Chapter 7 Reading on

Moment Calculation

Time Moments of Impulse Response h(t)

• Definition of moments

)()( sHth L

dttthsi

dtsti

thdtethsH

i

i

i

i

ist

00

00

0

)()(!

1

)(!

1)()()(

i-th moment dttthi

m iii

0)()1(

!

1

• Note that m1 = Elmore delay when h(t) is monotone voltage response of impulse input

Pade Approximation

• H(s) can be modeled by Pade approximation of type (p/q):

– where q < p << N

1)(ˆ

1

01,

sasa

bsbsbsH

qq

pp

qp

• Or modeled by q-th Pade approximation (q << N):

q

j j

jqqq ps

ksHsH

1,1 )(ˆ)(ˆ

• Formulate 2q constraints by matching 2q moments to compute ki’s & pi’s

General Moment Matching Technique

• Basic idea: match the moments m-(2q-r), …, m-1, m0, m1, …, mr-1

(i) initial condition matches, i.e.

)ˆ(or

)(lim)(ˆlimor ),0()0(ˆ

11

s

mm

ssHsHshhs

(ii) 22,,1,0for ˆ qkmm kk

)(ˆ of moments sH

q

q

ps

k

ps

k

ps

ksH

2

2

1

1)(ˆ

)(ˆˆˆ 1110

rrr ssmsmm

• When r = 2q-1:

Compute Residues & Poles

j

j ii

i

i

ii

i

i

p

s

p

k

ps

pk

ps

k)(

1 0

2212122

212

1

1

1222

221

1

02

2

1

1

121

)(

)(

)(

)0(

qqq

q

qq

q

q

q

q

q

mp

k

p

k

p

k

mp

k

p

k

p

k

mp

k

p

k

p

k

mhkkk

condition initial

))(ˆlim( sHss

match first 2q-1 momentsEQ1

Basic Steps for Moment Matching

Step 1: Compute 2q moments m-1, m0, m1, …, m(2q-2) of H(s)

Step 2: Solve 2q non-linear equations of EQ1 to get

residues :,,,

poles :,,,

21

21

q

q

kkk

ppp

Step 3: Get approximate waveform

Step 4: Increase q and repeat 1-4, if necessary, for better accuracy

tpq

tptp qekekekth 2121)(ˆ

Components of Moment Matching Model

• Moment computation– Iterative DC analysis on transformed equivalent DC

circuit

– Recursive computation based on tree traversal

– Incremental moment computation

• Moment matching methods– Asymptotic Waveform Evaluation (AWE) [Pillage-

Rohrer, TCAD’90]

– 2-pole method [Horowitz, 1984] [Gao-Zhou, ISCAS’93]...

Basis of Moment Computation by DC Analysis

• Applicable to general RLC networks

• Used in Asymptotic Waveform Evaluation (AWE) [Pillage-Rohrer, TCAD’90]

• Represent a lumped, linear, time-invariant circuit by a system of first-order differential equations:

xl

bGxxCTy

u

where x represents circuit variables (currents and voltages)

G represents memoryless elements (resistors)

C represents memory elements (capacitors and inductors)

bu represents excitations from independent sources

y is the output of interest

Transfer Function

• Assume zero initial conditions and perform Laplace transform:

Xl

bGXCXTY

Us

where X, U, Y denote Laplace transform of x, u, y, respectively

• Let s = s0 + , where s0 is an arbitrary, but fixed expansion point such that G+s0C is non-singular

• Transfer function:

bCGl 1)()()()( ssUsYsH T

rAIl 10 )()( TsH

bCGrCCGA 10

10 )( ,)( where ss

Taylor Expansion and Moments

• Expansion of H(s) about = 0:

k

kk

T

m

sH

0

220 )()( rAAIl

• Recursive moment computation:

,2,1 )(

)()(

10

10000

ks

ss

kk CuuCG

bCGubuCG

rAl kTkm where

rAu

Aru

ru

22

1

0

Taylor Expansion and Moments (Cont’d)

• Expansion of H(s) around

k

kk

T

m

sH

1

2211110 ))(()( rAAIAl

• Recursive moment computation:

,2,1 )(

)()(

10

10000

ks

ss

kk uCGCu

bCGubuCG

rAl kTkm where

rAu

rAu

ru

22

11

0

Interpretation of Moment Computation

• Compute: bCGu 100 )( s

1C

2L

1L

2C

inV

1Cm

2Lm

1Lm

2Cm

inV

V0

V0A0

A0

• Convert: Inductor Voltage source

Capacitor Current source

• When s0 = 0, equivalent to DC analysis:

– shorting inductors (0V) and opening capacitors (0A)

– compute currents through inductors and voltages across capacitors as moments

, setting 0x

Interpretation of Moment Computation (Cont’d)• Compute: 010 )( CuuCG s

1C

2L

1L

2C

inV11 CmC

22 LmL

11 LmL

22 CmC

V0

• When s0 = 0, equivalent to DC analysis:

– voltage sources of inductor L= LmL, current sources of capacitor C = CmC

– external excitations = 0

– compute currents through inductors and voltages across capacitors as moments

, setting 0ux

• Convert: Inductor Voltage source

Capacitor Current source

1Cm

2Lm

1Lm

2Cm

V0

Interpretation of Moment Computation (Cont’d)

• Compute: 001 )( uCGCu s

1C

2L

1L

2C

inV

• When s0 = 0, equivalent to DC analysis:

– moments as currents through inductors and voltages across capacitors

– external excitations = 0

– compute voltage sources of inductors and current sources of capacitors

, setting 0ux

• Convert: Inductor Voltage source

Capacitor Current source

Moment Computation by DC Analysis

DC analysis:

modified nodal analysis (used in original AWE )

sparse-tableau

……

Time complexity to compute moments up to the p-th

order: p time complexity of DC analysis

Perform DC analysis to compute the (i+1)-th order moments

voltage across Cj => the (i+1)-th order moment of Cj

current across Lj => the (i+1)-th order moment of Lj

Advantage and Disadvantage ofMoment Computation by DC Analysis

Computation of uk corresponds to vector iteration with matrix A

Converges to an eigenvector corresponding to an eigenvalue of A with largest absolute value

Recursive computation of vectors uk is efficient since the matrix (G+s0C) is LU-factored exactly once

Moment Computation for RLC Trees

• Most interconnects are RLC trees

• Exploit special structure of G and C matrices to compute moments

• Two basic approaches:– Recursive moment computation [Yu-Kuh, TVLSI’95]

– Incremental moment computation [Cong-Koh, ICCAD’97]

• Both papers used a slightly different definition of moment

dttthi

m ii

0)(

!

1

dttthi

m ii

i

0

)(!

)1(• We continue to use the same definition as before

Basis of Moment Computation for RLC Trees• Wire segment modeled as lumped RLC element

• Definition:

kpm

vsV

Cv

isI

kki

kC

kkL

kkR

kk

kT

pk

kk

kk

kk

k

k

k

k

k

node ofmoment order th -

of transformLaplace)(

across voltage

of transformLaplace)(

to fromcurrent

nodeat ecapacitanc

and nodesbetween inductance

and nodesbetween resistance

node of nodeparent

nodeat rooted subtree

kL

kR

To source

k

k

kT

kT

kC

ki

Transfer Function for RLC Trees

• Apply KCL at node k:

jkjkik

jkjk

jkjk

kjjk

k

sLRZ

PL

PR

PPP

kP

path on inductance total

path on resistance total

node root to frompath

)()( ssVCsI jTj

jk

k

• Let

)(

)()()()(

ssVCZ

ssVCsLRsVsV

kk

kik

jTj

jkPk

kiin

ki

• Voltage drop from root to node i:

)(1)( ssHCZsH kk

kiki • Transfer function:

Recursive Formula for Moments

• Expanding Hi(s), Hk(s) into Taylor series:

)(1)( ssHCZsH kk

kiki • Transfer function:

k

jkkik

jkkik

j

jk

kik

j

j

ji mCLmCRsCsRsm )(11 21

21

• Recursive formula for moments:

1 if

0 if

1 if

)(

1

0

21 p

p

p

CLCR

m

k

pkik

pkik

pi

• Define p-th order weighted capacitance of Ck: kpk

pk CmC

Recursive Formula for Moments (Cont’d)

• Can show that:

root and 1 if

root and 1 if

0 if

1 if

)(

0

1

0

21 kp

kp

p

p

CLCRm

m

pTk

pTk

pk

pk

kk

• Define p-th order weighted capacitance in subtree Tk

k

kTj

jpj

pT CmC

• Similarity with definition of Elmore delay

• Equivalently:

k

jjPj

pTj

pTj

pk CLCRm )( 21

Recursive Moment Computation for RLC Trees[Yu-Kuh, TVLSI’95]

• Bottom-up tree traversal phase: O(n) for n nodespkk mC– Compute all p-th order weighted capacitance

k

kTj

pkk

pT mCC

– Compute the p-th order weighted capacitance in subtree Tk

)( 111 pTk

pTk

pk

pk kk

CLCRmm

• Top-down tree traversal phase; O(n) for n nodes– Compute the moment at each capacitor node k

• Time complexity to compute moments up to the p-th order = O(np)

• Initialize (-1)-th and 0-th order moments

• Compute moments from order 1 to order p successively

Incremental Moment Computation[Cong-Koh, ICCAD’97]

• Iterative tree traversal approaches such as [Kuh-Yu, TVLSI’95] assume a static tree topology– More suitable for interconnect analysis

– Not suitable for tree topology optimization

– After each modification of tree topology, need to recompute all p-th order moments

• Incremental updates of sink moments – More suitable for tree optimization algorithm that construct

topology in a bottom-up fashion

– As we modify the tree topology, update the transfer function Hi(s) for sink si incrementally

Basis for Bottom-Up Moment Computation• Consider the computation of moments for sink w

uku

u

kp

T

up

k

vkv

v

kp

T

vpk

TksH

T

TpC

Tkpm

TksH

T

TpC

Tkpm

k

k

subtree newin node offunction transfer )(

subtree newin

subtree of ecapacitanc htedorder weigth -

subtree newin node ofmoment order th -

subtree originalin node offunction transfer )(

subtree original

in subtree of ecapacitanc htedorder weigth -

subtree originalin node ofmoment order th -

~

~

• Assume sink w is originally in subtree Tv

• Merge subtree Tv with another subtree and the new tree is rooted at node u

• Definition:

u

v

w

pwm

1pTv

C

pvm

Bottom-Up Moment Computation• Maintain transfer function Hv~w(s) for sink w in subtree Tv, and

moment-weighted capacitance of subtree:

• As we merge subtrees, compute new transfer function Hu~v(s) and weighted capacitance recursively:

u

vvv CLR ,,

v

w

up

um 1pTu

C

• New transfer function for node w

10for ,0for pjCpjm jT

jw v

pjCLCRm

CmCmC

jTv

jTv

jv

qT

j

q

qjvv

jv

jT

vv

vv

1for )(

,

21

1

0

111

pjmmmj

q

qw

qjv

jw 0for

0

pwm

pvm

1pTv

C

• New moment-weighted capacitance of Tu:

10for )(

pjCCuchildv

jT

jT vu

)()()( ~~~ sHsHsH wvvuwu

Complexity Analysis ofIncremental Moment Computation

• Assume topology has n nodes• Complexity due to each edge: O(p2)• Total time complexity: O(np2)• If there are k nodes of interest, time complexity =

O(knp2)

Moment Matching by AWE[Pillage-Rohrer, TCAD’90]

• Recall the transfer function obtained from a linear circuit

– Let s = s0 + , where s0 is an arbitrary, but fixed expansion point such that G+s0C is non-singular

rAIl 10 )()( TsH

bCGrCCGA 10

10 )( ,)( where ss

• When matrix A is diagonalizable1 SSΛA ),,,( 21 Ndiag Λ

rSΛISl 110 )()( TsH

Tf g

N

j j

jj gfsH

10 1

)(

pole of reciprocal :j

q-th Pade Approximation

• Pade approximation of type (p/q):

)()(

1)(

10

1

010,

qp

qq

pp

qp

sH

aa

bbbsH

• q-th Pade approximation (q << N):

q

j j

jqqq p

kHsH

1,10 )(

• Equivalent to finding a reduced-order matrix AR such that eigenvalues j of AR are reciprocals of the approximating poles pj for the original system

Asymptotic Waveform Evaluation

• Recall EQ1:

• Let

2212122

212

1

1

1222

221

1

02

2

1

1

121

)(

)(

)(

)0()(

qqq

q

qq

q

q

q

q

q

mp

k

p

k

p

k

mp

k

p

k

p

k

mp

k

p

k

p

k

mhkkk

),,,(

][

]1[

1

21

21

12

q

q

Tqjjjj

jj

diag

p

Λ

VVVV

V

Asymptotic Waveform Evaluation (Cont’d)

• Rewrite EQ1:

• Let

hq

l

mkΛ

mk

V

V

1 VVΛAR

where

Tqqqh

Tql

Tq

kkm

mmm

kkk

][

][

][

221

201

21

m

m

k

• Solving for k:

hlq

l

mmΛ

mk

1

1

VV

V

hlq mmAR

Need to compute all the poles first

Structure of Matrix AR

• Matrix:

• Therefore, AR could be a matrix of the above structure

121

1000

0100

0010

aaaa qqq

has characteristic equation: 011

1

qqqq aaa

Tqjjjj ]1[r eigenvecto has Eigenvalue 12

• Note that: 1

Characteristic equation becomes the denominator of Hq(s):01 1

11

qq

qq aaa

Solving for Matrix AR

• Consider multiplications of AR on ml

1

2

1

0

2

3

0

1

121 '

1000

0100

0010

q

q

q

q

qqq m

m

m

m

m

m

m

m

aaaa

produces 21120111' qqqqq mamamamam

q

q

q

q

qqq

l

m

m

m

m

m

m

m

m

aaaa '

'

'

1000

0100

0010

1

2

1

1

2

1

0

121

2

mAR

produces 1122110 '' qqqqq mamamamam

Solving for Matrix AR (Cont’d)• After q multiplications of AR on ml

h

q

q

q

q

q

q

q

q

qqq

lq

m

m

m

m

m

m

m

m

aaaa

mmAR

22

32

1

32

42

1

2

121 '

'

'

'

'

'

'

10000

0100

0010

produces 32121122 '''' qqqqqqqq mamamamam

• Equating m’ with m:

22

32

1

1

2

1

3212

42123

1210

2101

q

q

q

q

q

q

qqqq

qqqq

q

q

m

m

m

m

a

a

a

a

mmmm

mmmm

mmmm

mmmm

Summary of AWEStep 1: Compute 2q moments, choice of q depends on accuracy

requirement; in practice, q 5 is frequently used

Step 2: Solve a system of linear equations by Gaussian elimination to get aj

Step 3: Solve the characteristic equation of AR to determine the approximate poles pj

22

32

1

1

2

1

3212

42123

1210

2101

q

q

q

q

q

q

qqqq

qqqq

q

q

m

m

m

m

a

a

a

a

mmmm

mmmm

mmmm

mmmm

011

1

qqqq aaa

Step 4: Solve for residues kj

lmk 1 V

Numerical Limitations of AWE

Due to recursive computation of moments

Converges to an eigenvector corresponding to an eigenvalue of matrix A with largest absolute value

Moment matrix used in AWE becomes rapidly ill-conditioned

Increasing number of poles does not improve accuracy

Unable to estimate the accuracy of the approximating model

Remedial techniques are sometimes heuristic, hard to apply automatically, and may be computationally expensive