadays the integrationparadigm for

smart-phones, tablets and phablets. A ni e overview of the most ommon system

integration 2

te hnologies as a fun tion of form fa tor and ir uit-to- ir uit

density [3, 4℄ is depi ted in Figure 1.1. Planar integration te hnologies are

be omingmore hallenging as transistor hannellengthshit the range20-30

nm. Inordertomeettherequirementsofthemarket, 3Dsta kingte hniques

are emergingasapromisingworkaroundtoplanarintegrationlimitations[5℄.

1.2 Design hallenges

Compared with the design of nowadays lassi ICs, Radio Frequen y (RF)

SoC design is more involved due to the melt of heterogeneous ele troni

systems in a small pa kage [6℄. Moreover, for RF and mobile appli ations,

Analog Mixed Signal (AMS) methodologies are a must sin e Digital Signal

2

Systemintegrationisdenedasthe ombinationof ir uitsandIntelle tualProperty (IP)blo ksonthesamedie.

(12)

Figure 1.2: Fabri ation apital versus test apital based on Semi ondu tor

IndustryAsso iation(SIA)andInternationalTe hnologyRoadmapfor

Semi- ondu tors (ITRS), sour e [12℄.

Pro essing(DSP)blo ksarein lose onne tionwithanalogandRF

ompo-nents [7℄thus in reasing the overall design omplexity.

The main issues arising in RF SoC appli ations an be divided in two

ma ro groups.

1. Die and pa kage: At this level the growth in transistor ount and

op-erating frequen y has a dire t impa ton design omplexity leadingto

•

poormanufa turability: asthe miniaturizationpro essgets loser

tothe theoreti al limitsofCMOS (Complementary

Metal-Oxide-Semi ondu tor) te hnology [8℄ the design be omes very sensitive

topro essvariation. Thisae ts thethroughputyield 3

,reliability

and testability. In 1999 the Semi ondu tor Industry Asso iation

(SIA) proposed an International Te hnology Roadmap for

Semi- ondu tors(ITRS)showinghowthe ostoftestisgoingtosurpass

the ost of sili on manufa turing as depi ted in Figure 1.2. As a

onsequen e there is an in reasing interest in automati testing

methodologies [10℄ and adaptive design te hniques [11℄ to stem

the drawba ks related with pro ess toleran es;

•

power onsumption: fourarethemainsour esofpowerdissipation inCMOS te hnology [13℄.

P

dyn

: dynami swit hing powerdue to the harging and dis harging of ir uit apa itan es.

P

leak

: due to the leakage urrent from the reverse-biased diodes and

sub-threshold ondu tion.

P

short

: due to the nite signal rise/fall times.

P

bias

: stati biasing power. Those issues are addressed

by supply power s aling te hniques and Low Power (LP) CMOS

te hnologies[14℄;

3

(13)

•

power delivery issues: low-power onsumption onstraints trans-formedthedesignofPowerDeliveryNetworks(PDNs)intoavery

hallenging taskin omparison with previousIC te hnologies[15,

16℄. Multi-layer pa kages and grids are ommon to supply lean

power to the integrated ir uits. Two are the gures of merit

for PDNs: the target impedan e

4

[18℄ and the voltage IR drop.

Both a ount for two dierent phenomena: the stati IR voltage

drop 5

whi hisintrodu edbythe resistivenatureofthePDN

on-du tors, and the indu tive di/dt voltage drop whi h derivesfrom

lo alized power demand and swit hing patterns [19℄. Moreover,

large voltage drops in on- hip PDN due to large di/dt may lead

toEle tro-migration 6

(EM)that isone of the most riti al

inter- onne t failure me hanism in ICs [17℄. Besides Power Integrity

(PI) onsiderations, PDN should be also designed to aord

dy-nami power management methodologies meant for powersaving

modes driven by the ontrolrmware [21℄;

•

heat dissipation: the typi alrange of operating jun tion

temper-ature for modern VLSI designs is between

80 ◦

and

120 ◦

on the

sili on substrate [22℄. Su h boundaries are easily ex eeded due

to the umulative power dissipation of the transistors leading to

the generation of extreme amounts of heat in a relatively small

area. Highthermal density has a negative impa t on ir uit

per-forman es by in reasing the gate delay and shortening the life of

the devi e. Therefore the pa kages are arefully designed to

re-movethe heat from the IC substrate;

•

on- hip rosstalk: this ismainlyintrodu edby theinter-wire ou-pling apa itan ebetweenadja entsignallinesinon- hipbuses[23℄.

Both hardware (shielding via grounded ondu tors or parti ular

layout fabri s [24℄) and oding signalte hniques ( rosstalk

avoid-an e odes, CACs [25℄) are availableto opewith this problem;

•

noise: the ee t of thermal/white noise due to the in rease of

temperature be omes always more relevant and needs tobe

are-fully addressed. The i ker (1/f) noise istightly related with the

4

Thetargetimpedan eis al ulatedfrom:powersupplytoleran e, urrentand swit h-inga tivityand hastobesatisedbythePDNfrom DCtoatleasttherstharmoni of the lo kfrequen y[17℄.

5

Stati IRVoltagedrop: istheredu tionofthenominalreferen evoltagefortransistors due tothetransitionof urrent(I)intheresistive(R) powerdeliverynetwork.

6

Ele tro-migration: owofmetalatoms undertheinuen e ofhigh urrentdensities. Maybethe auseforin reasedresistan eandreliabilityproblems[20℄.

(14)

CMOSte hnologyandbe omesrelevantonlybelowaspe i

or-ner frequen y [13℄.

2. System and omponents: onsidering that portable devi es are meant

to support dierent ommuni ation standards like: Bluetooth (IEEE

802.15.1), Wi-Fi(IEEE802.11),GSM, GPRS, UMTSand manymore,

it is sensible that the same trans eiver has to be used for all the

om-muni ationsstandardstomeettheform-fa tor onstraintsofaportable

devi e. Asa onsequen etrans eiversand ommuni ationssystems

be- ome more omplex due to the advent of new standards and the need

to preserve retro- ompatibilityleading to

•

inter onne t delay: for o- hip buses the main bottlene k is rep-resented by the pa kage. Data rate limits are related with the

quality of the pa kage. Be ause of that the performan e of the

pa kage are ru ial for the assessment of Signal and Power

In-tegrity (SI,PI) analysis;

•

o- hip rosstalk: this is mainlydue tointer-symbolinterferen e (ISI) and indu tive rosstalk [26℄. Eye diagram analysis [27℄ is

usually adopted to study su h kindof problems.

Exploiting Sili on On Insulator (SOI) te hnology [28℄ the future of IC

inte-gration goes in the dire tion of 3D sta king [29℄. Integration density, power

onsumption and form-fa tor an be ee tivelyaddressed by 3DSoC design

methodologies[30℄ while Through Sili on Via (TSV) and Network on Chip

(NoC) are the emerging inter onne t paradigms [31℄.

Allthe design hallenges and methodologiesdes ribed inthis se tion are

fa ed relyingonadvan edmodellingte hniquesandawellestablisheddesign

ow. Next se tions will outline the state of the art on ma romodeling and

design ow forRF SoC.

1.3 Ma romodeling and Design ow

ComputerAidedDesign(CAD)te hniquesarewellestablishedandwidespread

intheele troni industriessin ede ades. Theintrodu tionofEle troni

De-sign Automation(EDA) dates ba k to 1980 when it be ame lear that the

gap advan esinengineeringprodu tivity(

P

1

) omparedwiththe in rease in sili on omplexity (

P

2

)was widening, as depi ted in Figure1.3. This trend, know as produ tivity gap [32℄, be ame more relevant due to the advent of

(15)

Figure 1.3: The bordeaux line represents the in rease for the number of

transistors per hip asa fun tionof years (

P

2

) whilethe green lineindi ates the advan es in engineering(

P

1

),sour e [32℄.

oping with the produ tivity gap in the ontext of SoC for mobile

appli a-tions. The following requirements should be met by anee tive SoC design

ow:

•

rapid development tosatisfy time-to-marketpressures;

•

quality of results: performan e, form-fa tor and power onsumption;

•

simple veri ation of omplex hips;

•

simple touse for teamswith dierent levels and areas of expertise. To satisfy the onstraints listed above modern design ows are heavily

re-lying on the on ept of IP (Intelle tual Property) reuse [33℄: ea h step in

the design ow depi ted in Figure 1.4 is now enhan ed and supported by

well established IP blo ks. In a similar fashion to the ode reuse pattern

widely used in Information Te hnology appli ations, the main idea behind

the IP reusestrategyrelies uponthe onstru tionofalibrary of omponents

(generally alledIP blo ksorma ros)tobeusedinseveraldierentproje ts.

More detailson this topi are provided in the next se tion.

Together with IP reuse, as ICs and design ows be ome more involved,

ma romodels and related tools must improve and a omplish new features.

A ma romodelisa high-levelmathemati aldes ription of the system under

analysis that a urately represents itsbehaviour. The prex ma ro

empha-sizes that only theinput/output response is des ribed, while noinformation

isretained onthe internalstru tureofthe physi alsystem. Besides the

(16)

Figure 1.4: The mains steps involved in the design ow of mobile devi es

are sket hed. Starting from spe i ations and standards the on ept of the

devi eisbuilt. Amodelprototypeis reated usingape uliarte hnologyina

CAD environment. Several EDA software areused inthe pre-tape-outphase

to address: fun tional spe i ations, manufa turability and physi al

onsis-ten y oftheprototypemodel. Inthetape-outphasefun tionalspe i ations

are he ked onphysi aldesigns. In aseofissuesthemodelprototypeisused

asatestben h. Of ourse,toredu eprodu tion osts,the minimumnumber

of tape-outs should be used tomeet allthe spe i ations.

•

parameterization: in order to speed up what-if analysis and

optimiza-tion pro edures ma romodels should admitsome of the most ommon

designparameters(temperature,

V

dd

and geometry)asinputvariables. Withsu hafeaturethereisnoneedtobuildanew ma romodelin ase

of variations of designparameters;

•

usability: ma romodels should be available in a standard format, like Spi enetlistorHDL(HardwareDes ritionLanguage). Thesamemodel

mustbeee tivefordierenttypeofanalyses(time/frequen ydomain,

noise). Inputs,parametersandoptionsmusten loseasimpleand lear

des riptiontogetherwithappli abilitybounds. Therebyindependently

of user's expertise the model an be used ee tively, in a short time

and in several dierent ontexts;

•

s alability: it is well known that SoCs omplexity, intended as dy-nami al order and elements/inter onne tions ount, grows really fast

with time. Modelling te hniques must opewith this trend, providing

a urate models with low omplexity ina short time.

(17)

providing low omplexity is the orner stone for a modern design ow

in-tended tomeettighttime-to-market onstraints. Ma romodels asso iated to

the sub-blo ks ofa omplexsystem an be ombinedtomimi the behaviour

of the whole system leading to a tremendous simpli ation in the analysis

of omplex devi es. In the following se tions the main features of IP reuse

and how to deliver adequate ma romodels for this new design paradigm is

dis ussed.

1.3.1 IP reuse

Design te hniquesbasedonIPreusewereborninthe beginningofthe1990's

[33℄. Two major eventsare onsidered asthe startingpointsfor theIP reuse

diusion:

•

Establishmentofthe VirtualSo ketInterfa e Allian e(VSIA):in1996 this ross-industry organization, fo used on IP reuse for SoC design,

was founded to help foster this new design pattern by ombining the

skills and knowledge of semi ondu tor ompanies, system ompanies

and EDA industry;

•

Register Transfer Level (RTL) IP reuse: in 1997 teams from Mentor Graphi s and Synopsis proposed the so alled Reuse Methodology for

soft IP. The di tatesof this design pattern are olle ted inthe widely

known Reuse Methodology Manual[34℄.

The ore idea behind IP-oriented SoC design relies upon the availability

of reusableIP blo ks thatsupportplug-and-playintegrationinapre-dened

ow. Assu hIP blo ksare the highestlevel buildingblo ksof anSoC,they

are olle ted in libraries with various timing, area and power ongurations

providingto designers simple touse IP ma ros.

The form of a reusable IP ore an vary depending on the IP

devel-oper/vendor; as ahigh level lassi ation, three are the following main

at-egories of IP blo ks [34℄:

•

softIP:blo ksdenedusingRTLorhigherleveldes riptions. Theyare typi ally used for digital ores relying ona pro ess-independent

hard-ware des riptionlanguage(HDL)that anbesynthesized togatelevel.

Advantages: exibility,portabilityandreusability;whilethedrawba ks

are: la k oftimingand power hara teristi s be auseperforman es are

tightly relatedwiththe te hnology usedto synthesizethe HDL. Those

ma ros an be en rypted to hide IP details and prevent the

(18)

Figure 1.5: The sele tion of the most suitable form to deliver and IP blo k

should take into a ountthe trade-odepi ted inthis plot [35℄.

•

hard IP: usually dened by means of faithful layouts tailored for a spe i appli ation based on a given te hnology. For those blo ks,

performan es are predi tablebut the onsequent drawba k is the la k

of portability;

•

rmIP:inthemiddlebetweenhardandsoftblo ks,rmIPisdelivered as parametrized analog ir uit meant tobe tailoredby designers for a

spe i appli ation. Blo k'sfeatures an betrimmedleveragingonthe

availableparameters while retainingpredi table performan es.

As a onsequen e, sele ting the most suited IP form for ea h blo k is of

paramountimportan einorder tobuild anee tiveand reliabledesign ow

for SoCappli ations. Todrive su h animportantde ision,the plot depi ted

in Figure1.5is suggested as a referen emap in [35℄.

WhentheIPreusestrategyisappliedtotheAMSdesignforRFSoC,one

problemarises [36℄, i.e. the sele tionofthe IP formmost suited foranAMS

blo k. Compared todigitaldesign,for whi ha ommondesign methodology

is available [37℄, AMS design usually relies on spe i design pro ess. This

issue anbeaddressedusinganee tivemixed-signalSoCow[38,39℄based

on the AMS IP blo ks in [40, 41, 42℄.

Currently, due to the omplexity of AMS designs, the soft and hard IP

forms are used for analog-mixed signal appli ations [40, 41, 42℄. Of ourse,

this hoi e restri ts the s ope of appli ations redu ing the overall SoC

de-sign ow e ien y [36℄. The migration of hard AMS IP blo ks to the rm

form alls for new features on the ma romodels used to derive netlists and

s hemati s. Indeed, as stated in Se tion 1.3, parametrizability and

(19)

order to provide a high level of usability for su h models ( onsequently for

thermAMSIPblo ks)a learandsimpletaxonomyisneeded; nextse tion

introdu essu h a lassi ationfor ma romodels.

1.3.2 Ma romodels taxonomy

A simple and lear taxonomy for ma romodels is needed in order to meet

the usability onstraints imposed by the IP reuse paradigmdetailed in

Se -tion 1.3.1. Considering that the main bottlene k in the design of

analog-mixed-signal omponentsisrepresentedby theanalogblo ks,twowillbethe

main riteriabehind the proposed taxonomy: allthe omponentsare analog

(indeed also digital blo ks are synthesized via analog elements), and their

level of non-linearity is the base for lassi ation. As a onsequen e of this

oarse lassi ationtheproposedtaxonomyisorthogonaltone

te hnolog-i al details attaining the degree of portability required by IP blo ks meant

to the rm IP form.

In the following for ea h level of lassi ation the state of the art on

ma romodeling and system identi ation will be briey outlined together

with a list of AMS omponentsbelongingtoea h level of the proposed

tax-onomy.

Linear Time Invariant (LTI) systems

There are several omponents that an be a urately modelled as Linear

Time Invariantsystems: pa kages [43℄, busesand inter onne ts[44℄,Printed

Cir uit Boards (PCB) [45℄, Power Distribution Networks (PDNs) [46℄ and

Through Sili onVia(TSV)for3DSoC [47℄. The onstru tionofLTImodels

forthose omponentsisusuallybasedontheworkowdepi tedinFigure1.6

from [43℄: S atteringparameters are extra ted fromthe layout or3D model

of the omponent under analysis using a full wave solver. Thus the LTI

model an be extra ted using the time or frequen y raw data leading to a

state-spa e[48℄ordes riptorrepresentation. Severalwellassessedte hniques

are available to onstru t LTI models fromtabulated data:

•

Nevanlinna-Pi k interpolation [49, 50℄ is a well known result of om-plex analysis. Two matrix versions exist for this problem: the

ma-trix Nevanlinna-Pi k problem[51℄ and the tangentialNevanlinna-Pi k

problem[52℄. Thismethodwasadopted forthe rsttimeinthesystem

identi ation ontextby [53℄and morere entappli ation an befound

in [54℄. A omprehensive des ription of the Nevanlinna-Pi k problem,

(20)

Figure 1.6: The typi alwork ow used for the reation of LTI models from

pa kages, PCB,TSV andrelated omponentsispresented. Startingfromthe

layout or the 3D modela full wave solver is applied in order to extra t the

S attering parameters. From S-parameters the LTI model is identied via

the standardte hniquessummarizedinSe tion1.3.2. On etheLTImodelis

availableit an besynthesized asaSpi enetworkand the resultsfromSpi e

(21)

is theoreti ally attra tive it is seldom used in pra ti e due to

ompu-tational omplexity and numeri alstability reasons;

•

Löwner interpolationdatesba ktothe workofLöwnerforthe interpo-lationof given dataonafullar ofthe unit ir le inthe omplexplane

[56℄. It was introdu ed in the ontext of ontrol theory and system

identi ationby Kalmanand Belevit h [57℄. More re ent appli ations

of this method an befound in[58, 59℄;

•

The Sanathanan-Koerneriterationwas originallyproposed in [60℄ and itis basedonthe omplex urve ttingproposedby Levy in[61℄. This

is a general strategy to re ast a non-linear interpolation problem to

the solutionof asequen e of linearoverdetermined systems. The most

popular evolution of the Sanathanan-Koerner iteration is the Ve tor

Fitting(VF)algorithm[62,63℄. Nowadaysthis isthedefa tostandard

for the identi ation of linear systems in the EDA ontext. Despite

VF has no guarantees of onvergen e when dealing with noisy data

[64℄, itoers the best trade-o between omputational omplexity and

robustness [65℄. As a onsequen e the Sanathanan-Koerner iteration

and VF are used in this work and are presented in more details in

Se tion 2.2;

•

Padé approximation, originally proposed by the mathemati ian Henri Padé [66℄,addresses the best approximationof afun tion undera

spe- i norm by a rational fun tion of a given order. It was introdu ed

in ontroltheory tomodelexponentialdelays[67℄. Re entappli ations

an befound insystem identi ationliterature [68℄. This methodwas

quitepopularbeforetheintrodu tionofVFand anbestill onsidered

a good alternative to the Sanathanan-Koerner iteration for low-order

systems [69, 70℄;

•

subspa e methods [71℄ are all omposed by three steps: estimation of the predi table subspa e from raw data, extra tion of the state

vari-ablefromthepredi tablesubspa eandttingtheestimatedstatestoa

statespa emodel. Severalalgorithmsareavailablebothfor ontinuous

[72℄ and dis rete[73℄ time models identi ation. Those te hniques are

numeri allystableande ient[74℄. Thela k ofaprioriphysi al

prop-erties impositions, likestability and passivity, prevents the systemati

appli ation of those methods onanalog ir uits.

It is worth noting that the te hniques listed above are meant for ele troni

(22)

networks,i.e. thepropagationdelay ofthesignal anbenegle ted,otherwise

dierent te hniques should be used, like[75, 76℄.

Parameterized LTI (P-LTI) systems

Although LTI models are helpful and their usage is widespread, the main

drawba k of the LTI approa h lies in the la k of exibility. Indeed several

omponents like: PCBs, inter onne ts, pa kages, RF indu tors and TSVs

are designed and tested onsidering dierent geometri al ongurations and

working temperatures. As a onsequen e, a onsiderable eort wasspent in

the last years to extend the identi ation algorithmsintrodu ed in Se tion

1.3.2 to obtainParameterized-LTImodels:

•

parameterized Nevanlinna-Pi kinterpolationwas rstproposed in[77℄ but found onlyfew appli ationsin robust ontrol appli ations[78℄;

•

parameterizedLöwnerinterpolationwasintrodu edby[79℄. Duetothe majormemory onsumption this method isnot used inpra ti e;

•

parameterized Sanathanan-Koerner (SK) iteration was rst proposed

by Triverio in [80℄ and then extended by the same author to a ount

for stability [81℄ and passivity [82℄. In a similar fashion VF was used

by Ferranti for the P-LTI identi ation [83℄ and then with passivity

onstraints [84℄. Currently those are the most diused te hniques for

the identi ation of P-LTI models. Some appli ations and advan es

are presented in Se tion3.2;

•

parameterized Padé approximation an be found in[85℄. Being a om-petitivealternativetoVFandSKiterationitfoundseveralappli ations

[86, 87℄;

•

parameterized subspa e methods were addressed re ently [88℄. Those methods suer from a urse of dimensionality leading to an ill-posed

parameter estimation problem; a re ent attempt to over ome su h a

limitation an befound in [89℄.

Despite the theoreti aleort, up tonow noneof the te hniques listed above

for the identi ation ofP-LTI systems has the robustness and the e ien y

to be ome part of auser-friendly EDA tool.

Small-Signal P-LTI

(23)

non-linear devi es that behave almost linearly in the neighbourhood of one

equilibrium point 7

. This is a ommon s enario in RF appli ations, indeed

omponents like: Low Dropout (LDO) regulators, Operational Ampliers

(Op Amp), Low Noise Ampliers (LNA), buers and a tive lters are

de-signed to behave almost linearly under spe i working onditions. In the

ontext ofRF appli ations,linear behaviourmeans thatthe devi edoesnot

generate spuriousharmoni s orthat the spuriousharmoni s are strongly

at-tenuated and thus negligible. For AMS high integration te hnologies, like

in SoC and SiP, the suppression of spuriousharmoni s isrelevant to ontrol

ouplingnoise and undesired mixingee ts.

Pie ewise linear P-LTI

The P-LTI method an be extended to modelstrong non-linear devi es like

drivers, mixers and Phase-Lo ked-Loops (PLLs) using apie ewise linear

in-ter onne tion of P-LTImodels. The rstwork dealingwithpie e wiselinear

(PWL)networks datesba k toSternin1956 [92℄. Amore rigorousstudy on

PWL models for non-linear devi es is due to Chua [93℄, while several PWL

te hniques are ompared in [94℄. The idea to use state-spa e models with

PWL states is quite re ent and found several appli ations for the modelling

ofnon-lineardevi es[87℄. Inthe ontextofAMS ir uitsPWLte hniques an

befound in: formalveri ation ofanalog ir uits[95℄,behavioural modelling

of nonlinear power ampliers[96℄ and mixed-signal ir uits[97℄.

1.4 Proposed advan es

Despite the resear h eort spent in the development of EDA tools and

al-gorithms, design and veri ation of AMS SoC is still an open issue, whi h

osts to mobile ommuni ations ompanies huge resour es [32℄. Therefore

the main obje tive of this do toralwork onsists in the development of new

methodologies to ope with the hallenges posed by SoC integration

high-lighted inSe tion 1.2. The proposed solutions, while advan ing the state of

the art for ma romodeling of ele troni devi es, arise from industrial

on-straints and real design test ases, providing an immediate ontribution to

pra ti alneeds.

Chapter 2 deals with the identi ation of linear ma romodels belonging

to the LTI taxonomy lass presented in Se tion 1.3.2. State-spa e models

7

InthisworkbyequilibriumorxedpointtheLyapunovdenitionoflo alstabilityis onsidered [91℄.

(24)

representation and basi identi ation tools are summarized in Se tion 2.1

and 2.2. The main ontributions of Chapter 2are:

•

a ompressedma romodelingalgorithmis introdu edinSe tion 2.3to over ome the limitationsof VFwhen dealing with omponentshaving

huge ports ount, from tens to hundreds. As dis ussed in Se tion 1.2,

atsystem levelthe main bottlene k forinter onne tionsis represented

by the pa kage, while at hip level3D te hnologies like TSV and NoC

are meanttoin rease the onne tivity. Theoriginal version of VF[62℄

andalsothemorere entadvan eslike[98,99℄arenotsuitedtoaddress

su h devi es be ause of the ex essive memory onsumption or due to

ill- onditioning. The proposed ompressed ma romodeling algorithm

over omesthoseissuesrelyingona leverredu tionofthedatasetused

for the identi ation of the model. A ura y and physi al properties

like ausality and passivity an beimposed dire tlyonthe ompressed

ma romodel, as presented in Se tion 2.3.4, leading to a tremendous

speedup on the overall identi ation pro edure (see Se tion 2.4.4) in

omparison with state of the art te hniques [100℄;

•

a parallel algorithm to verify the passivity of linear ma romodels is introdu ed in Se tion2.5. Sin e the most ommonalgorithmsfor

sys-tem identi ation (VF and SK) do not guarantee the generation of

passive models, passivity needs to be addressed independently [101℄.

Moreover, passivity hara terization is of ourse the rst step for the

passivity enfor ement [102℄, and needs to be repeated several times.

Several algorithmsareavailableforthe passivity hara terization [101,

103, 104℄. Someof themare alreadyavailableforparallelar hite tures

[105℄. The algorithm proposed in Se tion 2.5 is an e ient parallel

implementationof [104℄;

Chapter3dis usses theidenti ationofparameterizedLTI(P-LTI)

mod-els. Theavailabilityofparameterizedmodelsisthe ornerstoneforthe

devel-opment of a modern and ee tive design and veri ation ow. Considering

thatseveralmethodologiesfortheidenti ationofP-LTImodelsareavailable

as dis ussed in Se tion1.3.2, the main ontributions of Chapter 3 are

•

a Dire t Current (DC) orre tion strategy for small-signal models of non-linear ir uits,presented in Se tion3.1. This simple but ee tive

idea is the link between linear and small-signal models for non-linear

devi es. RF devi es like LDO and OpAmp are designed to behave

al-mostlinearlyunderappropriatebiasing. Theso alledsmall-signalLTI

(25)

but fail to reprodu e the DC response of the real non-linear devi e.

The proposed DC orre tion an be used to over ome this issue;

•

parameterized small-signal models are proposed in Se tion 3.2. A - ording to the taxonomy proposed in Se tion 1.3.2 models are sorted

depending on the level of non-linearity. The ombination of P-LTI

models with a parameterized DC orre tion strategy makes itpossible

tomodelfairlynon-lineardevi esusingasmooth ombinationoflinear

models parameterized by theoperatingpoint. Theee tiveness of this

strategy isdemonstrated inSe tion3.3by analysingsome test asesof

pra ti alrelevan e.

Chapter 4presentsthe synthesisofState-Spa emodelsaslinearlumped

net-works. As already noted in Se tion 1.3.1, the rst step for the migrationof

AMS IP blo ks towards the IP rm des ription relies on the availability of

exibleand e ientimplementationsofthe ma romodels. Therefore

anoni- alsynthesis 8

algorithminSpi e ompatibleformatare des ribed. Themain

ontributions of Chapter 4are:

•

modern presentation of anoni al synthesis methods for stati and dy-nami networks dis ussed in Se tion 4.2 and 4.3. For ea h synthesis

method: omplexity of the network and pra ti al relevan e are

de-tailed. In parti ular: stability,a ura y and noise analysis omplian e

are onsidered. Stati alnetworksynthesiste hniques are onsidered in

theirown be auseofpra ti alrelevan efor onne tivity,stati IRdrop

[106℄ and powerdistribution analysis;

•

a new synthesis method for dynami networks, based on Darlington

resistan e extra tion framework, is presented in Se tion 4.3.4. Ea h

stepofthisnewalgorithmisdes ribedfo usingonnumeri alrobustness

and noise omplian e of the resulting Spi enetlist.

Finally, on lusions are summarized in the last Chapter, highlighting both

theoreti al and pra ti alrelevan eof results and methodologiesdis ussed in

this work.

8

(26)

Linear Time Invariant

ma romodels

Ma romodelingte hniqueshavebe omeastandardpra ti einsystemdesign

and veri ation ows. Su h methods allowto onvert external

hara teriza-tions oflinearandtime-invariantstru tures su haspassivedevi es and

ele -tri al inter onne ts into ompa t losed-form mathemati al expressions or

ir uitequivalents. This onversionisneeded toallowsystem-leveltransient

simulations and veri ations starting from a native hara terization that is

typi ally available in the frequen y domain in form of tabulated s attering

responses,thelatterbeingdeterminedfromdire tmeasurementsorfull-wave

numeri al solutions.

This Chapter introdu es some advan es to the state of the art of Linear

TimeInvariants(LTI)ma romodelingte hniques. Thene essaryba kground

on state-spa e models and system identi ation is dis ussed in Se tion 2.1,

while two of the most popular algorithms for linear systems identi ation

are des ribed inSe tion2.2,i.e. the Sanathanan-Koerneriterationand

Ve -tor Fitting. Extensions and improvements for those identi ation methods

are the main ontributions of this Chapter. In Se tion 2.3, the Compressed

ma romodelingalgorithmisintrodu edasa leversystem identi ation

pro- edure based on Ve tor Fitting for systems having a large port ount. In

Se tion 2.5, a highly e ient parallel passivity veri ation method is

pre-sented.

2.1 State-spa e ma romodels

The state-spa e representation was introdu ed in ontrol engineering and

(27)

des rip-tionfordynami alsystems. State-spa eequations onstituteamathemati al

model of the physi al system under analysis as a set of input, output and

internal state variables related by oupled rst-order dierential equations.

Dealing with linear time-invariant systems the asso iated state-spa e

equa-tions read

˙x(t) = Ax(t) + Bu(t),

(2.1)

y(t) = Cx(t) + Du(t).

(2.2) with

A

∈ R

N ×N

,

B

∈ R

N ×P

,

C

∈ R

P ×N

and

D

∈ R

P ×P

onstant matri es.

Inputs are olle ted inve tor

u

,outputsinve tor

y

whilethe internalstates are inve tor

x

. Two featuresare of paramount importan efor astate-spa e system

•

observability: dened as the ability to always re onstru t the initial state

x(0)

observingthe outputsof the system for

t

≥ 0

provided that alsothe input evolutionis known for

t

≥ 0

;

•

ontrollability: dened as the possibility to always design an input sequen e that steers the system to a desirednal state.

Both onditions are guaranteed when the model(2.1)-(2.2) has minimal

dy-nami order, dened as the M Millan degree of the system [107℄. If the

state-spa e is not minimal, it an be onverted to a minimal one by means

of standard te hniques [108℄.

TakingnowtheLapla etransformof (2.1)and(2.2)andassuming

x(0) =

0

, itfollows

sX(s) = AX(s) + BU(s),

(2.3)

Y(s) = CX(s) + DU(s),

(2.4)

whi hleads tothe transfer fun tion matrix relating

U(s)

and

Y(s)

H(s) = D + C(sI

_{− A)}

−1

_B.

(2.5)

The transfer fun tion (2.5) is rational. In ase of poles (eigenvalues of

A

) with unit multipli ity,

H(s)

an alsobe writtenin the so alled pole-residue form, i.e.

H(s) = D +

N

X

n=1

R

n

s

_{− p}

n

,

(2.6)

(28)

I

1 I

2 +

−

V

1 +

−

V

2 b

1 a

1 b

2 a

2

2-port

Figure2.1: In ident

a

i

and ree ted

b

i

power waves fora two-port network.

and (2.6) asso iated to the minimal state-spa e (2.1) system are equivalent

toea hotherandoneispreferredtotheothersdependingontheappli ation

ontext.

Theidenti ationworkowdes ribed inSe tion1.3(Figure1.4)heavily

relies on the availability of a urate models in the formof (2.1). Su h

mod-els an be onverted to linear lumped networks to be solved using SPICE

based solvers using the synthesis te hniques dis ussed in Chapter 4. In

or-der toextra ta urate state-spa e modelsusing therawdata obtainedfrom

measurement or full-wave solvers an identi ation algorithm must be used.

In the following the raw data used for the identi ationare supposed to be

S attering (S)-parameters [109℄. Re all that the S-parameters for a 2-port

(theextensionto

P

-portisstraightforward)lineartime-invariantnetwork,as depi ted in Figure2.1, are dened as

b

1 b

2 =

S

11 S

12 S

21 S

22 a

1 a

2 → b = Sa ,

(2.7)

where

Z

0

is a pres ribed real referen e impedan e and the travelling waves

a

i

and

b

i

are dened as

a

1 =

V

1 + Z

0 I

1

2 √

Z

0 ,

a

2 =

V

2 + Z

0 I

2

2 √

Z

0 ,

(2.8)

b

1 =

V

1 − Z

0 I

1

2 √

Z

0 ,

b

2 =

V

2 − Z

0 I

2

2 √

Z

0 .

(2.9)

The main onstraint ommonto allidenti ationpro edures onsists in the

minimization of the dieren e between the linear identied model response

and the referen e raw data-set. Working with S-parameters (2.7) the raw

data for an LTI network is omposed of matri es

S

l

= S(s

l

)

∈ R

P ×P

, with

l = 1, . . . , L

number of dis rete frequen y samples

s

l

= ω

l

. In this ontext

the identi ation problem an be formulated as: nd a state-spa e model

S(s)

su h that

min

X

l

(29)

for a given norm.

Among all the identi ationalgorithms, listed in Se tion 1.3.2 of

Chap-ter 1, for the extra tion of a urate state-spa e models starting from raw

data the two most used in pra ti e are the Sanathanan-Koerner iteration

and the Ve tor Fitting pro edure. Those two methods are presented in the

followingse tion.

2.2 Sanathanan-Koerner and Ve tor Fitting

The minimization onstraint (2.10) asso iated to the identi ation problem

wasaddressedbySanathananandKoernerin[60℄,the resulting

Sanathanan-Koerner (SK) Iteration is briey summarized in this se tion together with

his most popular extension,i.e. the Ve tor Fitting(VF) algorithm [62℄.

Theidenti ationofas alar transferfun tion

h(s)

is onsidered,instead of the matrix ase (2.10), inorder to simplifyand fo us the presentation on

thealgorithm. Theextensiontomulti-portdevi esisstraightforward[60℄. In

the basi SK Iteration framework aset of frequen y-domain samples

(s

l

, h

l

)

for

l = 1, . . . , L

is used to identify a rationalmodel of the form

h(s; x) =

n(s; x)

d(s; x)

=

a

0 + a

1 s +

· · · + a

m

s

m

b

0 + b

1 s +

· · · + b

n−1

s

n−1

+ s

n

(2.11)

where

n(s; x)

and

d(s; x)

are respe tively numerator and denominator

poly-nomials of degree

m

and

n

. The unknown oe ients are olle ted in the

ve tor

x

= (a

0 , a

1 , . . . , a

m

, b

0 , b

1 , . . . , b

n−1

)

T

.

(2.12) The generalidenti ationproblemrequires tond oe ients

x

whi h min-imize in some norm the residualerror

r(x)

,whose omponents are

r

l

(x) = h

l

−

n(s

l

; x)

d(s

l

; x)

.

(2.13)

To avoidthesolutionof anon-linearinterpolationproblemthe strategy

pro-posed by Levy in[61℄ an be used, i.e. instead of minimizingthe non-linear

residual (2.13),the following modiedresidual isminimised

e

l

(x) = d(s

l

; x)r

l

(x) = d(s

l

; x)h

l

− n(s

l

; x)

(2.14)

by solving the linear least square problem

(30)

where

g

l

= h

l

s

n

l

,

F

= (V

m+1

,

− ˜

HV

n

)

with

H

˜

= diag(h

i

)

,

i = 1, . . . , L

and

V

_n

Vandermondematrix [110℄

V

n

=







1 s

2

1 . . . s

n

1 1 s

2 s

2

2 . . . s

n

2

. . . . . . . . . . . .

1 s

L

s

2 L

. . . s

n

L







(2.16)

based on the available frequen y points

s

l

with

n + 1

olumns. Minimizing

ke(x)k

or

kr(x)k

isnotequivalentduetotheweight

d(s

l

; x)

,thereforetheSK

iteration [60℄ tries to over ome this limitationusing an iteration-dependent

residual, dened as

r

_l

ν

(x

ν

) =

d(s

l

; x

ν

)h

l

− n(s

l

; x

ν

)

d(s

l

; x

ν−1

)

(2.17)

wherethe normalizationweight

d(s

l

; x

ν−1

)

isknownfromthe previous itera-tion

ν

− 1

. Theiteration-dependentve tor

x

ν

whi hminimizesthe iteration-dependent residual (2.17) an be found solving the overdetermined linear

system

M

_ν−1

Fx

ν

≃ M

ν−1

g

(2.18)

where

M

ν−1

= diag(m

(ν−1)

i

)

with

i = 1, . . . , L

and

m

(ν−1)

i

= d

−1

(s

i

; x

ν−1

)

.

In ase of onvergen e, as

ν

→ ∞

the minimization of (2.17) is

equiva-lent to minimizing (2.13). In pra ti e some numeri al issues arise: it is

well known that Vandermonde matri esand their ompositionsare very

ill- onditioned [111℄, moreover raw input data an be ae ted by noise thus

making the identi ationproblem moredi ult.

In order to avoid those issues a general basis expansion an be used for

the numerator and denominatorin (2.11), i.e.

h(s; x) =

n(s; x)

d(s; x)

=

m

P

j=0

c

j

φ

j

(s)

n

P

j=0

d

j

φ

j

(s)

(2.19)

with

x

olle ting the unknown oe ients

c

j

, d

j

leading to the so- alled Generalized-SK iteration [112℄. A typi al hoi e is to use partial fra tion

basis fun tions asso iated toa set of pres ribed poles

q

j

, j = 1, . . . , n

,i.e.

φ

0 (s) = 1,

and

φ

j

(s) =

1 s

− q

j

(31)

Substituting (2.20) into(2.19) leads to

h(s; x) =

n(s; x)

d(s; x)

=

c

0 +

n

P

j=1

c

j

s−q

j

1 +

n

P

j=1

d

j

s−q

j

(2.21)

whi h isequivalent tothe modelin (2.11). Indeed supposing that

c

j

, d

j

and the basis poles

q

j

are known, (2.21) an be onverted in astandard rational form with the zeros of the numerator

z

j

, and the zeros of the denominator

p

j

su h that

h(s; x) =

n(s; x)

d(s; x)

= c

0 n

Q

j=1

s−z

j

s−q

j

n

Q

j=1

s−p

j

s−q

j

= c

0 n

Q

j=1

(s

− z

j

)

n

Q

j=1

(s

_{− p}

j

)

(2.22)

where it is lear that the poles

q

j

an el out being ommon toboth numer-ator and denominator. The GSK iterationis thusobtained by repla ing the

monomials

s

j

l

in(2.16) with

φ

j

(s

l

)

.

A simple update on the basis poles and fun tions of ea hiteration leads

to the VF algorithm: starting from an arbitrary guess of the model poles

used to dene the basis fun tions (2.20), the non-linear problem (2.13) is

solved using one GSK; then the initial basis poles are improved by using,

at the se ond iteration,the set

p

j

dened in (2.22) to onstru t the partial fra tion basis fun tions. The pro ess is then iterated until onvergen e. A

more detailed des ription of VF algorithm an be found in Appendix B or

in[62℄. Nomoredetailsareprovided heresin einthefollowingVFisusedas

an identi ationengine, the main fo us willbein prepro essing of the data

and post-pro essing of the model.

Onedrawba k ofVFappears when dealingwithdevi es with large ports

ounts like TSV, pa kages and PDNs. Sin e the omplexity of VF in the

most advan edformulation[98℄s ales as

O (P

2 _LN

2 ₎

periteration,the

iden-ti ation of devi es having more than one hundred ports (

P

) and requir-ing several frequen y samples (

L

) for an a urate hara terization will run

out-of-memory on ommodity hardware, and will take a long time on high

performan e servers. Therefore a lever reformulation of the identi ation

problemaimedatredu ingtheimpa tofports(

P

) ountandnumberof sam-ples (

L

)onthe overall omplexityofVe tor Fitting(VF) isofgreatinterest. Next se tion introdu es an innovative te hnique, the so alled ompressed

(32)

of largeports ountdevi es, a urately sampledinfrequen y, on ommodity

hardware (laptop) and in a short time ompared to standard identi ation

pro edures.

2.3 Compressed ma romodeling

InthisSe tionanapproa hforimprovingthee ien yofrationalttingand

passivity enfor ement for medium and large-s ale stru tures is presented.

Problems hara terized by possibly hundreds of ports and requiring

thou-sands of internal states for their models are addressed. Requirements for

problems of su h omplexity arise, as dis ussed in Se tion1.2, inpower bus

modeling and optimization, hip-pa kage o-design, TSV and NoC

inter on-ne ts for 3D pa kages and mixed-signalsystem design.

The basi idea behind the proposed strategy an be easily understood

onsidering a generi

P

-port ele tri al inter onne t stru ture hara terized through tabulateds atteringfrequen ysamples

S

l

∈ C

P ×P

atfrequen ies

ω

l

,

with

l = 1, . . . , L

. Thisrawdataisusuallyavailablefromeldsimulationsor

dire t measurements. The VF algorithm from Se tion 2.2 is routinely used

to tthese data samples with arational model

S(s) = S

_∞

+

N

X

n=1

R

n

s

_{− p}

n

,

(2.23)

where

p

n

arethepolesofthe ma romodel,

R

n

aretheasso iatedresidue ma-tri es, and

S

∞

isthe dire t ouplingterm. Standardformulationsof theVF algorithm [62℄ minimize the global model error (2.10) through an iterative

sequen e of linear least squares solutions. Sin e the ompression strategy

presented here is omplementary to the VF implementation, a detailed

de-s ription of VF algorithm is not reported here, more details an be found

in Appendix B or[62℄.

The mainidea of the ompressions heme ispresented through an

exam-ple. Figure2.2depi tsseverals atteringresponsesofahigh-speed onne tor.

As it an be seen the various responses that are depi ted look very similar,

with only marginal dieren es. Of ourse, these dieren es may be

impor-tant, so they should be preserved in the nal ma romodel. However, it is

on eivable thatalltheseresponses anberepresented asalinear

superposi-tionofsele tedrepresentative responsesor,moreformally,basisfun tions.

Therefore expansions of the form

S

ij

(s)

≃

ρ

X

q=1

(33)

0

1

2

3

4

5

6

7

8

9

10 −80

−60

−40

−20

0 Frequency [GHz]

Magnitude, dB

Scattering matrix entries, magnitude (dB)

Figure 2.2: Various s attering responses of a high-speed onne tor (top

urves: ree tion oe ients, bottom urves: rosstalks).

with onstant oe ients

α

(i,j)

q

and frequen y-dependent basis fun tions

w

q

(s)

, are suited for a lever redu tion of the dataset. It is lear that if the number of required basis fun tions

w

q

(s)

is mu h smaller than the total numberofresponses,

ρ

≪ P

2

,itispossibletoa hieveasigni ant

omputa-tional ostredu tion byapplyingVF tothe fewfun tions

w

q

(s)

,ratherthan to the omplete set of

P

2

raw s attering responses. This idea is developed

in the followingSubse tion 2.3.1, relying onthe wellknown SingularValues

De omposition [110℄.

2.3.1 SVD-based ompression

Considerthesetofraws atteringsamples

S

l

,

∀l

. Forea hsele tedfrequen y

ω

l

, allelements of the s attering matrix are sta ked into a single row-ve tor

x

l

∈ C

P

2

, onstru ted as

x

l

= vec(S

l

)

T

. The

vec()

operator sta ks all

olumns of its matrix element into a single olumn ve tor. More pre isely,

element

(S

l

)

ij

with

1 ≤ i, j ≤ P

orrespondstoelement

(x

l

)

k

for

1 ≤ k ≤ P

2

through

k = i + (j

_{− 1)P}

i

= 1 + mod(k

− 1, P )

j =

_{⌈k/P ⌉}

(2.25)

where

mod(a, b)

returns the remainder of the integer division

a/b

and

⌈c⌉

is the eil operator that returns the smallest integer not less than

c

. The mapping

(i, j)

↔ k

in(2.25)willbeused onsistentlyduringthepresentation. All theve tors

x

l

orresponding todierent frequen ies

ω

l

are now olle ted

(34)

as rows ina matrix

X

∈ C

L×P

2

, i.e.

X

=







←− x

1 −→

. . . . . . . . .

←− x

L

−→





 =





z

↑ · · ·

₁

_{· · · z}

↑

_P

2 ↓ · · ·

↓



 .

(2.26)

Ea h row

x

l

of this matrix orresponds to a single frequen y

ω

l

, while ea h olumn

z

k

olle tsallfrequen y samples ofa singles attering response

(z

k

)

l

= S

ij

(ω

l

)

.

Assume that the

P

2

s attering responses an be represented as an

ap-proximate sum of few basis fun tions. This implies that the olumn span of

matrix

X

anbesafelyapproximatedby proje tion ontoasubspa e

W

hav-ing a dimension

ρ

≪ P

2

. Several alternatives are available for onstru ting

this subspa e. Inthiswork,theSingularValueDe omposition(SVD)isused

sin e it providesa full ontrol overthe approximationerror [113℄.

A dire tappli ation of SVD to matrix

X

leads to

X

= e

U e

Σ e

V

H

= f

W e

V

H

(2.27)

where

U

e

and

V

e

are omplex unitary matri es olle ting the left and right singular ve tors and

Σ

e

olle ts the sorted real and positive singular values

eσ

q

on its main diagonal. Matrix

W

f

= e

U e

Σ

is orthogonal with ea h olumn

e

w

q

s aledby the orrespondingsingularvalue,

k e

w

q

k = eσ

q

. The

k

-th olumn of

X

is thus represented, using (2.27), as

z

k

=

X

q

ev

kq

∗

w

e

q

.

(2.28)

This expression is exa t, with no approximation error, if all singular

val-ues/ve tors are onsidered in the expansion. Ea h sampled s attering

re-sponse is thus represented as a superposition of basis ve tors

w

e

q

, whose norm de reases uniformlywith in reasing

q

.

The oe ients

ev

∗

kq

are omplex-valued onstants. Sin e arealexpansion oe ient is needed in order to guarantee the ausality and the realness

of ea h element in the expansion (2.24), the SVD is slightly modied by

splitting real and imaginary parts

X

= X

′

_{+ X}

′′

where

X

′

_{, X}

′′

_{∈ R}

L×P

2

, or equivalently

X

=

I

_L

I

L

X

′

X

′′

(2.29)

where

I

L

is the identity matrix of size

L

. Then, atrun ated SVD de ompo-sition is performed, based onthe optimized implementationfor large

matri- es [114℄, whereonly the rst

ρ

singular values are retained

X

′

X

′′

(35)

where

U

¯

∈ R

2L×ρ

,

Σ

¯

∈ R

ρ×ρ

,

V

¯

∈ R

P

2 ×ρ

with

ρ

≪ r = min{2L, P

2 _}

,and

V

¯

is orthonormal,

V

¯

T

_¯

V

= I

. Dening now

¯

W

=

I

L

I

L

¯

U ¯

Σ

(2.31)

leads the low-rank approximation

X

_{≃ ¯}

X

= ¯

W ¯

V

T

.

(2.32) Equivalently,

z

_k

_≃

ρ

X

q=1

v

kq

w

¯

q

,

(2.33)

whi h is similar to (2.28) but has guaranteed real oe ients

v

kq

. The

q

-th olumn

w

¯

q

∈ C

L

of

W

¯

, olle ts all frequen y samples that dene the

q

-th basis fun tion. Sharp bounds, in dierent norms, an be provided for the errorbetween theoriginalmatrix

X

olle tingalls atteringdata and its low-rankapproximation

X

¯

. Usingthespe tralnorm,denedinAppendixA, leads to

E

2 =

¯

X

_{− X}

₂

=

I

_L

I

L

h ¯

U ¯

Σ ¯

V

T

− UΣV

T

i

2 ≤

I

L

I

L

₂

¯

U ¯

Σ ¯

V

T

_{− UΣV}

T

2 ≤

√

2σ

ρ+1

,

(2.34)

where the lastrow follows fromstandard properties of the SVD

de omposi-tion. It followsthat the a ura y ofthe approximationis fully ontrolled by

the rst negle ted singular value

σ

ρ+1

. Usingthe Frobenius norm the error

boundbe omes

E

F

=

¯

X

_{− X}

F

≤

√

2L

v

u

t

r

X

n=ρ+1

σ

2 n

,

(2.35)

in terms of the umulative energy of the negle ted singularvalues in(2.30).

The a ura y of the proposed ompression strategy is demonstrated in

Fig-ure2.3: Thetoppaneldepi tstwos atteringresponsesofthesame onne tor

already onsidered in Figure 2.2, together with the orresponding low-rank

approximation. The dieren e is hardly visible; while the bottom panel

re-ports the rst three basis ve tors

w

¯

q

inthe orrespondingexpansion (2.33).

2.3.2 Fitting the basis fun tions

On e expansion (2.33) is available, a rational approximation of ea h basis

ve tor

w

¯

q

isperformed. Considerarow-ve torofs alarfun tionsoffrequen y

(36)

0

2

4

6

8

10

12

14

16

18

20

0

0.2

0.4

0.6

0.8

1 Frequency [GHz]

Magnitude

S

14,7

S

1,1

raw data

compressed data

VF model

0

2

4

6

8

10

12

14

16

18

20

0

2

4

6

8 Frequency [GHz]

Magnitude

w1

w3

w2

basis vectors

VF model

Figure 2.3: Top: raw s attering responses of a high-speed onne tor before

ompression (red dashed line), its ompressed (

ρ = 30

) approximation (blue dashedline),anditslow-rankrationalapproximation omputedbyVF(bla k

line). Bottom: rst three ve tors

w

¯

q

(blue dashed lines)in expansion(2.33) and orrespondingVF approximation (bla k line).

(37)

with ea h element assumed in rationalform

w

q

(s) = w

q,∞

+

N

w

X

n=1

r

q,n

s

− p

n

.

(2.37)

The unknown poles

p

n

, residues

r

q,n

and dire t oupling onstants

w

q,∞

are

omputed by applying a standard VF run. Sin e only

ρ

independent

re-sponses are on urrentlyttedinsteadot

P

2

,itisexpe tedthat theruntime

oftheVFpro essisdrasti allyredu ed. Thisisindeedthe ase,asitisshown

in Se tion 2.3.3. Note that a set of ommon poles

p

n

for all basis fun tions is used in

w(s)

, sin e these will be used tore onstru t the original s atter-ing matrix through (2.33), thus obtaining a global rational ma romodel in